Integrand size = 23, antiderivative size = 430 \[ \int \frac {\tan ^{\frac {11}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {a^{7/2} \left (35 a^4+102 a^2 b^2+99 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{9/2} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (c+d x)}}{1+\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{4 b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \] Output:
-1/2*(a+b)*(a^2-4*a*b+b^2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)/(a^ 2+b^2)^3/d-1/2*(a+b)*(a^2-4*a*b+b^2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^ (1/2)/(a^2+b^2)^3/d+1/4*a^(7/2)*(35*a^4+102*a^2*b^2+99*b^4)*arctan(b^(1/2) *tan(d*x+c)^(1/2)/a^(1/2))/b^(9/2)/(a^2+b^2)^3/d-1/2*(a-b)*(a^2+4*a*b+b^2) *arctanh(2^(1/2)*tan(d*x+c)^(1/2)/(1+tan(d*x+c)))*2^(1/2)/(a^2+b^2)^3/d-1/ 4*a*(35*a^4+67*a^2*b^2+24*b^4)*tan(d*x+c)^(1/2)/b^4/(a^2+b^2)^2/d+1/12*(35 *a^4+67*a^2*b^2+8*b^4)*tan(d*x+c)^(3/2)/b^3/(a^2+b^2)^2/d-1/2*a^2*tan(d*x+ c)^(7/2)/b/(a^2+b^2)/d/(a+b*tan(d*x+c))^2-1/4*a^2*(7*a^2+15*b^2)*tan(d*x+c )^(5/2)/b^2/(a^2+b^2)^2/d/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 12.35 (sec) , antiderivative size = 890, normalized size of antiderivative = 2.07 \[ \int \frac {\tan ^{\frac {11}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \sqrt {\tan (c+d x)} \left (-\frac {a \left (35 a^4+69 a^2 b^2+24 b^4\right )}{4 (a-i b)^2 (a+i b)^2 b^4}+\frac {a^5}{2 (a-i b)^2 (a+i b)^2 b^2 (a \cos (c+d x)+b \sin (c+d x))^2}+\frac {3 \left (3 a^5 \sin (c+d x)+7 a^3 b^2 \sin (c+d x)\right )}{4 (a-i b)^2 (a+i b)^2 b^3 (a \cos (c+d x)+b \sin (c+d x))}+\frac {2 \tan (c+d x)}{3 b^3}\right )}{d (a+b \tan (c+d x))^3}+\frac {\sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \left (\frac {2 \left (35 a^6+67 a^4 b^2+28 a^2 b^4-4 b^6\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) \csc (c+d x) \sec ^3(c+d x) (a+b \tan (c+d x))}{\sqrt {a} \sqrt {b} (b+a \cot (c+d x)) \left (1+\tan ^2(c+d x)\right )^2}+\frac {2 a b^5 \csc ^2(c+d x) \left (-8 \sqrt {a} \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )+\sqrt {2} \left (-2 (a+b) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )+2 (a+b) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+(a-b) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )\right )\right ) \sec ^2(c+d x) \sin (2 (c+d x)) (a+b \tan (c+d x))}{\left (a^2+b^2\right ) (b+a \cot (c+d x)) \left (1+\tan ^2(c+d x)\right )}-\frac {\left (-4 a^2 b^4+4 b^6\right ) \cos (2 (c+d x)) \csc (c+d x) \left (\frac {4 \left (a^2-b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}+\sqrt {2} \left (2 (a-b) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-2 (a-b) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+(a+b) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )\right )\right ) \sec ^3(c+d x) (a+b \tan (c+d x))}{2 \left (a^2+b^2\right ) (b+a \cot (c+d x)) \left (1-\tan ^2(c+d x)\right ) \left (1+\tan ^2(c+d x)\right )}\right )}{8 (a-i b)^2 (a+i b)^2 b^4 d (a+b \tan (c+d x))^3} \] Input:
Integrate[Tan[c + d*x]^(11/2)/(a + b*Tan[c + d*x])^3,x]
Output:
(Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3*Sqrt[Tan[c + d*x]]*(-1 /4*(a*(35*a^4 + 69*a^2*b^2 + 24*b^4))/((a - I*b)^2*(a + I*b)^2*b^4) + a^5/ (2*(a - I*b)^2*(a + I*b)^2*b^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) + (3*( 3*a^5*Sin[c + d*x] + 7*a^3*b^2*Sin[c + d*x]))/(4*(a - I*b)^2*(a + I*b)^2*b ^3*(a*Cos[c + d*x] + b*Sin[c + d*x])) + (2*Tan[c + d*x])/(3*b^3)))/(d*(a + b*Tan[c + d*x])^3) + (Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3* ((2*(35*a^6 + 67*a^4*b^2 + 28*a^2*b^4 - 4*b^6)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Csc[c + d*x]*Sec[c + d*x]^3*(a + b*Tan[c + d*x]))/(Sqrt[ a]*Sqrt[b]*(b + a*Cot[c + d*x])*(1 + Tan[c + d*x]^2)^2) + (2*a*b^5*Csc[c + d*x]^2*(-8*Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] + Sqrt[2]*(-2*(a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] + 2*(a + b)*Ar cTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + (a - b)*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d* x]])))*Sec[c + d*x]^2*Sin[2*(c + d*x)]*(a + b*Tan[c + d*x]))/((a^2 + b^2)* (b + a*Cot[c + d*x])*(1 + Tan[c + d*x]^2)) - ((-4*a^2*b^4 + 4*b^6)*Cos[2*( c + d*x)]*Csc[c + d*x]*((4*(a^2 - b^2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]]) /Sqrt[a]])/(Sqrt[a]*Sqrt[b]) + Sqrt[2]*(2*(a - b)*ArcTan[1 - Sqrt[2]*Sqrt[ Tan[c + d*x]]] - 2*(a - b)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + (a + b )*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sq rt[Tan[c + d*x]] + Tan[c + d*x]])))*Sec[c + d*x]^3*(a + b*Tan[c + d*x])...
Time = 2.65 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.08, number of steps used = 30, number of rules used = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.261, Rules used = {3042, 4048, 27, 3042, 4128, 27, 3042, 4130, 27, 3042, 4130, 27, 3042, 4136, 27, 3042, 4017, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103, 4117, 73, 218}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\tan ^{\frac {11}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^{11/2}}{(a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {\int \frac {\tan ^{\frac {5}{2}}(c+d x) \left (7 a^2-4 b \tan (c+d x) a+\left (7 a^2+4 b^2\right ) \tan ^2(c+d x)\right )}{2 (a+b \tan (c+d x))^2}dx}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\tan ^{\frac {5}{2}}(c+d x) \left (7 a^2-4 b \tan (c+d x) a+\left (7 a^2+4 b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2}dx}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\tan (c+d x)^{5/2} \left (7 a^2-4 b \tan (c+d x) a+\left (7 a^2+4 b^2\right ) \tan (c+d x)^2\right )}{(a+b \tan (c+d x))^2}dx}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {\frac {\int \frac {\tan ^{\frac {3}{2}}(c+d x) \left (-16 a \tan (c+d x) b^3+\left (35 a^4+67 b^2 a^2+8 b^4\right ) \tan ^2(c+d x)+5 a^2 \left (7 a^2+15 b^2\right )\right )}{2 (a+b \tan (c+d x))}dx}{b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\tan ^{\frac {3}{2}}(c+d x) \left (-16 a \tan (c+d x) b^3+\left (35 a^4+67 b^2 a^2+8 b^4\right ) \tan ^2(c+d x)+5 a^2 \left (7 a^2+15 b^2\right )\right )}{a+b \tan (c+d x)}dx}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\tan (c+d x)^{3/2} \left (-16 a \tan (c+d x) b^3+\left (35 a^4+67 b^2 a^2+8 b^4\right ) \tan (c+d x)^2+5 a^2 \left (7 a^2+15 b^2\right )\right )}{a+b \tan (c+d x)}dx}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {\frac {2 \int -\frac {3 \sqrt {\tan (c+d x)} \left (-8 \left (a^2-b^2\right ) \tan (c+d x) b^3+a \left (35 a^4+67 b^2 a^2+24 b^4\right ) \tan ^2(c+d x)+a \left (35 a^4+67 b^2 a^2+8 b^4\right )\right )}{2 (a+b \tan (c+d x))}dx}{3 b}+\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\int \frac {\sqrt {\tan (c+d x)} \left (-8 \left (a^2-b^2\right ) \tan (c+d x) b^3+a \left (35 a^4+67 b^2 a^2+24 b^4\right ) \tan ^2(c+d x)+a \left (35 a^4+67 b^2 a^2+8 b^4\right )\right )}{a+b \tan (c+d x)}dx}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\int \frac {\sqrt {\tan (c+d x)} \left (-8 \left (a^2-b^2\right ) \tan (c+d x) b^3+a \left (35 a^4+67 b^2 a^2+24 b^4\right ) \tan (c+d x)^2+a \left (35 a^4+67 b^2 a^2+8 b^4\right )\right )}{a+b \tan (c+d x)}dx}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 \int -\frac {16 a \tan (c+d x) b^5+\left (35 a^6+67 b^2 a^4+32 b^4 a^2-8 b^6\right ) \tan ^2(c+d x)+a^2 \left (35 a^4+67 b^2 a^2+24 b^4\right )}{2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{b}+\frac {2 a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{b d}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\int \frac {16 a \tan (c+d x) b^5+\left (35 a^6+67 b^2 a^4+32 b^4 a^2-8 b^6\right ) \tan ^2(c+d x)+a^2 \left (35 a^4+67 b^2 a^2+24 b^4\right )}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\int \frac {16 a \tan (c+d x) b^5+\left (35 a^6+67 b^2 a^4+32 b^4 a^2-8 b^6\right ) \tan (c+d x)^2+a^2 \left (35 a^4+67 b^2 a^2+24 b^4\right )}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {\int -\frac {8 \left (a b^4 \left (a^2-3 b^2\right )-b^5 \left (3 a^2-b^2\right ) \tan (c+d x)\right )}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}+\frac {a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right ) \int \frac {\tan ^2(c+d x)+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right ) \int \frac {\tan ^2(c+d x)+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}-\frac {8 \int \frac {a b^4 \left (a^2-3 b^2\right )-b^5 \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}-\frac {8 \int \frac {a b^4 \left (a^2-3 b^2\right )-b^5 \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}-\frac {16 \int \frac {b^4 \left (a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \tan (c+d x)\right )}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}-\frac {16 b^4 \int \frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}-\frac {16 b^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d \left (a^2+b^2\right )}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}-\frac {16 b^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}-\frac {16 b^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}-\frac {16 b^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}-\frac {16 b^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}-\frac {16 b^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}-\frac {16 b^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}-\frac {16 b^4 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right ) \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}d\tan (c+d x)}{d \left (a^2+b^2\right )}-\frac {16 b^4 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {2 a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right ) \int \frac {1}{a+b \tan (c+d x)}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}-\frac {16 b^4 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {2 \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {2 a^{7/2} \left (35 a^4+102 a^2 b^2+99 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} d \left (a^2+b^2\right )}-\frac {16 b^4 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
Input:
Int[Tan[c + d*x]^(11/2)/(a + b*Tan[c + d*x])^3,x]
Output:
-1/2*(a^2*Tan[c + d*x]^(7/2))/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + ( -((a^2*(7*a^2 + 15*b^2)*Tan[c + d*x]^(5/2))/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))) + (-((-(((2*a^(7/2)*(35*a^4 + 102*a^2*b^2 + 99*b^4)*ArcTan[(Sqrt [b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[b]*(a^2 + b^2)*d) - (16*b^4*(((a + b)*(a^2 - 4*a*b + b^2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2]))/2 + ((a - b)*(a^2 + 4* a*b + b^2)*(-1/2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/Sqrt[2 ] + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2])))/2))/( (a^2 + b^2)*d))/b) + (2*a*(35*a^4 + 67*a^2*b^2 + 24*b^4)*Sqrt[Tan[c + d*x] ])/(b*d))/b) + (2*(35*a^4 + 67*a^2*b^2 + 8*b^4)*Tan[c + d*x]^(3/2))/(3*b*d ))/(2*b*(a^2 + b^2)))/(4*b*(a^2 + b^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2/f Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & & NeQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1 /(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c *(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) *Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*( n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 2] && LtQ [n, -1] && IntegerQ[2*m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*d^2 + c*(c*C - B*d))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Sim p[1/(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c*m + a*d* (n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b *(d*(B*c - A*d)*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ [a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_. ) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[ e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C *(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[ c, 0] && NeQ[a, 0])))
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^ n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] & & !GtQ[n, 0] && !LeQ[n, -1]
Time = 0.38 (sec) , antiderivative size = 373, normalized size of antiderivative = 0.87
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (-\frac {b \tan \left (d x +c \right )^{\frac {3}{2}}}{3}+3 a \sqrt {\tan \left (d x +c \right )}\right )}{b^{4}}+\frac {2 a^{4} \left (\frac {\left (-\frac {13}{8} b \,a^{4}-\frac {17}{4} a^{2} b^{3}-\frac {21}{8} b^{5}\right ) \tan \left (d x +c \right )^{\frac {3}{2}}-\frac {a \left (11 a^{4}+30 a^{2} b^{2}+19 b^{4}\right ) \sqrt {\tan \left (d x +c \right )}}{8}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (35 a^{4}+102 a^{2} b^{2}+99 b^{4}\right ) \arctan \left (\frac {b \sqrt {\tan \left (d x +c \right )}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{4} \left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(373\) |
| default | \(\frac {-\frac {2 \left (-\frac {b \tan \left (d x +c \right )^{\frac {3}{2}}}{3}+3 a \sqrt {\tan \left (d x +c \right )}\right )}{b^{4}}+\frac {2 a^{4} \left (\frac {\left (-\frac {13}{8} b \,a^{4}-\frac {17}{4} a^{2} b^{3}-\frac {21}{8} b^{5}\right ) \tan \left (d x +c \right )^{\frac {3}{2}}-\frac {a \left (11 a^{4}+30 a^{2} b^{2}+19 b^{4}\right ) \sqrt {\tan \left (d x +c \right )}}{8}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (35 a^{4}+102 a^{2} b^{2}+99 b^{4}\right ) \arctan \left (\frac {b \sqrt {\tan \left (d x +c \right )}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{4} \left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(373\) |
Input:
int(tan(d*x+c)^(11/2)/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(-2/b^4*(-1/3*b*tan(d*x+c)^(3/2)+3*a*tan(d*x+c)^(1/2))+2*a^4/b^4/(a^2+ b^2)^3*(((-13/8*b*a^4-17/4*a^2*b^3-21/8*b^5)*tan(d*x+c)^(3/2)-1/8*a*(11*a^ 4+30*a^2*b^2+19*b^4)*tan(d*x+c)^(1/2))/(a+b*tan(d*x+c))^2+1/8*(35*a^4+102* a^2*b^2+99*b^4)/(a*b)^(1/2)*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2)))+2/(a^2 +b^2)^3*(1/8*(-a^3+3*a*b^2)*2^(1/2)*(ln((tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/ 2)+1)/(tan(d*x+c)-2^(1/2)*tan(d*x+c)^(1/2)+1))+2*arctan(1+2^(1/2)*tan(d*x+ c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)))+1/8*(3*a^2*b-b^3)*2^(1/2) *(ln((tan(d*x+c)-2^(1/2)*tan(d*x+c)^(1/2)+1)/(tan(d*x+c)+2^(1/2)*tan(d*x+c )^(1/2)+1))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d *x+c)^(1/2)))))
Leaf count of result is larger than twice the leaf count of optimal. 2489 vs. \(2 (384) = 768\).
Time = 2.75 (sec) , antiderivative size = 5005, normalized size of antiderivative = 11.64 \[ \int \frac {\tan ^{\frac {11}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(tan(d*x+c)^(11/2)/(a+b*tan(d*x+c))^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\tan ^{\frac {11}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate(tan(d*x+c)**(11/2)/(a+b*tan(d*x+c))**3,x)
Output:
Timed out
