\(\int \frac {\sin ^2(a+b x) \tan (a+b x)}{c+d x} \, dx\) [225]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {\sin ^2(a+b x) \tan (a+b x)}{c+d x} \, dx=-\frac {\operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{2 d}-\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\text {Int}\left (\frac {\tan (a+b x)}{c+d x},x\right ) \]

[Out]

-1/2*cos(2*a-2*b*c/d)*Si(2*b*c/d+2*b*x)/d-1/2*Ci(2*b*c/d+2*b*x)*sin(2*a-2*b*c/d)/d+Unintegrable(tan(b*x+a)/(d*
x+c),x)

Rubi [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sin ^2(a+b x) \tan (a+b x)}{c+d x} \, dx=\int \frac {\sin ^2(a+b x) \tan (a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

-1/2*(CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/d - (Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2
*b*x])/(2*d) + Defer[Int][Tan[a + b*x]/(c + d*x), x]

Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx+\int \frac {\tan (a+b x)}{c+d x} \, dx \\ & = -\int \frac {\sin (2 a+2 b x)}{2 (c+d x)} \, dx+\int \frac {\tan (a+b x)}{c+d x} \, dx \\ & = -\left (\frac {1}{2} \int \frac {\sin (2 a+2 b x)}{c+d x} \, dx\right )+\int \frac {\tan (a+b x)}{c+d x} \, dx \\ & = -\left (\frac {1}{2} \cos \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx\right )-\frac {1}{2} \sin \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx+\int \frac {\tan (a+b x)}{c+d x} \, dx \\ & = -\frac {\operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{2 d}-\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\int \frac {\tan (a+b x)}{c+d x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.89 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sin ^2(a+b x) \tan (a+b x)}{c+d x} \, dx=\int \frac {\sin ^2(a+b x) \tan (a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

\[\int \frac {\sec \left (x b +a \right ) \sin \left (x b +a \right )^{3}}{d x +c}d x\]

[In]

int(sec(b*x+a)*sin(b*x+a)^3/(d*x+c),x)

[Out]

int(sec(b*x+a)*sin(b*x+a)^3/(d*x+c),x)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {\sin ^2(a+b x) \tan (a+b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right ) \sin \left (b x + a\right )^{3}}{d x + c} \,d x } \]

[In]

integrate(sec(b*x+a)*sin(b*x+a)^3/(d*x+c),x, algorithm="fricas")

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {\sin ^2(a+b x) \tan (a+b x)}{c+d x} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

integrate(sec(b*x+a)*sin(b*x+a)**3/(d*x+c),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 182, normalized size of antiderivative = 8.27 \[ \int \frac {\sin ^2(a+b x) \tan (a+b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right ) \sin \left (b x + a\right )^{3}}{d x + c} \,d x } \]

[In]

integrate(sec(b*x+a)*sin(b*x+a)^3/(d*x+c),x, algorithm="maxima")

[Out]

1/4*((-I*exp_integral_e(1, 2*(-I*b*d*x - I*b*c)/d) + I*exp_integral_e(1, -2*(-I*b*d*x - I*b*c)/d))*cos(-2*(b*c
- a*d)/d) + 8*d*integrate(sin(2*b*x + 2*a)/((d*x + c)*cos(2*b*x + 2*a)^2 + (d*x + c)*sin(2*b*x + 2*a)^2 + d*x
+ 2*(d*x + c)*cos(2*b*x + 2*a) + c), x) + (exp_integral_e(1, 2*(-I*b*d*x - I*b*c)/d) + exp_integral_e(1, -2*(
-I*b*d*x - I*b*c)/d))*sin(-2*(b*c - a*d)/d))/d

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sin ^2(a+b x) \tan (a+b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right ) \sin \left (b x + a\right )^{3}}{d x + c} \,d x } \]

[In]

integrate(sec(b*x+a)*sin(b*x+a)^3/(d*x+c),x, algorithm="giac")

[Out]

integrate(sec(b*x + a)*sin(b*x + a)^3/(d*x + c), x)

Mupad [N/A]

Not integrable

Time = 25.53 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {\sin ^2(a+b x) \tan (a+b x)}{c+d x} \, dx=\int \frac {{\sin \left (a+b\,x\right )}^3}{\cos \left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]

[In]

int(sin(a + b*x)^3/(cos(a + b*x)*(c + d*x)),x)

[Out]

int(sin(a + b*x)^3/(cos(a + b*x)*(c + d*x)), x)