Optimal. Leaf size=21 \[ \text {Int}\left (\frac {a+b \cot (e+f x)}{(c+d x)^2},x\right ) \]
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Rubi [A] time = 0.03, 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 = {} \[ \int \frac {a+b \cot (e+f x)}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {a+b \cot (e+f x)}{(c+d x)^2} \, dx &=\int \frac {a+b \cot (e+f x)}{(c+d x)^2} \, dx\\ \end {align*}
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Mathematica [A] time = 8.19, size = 0, normalized size = 0.00 \[ \int \frac {a+b \cot (e+f x)}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \cot \left (f x + e\right ) + a}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]
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 \[ \int \frac {b \cot \left (f x + e\right ) + a}{{\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.80, size = 0, normalized size = 0.00 \[ \int \frac {a +b \cot \left (f x +e \right )}{\left (d x +c \right )^{2}}\, dx \]
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 \[ -\frac {{\left (b d^{2} x + b c d\right )} \int \frac {\sin \left (f x + e\right )}{{\left (d x + c\right )}^{2} {\left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right )}}\,{d x} - {\left (b d^{2} x + b c d\right )} \int \frac {\sin \left (f x + e\right )}{{\left (d x + c\right )}^{2} {\left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1\right )}}\,{d x} + a}{d^{2} x + c d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {a+b\,\mathrm {cot}\left (e+f\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \]
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 \[ \int \frac {a + b \cot {\left (e + f x \right )}}{\left (c + d x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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