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