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

Optimal. Leaf size=78 \[ -\text {Int}\left (\frac {\tan (a+b x) \sec (a+b x)}{c+d x},x\right )+\frac {4 \sin \left (a-\frac {b c}{d}\right ) \text {Ci}\left (\frac {b c}{d}+b x\right )}{d}+\frac {4 \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d} \]

[Out]

-CannotIntegrate(sec(b*x+a)*tan(b*x+a)/(d*x+c),x)+4*cos(a-b*c/d)*Si(b*c/d+b*x)/d+4*Ci(b*c/d+b*x)*sin(a-b*c/d)/
d

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Rubi [A]  time = 0.27, 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 = {} \[ \int \frac {\sec ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx \]

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*}

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Mathematica [A]  time = 13.73, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx \]

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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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sec \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{d x + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{d x + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^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)

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maple [A]  time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (\sec ^{2}\left (b x +a \right )\right ) \sin \left (3 b x +3 a \right )}{d x +c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x+a)^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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sin \left (3\,a+3\,b\,x\right )}{{\cos \left (a+b\,x\right )}^2\,\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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