Optimal. Leaf size=110 \[ -\frac{c d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{d \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac{b c^2 d}{6 x^2}-\frac{b c^3 d}{4 x}+\frac{1}{3} b c^4 d \log (x)-\frac{7}{24} b c^4 d \log (1-c x)-\frac{1}{24} b c^4 d \log (c x+1)-\frac{b c d}{12 x^3} \]
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Rubi [A] time = 0.0924434, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {43, 5936, 12, 801} \[ -\frac{c d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{d \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac{b c^2 d}{6 x^2}-\frac{b c^3 d}{4 x}+\frac{1}{3} b c^4 d \log (x)-\frac{7}{24} b c^4 d \log (1-c x)-\frac{1}{24} b c^4 d \log (c x+1)-\frac{b c d}{12 x^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 5936
Rule 12
Rule 801
Rubi steps
\begin{align*} \int \frac{(d+c d x) \left (a+b \tanh ^{-1}(c x)\right )}{x^5} \, dx &=-\frac{d \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac{c d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-(b c) \int \frac{d (-3-4 c x)}{12 x^4 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac{d \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac{c d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{1}{12} (b c d) \int \frac{-3-4 c x}{x^4 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac{d \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac{c d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{1}{12} (b c d) \int \left (-\frac{3}{x^4}-\frac{4 c}{x^3}-\frac{3 c^2}{x^2}-\frac{4 c^3}{x}+\frac{7 c^4}{2 (-1+c x)}+\frac{c^4}{2 (1+c x)}\right ) \, dx\\ &=-\frac{b c d}{12 x^3}-\frac{b c^2 d}{6 x^2}-\frac{b c^3 d}{4 x}-\frac{d \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac{c d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}+\frac{1}{3} b c^4 d \log (x)-\frac{7}{24} b c^4 d \log (1-c x)-\frac{1}{24} b c^4 d \log (1+c x)\\ \end{align*}
Mathematica [A] time = 0.0649337, size = 94, normalized size = 0.85 \[ -\frac{d \left (8 a c x+6 a+6 b c^3 x^3+4 b c^2 x^2-8 b c^4 x^4 \log (x)+7 b c^4 x^4 \log (1-c x)+b c^4 x^4 \log (c x+1)+2 b c x+2 b (4 c x+3) \tanh ^{-1}(c x)\right )}{24 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 105, normalized size = 1. \begin{align*} -{\frac{da}{4\,{x}^{4}}}-{\frac{cda}{3\,{x}^{3}}}-{\frac{db{\it Artanh} \left ( cx \right ) }{4\,{x}^{4}}}-{\frac{cdb{\it Artanh} \left ( cx \right ) }{3\,{x}^{3}}}-{\frac{7\,{c}^{4}db\ln \left ( cx-1 \right ) }{24}}-{\frac{cdb}{12\,{x}^{3}}}-{\frac{b{c}^{2}d}{6\,{x}^{2}}}-{\frac{b{c}^{3}d}{4\,x}}+{\frac{{c}^{4}db\ln \left ( cx \right ) }{3}}-{\frac{b{c}^{4}d\ln \left ( cx+1 \right ) }{24}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965634, size = 154, normalized size = 1.4 \begin{align*} -\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{artanh}\left (c x\right )}{x^{3}}\right )} b c d + \frac{1}{24} \,{\left ({\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac{2 \,{\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c - \frac{6 \, \operatorname{artanh}\left (c x\right )}{x^{4}}\right )} b d - \frac{a c d}{3 \, x^{3}} - \frac{a d}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.10247, size = 267, normalized size = 2.43 \begin{align*} -\frac{b c^{4} d x^{4} \log \left (c x + 1\right ) + 7 \, b c^{4} d x^{4} \log \left (c x - 1\right ) - 8 \, b c^{4} d x^{4} \log \left (x\right ) + 6 \, b c^{3} d x^{3} + 4 \, b c^{2} d x^{2} + 2 \,{\left (4 \, a + b\right )} c d x + 6 \, a d +{\left (4 \, b c d x + 3 \, b d\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{24 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.09623, size = 129, normalized size = 1.17 \begin{align*} \begin{cases} - \frac{a c d}{3 x^{3}} - \frac{a d}{4 x^{4}} + \frac{b c^{4} d \log{\left (x \right )}}{3} - \frac{b c^{4} d \log{\left (x - \frac{1}{c} \right )}}{3} - \frac{b c^{4} d \operatorname{atanh}{\left (c x \right )}}{12} - \frac{b c^{3} d}{4 x} - \frac{b c^{2} d}{6 x^{2}} - \frac{b c d \operatorname{atanh}{\left (c x \right )}}{3 x^{3}} - \frac{b c d}{12 x^{3}} - \frac{b d \operatorname{atanh}{\left (c x \right )}}{4 x^{4}} & \text{for}\: c \neq 0 \\- \frac{a d}{4 x^{4}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25656, size = 146, normalized size = 1.33 \begin{align*} -\frac{1}{24} \, b c^{4} d \log \left (c x + 1\right ) - \frac{7}{24} \, b c^{4} d \log \left (c x - 1\right ) + \frac{1}{3} \, b c^{4} d \log \left (x\right ) - \frac{{\left (4 \, b c d x + 3 \, b d\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{24 \, x^{4}} - \frac{3 \, b c^{3} d x^{3} + 2 \, b c^{2} d x^{2} + 4 \, a c d x + b c d x + 3 \, a d}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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