Optimal. Leaf size=88 \[ \text {Int}\left (\frac {\csc (a+b x)}{(c+d x)^2},x\right )-\frac {b \cos \left (a-\frac {b c}{d}\right ) \text {Ci}\left (\frac {b c}{d}+b x\right )}{d^2}+\frac {b \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^2}+\frac {\sin (a+b x)}{d (c+d x)} \]
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Rubi [A] time = 0.13, 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 = {} \[ \int \frac {\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx &=\int \frac {\csc (a+b x)}{(c+d x)^2} \, dx-\int \frac {\sin (a+b x)}{(c+d x)^2} \, dx\\ &=\frac {\sin (a+b x)}{d (c+d x)}-\frac {b \int \frac {\cos (a+b x)}{c+d x} \, dx}{d}+\int \frac {\csc (a+b x)}{(c+d x)^2} \, dx\\ &=\frac {\sin (a+b x)}{d (c+d x)}-\frac {\left (b \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \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 {\sin \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{d}+\int \frac {\csc (a+b x)}{(c+d x)^2} \, dx\\ &=-\frac {b \cos \left (a-\frac {b c}{d}\right ) \text {Ci}\left (\frac {b c}{d}+b x\right )}{d^2}+\frac {\sin (a+b x)}{d (c+d x)}+\frac {b \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^2}+\int \frac {\csc (a+b x)}{(c+d x)^2} \, dx\\ \end {align*}
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Mathematica [A] time = 4.02, size = 0, normalized size = 0.00 \[ \int \frac {\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]
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 \[ \int \frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )}{{\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b x +a \right ) \cot \left (b x +a \right )}{\left (d x +c \right )^{2}}\, dx \]
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 \[ \frac {{\left (i \, E_{2}\left (\frac {i \, b d x + i \, b c}{d}\right ) - i \, E_{2}\left (-\frac {i \, b d x + i \, b c}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + 2 \, {\left (d^{2} x + c d\right )} \int \frac {\sin \left (b x + a\right )}{{\left (d x + c\right )}^{2} {\left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right )}}\,{d x} + 2 \, {\left (d^{2} x + c d\right )} \int \frac {\sin \left (b x + a\right )}{{\left (d x + c\right )}^{2} {\left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right )}}\,{d x} + {\left (E_{2}\left (\frac {i \, b d x + i \, b c}{d}\right ) + E_{2}\left (-\frac {i \, b d x + i \, b c}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )}{2 \, {\left (d^{2} x + c d\right )}} \]
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 \[ \int \frac {\cos \left (a+b\,x\right )\,\mathrm {cot}\left (a+b\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (a + b x \right )} \cot {\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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