Time = 0.16 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.04 \[ \int \frac {\tan ^{\frac {11}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {3 \, {\left (35 \, a^{8} + 102 \, a^{6} b^{2} + 99 \, a^{4} b^{4}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}\right )} \sqrt {a b}} - \frac {3 \, {\left (2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {3 \, {\left ({\left (13 \, a^{6} b + 21 \, a^{4} b^{3}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (11 \, a^{7} + 19 \, a^{5} b^{2}\right )} \sqrt {\tan \left (d x + c\right )}\right )}}{a^{6} b^{4} + 2 \, a^{4} b^{6} + a^{2} b^{8} + {\left (a^{4} b^{6} + 2 \, a^{2} b^{8} + b^{10}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{5} + 2 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )} + \frac {8 \, {\left (b \tan \left (d x + c\right )^{\frac {3}{2}} - 9 \, a \sqrt {\tan \left (d x + c\right )}\right )}}{b^{4}}}{12 \, d} \] Input:
integrate(tan(d*x+c)^(11/2)/(a+b*tan(d*x+c))^3,x, algorithm="maxima")
Output:
1/12*(3*(35*a^8 + 102*a^6*b^2 + 99*a^4*b^4)*arctan(b*sqrt(tan(d*x + c))/sq rt(a*b))/((a^6*b^4 + 3*a^4*b^6 + 3*a^2*b^8 + b^10)*sqrt(a*b)) - 3*(2*sqrt( 2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(ta n(d*x + c)))) + 2*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*arctan(-1/2*sqrt (2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) + sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 3*((13*a^6*b + 21*a^4*b^3)*tan(d *x + c)^(3/2) + (11*a^7 + 19*a^5*b^2)*sqrt(tan(d*x + c)))/(a^6*b^4 + 2*a^4 *b^6 + a^2*b^8 + (a^4*b^6 + 2*a^2*b^8 + b^10)*tan(d*x + c)^2 + 2*(a^5*b^5 + 2*a^3*b^7 + a*b^9)*tan(d*x + c)) + 8*(b*tan(d*x + c)^(3/2) - 9*a*sqrt(ta n(d*x + c)))/b^4)/d
\[ \int \frac {\tan ^{\frac {11}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\int { \frac {\tan \left (d x + c\right )^{\frac {11}{2}}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:
integrate(tan(d*x+c)^(11/2)/(a+b*tan(d*x+c))^3,x, algorithm="giac")
Output:
undef
Time = 18.18 (sec) , antiderivative size = 18832, normalized size of antiderivative = 43.80 \[ \int \frac {\tan ^{\frac {11}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \] Input:
int(tan(c + d*x)^(11/2)/(a + b*tan(c + d*x))^3,x)
Output:
(2*tan(c + d*x)^(3/2))/(3*b^3*d) - atan((((((32*a*b^22*d^2 - 2450*a^23*d^2 + 192*a^3*b^20*d^2 + 19488*a^5*b^18*d^2 - 24128*a^7*b^16*d^2 - 180858*a^9 *b^14*d^2 - 146126*a^11*b^12*d^2 + 208974*a^13*b^10*d^2 + 452586*a^15*b^8* d^2 + 330770*a^17*b^6*d^2 + 106102*a^19*b^4*d^2 + 7770*a^21*b^2*d^2)/(2*(b ^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15 *d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*d^5 )) - (((((640*a^2*b^27*d^4 + 9536*a^4*b^25*d^4 + 55488*a^6*b^23*d^4 + 1774 08*a^8*b^21*d^4 + 354816*a^10*b^19*d^4 + 470400*a^12*b^17*d^4 + 422016*a^1 4*b^15*d^4 + 254208*a^16*b^13*d^4 + 98688*a^18*b^11*d^4 + 22336*a^20*b^9*d ^4 + 2240*a^22*b^7*d^4)/(2*(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*d^5)) - (tan(c + d*x)^(1/2)*(1/(b^6*d^2*1i - a^6 *d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a ^4*b^2*d^2*15i))^(1/2)*(512*b^32*d^4 + 4608*a^2*b^30*d^4 + 17920*a^4*b^28* d^4 + 38400*a^6*b^26*d^4 + 46080*a^8*b^24*d^4 + 21504*a^10*b^22*d^4 - 2150 4*a^12*b^20*d^4 - 46080*a^14*b^18*d^4 - 38400*a^16*b^16*d^4 - 17920*a^18*b ^14*d^4 - 4608*a^20*b^12*d^4 - 512*a^22*b^10*d^4))/(4*(b^23*d^4 + 8*a^2*b^ 21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^1 3*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*d^4)))*(1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^...
\[ \int \frac {\tan ^{\frac {11}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\int \frac {\sqrt {\tan \left (d x +c \right )}\, \tan \left (d x +c \right )^{5}}{\tan \left (d x +c \right )^{3} b^{3}+3 \tan \left (d x +c \right )^{2} a \,b^{2}+3 \tan \left (d x +c \right ) a^{2} b +a^{3}}d x \] Input:
int(tan(d*x+c)^(11/2)/(a+b*tan(d*x+c))^3,x)
Output:
int((sqrt(tan(c + d*x))*tan(c + d*x)**5)/(tan(c + d*x)**3*b**3 + 3*tan(c + d*x)**2*a*b**2 + 3*tan(c + d*x)*a**2*b + a**3),x)