Integrand size = 23, antiderivative size = 493 \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {b^{7/2} \left (99 a^4+102 a^2 b^2+35 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{9/2} \left (a^2+b^2\right )^3 d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {8 a^4+67 a^2 b^2+35 b^4}{12 a^3 \left (a^2+b^2\right )^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{4 a^4 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}+\frac {b^2 \left (15 a^2+7 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))} \]
1/4*b^(7/2)*(99*a^4+102*a^2*b^2+35*b^4)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^ (1/2))/a^(9/2)/(a^2+b^2)^3/d-1/2*(a+b)*(a^2-4*a*b+b^2)*arctan(-1+2^(1/2)*t an(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1/2*(a+b)*(a^2-4*a*b+b^2)*arctan(1+ 2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)+1/4*(a-b)*(a^2+4*a*b+b^2)* ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^3/d*2^(1/2)-1/4*(a-b)* (a^2+4*a*b+b^2)*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^3/d*2^ (1/2)+1/4*b*(24*a^4+67*a^2*b^2+35*b^4)/a^4/(a^2+b^2)^2/d/tan(d*x+c)^(1/2)+ 1/12*(-8*a^4-67*a^2*b^2-35*b^4)/a^3/(a^2+b^2)^2/d/tan(d*x+c)^(3/2)+1/2*b^2 /a/(a^2+b^2)/d/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^2+1/4*b^2*(15*a^2+7*b^2)/ a^2/(a^2+b^2)^2/d/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 6.24 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\frac {b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}+\frac {\frac {-\frac {2 \left (-\frac {2 \left (\frac {2 \left (-3 a^6 b^2+\frac {3}{16} a^2 b^2 \left (24 a^4+67 a^2 b^2+35 b^4\right )-\frac {3}{16} b^2 \left (8 a^6-32 a^4 b^2-67 a^2 b^4-35 b^6\right )\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \left (a^2+b^2\right ) d}+\frac {-\frac {\sqrt [4]{-1} \left (-\frac {3}{2} a^5 \left (a^2-3 b^2\right )-\frac {3}{2} i a^4 b \left (3 a^2-b^2\right )\right ) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}-\frac {\sqrt [4]{-1} \left (-\frac {3}{2} a^5 \left (a^2-3 b^2\right )+\frac {3}{2} i a^4 b \left (3 a^2-b^2\right )\right ) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}}{a^2+b^2}\right )}{a}-\frac {3 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{4 a d \sqrt {\tan (c+d x)}}\right )}{3 a}-\frac {8 a^4+67 a^2 b^2+35 b^4}{6 a d \tan ^{\frac {3}{2}}(c+d x)}}{a \left (a^2+b^2\right )}+\frac {\frac {11 a^2 b^2}{2}+\frac {1}{2} b^2 \left (4 a^2+7 b^2\right )}{a \left (a^2+b^2\right ) d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{2 a \left (a^2+b^2\right )} \]
b^2/(2*a*(a^2 + b^2)*d*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2) + (((-2* ((-2*((2*(-3*a^6*b^2 + (3*a^2*b^2*(24*a^4 + 67*a^2*b^2 + 35*b^4))/16 - (3* b^2*(8*a^6 - 32*a^4*b^2 - 67*a^2*b^4 - 35*b^6))/16)*ArcTan[(Sqrt[b]*Sqrt[T an[c + d*x]])/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(a^2 + b^2)*d) + (-(((-1)^(1/4)*( (-3*a^5*(a^2 - 3*b^2))/2 - ((3*I)/2)*a^4*b*(3*a^2 - b^2))*ArcTan[(-1)^(3/4 )*Sqrt[Tan[c + d*x]]])/d) - ((-1)^(1/4)*((-3*a^5*(a^2 - 3*b^2))/2 + ((3*I) /2)*a^4*b*(3*a^2 - b^2))*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d)/(a^2 + b^2)))/a - (3*b*(24*a^4 + 67*a^2*b^2 + 35*b^4))/(4*a*d*Sqrt[Tan[c + d*x]] )))/(3*a) - (8*a^4 + 67*a^2*b^2 + 35*b^4)/(6*a*d*Tan[c + d*x]^(3/2)))/(a*( a^2 + b^2)) + ((11*a^2*b^2)/2 + (b^2*(4*a^2 + 7*b^2))/2)/(a*(a^2 + b^2)*d* Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])))/(2*a*(a^2 + b^2))
Time = 2.70 (sec) , antiderivative size = 463, normalized size of antiderivative = 0.94, number of steps used = 30, number of rules used = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.261, Rules used = {3042, 4052, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4132, 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.
