Optimal. Leaf size=197 \[ -\frac {c^3 e \left (a+b \coth ^{-1}(c x)\right )^2}{3 b}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{3 x^3}+\frac {2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{3 x}-b c^3 e \log (x)-\frac {b c \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{6 x^2}+\frac {1}{6} b c^3 \log \left (1-\frac {1}{1-c^2 x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac {1}{6} b c^3 e \text {Li}_2\left (\frac {1}{1-c^2 x^2}\right )+\frac {1}{3} b c^3 e \log \left (1-c^2 x^2\right ) \]
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Rubi [A] time = 0.46, antiderivative size = 191, normalized size of antiderivative = 0.97, number of steps used = 17, number of rules used = 16, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {6082, 2475, 2411, 2347, 2344, 2301, 2316, 2315, 2314, 31, 5983, 5917, 266, 36, 29, 5949} \[ -\frac {1}{6} b c^3 e \text {PolyLog}\left (2,c^2 x^2\right )-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{3 x^3}-\frac {c^3 e \left (a+b \coth ^{-1}(c x)\right )^2}{3 b}+\frac {2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{3 x}-\frac {b c^3 \left (e \log \left (1-c^2 x^2\right )+d\right )^2}{12 e}-\frac {b c \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{6 x^2}+\frac {1}{3} b c^3 d \log (x)+\frac {1}{3} b c^3 e \log \left (1-c^2 x^2\right )-b c^3 e \log (x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 2301
Rule 2314
Rule 2315
Rule 2316
Rule 2344
Rule 2347
Rule 2411
Rule 2475
Rule 5917
Rule 5949
Rule 5983
Rule 6082
Rubi steps
\begin {align*} \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^4} \, dx &=-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{3 x^3}+\frac {1}{3} (b c) \int \frac {d+e \log \left (1-c^2 x^2\right )}{x^3 \left (1-c^2 x^2\right )} \, dx-\frac {1}{3} \left (2 c^2 e\right ) \int \frac {a+b \coth ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{3 x^3}+\frac {1}{6} (b c) \operatorname {Subst}\left (\int \frac {d+e \log \left (1-c^2 x\right )}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )-\frac {1}{3} \left (2 c^2 e\right ) \int \frac {a+b \coth ^{-1}(c x)}{x^2} \, dx-\frac {1}{3} \left (2 c^4 e\right ) \int \frac {a+b \coth ^{-1}(c x)}{1-c^2 x^2} \, dx\\ &=\frac {2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{3 x}-\frac {c^3 e \left (a+b \coth ^{-1}(c x)\right )^2}{3 b}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{3 x^3}-\frac {b \operatorname {Subst}\left (\int \frac {d+e \log (x)}{x \left (\frac {1}{c^2}-\frac {x}{c^2}\right )^2} \, dx,x,1-c^2 x^2\right )}{6 c}-\frac {1}{3} \left (2 b c^3 e\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx\\ &=\frac {2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{3 x}-\frac {c^3 e \left (a+b \coth ^{-1}(c x)\right )^2}{3 b}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{3 x^3}-\frac {b \operatorname {Subst}\left (\int \frac {d+e \log (x)}{\left (\frac {1}{c^2}-\frac {x}{c^2}\right )^2} \, dx,x,1-c^2 x^2\right )}{6 c}-\frac {1}{6} (b c) \operatorname {Subst}\left (\int \frac {d+e \log (x)}{x \left (\frac {1}{c^2}-\frac {x}{c^2}\right )} \, dx,x,1-c^2 x^2\right )-\frac {1}{3} \left (b c^3 e\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )\\ &=\frac {2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{3 x}-\frac {c^3 e \left (a+b \coth ^{-1}(c x)\right )^2}{3 b}-\frac {b c \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{6 x^2}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{3 x^3}-\frac {1}{6} (b c) \operatorname {Subst}\left (\int \frac {d+e \log (x)}{\frac {1}{c^2}-\frac {x}{c^2}} \, dx,x,1-c^2 x^2\right )-\frac {1}{6} \left (b c^3\right ) \operatorname {Subst}\left (\int \frac {d+e \log (x)}{x} \, dx,x,1-c^2 x^2\right )+\frac {1}{6} (b c e) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x}{c^2}} \, dx,x,1-c^2 x^2\right )-\frac {1}{3} \left (b c^3 e\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{3} \left (b c^5 e\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )\\ &=\frac {2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{3 x}-\frac {c^3 e \left (a+b \coth ^{-1}(c x)\right )^2}{3 b}+\frac {1}{3} b c^3 d \log (x)-b c^3 e \log (x)+\frac {1}{3} b c^3 e \log \left (1-c^2 x^2\right )-\frac {b c \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{6 x^2}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{3 x^3}-\frac {b c^3 \left (d+e \log \left (1-c^2 x^2\right )\right )^2}{12 e}-\frac {1}{6} (b c e) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {1}{c^2}-\frac {x}{c^2}} \, dx,x,1-c^2 x^2\right )\\ &=\frac {2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{3 x}-\frac {c^3 e \left (a+b \coth ^{-1}(c x)\right )^2}{3 b}+\frac {1}{3} b c^3 d \log (x)-b c^3 e \log (x)+\frac {1}{3} b c^3 e \log \left (1-c^2 x^2\right )-\frac {b c \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{6 x^2}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{3 x^3}-\frac {b c^3 \left (d+e \log \left (1-c^2 x^2\right )\right )^2}{12 e}-\frac {1}{6} b c^3 e \text {Li}_2\left (c^2 x^2\right )\\ \end {align*}
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Mathematica [B] time = 0.37, size = 457, normalized size = 2.32 \[ \frac {1}{6} \left (-4 a c^3 e \tanh ^{-1}(c x)-\frac {2 a e \log \left (1-c^2 x^2\right )}{x^3}+\frac {4 a c^2 e}{x}-\frac {2 a d}{x^3}+2 b c^3 d \log (x)-2 b c^3 e \text {Li}_2(-c x)-2 b c^3 e \text {Li}_2(c x)+b c^3 e \text {Li}_2\left (\frac {1}{2}-\frac {c x}{2}\right )+b c^3 e \text {Li}_2\left (\frac {1}{2} (c x+1)\right )+\frac {1}{2} b c^3 e \log ^2\left (x-\frac {1}{c}\right )+\frac {1}{2} b c^3 e \log ^2\left (\frac {1}{c}+x\right )-2 b c^3 e \log (x)+b c^3 e \log \left (\frac {1}{c}+x\right ) \log \left (\frac {1}{2} (1-c x)\right )-2 b c^3 e \log (x) \log (1-c x)+b c^3 e \log \left (x-\frac {1}{c}\right ) \log \left (\frac {1}{2} (c x+1)\right )-2 b c^3 e \log (x) \log (c x+1)-2 b c^3 e \coth ^{-1}(c x)^2-\frac {b c e \log \left (1-c^2 x^2\right )}{x^2}-\frac {2 b e \log \left (1-c^2 x^2\right ) \coth ^{-1}(c x)}{x^3}+\frac {4 b c^2 e \coth ^{-1}(c x)}{x}-b c^3 d \log \left (1-c^2 x^2\right )-4 b c^3 e \log \left (\frac {1}{\sqrt {1-\frac {1}{c^2 x^2}}}\right )+b c^3 e \log \left (1-c^2 x^2\right )+2 b c^3 e \log (x) \log \left (1-c^2 x^2\right )-b c^3 e \log \left (x-\frac {1}{c}\right ) \log \left (1-c^2 x^2\right )-b c^3 e \log \left (\frac {1}{c}+x\right ) \log \left (1-c^2 x^2\right )-\frac {2 b d \coth ^{-1}(c x)}{x^3}-\frac {b c d}{x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b d \operatorname {arcoth}\left (c x\right ) + a d + {\left (b e \operatorname {arcoth}\left (c x\right ) + a e\right )} \log \left (-c^{2} x^{2} + 1\right )}{x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccoth}\left (c x \right )\right ) \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{6} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c + \frac {2 \, \operatorname {arcoth}\left (c x\right )}{x^{3}}\right )} b d - \frac {1}{3} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c^{2} + \frac {\log \left (-c^{2} x^{2} + 1\right )}{x^{3}}\right )} a e - \frac {1}{6} \, b e {\left (\frac {\log \left (c x + 1\right )^{2}}{x^{3}} - 3 \, \int -\frac {3 \, {\left (c x + 1\right )} \log \left (c x - 1\right )^{2} - {\left (3 i \, \pi + {\left (3 i \, \pi c + 2 \, c\right )} x\right )} \log \left (c x + 1\right ) - {\left (-3 i \, \pi - 3 i \, \pi c x\right )} \log \left (c x - 1\right )}{3 \, {\left (c x^{5} + x^{4}\right )}}\,{d x}\right )} - \frac {a d}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (1-c^2\,x^2\right )\right )}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acoth}{\left (c x \right )}\right ) \left (d + e \log {\left (- c^{2} x^{2} + 1 \right )}\right )}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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