Integrand size = 23, antiderivative size = 322 \[ \int \frac {\tan ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left (3 a^4-26 a^2 b^2+3 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 \sqrt {a} \sqrt {b} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (c+d x)}}{1+\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {a \sqrt {\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \] Output:
-1/2*(a+b)*(a^2-4*a*b+b^2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)/(a^ 2+b^2)^3/d-1/2*(a+b)*(a^2-4*a*b+b^2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^ (1/2)/(a^2+b^2)^3/d+1/4*(3*a^4-26*a^2*b^2+3*b^4)*arctan(b^(1/2)*tan(d*x+c) ^(1/2)/a^(1/2))/a^(1/2)/b^(1/2)/(a^2+b^2)^3/d-1/2*(a-b)*(a^2+4*a*b+b^2)*ar ctanh(2^(1/2)*tan(d*x+c)^(1/2)/(1+tan(d*x+c)))*2^(1/2)/(a^2+b^2)^3/d+1/2*a *tan(d*x+c)^(1/2)/(a^2+b^2)/d/(a+b*tan(d*x+c))^2+1/4*(3*a^2-5*b^2)*tan(d*x +c)^(1/2)/(a^2+b^2)^2/d/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 2.46 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.01 \[ \int \frac {\tan ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {b^{9/2} \tan ^{\frac {5}{2}}(c+d x)-b^{7/2} \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))+\frac {(a+b \tan (c+d x)) \left (a^{5/2} b^{5/2} \left (a^2+b^2\right )^2 \sqrt {\tan (c+d x)}+\frac {1}{2} \left (a^{5/2} b^{5/2} \left (3 a^2-5 b^2\right ) \left (a^2+b^2\right ) \sqrt {\tan (c+d x)}+\left (4 \sqrt [4]{-1} a^{5/2} (a+i b)^3 b^{5/2} \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+a^2 b^2 \left (3 a^4-26 a^2 b^2+3 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )+4 \sqrt [4]{-1} a^{5/2} (a-i b)^3 b^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )\right ) (a+b \tan (c+d x))\right )\right )}{a^{3/2} \left (a^2+b^2\right )^2}}{2 a b^{5/2} \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2} \] Input:
Integrate[Tan[c + d*x]^(3/2)/(a + b*Tan[c + d*x])^3,x]
Output:
(b^(9/2)*Tan[c + d*x]^(5/2) - b^(7/2)*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d* x]) + ((a + b*Tan[c + d*x])*(a^(5/2)*b^(5/2)*(a^2 + b^2)^2*Sqrt[Tan[c + d* x]] + (a^(5/2)*b^(5/2)*(3*a^2 - 5*b^2)*(a^2 + b^2)*Sqrt[Tan[c + d*x]] + (4 *(-1)^(1/4)*a^(5/2)*(a + I*b)^3*b^(5/2)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x ]]] + a^2*b^2*(3*a^4 - 26*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*b^(5/2)*ArcTanh[(-1)^(3/4 )*Sqrt[Tan[c + d*x]]])*(a + b*Tan[c + d*x]))/2))/(a^(3/2)*(a^2 + b^2)^2))/ (2*a*b^(5/2)*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2)
Time = 1.58 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.12, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4050, 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.