\(\displaystyle \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (c+d x)^{5/2} (a+b \tan (c+d x))^3}dx\) |
\(\Big \downarrow \) 4052 |
\(\displaystyle \frac {\int \frac {4 a^2-4 b \tan (c+d x) a+7 b^2+7 b^2 \tan ^2(c+d x)}{2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2}dx}{2 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {4 a^2-4 b \tan (c+d x) a+7 b^2+7 b^2 \tan ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2}dx}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {4 a^2-4 b \tan (c+d x) a+7 b^2+7 b^2 \tan (c+d x)^2}{\tan (c+d x)^{5/2} (a+b \tan (c+d x))^2}dx}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 4132 |
\(\displaystyle \frac {\frac {\int \frac {8 a^4-16 b \tan (c+d x) a^3+67 b^2 a^2+35 b^4+5 b^2 \left (15 a^2+7 b^2\right ) \tan ^2(c+d x)}{2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}dx}{a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\int \frac {8 a^4-16 b \tan (c+d x) a^3+67 b^2 a^2+35 b^4+5 b^2 \left (15 a^2+7 b^2\right ) \tan ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}dx}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\int \frac {8 a^4-16 b \tan (c+d x) a^3+67 b^2 a^2+35 b^4+5 b^2 \left (15 a^2+7 b^2\right ) \tan (c+d x)^2}{\tan (c+d x)^{5/2} (a+b \tan (c+d x))}dx}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 4132 |
\(\displaystyle \frac {\frac {-\frac {2 \int \frac {3 \left (8 \left (a^2-b^2\right ) \tan (c+d x) a^3+b \left (8 a^4+67 b^2 a^2+35 b^4\right ) \tan ^2(c+d x)+b \left (24 a^4+67 b^2 a^2+35 b^4\right )\right )}{2 \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}dx}{3 a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {-\frac {\int \frac {8 \left (a^2-b^2\right ) \tan (c+d x) a^3+b \left (8 a^4+67 b^2 a^2+35 b^4\right ) \tan ^2(c+d x)+b \left (24 a^4+67 b^2 a^2+35 b^4\right )}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}dx}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {-\frac {\int \frac {8 \left (a^2-b^2\right ) \tan (c+d x) a^3+b \left (8 a^4+67 b^2 a^2+35 b^4\right ) \tan (c+d x)^2+b \left (24 a^4+67 b^2 a^2+35 b^4\right )}{\tan (c+d x)^{3/2} (a+b \tan (c+d x))}dx}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 4132 |
\(\displaystyle \frac {\frac {-\frac {-\frac {2 \int -\frac {8 a^6-16 b \tan (c+d x) a^5-32 b^2 a^4-67 b^4 a^2-35 b^6-b^2 \left (24 a^4+67 b^2 a^2+35 b^4\right ) \tan ^2(c+d x)}{2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a}-\frac {2 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {-\frac {\frac {\int \frac {8 a^6-16 b \tan (c+d x) a^5-32 b^2 a^4-67 b^4 a^2-35 b^6-b^2 \left (24 a^4+67 b^2 a^2+35 b^4\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a}-\frac {2 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {-\frac {\frac {\int \frac {8 a^6-16 b \tan (c+d x) a^5-32 b^2 a^4-67 b^4 a^2-35 b^6-b^2 \left (24 a^4+67 b^2 a^2+35 b^4\right ) \tan (c+d x)^2}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a}-\frac {2 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 4136 |
