Optimal. Leaf size=23 \[ \text {Int}\left (\frac {(a+b \coth (e+f x))^2}{c+d x},x\right ) \]
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Rubi [A] time = 0.06, 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 = {} \[ \int \frac {(a+b \coth (e+f x))^2}{c+d x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(a+b \coth (e+f x))^2}{c+d x} \, dx &=\int \frac {(a+b \coth (e+f x))^2}{c+d x} \, dx\\ \end {align*}
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Mathematica [A] time = 45.49, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \coth (e+f x))^2}{c+d x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \coth \left (f x + e\right )^{2} + 2 \, a b \coth \left (f x + e\right ) + a^{2}}{d x + c}, x\right ) \]
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 \[ \int \frac {{\left (b \coth \left (f x + e\right ) + a\right )}^{2}}{d x + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \coth \left (f x +e \right )\right )^{2}}{d x +c}\, dx \]
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 \[ \frac {a^{2} \log \left (d x + c\right )}{d} + \frac {2 \, b^{2}}{d f x + c f - {\left (d f x e^{\left (2 \, e\right )} + c f e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}} + \frac {{\left (2 \, a b + b^{2}\right )} \log \left (d x + c\right )}{d} - \int \frac {2 \, a b d f x + 2 \, a b c f - b^{2} d}{d^{2} f x^{2} + 2 \, c d f x + c^{2} f + {\left (d^{2} f x^{2} e^{e} + 2 \, c d f x e^{e} + c^{2} f e^{e}\right )} e^{\left (f x\right )}}\,{d x} + \int -\frac {2 \, a b d f x + 2 \, a b c f - b^{2} d}{d^{2} f x^{2} + 2 \, c d f x + c^{2} f - {\left (d^{2} f x^{2} e^{e} + 2 \, c d f x e^{e} + c^{2} f e^{e}\right )} e^{\left (f x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a+b\,\mathrm {coth}\left (e+f\,x\right )\right )}^2}{c+d\,x} \,d x \]
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 \[ \int \frac {\left (a + b \coth {\left (e + f x \right )}\right )^{2}}{c + d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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