Optimal. Leaf size=152 \[ \frac{b c^2 \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{d^2}-\frac{b c^2 \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{d^2}+\frac{c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (1-c^2 x^2\right )}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac{4 c^2 \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2}-\frac{b c}{2 d^2 x \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.261622, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5746, 103, 12, 39, 5754, 5721, 5461, 4182, 2279, 2391} \[ \frac{b c^2 \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{d^2}-\frac{b c^2 \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{d^2}+\frac{c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (1-c^2 x^2\right )}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac{4 c^2 \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2}-\frac{b c}{2 d^2 x \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5746
Rule 103
Rule 12
Rule 39
Rule 5754
Rule 5721
Rule 5461
Rule 4182
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{x^3 \left (d-c^2 d x^2\right )^2} \, dx &=-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\left (2 c^2\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx-\frac{(b c) \int \frac{1}{x^2 (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2}\\ &=-\frac{b c}{2 d^2 x \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (1-c^2 x^2\right )}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac{(b c) \int \frac{2 c^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2}+\frac{\left (b c^3\right ) \int \frac{1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d^2}+\frac{\left (2 c^2\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x \left (d-c^2 d x^2\right )} \, dx}{d}\\ &=-\frac{b c}{2 d^2 x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 x}{d^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (1-c^2 x^2\right )}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac{\left (2 c^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2}-\frac{\left (b c^3\right ) \int \frac{1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d^2}\\ &=-\frac{b c}{2 d^2 x \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (1-c^2 x^2\right )}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac{\left (4 c^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2}\\ &=-\frac{b c}{2 d^2 x \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (1-c^2 x^2\right )}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac{4 c^2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^2}+\frac{\left (2 b c^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2}-\frac{\left (2 b c^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2}\\ &=-\frac{b c}{2 d^2 x \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (1-c^2 x^2\right )}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac{4 c^2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^2}+\frac{\left (b c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{d^2}-\frac{\left (b c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{d^2}\\ &=-\frac{b c}{2 d^2 x \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (1-c^2 x^2\right )}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac{4 c^2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^2}+\frac{b c^2 \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{d^2}-\frac{b c^2 \text{Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^2}\\ \end{align*}
Mathematica [B] time = 0.561113, size = 319, normalized size = 2.1 \[ \frac{-2 b c^2 x^2 \left (c^2 x^2-1\right ) \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+2 b c^2 x^2 \left (c^2 x^2-1\right ) \text{PolyLog}\left (2,e^{-2 \cosh ^{-1}(c x)}\right )-2 a c^2 x^2+4 a c^4 x^4 \log (x)-2 a c^4 x^4 \log \left (1-c^2 x^2\right )-4 a c^2 x^2 \log (x)+2 a c^2 x^2 \log \left (1-c^2 x^2\right )+a-b c^2 x^2 \sqrt{\frac{c x-1}{c x+1}}-2 b c^2 x^2 \cosh ^{-1}(c x)-4 b c^4 x^4 \cosh ^{-1}(c x) \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )+4 b c^4 x^4 \cosh ^{-1}(c x) \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )+4 b c^2 x^2 \cosh ^{-1}(c x) \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )-4 b c^2 x^2 \cosh ^{-1}(c x) \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )-b c x \sqrt{\frac{c x-1}{c x+1}}+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (c^2 x^2-1\right )} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.112, size = 371, normalized size = 2.4 \begin{align*} -{\frac{{c}^{2}a}{4\,{d}^{2} \left ( cx-1 \right ) }}-{\frac{{c}^{2}a\ln \left ( cx-1 \right ) }{{d}^{2}}}-{\frac{a}{2\,{d}^{2}{x}^{2}}}+2\,{\frac{{c}^{2}a\ln \left ( cx \right ) }{{d}^{2}}}+{\frac{{c}^{2}a}{4\,{d}^{2} \left ( cx+1 \right ) }}-{\frac{{c}^{2}a\ln \left ( cx+1 \right ) }{{d}^{2}}}-{\frac{{c}^{2}b{\rm arccosh} \left (cx\right )}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{bc}{2\,{d}^{2}x \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}+{\frac{b{\rm arccosh} \left (cx\right )}{2\,{d}^{2}{x}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}+2\,{\frac{{c}^{2}b{\rm arccosh} \left (cx\right )\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}+1 \right ) }{{d}^{2}}}+{\frac{{c}^{2}b}{{d}^{2}}{\it polylog} \left ( 2,- \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2} \right ) }-2\,{\frac{{c}^{2}b{\rm arccosh} \left (cx\right )\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }{{d}^{2}}}-2\,{\frac{{c}^{2}b{\it polylog} \left ( 2,-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }{{d}^{2}}}-2\,{\frac{{c}^{2}b{\rm arccosh} \left (cx\right )\ln \left ( 1-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }{{d}^{2}}}-2\,{\frac{{c}^{2}b{\it polylog} \left ( 2,cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }{{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, a{\left (\frac{2 \, c^{2} \log \left (c x + 1\right )}{d^{2}} + \frac{2 \, c^{2} \log \left (c x - 1\right )}{d^{2}} - \frac{4 \, c^{2} \log \left (x\right )}{d^{2}} + \frac{2 \, c^{2} x^{2} - 1}{c^{2} d^{2} x^{4} - d^{2} x^{2}}\right )} + b \int \frac{\log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{c^{4} d^{2} x^{7} - 2 \, c^{2} d^{2} x^{5} + d^{2} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{arcosh}\left (c x\right ) + a}{c^{4} d^{2} x^{7} - 2 \, c^{2} d^{2} x^{5} + d^{2} x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac{b \operatorname{acosh}{\left (c x \right )}}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx}{d^{2}} \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{arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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