Integrand size = 23, antiderivative size = 396 \[ \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^3} \, dx=-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {b^{3/2} \left (35 a^4+6 a^2 b^2+3 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
1/4*b^(3/2)*(35*a^4+6*a^2*b^2+3*b^4)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/ 2))/a^(5/2)/(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))/(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)+1/2*b^2*tan(d*x+c)^(1/2)/a/(a^2+b^2)/d/(a+b*tan(d*x+c))^2+1/4*b^2*(11*a ^2+3*b^2)*tan(d*x+c)^(1/2)/a^2/(a^2+b^2)^2/d/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 3.15 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.59 \[ \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^3} \, dx=\frac {\frac {-4 \sqrt [4]{-1} a^{5/2} (a+i b)^3 \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+b^{3/2} \left (35 a^4+6 a^2 b^2+3 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )-4 \sqrt [4]{-1} a^{5/2} (a-i b)^3 \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{a^{3/2} \left (a^2+b^2\right )^2}+\frac {2 b^2 \sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^2}+\frac {\left (11 a^2 b^2+3 b^4\right ) \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right ) d} \]
((-4*(-1)^(1/4)*a^(5/2)*(a + I*b)^3*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + b^(3/2)*(35*a^4 + 6*a^2*b^2 + 3*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]]) /Sqrt[a]] - 4*(-1)^(1/4)*a^(5/2)*(a - I*b)^3*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/(a^(3/2)*(a^2 + b^2)^2) + (2*b^2*Sqrt[Tan[c + d*x]])/(a + b*Tan [c + d*x])^2 + ((11*a^2*b^2 + 3*b^4)*Sqrt[Tan[c + d*x]])/(a*(a^2 + b^2)*(a + b*Tan[c + d*x])))/(4*a*(a^2 + b^2)*d)
Time = 1.72 (sec) , antiderivative size = 375, normalized size of antiderivative = 0.95, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4052, 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.
| \(\displaystyle \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4052 |
| \(\displaystyle \frac {\int \frac {4 a^2-4 b \tan (c+d x) a+3 b^2+3 b^2 \tan ^2(c+d x)}{2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}dx}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {4 a^2-4 b \tan (c+d x) a+3 b^2+3 b^2 \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}dx}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {4 a^2-4 b \tan (c+d x) a+3 b^2+3 b^2 \tan (c+d x)^2}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}dx}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {\frac {\int \frac {8 a^4-16 b \tan (c+d x) a^3+3 b^2 a^2+3 b^4+b^2 \left (11 a^2+3 b^2\right ) \tan ^2(c+d x)}{2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {8 a^4-16 b \tan (c+d x) a^3+3 b^2 a^2+3 b^4+b^2 \left (11 a^2+3 b^2\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {8 a^4-16 b \tan (c+d x) a^3+3 b^2 a^2+3 b^4+b^2 \left (11 a^2+3 b^2\right ) \tan (c+d x)^2}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {\frac {\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan ^2(c+d x)+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {\int \frac {8 \left (a^3 \left (a^2-3 b^2\right )-a^2 b \left (3 a^2-b^2\right ) \tan (c+d x)\right )}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan ^2(c+d x)+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {8 \int \frac {a^3 \left (a^2-3 b^2\right )-a^2 b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {8 \int \frac {a^3 \left (a^2-3 b^2\right )-a^2 b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {\frac {\frac {16 \int \frac {a^2 \left (a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \tan (c+d x)\right )}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}+\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {16 a^2 \int \frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}+\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {16 a^2 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {\frac {\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}d\tan (c+d x)}{d \left (a^2+b^2\right )}+\frac {16 a^2 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {\frac {2 b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {1}{a+b \tan (c+d x)}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}+\frac {16 a^2 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\frac {16 a^2 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {2 b^{3/2} \left (35 a^4+6 a^2 b^2+3 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}}{4 a \left (a^2+b^2\right )}\) |
(b^2*Sqrt[Tan[c + d*x]])/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (((2 *b^(3/2)*(35*a^4 + 6*a^2*b^2 + 3*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/ Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)*d) + (16*a^2*(((a + b)*(a^2 - 4*a*b + b^2)* (-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sq rt[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))/(2*a*(a^ 2 + b^2)) + (b^2*(11*a^2 + 3*b^2)*Sqrt[Tan[c + d*x]])/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])))/(4*a*(a^2 + b^2))
3.7.5.3.1 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 = 347, normalized size of antiderivative = 0.88
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}+\frac {2 b^{2} \left (\frac {\frac {b \left (11 a^{4}+14 a^{2} b^{2}+3 b^{4}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{8 a^{2}}+\frac {\left (13 a^{4}+18 a^{2} b^{2}+5 b^{4}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{8 a}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (35 a^{4}+6 a^{2} b^{2}+3 b^{4}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{8 a^{2} \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(347\) |
| default | \(\frac {\frac {\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}+\frac {2 b^{2} \left (\frac {\frac {b \left (11 a^{4}+14 a^{2} b^{2}+3 b^{4}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{8 a^{2}}+\frac {\left (13 a^{4}+18 a^{2} b^{2}+5 b^{4}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{8 a}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (35 a^{4}+6 a^{2} b^{2}+3 b^{4}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{8 a^{2} \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(347\) |
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^2/(a^2+b^2)^3*((1/8*b*(11*a^4+14*a^2*b^2+3*b ^4)/a^2*tan(d*x+c)^(3/2)+1/8*(13*a^4+18*a^2*b^2+5*b^4)/a*tan(d*x+c)^(1/2)) /(a+b*tan(d*x+c))^2+1/8*(35*a^4+6*a^2*b^2+3*b^4)/a^2/(a*b)^(1/2)*arctan(b* tan(d*x+c)^(1/2)/(a*b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 4126 vs. \(2 (348) = 696\).
