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

Optimal. Leaf size=101 \[ \text{Unintegrable}\left (\frac{\tan (a+b x)}{(c+d x)^2},x\right )-\frac{b \cos \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{d^2}+\frac{b \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{d^2}+\frac{\sin (2 a+2 b x)}{2 d (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.176326, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 2.50299, size = 0, normalized size = 0. \[ \int \frac{\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.812, size = 0, normalized size = 0. \begin{align*} \int{\frac{\sec \left ( bx+a \right ) \left ( \sin \left ( bx+a \right ) \right ) ^{3}}{ \left ( dx+c \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x+a)*sin(b*x+a)^3/(d*x+c)^2,x)

[Out]

int(sec(b*x+a)*sin(b*x+a)^3/(d*x+c)^2,x)

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)*sin(b*x+a)^3/(d*x+c)^2,x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sec \left (b x + a\right ) \sin \left (b x + a\right )}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)*sin(b*x+a)^3/(d*x+c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)*sin(b*x+a)**3/(d*x+c)**2,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (b x + a\right ) \sin \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)*sin(b*x+a)^3/(d*x+c)^2,x, algorithm="giac")

[Out]

integrate(sec(b*x + a)*sin(b*x + a)^3/(d*x + c)^2, x)