Optimal. Leaf size=101 \[ \text{Unintegrable}\left (\frac{\cot (a+b x)}{(c+d x)^2},x\right )-\frac{b \cos \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{d^2}+\frac{b \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{d^2}+\frac{\sin (2 a+2 b x)}{2 d (c+d x)} \]
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Rubi [A] time = 0.1703, 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 = {} \[ \int \frac{\cos ^2(a+b x) \cot (a+b x)}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\cos ^2(a+b x) \cot (a+b x)}{(c+d x)^2} \, dx &=\int \frac{\cot (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{\cot (a+b x)}{(c+d x)^2} \, dx-\int \frac{\sin (2 a+2 b x)}{2 (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{\cot (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{\cot (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{\cot (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{\cot (a+b x)}{(c+d x)^2} \, dx\\ \end{align*}
Mathematica [A] time = 2.49899, size = 0, normalized size = 0. \[ \int \frac{\cos ^2(a+b x) \cot (a+b x)}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.63, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{2}\cot \left ( bx+a \right ) }{ \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{\cos \left (b x + a\right )^{2} \cot \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{\cos ^{2}{\left (a + b x \right )} \cot{\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{\cos \left (b x + a\right )^{2} \cot \left (b x + a\right )}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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