Optimal. Leaf size=167 \[ -\frac{a+b \coth ^{-1}(c+d x)}{2 f (e+f x)^2}-\frac{b d^2 \log (-c-d x+1)}{4 f (-c f+d e+f)^2}+\frac{b d^2 \log (c+d x+1)}{4 f (-c f+d e-f)^2}-\frac{b d^2 (d e-c f) \log (e+f x)}{(-c f+d e+f)^2 (d e-(c+1) f)^2}+\frac{b d}{2 (e+f x) (-c f+d e+f) (d e-(c+1) f)} \]
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Rubi [A] time = 0.234421, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6110, 1982, 709, 800} \[ -\frac{a+b \coth ^{-1}(c+d x)}{2 f (e+f x)^2}-\frac{b d^2 \log (-c-d x+1)}{4 f (-c f+d e+f)^2}+\frac{b d^2 \log (c+d x+1)}{4 f (-c f+d e-f)^2}-\frac{b d^2 (d e-c f) \log (e+f x)}{(-c f+d e+f)^2 (d e-(c+1) f)^2}+\frac{b d}{2 (e+f x) (-c f+d e+f) (d e-(c+1) f)} \]
Antiderivative was successfully verified.
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Rule 6110
Rule 1982
Rule 709
Rule 800
Rubi steps
\begin{align*} \int \frac{a+b \coth ^{-1}(c+d x)}{(e+f x)^3} \, dx &=-\frac{a+b \coth ^{-1}(c+d x)}{2 f (e+f x)^2}+\frac{(b d) \int \frac{1}{(e+f x)^2 \left (1-(c+d x)^2\right )} \, dx}{2 f}\\ &=-\frac{a+b \coth ^{-1}(c+d x)}{2 f (e+f x)^2}+\frac{(b d) \int \frac{1}{(e+f x)^2 \left (1-c^2-2 c d x-d^2 x^2\right )} \, dx}{2 f}\\ &=\frac{b d}{2 (d e+f-c f) (d e-(1+c) f) (e+f x)}-\frac{a+b \coth ^{-1}(c+d x)}{2 f (e+f x)^2}+\frac{(b d) \int \frac{-d (d e-2 c f)+d^2 f x}{(e+f x) \left (1-c^2-2 c d x-d^2 x^2\right )} \, dx}{2 f \left (-d^2 e^2+2 c d e f+\left (1-c^2\right ) f^2\right )}\\ &=\frac{b d}{2 (d e+f-c f) (d e-(1+c) f) (e+f x)}-\frac{a+b \coth ^{-1}(c+d x)}{2 f (e+f x)^2}+\frac{(b d) \int \left (\frac{d^2 (-d e+(1+c) f)}{2 (d e+f-c f) (1-c-d x)}+\frac{d^2 (-d e-f+c f)}{2 (d e-(1+c) f) (1+c+d x)}+\frac{2 d f^2 (d e-c f)}{(d e+(1-c) f) (d e-f-c f) (e+f x)}\right ) \, dx}{2 f \left (-d^2 e^2+2 c d e f+\left (1-c^2\right ) f^2\right )}\\ &=\frac{b d}{2 (d e+f-c f) (d e-(1+c) f) (e+f x)}-\frac{a+b \coth ^{-1}(c+d x)}{2 f (e+f x)^2}-\frac{b d^2 \log (1-c-d x)}{4 f (d e+f-c f)^2}+\frac{b d^2 \log (1+c+d x)}{4 f (d e-f-c f)^2}-\frac{b d^2 (d e-c f) \log (e+f x)}{(d e+f-c f)^2 (d e-(1+c) f)^2}\\ \end{align*}
Mathematica [A] time = 0.346272, size = 174, normalized size = 1.04 \[ \frac{1}{4} \left (-\frac{2 a}{f (e+f x)^2}+\frac{2 b d}{(e+f x) \left (\left (c^2-1\right ) f^2-2 c d e f+d^2 e^2\right )}-\frac{4 b d^2 (d e-c f) \log (e+f x)}{\left (\left (c^2-1\right ) f^2-2 c d e f+d^2 e^2\right )^2}-\frac{b d^2 \log (-c-d x+1)}{f (-c f+d e+f)^2}+\frac{b d^2 \log (c+d x+1)}{f (c f-d e+f)^2}-\frac{2 b \coth ^{-1}(c+d x)}{f (e+f x)^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 236, normalized size = 1.4 \begin{align*} -{\frac{a{d}^{2}}{2\, \left ( dfx+de \right ) ^{2}f}}-{\frac{b{d}^{2}{\rm arccoth} \left (dx+c\right )}{2\, \left ( dfx+de \right ) ^{2}f}}-{\frac{b{d}^{2}\ln \left ( dx+c-1 \right ) }{4\,f \left ( cf-de-f \right ) ^{2}}}+{\frac{b{d}^{2}\ln \left ( dx+c+1 \right ) }{4\,f \left ( cf-de+f \right ) ^{2}}}+{\frac{b{d}^{2}}{ \left ( 2\,cf-2\,de-2\,f \right ) \left ( cf-de+f \right ) \left ( dfx+de \right ) }}+{\frac{b{d}^{2}f\ln \left ( \left ( dx+c \right ) f-cf+de \right ) c}{ \left ( cf-de-f \right ) ^{2} \left ( cf-de+f \right ) ^{2}}}-{\frac{b{d}^{3}\ln \left ( \left ( dx+c \right ) f-cf+de \right ) e}{ \left ( cf-de-f \right ) ^{2} \left ( cf-de+f \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04528, size = 393, normalized size = 2.35 \begin{align*} \frac{1}{4} \,{\left (d{\left (\frac{d \log \left (d x + c + 1\right )}{d^{2} e^{2} f - 2 \,{\left (c + 1\right )} d e f^{2} +{\left (c^{2} + 2 \, c + 1\right )} f^{3}} - \frac{d \log \left (d x + c - 1\right )}{d^{2} e^{2} f - 2 \,{\left (c - 1\right )} d e f^{2} +{\left (c^{2} - 2 \, c + 1\right )} f^{3}} - \frac{4 \,{\left (d^{2} e - c d f\right )} \log \left (f x + e\right )}{d^{4} e^{4} - 4 \, c d^{3} e^{3} f + 2 \,{\left (3 \, c^{2} - 1\right )} d^{2} e^{2} f^{2} - 4 \,{\left (c^{3} - c\right )} d e f^{3} +{\left (c^{4} - 2 \, c^{2} + 1\right )} f^{4}} + \frac{2}{d^{2} e^{3} - 2 \, c d e^{2} f +{\left (c^{2} - 1\right )} e f^{2} +{\left (d^{2} e^{2} f - 2 \, c d e f^{2} +{\left (c^{2} - 1\right )} f^{3}\right )} x}\right )} - \frac{2 \, \operatorname{arcoth}\left (d x + c\right )}{f^{3} x^{2} + 2 \, e f^{2} x + e^{2} f}\right )} b - \frac{a}{2 \,{\left (f^{3} x^{2} + 2 \, e f^{2} x + e^{2} f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 9.18629, size = 1766, normalized size = 10.57 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arcoth}\left (d x + c\right ) + a}{{\left (f x + e\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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