Integrand size = 22, antiderivative size = 22 \[ \int \frac {\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=-\frac {b \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}+\frac {\sin (2 a+2 b x)}{2 d (c+d x)}+\frac {b \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}+\text {Int}\left (\frac {\tan (a+b x)}{(c+d x)^2},x\right ) \]
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Not integrable
Time = 0.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int \frac {\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\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 ) \operatorname {CosIntegral}\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*}
Not integrable
Time = 2.96 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int \frac {\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx \]
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Not integrable
Time = 1.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
\[\int \frac {\sec \left (x b +a \right ) \sin \left (x b +a \right )^{3}}{\left (d x +c \right )^{2}}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.00 \[ \int \frac {\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\sec \left (b x + a\right ) \sin \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{2}} \,d x } \]
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Exception generated. \[ \int \frac {\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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Not integrable
Time = 0.77 (sec) , antiderivative size = 242, normalized size of antiderivative = 11.00 \[ \int \frac {\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\sec \left (b x + a\right ) \sin \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{2}} \,d x } \]
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Not integrable
Time = 2.80 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\sec \left (b x + a\right ) \sin \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{2}} \,d x } \]
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Not integrable
Time = 25.57 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int \frac {{\sin \left (a+b\,x\right )}^3}{\cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^2} \,d x \]
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