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

Optimal. Leaf size=75 \[ \text{CannotIntegrate}\left (\frac{\tan (a+b x) \sec (a+b x)}{c+d x},x\right )-\frac{\sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{b c}{d}+b x\right )}{d}-\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{d} \]

[Out]

CannotIntegrate[(Sec[a + b*x]*Tan[a + b*x])/(c + d*x), x] - (CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d -
(Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d

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Rubi [A]  time = 0.150145, 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 = {} \[ \int \frac{\sin (a+b x) \tan ^2(a+b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-((CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d) - (Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d + Defer[I
nt][(Sec[a + b*x]*Tan[a + b*x])/(c + d*x), x]

Rubi steps

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

Mathematica [A]  time = 3.81839, size = 0, normalized size = 0. \[ \int \frac{\sin (a+b x) \tan ^2(a+b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.299, size = 0, normalized size = 0. \begin{align*} \int{\frac{\sin \left ( bx+a \right ) \left ( \tan \left ( bx+a \right ) \right ) ^{2}}{dx+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(sin(b*x+a)*tan(b*x+a)^2/(d*x+c),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(sin(b*x+a)*tan(b*x+a)^2/(d*x+c),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{\sin \left (b x + a\right ) \tan \left (b x + a\right )^{2}}{d x + c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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