Integrand size = 28, antiderivative size = 157 \[ \int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^n \, dx=\frac {a^3 (i c-d (5+2 n)) (c+d \tan (e+f x))^{1+n}}{d^2 f (1+n) (2+n)}+\frac {4 a^3 \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c+d \tan (e+f x)}{c-i d}\right ) (c+d \tan (e+f x))^{1+n}}{(i c+d) f (1+n)}-\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{1+n}}{d f (2+n)} \]
[Out]
Time = 0.41 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3637, 3673, 3618, 70} \[ \int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^n \, dx=\frac {a^3 (-d (2 n+5)+i c) (c+d \tan (e+f x))^{n+1}}{d^2 f (n+1) (n+2)}+\frac {4 a^3 (c+d \tan (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {c+d \tan (e+f x)}{c-i d}\right )}{f (n+1) (d+i c)}-\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{n+1}}{d f (n+2)} \]
[In]
[Out]
Rule 70
Rule 3618
Rule 3637
Rule 3673
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{1+n}}{d f (2+n)}+\frac {a \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^n (a (i c+d (3+2 n))+a (c+i d (5+2 n)) \tan (e+f x)) \, dx}{d (2+n)} \\ & = \frac {a^3 (i c-d (5+2 n)) (c+d \tan (e+f x))^{1+n}}{d^2 f (1+n) (2+n)}-\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{1+n}}{d f (2+n)}+\frac {a \int (c+d \tan (e+f x))^n \left (4 a^2 d (2+n)+4 i a^2 d (2+n) \tan (e+f x)\right ) \, dx}{d (2+n)} \\ & = \frac {a^3 (i c-d (5+2 n)) (c+d \tan (e+f x))^{1+n}}{d^2 f (1+n) (2+n)}-\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{1+n}}{d f (2+n)}+\frac {\left (16 i a^5 d (2+n)\right ) \text {Subst}\left (\int \frac {\left (c-\frac {i x}{4 a^2 (2+n)}\right )^n}{-16 a^4 d^2 (2+n)^2+4 a^2 d (2+n) x} \, dx,x,4 i a^2 d (2+n) \tan (e+f x)\right )}{f} \\ & = \frac {a^3 (i c-d (5+2 n)) (c+d \tan (e+f x))^{1+n}}{d^2 f (1+n) (2+n)}+\frac {4 a^3 \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c+d \tan (e+f x)}{c-i d}\right ) (c+d \tan (e+f x))^{1+n}}{(i c+d) f (1+n)}-\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{1+n}}{d f (2+n)} \\ \end{align*}
Time = 1.76 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.71 \[ \int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^n \, dx=-\frac {a^3 (c+d \tan (e+f x))^{1+n} \left (-4 d^2 (2+n) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c+d \tan (e+f x)}{c-i d}\right )+(c-i d) (c+3 i d (2+n)-d (1+n) \tan (e+f x))\right )}{d^2 (i c+d) f (1+n) (2+n)} \]
[In]
[Out]
\[\int \left (a +i a \tan \left (f x +e \right )\right )^{3} \left (c +d \tan \left (f x +e \right )\right )^{n}d x\]
[In]
[Out]
\[ \int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^n \, dx=\int { {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3} {\left (d \tan \left (f x + e\right ) + c\right )}^{n} \,d x } \]
[In]
[Out]
\[ \int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^n \, dx=- i a^{3} \left (\int i \left (c + d \tan {\left (e + f x \right )}\right )^{n}\, dx + \int \left (- 3 \left (c + d \tan {\left (e + f x \right )}\right )^{n} \tan {\left (e + f x \right )}\right )\, dx + \int \left (c + d \tan {\left (e + f x \right )}\right )^{n} \tan ^{3}{\left (e + f x \right )}\, dx + \int \left (- 3 i \left (c + d \tan {\left (e + f x \right )}\right )^{n} \tan ^{2}{\left (e + f x \right )}\right )\, dx\right ) \]
[In]
[Out]
\[ \int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^n \, dx=\int { {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3} {\left (d \tan \left (f x + e\right ) + c\right )}^{n} \,d x } \]
[In]
[Out]
\[ \int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^n \, dx=\int { {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3} {\left (d \tan \left (f x + e\right ) + c\right )}^{n} \,d x } \]
[In]
[Out]
Timed out. \[ \int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^n \, dx=\int {\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n \,d x \]
[In]
[Out]