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

Optimal. Leaf size=79 \[ -\text{Unintegrable}\left (\frac{\tan (a+b x)}{c+d x},x\right )+\frac{2 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{d}+\frac{2 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.299071, 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 = {} \[ \int \frac{\sec (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\sec (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx &=\int \left (\frac{3 \cos (a+b x) \sin (a+b x)}{c+d x}-\frac{\sin ^2(a+b x) \tan (a+b x)}{c+d x}\right ) \, dx\\ &=3 \int \frac{\cos (a+b x) \sin (a+b x)}{c+d x} \, dx-\int \frac{\sin ^2(a+b x) \tan (a+b x)}{c+d x} \, dx\\ &=3 \int \frac{\sin (2 a+2 b x)}{2 (c+d x)} \, dx+\int \frac{\cos (a+b x) \sin (a+b x)}{c+d x} \, dx-\int \frac{\tan (a+b x)}{c+d x} \, dx\\ &=\frac{3}{2} \int \frac{\sin (2 a+2 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\\ &=\frac{1}{2} \int \frac{\sin (2 a+2 b x)}{c+d x} \, dx+\frac{1}{2} \left (3 \cos \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+\frac{1}{2} \left (3 \sin \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-\int \frac{\tan (a+b x)}{c+d x} \, dx\\ &=\frac{3 \text{Ci}\left (\frac{2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac{2 b c}{d}\right )}{2 d}+\frac{3 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{2 d}+\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+\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{2 \text{Ci}\left (\frac{2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac{2 b c}{d}\right )}{d}+\frac{2 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{d}-\int \frac{\tan (a+b x)}{c+d x} \, dx\\ \end{align*}

Mathematica [A]  time = 3.15739, size = 0, normalized size = 0. \[ \int \frac{\sec (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.371, size = 0, normalized size = 0. \begin{align*} \int{\frac{\sec \left ( bx+a \right ) \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)*sin(3*b*x+3*a)/(d*x+c),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

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