Integrand size = 23, antiderivative size = 299 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {2 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d} \]
-1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/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))/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))/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))/d*2^(1 /2)+2*a*(a^2-3*b^2)*tan(d*x+c)^(1/2)/d+2/3*b*(3*a^2-b^2)*tan(d*x+c)^(3/2)/ d+32/35*a*b^2*tan(d*x+c)^(5/2)/d+2/7*b^2*tan(d*x+c)^(5/2)*(a+b*tan(d*x+c)) /d
Result contains complex when optimal does not.
Time = 1.48 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.48 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {105 \sqrt [4]{-1} (a-i b)^3 \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+105 \sqrt [4]{-1} (a+i b)^3 \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+2 \sqrt {\tan (c+d x)} \left (105 \left (a^3-3 a b^2\right )-35 b \left (-3 a^2+b^2\right ) \tan (c+d x)+63 a b^2 \tan ^2(c+d x)+15 b^3 \tan ^3(c+d x)\right )}{105 d} \]
(105*(-1)^(1/4)*(a - I*b)^3*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 105*(- 1)^(1/4)*(a + I*b)^3*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]*(105*(a^3 - 3*a*b^2) - 35*b*(-3*a^2 + b^2)*Tan[c + d*x] + 63*a*b^ 2*Tan[c + d*x]^2 + 15*b^3*Tan[c + d*x]^3))/(105*d)
Time = 1.06 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.93, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.870, Rules used = {3042, 4049, 27, 3042, 4113, 3042, 4011, 3042, 4011, 3042, 4017, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (c+d x)^{3/2} (a+b \tan (c+d x))^3dx\) |
| \(\Big \downarrow \) 4049 |
| \(\displaystyle \frac {2}{7} \int \frac {1}{2} \tan ^{\frac {3}{2}}(c+d x) \left (16 a b^2 \tan ^2(c+d x)+7 b \left (3 a^2-b^2\right ) \tan (c+d x)+a \left (7 a^2-5 b^2\right )\right )dx+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \tan ^{\frac {3}{2}}(c+d x) \left (16 a b^2 \tan ^2(c+d x)+7 b \left (3 a^2-b^2\right ) \tan (c+d x)+a \left (7 a^2-5 b^2\right )\right )dx+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \tan (c+d x)^{3/2} \left (16 a b^2 \tan (c+d x)^2+7 b \left (3 a^2-b^2\right ) \tan (c+d x)+a \left (7 a^2-5 b^2\right )\right )dx+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {1}{7} \left (\int \tan ^{\frac {3}{2}}(c+d x) \left (7 a \left (a^2-3 b^2\right )+7 b \left (3 a^2-b^2\right ) \tan (c+d x)\right )dx+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\int \tan (c+d x)^{3/2} \left (7 a \left (a^2-3 b^2\right )+7 b \left (3 a^2-b^2\right ) \tan (c+d x)\right )dx+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {1}{7} \left (\int \sqrt {\tan (c+d x)} \left (7 a \left (a^2-3 b^2\right ) \tan (c+d x)-7 b \left (3 a^2-b^2\right )\right )dx+\frac {14 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\int \sqrt {\tan (c+d x)} \left (7 a \left (a^2-3 b^2\right ) \tan (c+d x)-7 b \left (3 a^2-b^2\right )\right )dx+\frac {14 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {1}{7} \left (\int \frac {-7 a \left (a^2-3 b^2\right )-7 b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx+\frac {14 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\int \frac {-7 a \left (a^2-3 b^2\right )-7 b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx+\frac {14 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {1}{7} \left (\frac {2 \int -\frac {7 \left (a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan (c+d x)\right )}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}+\frac {14 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \int \frac {a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}+\frac {14 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}+\frac {14 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d}+\frac {14 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {14 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {14 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {14 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {14 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {14 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}+\frac {14 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\) |
(2*b^2*Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x]))/(7*d) + ((-14*(((a - b)*(a ^2 + 4*a*b + b^2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2]) + Arc Tan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2]))/2 + ((a + b)*(a^2 - 4*a*b + b^2)*(-1/2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/Sqrt[2] + Lo g[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2])))/2))/d + (14 *a*(a^2 - 3*b^2)*Sqrt[Tan[c + d*x]])/d + (14*b*(3*a^2 - b^2)*Tan[c + d*x]^ (3/2))/(3*d) + (32*a*b^2*Tan[c + d*x]^(5/2))/(5*d))/7
3.6.70.3.1 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_)^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_) + (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[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
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 - 2)*((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 - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*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] && GtQ[m, 2] && (GeQ[n, -1] || I ntegerQ[m]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) )
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && !LeQ[m, -1]
Time = 0.06 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.93
| method | result | size |
| derivativedivides | \(\frac {\frac {2 b^{3} \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right )}{7}+\frac {6 a \,b^{2} \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+2 a^{2} b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )-\frac {2 b^{3} \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+2 a^{3} \left (\sqrt {\tan }\left (d x +c \right )\right )-6 a \,b^{2} \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\) | \(279\) |
| default | \(\frac {\frac {2 b^{3} \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right )}{7}+\frac {6 a \,b^{2} \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+2 a^{2} b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )-\frac {2 b^{3} \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+2 a^{3} \left (\sqrt {\tan }\left (d x +c \right )\right )-6 a \,b^{2} \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\) | \(279\) |
| parts | \(\frac {a^{3} \left (2 \left (\sqrt {\tan }\left (d x +c \right )\right )-\frac {\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}\right )}{d}+\frac {b^{3} \left (\frac {2 \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right )}{7}-\frac {2 \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+\frac {\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}\right )}{d}+\frac {3 a \,b^{2} \left (\frac {2 \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}-2 \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {\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}\right )}{d}+\frac {3 a^{2} b \left (\frac {2 \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-\frac {\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}\right )}{d}\) | \(438\) |
1/d*(2/7*b^3*tan(d*x+c)^(7/2)+6/5*a*b^2*tan(d*x+c)^(5/2)+2*a^2*b*tan(d*x+c )^(3/2)-2/3*b^3*tan(d*x+c)^(3/2)+2*a^3*tan(d*x+c)^(1/2)-6*a*b^2*tan(d*x+c) ^(1/2)+1/4*(-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)))+1/4*(-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)+ta n(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. 1427 vs. \(2 (257) = 514\).