Time = 1.26 (sec) , antiderivative size = 8279, normalized size of antiderivative = 20.91 \[ \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]
\[ \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^3} \, dx=\int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right )^{3} \sqrt {\tan {\left (c + d x \right )}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^3} \, dx=\frac {\frac {{\left (35 \, a^{4} b^{2} + 6 \, a^{2} b^{4} + 3 \, b^{6}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} \sqrt {a b}} + \frac {2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (11 \, a^{2} b^{3} + 3 \, b^{5}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (13 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \sqrt {\tan \left (d x + c\right )}}{a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4} + {\left (a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \tan \left (d x + c\right )}}{4 \, d} \]
1/4*((35*a^4*b^2 + 6*a^2*b^4 + 3*b^6)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b ))/((a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6)*sqrt(a*b)) + (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*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) + ((11*a^2*b^3 + 3*b^5)*tan(d*x + c)^(3/2) + (13*a^3*b^2 + 5*a*b^4)*sqrt(tan(d*x + c)))/(a^8 + 2*a^6*b^2 + a^4*b^4 + (a^6*b^2 + 2*a^4*b^4 + a^2*b^6)*tan(d*x + c)^2 + 2*(a^7*b + 2*a^5*b^3 + a^ 3*b^5)*tan(d*x + c)))/d
Timed out. \[ \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^3} \, dx=\text {Timed out} \]
Time = 11.36 (sec) , antiderivative size = 17833, normalized size of antiderivative = 45.03 \[ \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]
((tan(c + d*x)^(1/2)*(5*b^4 + 13*a^2*b^2))/(4*a*(a^4 + b^4 + 2*a^2*b^2)) + (b*tan(c + d*x)^(3/2)*(3*b^4 + 11*a^2*b^2))/(4*a^2*(a^4 + b^4 + 2*a^2*b^2 )))/(a^2*d + b^2*d*tan(c + d*x)^2 + 2*a*b*d*tan(c + d*x)) - atan((((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)*(((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)*((192*a^2*b^24*d^4 + 1728*a^4*b^22*d^4 + 8320*a^6*b^20*d^4 + 27264* a^8*b^18*d^4 + 62592*a^10*b^16*d^4 + 99456*a^12*b^14*d^4 + 107520*a^14*b^1 2*d^4 + 76800*a^16*b^10*d^4 + 33984*a^18*b^8*d^4 + 7872*a^20*b^6*d^4 + 384 *a^22*b^4*d^4 - 128*a^24*b^2*d^4)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^ 5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) - (tan(c + d*x)^(1/2)*(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*20 i + 15*a^4*b^2*d^2)))^(1/2)*(512*a^4*b^25*d^4 + 4608*a^6*b^23*d^4 + 17920* a^8*b^21*d^4 + 38400*a^10*b^19*d^4 + 46080*a^12*b^17*d^4 + 21504*a^14*b^15 *d^4 - 21504*a^16*b^13*d^4 - 46080*a^18*b^11*d^4 - 38400*a^20*b^9*d^4 - 17 920*a^22*b^7*d^4 - 4608*a^24*b^5*d^4 - 512*a^26*b^3*d^4))/(a^20*d^4 + a^4* b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b ^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)) + (tan(c + d *x)^(1/2)*(72*a*b^22*d^2 + 576*a^3*b^20*d^2 + 5024*a^5*b^18*d^2 + 14272...