| \(\displaystyle \int \frac {\tan ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^{3/2}}{(a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4050 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\int \frac {-3 a \tan ^2(c+d x)-4 b \tan (c+d x)+a}{2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}dx}{2 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\int \frac {-3 a \tan ^2(c+d x)-4 b \tan (c+d x)+a}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}dx}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\int \frac {-3 a \tan (c+d x)^2-4 b \tan (c+d x)+a}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}dx}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\int \frac {-16 b \tan (c+d x) a^2-\left (3 a^2-5 b^2\right ) \tan ^2(c+d x) a+\left (5 a^2-3 b^2\right ) a}{2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a \left (a^2+b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\int \frac {-16 b \tan (c+d x) a^2-\left (3 a^2-5 b^2\right ) \tan ^2(c+d x) a+\left (5 a^2-3 b^2\right ) a}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{2 a \left (a^2+b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\int \frac {-16 b \tan (c+d x) a^2-\left (3 a^2-5 b^2\right ) \tan (c+d x)^2 a+\left (5 a^2-3 b^2\right ) a}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{2 a \left (a^2+b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {\int \frac {8 \left (a^2 \left (a^2-3 b^2\right )-a b \left (3 a^2-b^2\right ) \tan (c+d x)\right )}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}-\frac {a \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan ^2(c+d x)+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {8 \int \frac {a^2 \left (a^2-3 b^2\right )-a b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}-\frac {a \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan ^2(c+d x)+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {8 \int \frac {a^2 \left (a^2-3 b^2\right )-a b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}-\frac {a \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {16 \int \frac {a \left (a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \tan (c+d x)\right )}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}-\frac {a \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {16 a \int \frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}-\frac {a \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {16 a \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d \left (a^2+b^2\right )}-\frac {a \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {16 a \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}-\frac {a \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {16 a \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {a \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {16 a \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {a \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {16 a \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {a \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {16 a \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {a \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {16 a \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {a \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {16 a \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {a \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {16 a \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {a \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}d\tan (c+d x)}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {16 a \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {2 a \left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {1}{a+b \tan (c+d x)}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\frac {16 a \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {2 \sqrt {a} \left (3 a^4-26 a^2 b^2+3 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}-\frac {\left (3 a^2-5 b^2\right ) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 \left (a^2+b^2\right )}\) |
Input:
Int[Tan[c + d*x]^(3/2)/(a + b*Tan[c + d*x])^3,x]
Output:
(a*Sqrt[Tan[c + d*x]])/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (((-2*Sq rt[a]*(3*a^4 - 26*a^2*b^2 + 3*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqr t[a]])/(Sqrt[b]*(a^2 + b^2)*d) + (16*a*(((a + b)*(a^2 - 4*a*b + b^2)*(-(Ar cTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Ta n[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)) - ((3*a^2 - 5*b^2)*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*d*(a + b*Tan[c + d*x])))/(4*(a^2 + b^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2/f Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & & NeQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n - 1)/(f*(m + 1)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^ (n - 2)*Simp[a*c^2*(m + 1) + a*d^2*(n - 1) + b*c*d*(m - n + 2) - (b*c^2 - 2 *a*c*d - b*d^2)*(m + 1)*Tan[e + f*x] - d*(b*c - a*d)*(m + n)*Tan[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^ 2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && LtQ[1, n, 2] && IntegerQ[ 2*m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*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 = 0.14 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.05
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {2 \left (\left (\frac {3}{8} b \,a^{4}-\frac {1}{4} a^{2} b^{3}-\frac {5}{8} b^{5}\right ) \tan \left (d x +c \right )^{\frac {3}{2}}+\frac {a \left (5 a^{4}+2 a^{2} b^{2}-3 b^{4}\right ) \sqrt {\tan \left (d x +c \right )}}{8}\right )}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (3 a^{4}-26 a^{2} b^{2}+3 b^{4}\right ) \arctan \left (\frac {b \sqrt {\tan \left (d x +c \right )}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(339\) |
| default | \(\frac {\frac {\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {2 \left (\left (\frac {3}{8} b \,a^{4}-\frac {1}{4} a^{2} b^{3}-\frac {5}{8} b^{5}\right ) \tan \left (d x +c \right )^{\frac {3}{2}}+\frac {a \left (5 a^{4}+2 a^{2} b^{2}-3 b^{4}\right ) \sqrt {\tan \left (d x +c \right )}}{8}\right )}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (3 a^{4}-26 a^{2} b^{2}+3 b^{4}\right ) \arctan \left (\frac {b \sqrt {\tan \left (d x +c \right )}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(339\) |
Input:
int(tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(2/(a^2+b^2)^3*(1/8*(-a^3+3*a*b^2)*2^(1/2)*(ln((tan(d*x+c)+2^(1/2)*tan (d*x+c)^(1/2)+1)/(tan(d*x+c)-2^(1/2)*tan(d*x+c)^(1/2)+1))+2*arctan(1+2^(1/ 2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)))+1/8*(3*a^2*b-b ^3)*2^(1/2)*(ln((tan(d*x+c)-2^(1/2)*tan(d*x+c)^(1/2)+1)/(tan(d*x+c)+2^(1/2 )*tan(d*x+c)^(1/2)+1))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^ (1/2)*tan(d*x+c)^(1/2))))+2/(a^2+b^2)^3*(((3/8*b*a^4-1/4*a^2*b^3-5/8*b^5)* tan(d*x+c)^(3/2)+1/8*a*(5*a^4+2*a^2*b^2-3*b^4)*tan(d*x+c)^(1/2))/(a+b*tan( d*x+c))^2+1/8*(3*a^4-26*a^2*b^2+3*b^4)/(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. 2398 vs. \(2 (284) = 568\).
Time = 0.45 (sec) , antiderivative size = 4822, normalized size of antiderivative = 14.98 \[ \int \frac {\tan ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\tan ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\int \frac {\tan ^{\frac {3}{2}}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \] Input:
integrate(tan(d*x+c)**(3/2)/(a+b*tan(d*x+c))**3,x)
Output:
Integral(tan(c + d*x)**(3/2)/(a + b*tan(c + d*x))**3, x)
Time = 0.13 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.25 \[ \int \frac {\tan ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {{\left (3 \, a^{4} - 26 \, a^{2} b^{2} + 3 \, b^{4}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (3 \, a^{2} b - 5 \, b^{3}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (5 \, a^{3} - 3 \, a b^{2}\right )} \sqrt {\tan \left (d x + c\right )}}{a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4} + {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \tan \left (d x + c\right )}}{4 \, d} \] Input:
integrate(tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x, algorithm="maxima")
Output:
1/4*((3*a^4 - 26*a^2*b^2 + 3*b^4)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/( (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + 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) + ((3*a^2*b - 5*b^3)*tan(d*x + c)^(3/2) + (5*a^3 - 3*a*b^2)*sqrt(tan(d*x + c)))/(a^6 + 2*a^4*b^2 + a^2*b^4 + (a^4*b^2 + 2*a^2 *b^4 + b^6)*tan(d*x + c)^2 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*tan(d*x + c)))/ d
\[ \int \frac {\tan ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\int { \frac {\tan \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:
integrate(tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x, algorithm="giac")
Output:
undef
Time = 6.80 (sec) , antiderivative size = 17010, normalized size of antiderivative = 52.83 \[ \int \frac {\tan ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \] Input:
int(tan(c + d*x)^(3/2)/(a + b*tan(c + d*x))^3,x)
Output:
atan((((((518*a*b^15*d^2 - 18*a^15*b*d^2 - 4494*a^3*b^13*d^2 + 3022*a^5*b^ 11*d^2 + 17194*a^7*b^9*d^2 + 5298*a^9*b^7*d^2 - 3338*a^11*b^5*d^2 + 506*a^ 13*b^3*d^2)/(2*(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 5 6*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^ 14*b^2*d^5)) + (((((4224*a^4*b^18*d^4 - 320*a^2*b^20*d^4 - 192*b^22*d^4 + 22272*a^6*b^16*d^4 + 51072*a^8*b^14*d^4 + 67200*a^10*b^12*d^4 + 53760*a^12 *b^10*d^4 + 25344*a^14*b^8*d^4 + 5952*a^16*b^6*d^4 + 192*a^18*b^4*d^4 - 12 8*a^20*b^2*d^4)/(2*(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5)) - (tan(c + d*x)^(1/2)*(1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b ^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i) )^(1/2)*(512*b^25*d^4 + 4608*a^2*b^23*d^4 + 17920*a^4*b^21*d^4 + 38400*a^6 *b^19*d^4 + 46080*a^8*b^17*d^4 + 21504*a^10*b^15*d^4 - 21504*a^12*b^13*d^4 - 46080*a^14*b^11*d^4 - 38400*a^16*b^9*d^4 - 17920*a^18*b^7*d^4 - 4608*a^ 20*b^5*d^4 - 512*a^22*b^3*d^4))/(4*(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28 *a^12*b^4*d^4 + 8*a^14*b^2*d^4)))*(1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^ 2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/ 2))/2 + (tan(c + d*x)^(1/2)*(1544*a*b^18*d^2 + 64*a^3*b^16*d^2 - 7456*a^5* b^14*d^2 - 576*a^7*b^12*d^2 + 19504*a^9*b^10*d^2 + 18880*a^11*b^8*d^2 +...
\[ \int \frac {\tan ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\int \frac {\sqrt {\tan \left (d x +c \right )}\, \tan \left (d x +c \right )}{\tan \left (d x +c \right )^{3} b^{3}+3 \tan \left (d x +c \right )^{2} a \,b^{2}+3 \tan \left (d x +c \right ) a^{2} b +a^{3}}d x \] Input:
int(tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x)
Output:
int((sqrt(tan(c + d*x))*tan(c + d*x))/(tan(c + d*x)**3*b**3 + 3*tan(c + d* x)**2*a*b**2 + 3*tan(c + d*x)*a**2*b + a**3),x)