Integrand size = 33, antiderivative size = 600 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {a^{3/2} \left (3 a^4 A b+6 a^2 A b^3+35 A b^5-15 a^5 B-46 a^3 b^2 B-63 a b^4 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{7/2} \left (a^2+b^2\right )^3 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left (3 a^3 A b+11 a A b^3-15 a^4 B-31 a^2 b^2 B-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{4 b^3 \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right ) \tan ^{\frac {3}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
1/4*a^(3/2)*(3*A*a^4*b+6*A*a^2*b^3+35*A*b^5-15*B*a^5-46*B*a^3*b^2-63*B*a*b ^4)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/b^(7/2)/(a^2+b^2)^3/d-1/2*(3* a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*arctan(-1+2^(1/2)*tan(d*x+c )^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1/2*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a* b^2*(A+B))*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1/4*(a ^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*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^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b *(A+B)-b^3*(A+B))*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*(3*A*a^3*b+11*A*a*b^3-15*B*a^4-31*B*a^2*b^2-8*B*b^4)*tan(d*x+c )^(1/2)/b^3/(a^2+b^2)^2/d+1/2*a*(A*b-B*a)*tan(d*x+c)^(5/2)/b/(a^2+b^2)/d/( a+b*tan(d*x+c))^2+1/4*a*(A*a^2*b+9*A*b^3-5*B*a^3-13*B*a*b^2)*tan(d*x+c)^(3 /2)/b^2/(a^2+b^2)^2/d/(a+b*tan(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(1351\) vs. \(2(600)=1200\).
Time = 6.45 (sec) , antiderivative size = 1351, normalized size of antiderivative = 2.25 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx =\text {Too large to display} \]
(2*B*Tan[c + d*x]^(5/2))/(b*d*(a + b*Tan[c + d*x])^2) + (2*(-(((A*b - 5*a* B)*Tan[c + d*x]^(3/2))/(b*d*(a + b*Tan[c + d*x])^2)) - (2*(-1/6*((-3*a*A*b + 15*a^2*B + b^2*B)*Sqrt[Tan[c + d*x]])/(b*d*(a + b*Tan[c + d*x])^2) - (2 *((((a*b^2*(3*a*A*b - 15*a^2*B - b^2*B))/8 - a*((3*b^4*B)/8 - (a*(3*a^2*A* b - 3*A*b^3 - 15*a^3*B - a*b^2*B))/8))*Sqrt[Tan[c + d*x]])/(2*a*(a^2 + b^2 )*d*(a + b*Tan[c + d*x])^2) + (((2*(a^2*b*((3*a^2*b^3*(A*b - a*B))/4 + (3* a^2*b*(3*a^2*A*b + 4*A*b^3 - 15*a^3*B - 16*a*b^2*B))/16 - (3*a*b*(3*a^3*A* b - 15*a^4*B - 16*a^2*b^2*B - 4*b^4*B))/16) + (a^2*((3*a^2*b^2*(3*a^2*A*b + 4*A*b^3 - 15*a^3*B - 16*a*b^2*B))/16 - a*((-3*a*b^4*(A*b - a*B))/4 - (3* a^2*(3*a^3*A*b - 15*a^4*B - 16*a^2*b^2*B - 4*b^4*B))/16)))/2 + b^2*((3*a^2 *(a^2 + b^2/2)*(3*a^2*A*b + 4*A*b^3 - 15*a^3*B - 16*a*b^2*B))/16 + (a*((-3 *a*b^4*(A*b - a*B))/4 - (3*a^2*(3*a^3*A*b - 15*a^4*B - 16*a^2*b^2*B - 4*b^ 4*B))/16))/2))*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*Sqrt [b]*(a^2 + b^2)*d) + ((-3*(-1)^(1/4)*a^5*A*b^3*ArcTan[(-1)^(3/4)*Sqrt[Tan[ c + d*x]]])/(4*d) - (9*(-1)^(3/4)*a^4*A*b^4*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/(4*d) + (9*(-1)^(1/4)*a^3*A*b^5*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d* x]]])/(4*d) + (3*(-1)^(3/4)*a^2*A*b^6*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]] ])/(4*d) + (3*(-1)^(3/4)*a^5*b^3*B*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/ (4*d) - (9*(-1)^(1/4)*a^4*b^4*B*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/(4* d) - (9*(-1)^(3/4)*a^3*b^5*B*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/(4*...
