Optimal. Leaf size=102 \[ \text {Int}\left (\frac {\tan (a+b x)}{(c+d x)^2},x\right )-\frac {b \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Ci}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}+\frac {b \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}+\frac {\sin (2 a+2 b x)}{2 d (c+d x)} \]
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Rubi [A] time = 0.18, 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 = {} \[ \int \frac {\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx &=-\int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^2} \, dx+\int \frac {\tan (a+b x)}{(c+d x)^2} \, dx\\ &=-\int \frac {\sin (2 a+2 b x)}{2 (c+d x)^2} \, dx+\int \frac {\tan (a+b x)}{(c+d x)^2} \, dx\\ &=-\left (\frac {1}{2} \int \frac {\sin (2 a+2 b x)}{(c+d x)^2} \, dx\right )+\int \frac {\tan (a+b x)}{(c+d x)^2} \, dx\\ &=\frac {\sin (2 a+2 b x)}{2 d (c+d x)}-\frac {b \int \frac {\cos (2 a+2 b x)}{c+d x} \, dx}{d}+\int \frac {\tan (a+b x)}{(c+d x)^2} \, dx\\ &=\frac {\sin (2 a+2 b x)}{2 d (c+d x)}-\frac {\left (b \cos \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}{d}+\frac {\left (b \sin \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}{d}+\int \frac {\tan (a+b x)}{(c+d x)^2} \, dx\\ &=-\frac {b \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Ci}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}+\frac {\sin (2 a+2 b x)}{2 d (c+d x)}+\frac {b \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}+\int \frac {\tan (a+b x)}{(c+d x)^2} \, dx\\ \end {align*}
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Mathematica [A] time = 2.40, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sec \left (b x + a\right ) \sin \left (b x + a\right )}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]
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 \[ \int \frac {\sec \left (b x + a\right ) \sin \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.69, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (b x +a \right ) \left (\sin ^{3}\left (b x +a \right )\right )}{\left (d x +c \right )^{2}}\, dx \]
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 \[ \frac {{\left (i \, E_{2}\left (\frac {2 i \, b d x + 2 i \, b c}{d}\right ) - i \, E_{2}\left (-\frac {2 i \, b d x + 2 i \, b c}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 8 \, {\left (d^{2} x + c d\right )} \int \frac {\sin \left (2 \, b x + 2 \, a\right )}{{\left (d x + c\right )}^{2} {\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )}}\,{d x} + {\left (E_{2}\left (\frac {2 i \, b d x + 2 i \, b c}{d}\right ) + E_{2}\left (-\frac {2 i \, b d x + 2 i \, b c}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )}{4 \, {\left (d^{2} x + c d\right )}} \]
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 \[ \int \frac {{\sin \left (a+b\,x\right )}^3}{\cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^2} \,d x \]
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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