3.392 \(\int \frac{\sec ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx\)

Optimal. Leaf size=77 \[ -\text{CannotIntegrate}\left (\frac{\tan (a+b x) \sec (a+b x)}{c+d x},x\right )+\frac{4 \sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{b c}{d}+b x\right )}{d}+\frac{4 \cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{d} \]

[Out]

-CannotIntegrate[(Sec[a + b*x]*Tan[a + b*x])/(c + d*x), x] + (4*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d
+ (4*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d

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Rubi [A]  time = 0.266208, 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 = {} \[ \int \frac{\sec ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Sec[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x),x]

[Out]

(4*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d + (4*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d - Defer[
Int][(Sec[a + b*x]*Tan[a + b*x])/(c + d*x), x]

Rubi steps

\begin{align*} \int \frac{\sec ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx &=\int \left (\frac{3 \sin (a+b x)}{c+d x}-\frac{\sin (a+b x) \tan ^2(a+b x)}{c+d x}\right ) \, dx\\ &=3 \int \frac{\sin (a+b x)}{c+d x} \, dx-\int \frac{\sin (a+b x) \tan ^2(a+b x)}{c+d x} \, dx\\ &=\left (3 \cos \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sin \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx+\left (3 \sin \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cos \left (\frac{b c}{d}+b x\right )}{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\\ &=\frac{3 \text{Ci}\left (\frac{b c}{d}+b x\right ) \sin \left (a-\frac{b c}{d}\right )}{d}+\frac{3 \cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{d}+\cos \left (a-\frac{b c}{d}\right ) \int \frac{\sin \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx+\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{4 \text{Ci}\left (\frac{b c}{d}+b x\right ) \sin \left (a-\frac{b c}{d}\right )}{d}+\frac{4 \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 = 13.5578, size = 0, normalized size = 0. \[ \int \frac{\sec ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(Sec[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x),x]

[Out]

Integrate[(Sec[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x), x]

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Maple [A]  time = 0.388, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \sec \left ( bx+a \right ) \right ) ^{2}\sin \left ( 3\,bx+3\,a \right ) }{dx+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(sec(b*x+a)^2*sin(3*b*x+3*a)/(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(sec(b*x+a)^2*sin(3*b*x+3*a)/(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{\sec \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{d x + c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sec(b*x + a)^2*sin(3*b*x + 3*a)/(d*x + c), 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)**2*sin(3*b*x+3*a)/(d*x+c),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 )^{2} \sin \left (3 \, b x + 3 \, a\right )}{d x + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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