Time = 2.75 (sec) , antiderivative size = 528, normalized size of antiderivative = 0.88, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.788, Rules used = {3042, 4088, 27, 3042, 4128, 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.
| \(\displaystyle \int \frac {\tan ^{\frac {7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^{7/2} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4088 |
| \(\displaystyle \frac {\int -\frac {\tan ^{\frac {3}{2}}(c+d x) \left (\left (-5 B a^2+A b a-4 b^2 B\right ) \tan ^2(c+d x)-4 b (A b-a B) \tan (c+d x)+5 a (A b-a B)\right )}{2 (a+b \tan (c+d x))^2}dx}{2 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\int \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\left (-5 B a^2+A b a-4 b^2 B\right ) \tan ^2(c+d x)-4 b (A b-a B) \tan (c+d x)+5 a (A b-a B)\right )}{(a+b \tan (c+d x))^2}dx}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\int \frac {\tan (c+d x)^{3/2} \left (\left (-5 B a^2+A b a-4 b^2 B\right ) \tan (c+d x)^2-4 b (A b-a B) \tan (c+d x)+5 a (A b-a B)\right )}{(a+b \tan (c+d x))^2}dx}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\int \frac {\sqrt {\tan (c+d x)} \left (8 \left (A a^2+2 b B a-A b^2\right ) \tan (c+d x) b^2+\left (-15 B a^4+3 A b a^3-31 b^2 B a^2+11 A b^3 a-8 b^4 B\right ) \tan ^2(c+d x)+3 a \left (-5 B a^3+A b a^2-13 b^2 B a+9 A b^3\right )\right )}{2 (a+b \tan (c+d x))}dx}{b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\int \frac {\sqrt {\tan (c+d x)} \left (8 \left (A a^2+2 b B a-A b^2\right ) \tan (c+d x) b^2+\left (-15 B a^4+3 A b a^3-31 b^2 B a^2+11 A b^3 a-8 b^4 B\right ) \tan ^2(c+d x)+3 a \left (-5 B a^3+A b a^2-13 b^2 B a+9 A b^3\right )\right )}{a+b \tan (c+d x)}dx}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\int \frac {\sqrt {\tan (c+d x)} \left (8 \left (A a^2+2 b B a-A b^2\right ) \tan (c+d x) b^2+\left (-15 B a^4+3 A b a^3-31 b^2 B a^2+11 A b^3 a-8 b^4 B\right ) \tan (c+d x)^2+3 a \left (-5 B a^3+A b a^2-13 b^2 B a+9 A b^3\right )\right )}{a+b \tan (c+d x)}dx}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {2 \int -\frac {-8 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x) b^3+\left (-15 B a^5+3 A b a^4-31 b^2 B a^3+3 A b^3 a^2-24 b^4 B a+8 A b^5\right ) \tan ^2(c+d x)+a \left (-15 B a^4+3 A b a^3-31 b^2 B a^2+11 A b^3 a-8 b^4 B\right )}{2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{b}+\frac {2 \left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\int \frac {-8 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x) b^3+\left (-15 B a^5+3 A b a^4-31 b^2 B a^3+3 A b^3 a^2-24 b^4 B a+8 A b^5\right ) \tan ^2(c+d x)+a \left (-15 B a^4+3 A b a^3-31 b^2 B a^2+11 A b^3 a-8 b^4 B\right )}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{b}}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\int \frac {-8 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x) b^3+\left (-15 B a^5+3 A b a^4-31 b^2 B a^3+3 A b^3 a^2-24 b^4 B a+8 A b^5\right ) \tan (c+d x)^2+a \left (-15 B a^4+3 A b a^3-31 b^2 B a^2+11 A b^3 a-8 b^4 B\right )}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{b}}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {\int \frac {8 \left (b^3 \left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right )-b^3 \left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \tan (c+d x)\right )}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}+\frac {a^2 \left (-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\tan ^2(c+d x)+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {8 \int \frac {b^3 \left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right )-b^3 \left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}+\frac {a^2 \left (-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\tan ^2(c+d x)+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {8 \int \frac {b^3 \left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right )-b^3 \left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}+\frac {a^2 \left (-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {16 \int \frac {b^3 \left (A a^3+3 b B a^2-3 A b^2 a-b^3 B-\left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \tan (c+d x)\right )}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}+\frac {a^2 \left (-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {16 b^3 \int \frac {A a^3+3 b B a^2-3 A b^2 a-b^3 B-\left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}+\frac {a^2 \left (-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {16 b^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {a^2 \left (-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {16 b^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^2 \left (-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {16 b^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^2 \left (-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {16 b^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^2 \left (-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {16 b^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^2 \left (-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {16 b^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^2 \left (-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {16 b^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^2 \left (-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {a^2 \left (-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {16 b^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {a^2 \left (-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}d\tan (c+d x)}{d \left (a^2+b^2\right )}+\frac {16 b^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {2 a^2 \left (-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {1}{a+b \tan (c+d x)}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}+\frac {16 b^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {16 b^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {2 a^{3/2} \left (-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} d \left (a^2+b^2\right )}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
(a*(A*b - a*B)*Tan[c + d*x]^(5/2))/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]) ^2) - ((-(((2*a^(3/2)*(3*a^4*A*b + 6*a^2*A*b^3 + 35*A*b^5 - 15*a^5*B - 46* a^3*b^2*B - 63*a*b^4*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqr t[b]*(a^2 + b^2)*d) + (16*b^3*(-1/2*((3*a^2*b*(A - B) - b^3*(A - B) - a^3* (A + B) + 3*a*b^2*(A + B))*(-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[ 2]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2])) + ((a^3*(A - B) - 3 *a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*(-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*(3*a^3*A*b + 11*a*A*b^3 - 15*a^4*B - 31*a^2*b^2*B - 8*b^4*B)*Sqrt[Tan[c + d*x]])/(b*d)) /(2*b*(a^2 + b^2)) - (a*(a^2*A*b + 9*A*b^3 - 5*a^3*B - 13*a*b^2*B)*Tan[c + d*x]^(3/2))/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])))/(4*b*(a^2 + b^2))
3.5.10.3.1 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_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(b*c - a*d)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((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 - 2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a*A*d* (b*d*(m - 1) - a*c*(n + 1)) + (b*B*c - (A*b + a*B)*d)*(b*c*(m - 1) + a*d*(n + 1)) - d*((a*A - b*B)*(b*c - a*d) + (A*b + a*B)*(a*c + b*d))*(n + 1)*Tan[ e + f*x] - b*(d*(A*b*c + a*B*c - a*A*d)*(m + n) - b*B*(c^2*(m - 1) - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] & & LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
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 = 0.16 (sec) , antiderivative size = 466, normalized size of antiderivative = 0.78
| method | result | size |
| derivativedivides | \(\frac {\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right ) B}{b^{3}}+\frac {2 a^{2} \left (\frac {\left (-\frac {5}{8} A \,a^{4} b^{2}-\frac {9}{4} A \,a^{2} b^{4}-\frac {13}{8} A \,b^{6}+\frac {9}{8} B \,a^{5} b +\frac {13}{4} B \,a^{3} b^{3}+\frac {17}{8} B a \,b^{5}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )-\frac {a \left (3 A \,a^{4} b +14 A \,a^{2} b^{3}+11 A \,b^{5}-7 B \,a^{5}-22 B \,a^{3} b^{2}-15 B a \,b^{4}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{8}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (3 A \,a^{4} b +6 A \,a^{2} b^{3}+35 A \,b^{5}-15 B \,a^{5}-46 B \,a^{3} b^{2}-63 B a \,b^{4}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{3} \left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-3 A \,a^{2} b +A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(466\) |
| default | \(\frac {\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right ) B}{b^{3}}+\frac {2 a^{2} \left (\frac {\left (-\frac {5}{8} A \,a^{4} b^{2}-\frac {9}{4} A \,a^{2} b^{4}-\frac {13}{8} A \,b^{6}+\frac {9}{8} B \,a^{5} b +\frac {13}{4} B \,a^{3} b^{3}+\frac {17}{8} B a \,b^{5}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )-\frac {a \left (3 A \,a^{4} b +14 A \,a^{2} b^{3}+11 A \,b^{5}-7 B \,a^{5}-22 B \,a^{3} b^{2}-15 B a \,b^{4}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{8}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (3 A \,a^{4} b +6 A \,a^{2} b^{3}+35 A \,b^{5}-15 B \,a^{5}-46 B \,a^{3} b^{2}-63 B a \,b^{4}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{3} \left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-3 A \,a^{2} b +A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(466\) |
1/d*(2*tan(d*x+c)^(1/2)*B/b^3+2*a^2/b^3/(a^2+b^2)^3*(((-5/8*A*a^4*b^2-9/4* A*a^2*b^4-13/8*A*b^6+9/8*B*a^5*b+13/4*B*a^3*b^3+17/8*B*a*b^5)*tan(d*x+c)^( 3/2)-1/8*a*(3*A*a^4*b+14*A*a^2*b^3+11*A*b^5-7*B*a^5-22*B*a^3*b^2-15*B*a*b^ 4)*tan(d*x+c)^(1/2))/(a+b*tan(d*x+c))^2+1/8*(3*A*a^4*b+6*A*a^2*b^3+35*A*b^ 5-15*B*a^5-46*B*a^3*b^2-63*B*a*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*a^3-3*A*a*b^2+3*B*a^2*b-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)+t an(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*a^2*b+A*b^3+B*a^3-3*B*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)))) )
Leaf count of result is larger than twice the leaf count of optimal. 8982 vs. \(2 (545) = 1090\).