Time = 0.30 (sec) , antiderivative size = 1427, normalized size of antiderivative = 4.77 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\text {Too large to display} \]
1/210*(105*d*sqrt(-(6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*sqrt(-(a^12 - 30* a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d ^4))/d^2)*log(((3*a^2*b - b^3)*d^3*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4) + (a^9 - 18*a^7*b^ 2 + 60*a^5*b^4 - 46*a^3*b^6 + 3*a*b^8)*d)*sqrt(-(6*a^5*b - 20*a^3*b^3 + 6* a*b^5 + d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^ 4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2) - (a^12 - 12*a^10*b^2 - 27*a^8*b^4 + 27*a^4*b^8 + 12*a^2*b^10 - b^12)*sqrt(tan(d*x + c))) - 105*d*sqrt(-(6*a^ 5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2)*log(-((3*a^2*b - b^3)*d^3*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^ 4*b^8 - 30*a^2*b^10 + b^12)/d^4) + (a^9 - 18*a^7*b^2 + 60*a^5*b^4 - 46*a^3 *b^6 + 3*a*b^8)*d)*sqrt(-(6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b ^12)/d^4))/d^2) - (a^12 - 12*a^10*b^2 - 27*a^8*b^4 + 27*a^4*b^8 + 12*a^2*b ^10 - b^12)*sqrt(tan(d*x + c))) - 105*d*sqrt(-(6*a^5*b - 20*a^3*b^3 + 6*a* b^5 - d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4* b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2)*log(((3*a^2*b - b^3)*d^3*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^ 12)/d^4) - (a^9 - 18*a^7*b^2 + 60*a^5*b^4 - 46*a^3*b^6 + 3*a*b^8)*d)*sq...
\[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{3} \tan ^{\frac {3}{2}}{\left (c + d x \right )}\, dx \]
Time = 0.33 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.86 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {120 \, b^{3} \tan \left (d x + c\right )^{\frac {7}{2}} + 504 \, a b^{2} \tan \left (d x + c\right )^{\frac {5}{2}} - 210 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 210 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 105 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 105 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 280 \, {\left (3 \, a^{2} b - b^{3}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + 840 \, {\left (a^{3} - 3 \, a b^{2}\right )} \sqrt {\tan \left (d x + c\right )}}{420 \, d} \]
1/420*(120*b^3*tan(d*x + c)^(7/2) + 504*a*b^2*tan(d*x + c)^(5/2) - 210*sqr t(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)))) - 210*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)))) - 105*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) + 105*sqr t(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) + 280*(3*a^2*b - b^3)*tan(d*x + c)^(3/2) + 840*(a^3 - 3*a*b ^2)*sqrt(tan(d*x + c)))/d
Timed out. \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\text {Timed out} \]
Time = 9.67 (sec) , antiderivative size = 1729, normalized size of antiderivative = 5.78 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\text {Too large to display} \]
tan(c + d*x)^(1/2)*((2*a^3)/d - (6*a*b^2)/d) - tan(c + d*x)^(3/2)*((2*b^3) /(3*d) - (2*a^2*b)/d) - atan((((8*(4*a^3*d^2 - 12*a*b^2*d^2)*(-(6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*d^2 ))^(1/2))/d^3 - (16*tan(c + d*x)^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^ 2))/d^2)*(-(6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*d^2))^(1/2)*1i - ((8*(4*a^3*d^2 - 12*a*b^2*d^2)*(-(6*a* b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/ (4*d^2))^(1/2))/d^3 + (16*tan(c + d*x)^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15* a^4*b^2))/d^2)*(-(6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a ^3*b^3 + a^4*b^2*15i)/(4*d^2))^(1/2)*1i)/((16*(3*a^8*b - b^9 + 6*a^4*b^5 + 8*a^6*b^3))/d^3 + ((8*(4*a^3*d^2 - 12*a*b^2*d^2)*(-(6*a*b^5 + 6*a^5*b - a ^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*d^2))^(1/2))/d ^3 - (16*tan(c + d*x)^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2)*(- (6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2* 15i)/(4*d^2))^(1/2) + ((8*(4*a^3*d^2 - 12*a*b^2*d^2)*(-(6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*d^2))^(1/2) )/d^3 + (16*tan(c + d*x)^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2) *(-(6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b ^2*15i)/(4*d^2))^(1/2)))*(-(6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4* 15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*d^2))^(1/2)*2i - atan((((8*(4*a^3*d^...