\(\displaystyle \frac {\frac {-\frac {\frac {\frac {\int \frac {8 \left (a^5 \left (a^2-3 b^2\right )-a^4 b \left (3 a^2-b^2\right ) \tan (c+d x)\right )}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}-\frac {b^4 \left (99 a^4+102 a^2 b^2+35 b^4\right ) \int \frac {\tan ^2(c+d x)+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {-\frac {\frac {\frac {8 \int \frac {a^5 \left (a^2-3 b^2\right )-a^4 b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}-\frac {b^4 \left (99 a^4+102 a^2 b^2+35 b^4\right ) \int \frac {\tan ^2(c+d x)+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {-\frac {\frac {\frac {8 \int \frac {a^5 \left (a^2-3 b^2\right )-a^4 b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}-\frac {b^4 \left (99 a^4+102 a^2 b^2+35 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 4017 |
\(\displaystyle \frac {\frac {-\frac {\frac {\frac {16 \int \frac {a^4 \left (a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \tan (c+d x)\right )}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}-\frac {b^4 \left (99 a^4+102 a^2 b^2+35 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {-\frac {\frac {\frac {16 a^4 \int \frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}-\frac {b^4 \left (99 a^4+102 a^2 b^2+35 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 1482 |
\(\displaystyle \frac {\frac {-\frac {\frac {\frac {16 a^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d \left (a^2+b^2\right )}-\frac {b^4 \left (99 a^4+102 a^2 b^2+35 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {\frac {-\frac {\frac {\frac {16 a^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^4 \left (99 a^4+102 a^2 b^2+35 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {\frac {-\frac {\frac {\frac {16 a^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^4 \left (99 a^4+102 a^2 b^2+35 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\frac {-\frac {\frac {\frac {16 a^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^4 \left (99 a^4+102 a^2 b^2+35 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {\frac {-\frac {\frac {\frac {16 a^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^4 \left (99 a^4+102 a^2 b^2+35 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {-\frac {\frac {\frac {16 a^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^4 \left (99 a^4+102 a^2 b^2+35 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {-\frac {\frac {\frac {16 a^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^4 \left (99 a^4+102 a^2 b^2+35 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {\frac {-\frac {\frac {\frac {16 a^4 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^4 \left (99 a^4+102 a^2 b^2+35 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 4117 |
\(\displaystyle \frac {\frac {-\frac {\frac {\frac {16 a^4 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^4 \left (99 a^4+102 a^2 b^2+35 b^4\right ) \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}d\tan (c+d x)}{d \left (a^2+b^2\right )}}{a}-\frac {2 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {-\frac {\frac {\frac {16 a^4 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {2 b^4 \left (99 a^4+102 a^2 b^2+35 b^4\right ) \int \frac {1}{a+b \tan (c+d x)}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}}{a}-\frac {2 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}+\frac {\frac {b^2 \left (15 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}+\frac {-\frac {\frac {\frac {16 a^4 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {2 b^{7/2} \left (99 a^4+102 a^2 b^2+35 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}}{a}-\frac {2 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (8 a^4+67 a^2 b^2+35 b^4\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}}{4 a \left (a^2+b^2\right )}\) |
