\(\int \sin ^2(a+b x) \tan (c+d x) \, dx\) [259]

Optimal result

Integrand size = 15, antiderivative size = 142 \[ \int \sin ^2(a+b x) \tan (c+d x) \, dx=\frac {i x}{2}+\frac {e^{-2 i (a+b x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {b}{d},1-\frac {b}{d},-e^{2 i (c+d x)}\right )}{4 b}-\frac {e^{2 i (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b}{d},\frac {b+d}{d},-e^{2 i (c+d x)}\right )}{4 b}-\frac {\log \left (1+e^{2 i (c+d x)}\right )}{2 d}+\frac {i \cos (a+b x) \sin (a+b x)}{2 b} \] Output:

1/2*I*x+1/4*hypergeom([1, -b/d],[1-b/d],-exp(2*I*(d*x+c)))/b/exp(2*I*(b*x+ 
a))-1/4*exp(2*I*(b*x+a))*hypergeom([1, b/d],[(b+d)/d],-exp(2*I*(d*x+c)))/b 
-1/2*ln(1+exp(2*I*(d*x+c)))/d+1/2*I*cos(b*x+a)*sin(b*x+a)/b
 

Mathematica [A] (verified)

Time = 5.77 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.10 \[ \int \sin ^2(a+b x) \tan (c+d x) \, dx=-\frac {e^{-2 i (a+b x)} \left (d-d e^{4 i (a+b x)}-4 i b d e^{2 i (a+b x)} x-2 d \operatorname {Hypergeometric2F1}\left (1,-\frac {b}{d},1-\frac {b}{d},-e^{2 i (c+d x)}\right )+2 d e^{4 i (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b}{d},\frac {b+d}{d},-e^{2 i (c+d x)}\right )+4 b e^{2 i (a+b x)} \log \left (1+e^{2 i (c+d x)}\right )\right )}{8 b d} \] Input:

Integrate[Sin[a + b*x]^2*Tan[c + d*x],x]
 

Output:

-1/8*(d - d*E^((4*I)*(a + b*x)) - (4*I)*b*d*E^((2*I)*(a + b*x))*x - 2*d*Hy 
pergeometric2F1[1, -(b/d), 1 - b/d, -E^((2*I)*(c + d*x))] + 2*d*E^((4*I)*( 
a + b*x))*Hypergeometric2F1[1, b/d, (b + d)/d, -E^((2*I)*(c + d*x))] + 4*b 
*E^((2*I)*(a + b*x))*Log[1 + E^((2*I)*(c + d*x))])/(b*d*E^((2*I)*(a + b*x) 
))
 

Rubi [F]

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.

\(\displaystyle \int \sin ^2(a+b x) \tan (c+d x) \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \sin ^2(a+b x) \tan (c+d x)dx\)

Input:

Int[Sin[a + b*x]^2*Tan[c + d*x],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [F]

\[\int \sin \left (b x +a \right )^{2} \tan \left (d x +c \right )d x\]

Input:

int(sin(b*x+a)^2*tan(d*x+c),x)
 

Output:

int(sin(b*x+a)^2*tan(d*x+c),x)
 

Fricas [F]

\[ \int \sin ^2(a+b x) \tan (c+d x) \, dx=\int { \sin \left (b x + a\right )^{2} \tan \left (d x + c\right ) \,d x } \] Input:

integrate(sin(b*x+a)^2*tan(d*x+c),x, algorithm="fricas")
 

Output:

integral(-(cos(b*x + a)^2 - 1)*tan(d*x + c), x)
 

Sympy [F]

\[ \int \sin ^2(a+b x) \tan (c+d x) \, dx=\int \sin ^{2}{\left (a + b x \right )} \tan {\left (c + d x \right )}\, dx \] Input:

integrate(sin(b*x+a)**2*tan(d*x+c),x)
 

Output:

Integral(sin(a + b*x)**2*tan(c + d*x), x)
 

Maxima [F]

\[ \int \sin ^2(a+b x) \tan (c+d x) \, dx=\int { \sin \left (b x + a\right )^{2} \tan \left (d x + c\right ) \,d x } \] Input:

integrate(sin(b*x+a)^2*tan(d*x+c),x, algorithm="maxima")
 

Output:

integrate(sin(b*x + a)^2*tan(d*x + c), x)
 

Giac [F]

\[ \int \sin ^2(a+b x) \tan (c+d x) \, dx=\int { \sin \left (b x + a\right )^{2} \tan \left (d x + c\right ) \,d x } \] Input:

integrate(sin(b*x+a)^2*tan(d*x+c),x, algorithm="giac")
 

Output:

integrate(sin(b*x + a)^2*tan(d*x + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sin ^2(a+b x) \tan (c+d x) \, dx=\int {\sin \left (a+b\,x\right )}^2\,\mathrm {tan}\left (c+d\,x\right ) \,d x \] Input:

int(sin(a + b*x)^2*tan(c + d*x),x)
 

Output:

int(sin(a + b*x)^2*tan(c + d*x), x)
 

Reduce [F]

\[ \int \sin ^2(a+b x) \tan (c+d x) \, dx=\int \sin \left (b x +a \right )^{2} \tan \left (d x +c \right )d x \] Input:

int(sin(b*x+a)^2*tan(d*x+c),x)
 

Output:

int(sin(a + b*x)**2*tan(c + d*x),x)