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

Optimal. Leaf size=80 \[ -\text {Int}\left (\frac {\tan (a+b x)}{c+d x},x\right )+\frac {2 \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Ci}\left (\frac {2 b c}{d}+2 b x\right )}{d}+\frac {2 \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d} \]

[Out]

2*cos(2*a-2*b*c/d)*Si(2*b*c/d+2*b*x)/d+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)

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Rubi [A]  time = 0.30, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 3.23, size = 0, normalized size = 0.00 \[ \int \frac {\sec (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx \]

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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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sec \left (b x + a\right ) \sin \left (3 \, b x + 3 \, a\right )}{d x + c}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (b x + a\right ) \sin \left (3 \, b x + 3 \, a\right )}{d x + c}\,{d x} \]

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)

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maple [A]  time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (b x +a \right ) \sin \left (3 b x +3 a \right )}{d x +c}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (i \, E_{1}\left (\frac {2 i \, b d x + 2 i \, b c}{d}\right ) - i \, E_{1}\left (-\frac {2 i \, b d x + 2 i \, b c}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 2 \, d \int \frac {\sin \left (2 \, b x + 2 \, a\right )}{{\left (d x + c\right )} {\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )}}\,{d x} + {\left (E_{1}\left (\frac {2 i \, b d x + 2 i \, b c}{d}\right ) + E_{1}\left (-\frac {2 i \, b d x + 2 i \, b c}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )}{d} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sin \left (3\,a+3\,b\,x\right )}{\cos \left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]

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]

Exception raised: HeuristicGCDFailed

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