b^2/(2*a*(a^2 + b^2)*d*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2) + ((-((( (-2*b^(7/2)*(99*a^4 + 102*a^2*b^2 + 35*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d *x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)*d) + (16*a^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*L og[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))/a - (2*b*(24*a^4 + 67*a^2*b^2 + 35*b^4))/(a*d*Sqrt[Tan[c + d*x]]))/a) - (2*( 8*a^4 + 67*a^2*b^2 + 35*b^4))/(3*a*d*Tan[c + d*x]^(3/2)))/(2*a*(a^2 + b^2) ) + (b^2*(15*a^2 + 7*b^2))/(a*(a^2 + b^2)*d*Tan[c + d*x]^(3/2)*(a + b*Tan[ c + d*x])))/(4*a*(a^2 + b^2))
3.7.7.3.1 Defintions of rubi rules used
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^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + Simp[1 /((m + 1)*(a^2 + b^2)*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x], x], x] / ; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || Integ erQ[m]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
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*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[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] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !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 = 0.09 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}+\frac {2 b^{4} \left (\frac {\left (\frac {19}{8} b \,a^{4}+\frac {15}{4} a^{2} b^{3}+\frac {11}{8} b^{5}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )+\frac {a \left (21 a^{4}+34 a^{2} b^{2}+13 b^{4}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{8}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (99 a^{4}+102 a^{2} b^{2}+35 b^{4}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {2}{3 a^{3} \tan \left (d x +c \right )^{\frac {3}{2}}}+\frac {6 b}{a^{4} \sqrt {\tan \left (d x +c \right )}}}{d}\) | \(372\) |
default | \(\frac {\frac {\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}+\frac {2 b^{4} \left (\frac {\left (\frac {19}{8} b \,a^{4}+\frac {15}{4} a^{2} b^{3}+\frac {11}{8} b^{5}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )+\frac {a \left (21 a^{4}+34 a^{2} b^{2}+13 b^{4}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{8}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (99 a^{4}+102 a^{2} b^{2}+35 b^{4}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {2}{3 a^{3} \tan \left (d x +c \right )^{\frac {3}{2}}}+\frac {6 b}{a^{4} \sqrt {\tan \left (d x +c \right )}}}{d}\) | \(372\) |
1/d*(2/(a^2+b^2)^3*(1/8*(-a^3+3*a*b^2)*2^(1/2)*(ln((1+2^(1/2)*tan(d*x+c)^( 1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))+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((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x +c)^(1/2)+tan(d*x+c)))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^ (1/2)*tan(d*x+c)^(1/2))))+2*b^4/a^4/(a^2+b^2)^3*(((19/8*b*a^4+15/4*a^2*b^3 +11/8*b^5)*tan(d*x+c)^(3/2)+1/8*a*(21*a^4+34*a^2*b^2+13*b^4)*tan(d*x+c)^(1 /2))/(a+b*tan(d*x+c))^2+1/8*(99*a^4+102*a^2*b^2+35*b^4)/(a*b)^(1/2)*arctan (b*tan(d*x+c)^(1/2)/(a*b)^(1/2)))-2/3/a^3/tan(d*x+c)^(3/2)+6/a^4*b/tan(d*x +c)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 4286 vs. \(2 (437) = 874\).