Time = 166.93 (sec) , antiderivative size = 17991, normalized size of antiderivative = 29.98 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\text {Timed out} \]
Time = 0.37 (sec) , antiderivative size = 571, normalized size of antiderivative = 0.95 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {{\left (15 \, B a^{7} - 3 \, A a^{6} b + 46 \, B a^{5} b^{2} - 6 \, A a^{4} b^{3} + 63 \, B a^{3} b^{4} - 35 \, A a^{2} b^{5}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} 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 ({\left (A + B\right )} a^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} 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 ({\left (A - B\right )} a^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} 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 ({\left (A - B\right )} a^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (9 \, B a^{5} b - 5 \, A a^{4} b^{2} + 17 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (7 \, B a^{6} - 3 \, A a^{5} b + 15 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} \sqrt {\tan \left (d x + c\right )}}{a^{6} b^{3} + 2 \, a^{4} b^{5} + a^{2} b^{7} + {\left (a^{4} b^{5} + 2 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{4} + 2 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )} - \frac {8 \, B \sqrt {\tan \left (d x + c\right )}}{b^{3}}}{4 \, d} \]
-1/4*((15*B*a^7 - 3*A*a^6*b + 46*B*a^5*b^2 - 6*A*a^4*b^3 + 63*B*a^3*b^4 - 35*A*a^2*b^5)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*sqrt(a*b)) - (2*sqrt(2)*((A + B)*a^3 - 3*(A - B)*a^2*b - 3*(A + B)*a*b^2 + (A - B)*b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan (d*x + c)))) + 2*sqrt(2)*((A + B)*a^3 - 3*(A - B)*a^2*b - 3*(A + B)*a*b^2 + (A - B)*b^3)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) + sqr t(2)*((A - B)*a^3 + 3*(A + B)*a^2*b - 3*(A - B)*a*b^2 - (A + B)*b^3)*log(s qrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - sqrt(2)*((A - B)*a^3 + 3*( A + B)*a^2*b - 3*(A - B)*a*b^2 - (A + B)*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) - ((9*B*a^5*b - 5*A*a^4*b^2 + 17*B*a^3*b^3 - 13*A*a^2*b^4)*tan(d*x + c)^(3/2) + (7*B*a^ 6 - 3*A*a^5*b + 15*B*a^4*b^2 - 11*A*a^3*b^3)*sqrt(tan(d*x + c)))/(a^6*b^3 + 2*a^4*b^5 + a^2*b^7 + (a^4*b^5 + 2*a^2*b^7 + b^9)*tan(d*x + c)^2 + 2*(a^ 5*b^4 + 2*a^3*b^6 + a*b^8)*tan(d*x + c)) - 8*B*sqrt(tan(d*x + c))/b^3)/d
Timed out. \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\text {Timed out} \]
Time = 71.27 (sec) , antiderivative size = 27429, normalized size of antiderivative = 45.72 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]
(log(((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^ 4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*d^2 - 24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2) - (64*B* a*b*(15*a^4 + 2*b^4 + 41*a^2*b^2))/d)*((4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^ 4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*d^2 - 24*B^2*a*b^5*d^2 - 24*B^2* a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (8*B^2*a*tan(c + d*x)^(1/2)*(22 5*a^14 - 184*b^14 + 608*a^2*b^12 - 272*a^4*b^10 + 3937*a^6*b^8 + 5804*a^8* b^6 + 4006*a^10*b^4 + 1380*a^12*b^2))/(b^4*d^2*(a^2 + b^2)^4))*((4*(-B^4*d ^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*d^2 - 2 4*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (2*B^3 *a^2*(1125*a^14 + 16*b^14 + 6112*a^2*b^12 - 17727*a^4*b^10 - 23239*a^6*b^8 - 11174*a^8*b^6 + 2930*a^10*b^4 + 3525*a^12*b^2))/(b^4*d^3*(a^2 + b^2)^6) )*((4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2*a^ 3*b^3*d^2 - 24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2 ))/4 - (B^4*tan(c + d*x)^(1/2)*(32*b^18 - 225*a^18 + 128*a^2*b^16 + 192*a^ 4*b^14 - 3841*a^6*b^12 + 18050*a^8*b^10 + 26801*a^10*b^8 + 16860*a^12*b^6 + 4049*a^14*b^4 - 30*a^16*b^2))/(b^5*d^4*(a^2 + b^2)^8))*((4*(-B^4*d^4*(a^ 6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*d^2 - 24*B^2* a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (B^5*a^3*(22 5*a^12 + 504*b^12 + 872*a^2*b^10 + 4457*a^4*b^8 + 5916*a^6*b^6 + 4006*a...