Optimal. Leaf size=86 \[ \text{Unintegrable}\left (\frac{\sec (a+b x)}{(c+d x)^2},x\right )+\frac{b \sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{b c}{d}+b x\right )}{d^2}+\frac{b \cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{d^2}+\frac{\cos (a+b x)}{d (c+d x)} \]
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Rubi [A] time = 0.138904, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sin (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sin (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx &=-\int \frac{\cos (a+b x)}{(c+d x)^2} \, dx+\int \frac{\sec (a+b x)}{(c+d x)^2} \, dx\\ &=\frac{\cos (a+b x)}{d (c+d x)}+\frac{b \int \frac{\sin (a+b x)}{c+d x} \, dx}{d}+\int \frac{\sec (a+b x)}{(c+d x)^2} \, dx\\ &=\frac{\cos (a+b x)}{d (c+d x)}+\frac{\left (b \cos \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sin \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx}{d}+\frac{\left (b \sin \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cos \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx}{d}+\int \frac{\sec (a+b x)}{(c+d x)^2} \, dx\\ &=\frac{\cos (a+b x)}{d (c+d x)}+\frac{b \text{Ci}\left (\frac{b c}{d}+b x\right ) \sin \left (a-\frac{b c}{d}\right )}{d^2}+\frac{b \cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{d^2}+\int \frac{\sec (a+b x)}{(c+d x)^2} \, dx\\ \end{align*}
Mathematica [A] time = 7.23992, size = 0, normalized size = 0. \[ \int \frac{\sin (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.607, size = 0, normalized size = 0. \begin{align*} \int{\frac{\sec \left ( bx+a \right ) \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{ \left ( dx+c \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sec \left (b x + a\right )}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin ^{2}{\left (a + b x \right )} \sec{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (b x + a\right ) \sin \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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