Time = 3.37 (sec) , antiderivative size = 8599, normalized size of antiderivative = 17.44 \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]
\[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right )^{3} \tan ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\frac {\frac {3 \, {\left (99 \, a^{4} b^{4} + 102 \, a^{2} b^{6} + 35 \, b^{8}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6}\right )} \sqrt {a b}} - \frac {8 \, a^{7} + 16 \, a^{5} b^{2} + 8 \, a^{3} b^{4} - 3 \, {\left (24 \, a^{4} b^{3} + 67 \, a^{2} b^{5} + 35 \, b^{7}\right )} \tan \left (d x + c\right )^{3} - {\left (136 \, a^{5} b^{2} + 335 \, a^{3} b^{4} + 175 \, a b^{6}\right )} \tan \left (d x + c\right )^{2} - 56 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{8} b^{2} + 2 \, a^{6} b^{4} + a^{4} b^{6}\right )} \tan \left (d x + c\right )^{\frac {7}{2}} + 2 \, {\left (a^{9} b + 2 \, a^{7} b^{3} + a^{5} b^{5}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} + {\left (a^{10} + 2 \, a^{8} b^{2} + a^{6} b^{4}\right )} \tan \left (d x + c\right )^{\frac {3}{2}}} - \frac {3 \, {\left (2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}}}{12 \, d} \]
1/12*(3*(99*a^4*b^4 + 102*a^2*b^6 + 35*b^8)*arctan(b*sqrt(tan(d*x + c))/sq rt(a*b))/((a^10 + 3*a^8*b^2 + 3*a^6*b^4 + a^4*b^6)*sqrt(a*b)) - (8*a^7 + 1 6*a^5*b^2 + 8*a^3*b^4 - 3*(24*a^4*b^3 + 67*a^2*b^5 + 35*b^7)*tan(d*x + c)^ 3 - (136*a^5*b^2 + 335*a^3*b^4 + 175*a*b^6)*tan(d*x + c)^2 - 56*(a^6*b + 2 *a^4*b^3 + a^2*b^5)*tan(d*x + c))/((a^8*b^2 + 2*a^6*b^4 + a^4*b^6)*tan(d*x + c)^(7/2) + 2*(a^9*b + 2*a^7*b^3 + a^5*b^5)*tan(d*x + c)^(5/2) + (a^10 + 2*a^8*b^2 + a^6*b^4)*tan(d*x + c)^(3/2)) - 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(tan(d*x + c)))) + 2*sq rt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqr t(tan(d*x + c)))) + sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*log(sqrt(2)*sq rt(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))/d
Timed out. \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\text {Timed out} \]
Time = 21.83 (sec) , antiderivative size = 20088, normalized size of antiderivative = 40.75 \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]
atan(((tan(c + d*x)^(1/2)*(47691333632*a^34*b^35*d^5 - 3156213760*a^30*b^3 9*d^5 - 7535067136*a^32*b^37*d^5 - 321126400*a^28*b^41*d^5 + 451224272896* a^36*b^33*d^5 + 1855390220288*a^38*b^31*d^5 + 4902111674368*a^40*b^29*d^5 + 9182617010176*a^42*b^27*d^5 + 12661071282176*a^44*b^25*d^5 + 12996430528 512*a^46*b^23*d^5 + 9861255397376*a^48*b^21*d^5 + 5375636013056*a^50*b^19* d^5 + 1966525644800*a^52*b^17*d^5 + 396976193536*a^54*b^15*d^5 + 185178521 6*a^56*b^13*d^5 - 18624806912*a^58*b^11*d^5 - 3332636672*a^60*b^9*d^5 - 11 7440512*a^62*b^7*d^5 - 8388608*a^64*b^5*d^5) + (1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b ^2*d^2)))^(1/2)*((tan(c + d*x)^(1/2)*(2569011200*a^29*b^46*d^7 + 509398220 80*a^31*b^44*d^7 + 479763365888*a^33*b^42*d^7 + 2849006157824*a^35*b^40*d^ 7 + 11943926562816*a^37*b^38*d^7 + 37510046547968*a^39*b^36*d^7 + 91385554 272256*a^41*b^34*d^7 + 176470173417472*a^43*b^32*d^7 + 273612095356928*a^4 5*b^30*d^7 + 342917730271232*a^47*b^28*d^7 + 347997439262720*a^49*b^26*d^7 + 285130161651712*a^51*b^24*d^7 + 187202969534464*a^53*b^22*d^7 + 9724576 0323584*a^55*b^20*d^7 + 39238359842816*a^57*b^18*d^7 + 12009902964736*a^59 *b^16*d^7 + 2725695717376*a^61*b^14*d^7 + 460706545664*a^63*b^12*d^7 + 626 31444480*a^65*b^10*d^7 + 6710886400*a^67*b^8*d^7 - 16777216*a^69*b^6*d^7 - 167772160*a^71*b^4*d^7 - 16777216*a^73*b^2*d^7) - (1i/(4*(b^6*d^2 - a^6*d ^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 1...