Optimal. Leaf size=193 \[ -\frac {b c x \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt {c^2 x^2+1}}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (c^2 x^2+1\right )}-\frac {b \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}+\frac {b \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}-\frac {2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2}+\frac {b^2 \log \left (c^2 x^2+1\right )}{2 d^2}+\frac {b^2 \text {Li}_3\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}-\frac {b^2 \text {Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2} \]
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Rubi [A] time = 0.34, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5755, 5720, 5461, 4182, 2531, 2282, 6589, 5687, 260} \[ -\frac {b \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}+\frac {b \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}+\frac {b^2 \text {PolyLog}\left (3,-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}-\frac {b^2 \text {PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}-\frac {b c x \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt {c^2 x^2+1}}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (c^2 x^2+1\right )}-\frac {2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2}+\frac {b^2 \log \left (c^2 x^2+1\right )}{2 d^2} \]
Antiderivative was successfully verified.
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Rule 260
Rule 2282
Rule 2531
Rule 4182
Rule 5461
Rule 5687
Rule 5720
Rule 5755
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )^2} \, dx &=\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac {(b c) \int \frac {a+b \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{d^2}+\frac {\int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )} \, dx}{d}\\ &=-\frac {b c x \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt {1+c^2 x^2}}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}+\frac {\operatorname {Subst}\left (\int (a+b x)^2 \text {csch}(x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}+\frac {\left (b^2 c^2\right ) \int \frac {x}{1+c^2 x^2} \, dx}{d^2}\\ &=-\frac {b c x \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt {1+c^2 x^2}}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}+\frac {b^2 \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {2 \operatorname {Subst}\left (\int (a+b x)^2 \text {csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}\\ &=-\frac {b c x \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt {1+c^2 x^2}}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac {b^2 \log \left (1+c^2 x^2\right )}{2 d^2}-\frac {(2 b) \operatorname {Subst}\left (\int (a+b x) \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}+\frac {(2 b) \operatorname {Subst}\left (\int (a+b x) \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}\\ &=-\frac {b c x \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt {1+c^2 x^2}}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac {b^2 \log \left (1+c^2 x^2\right )}{2 d^2}-\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac {b^2 \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}-\frac {b^2 \operatorname {Subst}\left (\int \text {Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}\\ &=-\frac {b c x \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt {1+c^2 x^2}}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac {b^2 \log \left (1+c^2 x^2\right )}{2 d^2}-\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}\\ &=-\frac {b c x \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt {1+c^2 x^2}}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac {b^2 \log \left (1+c^2 x^2\right )}{2 d^2}-\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac {b^2 \text {Li}_3\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}-\frac {b^2 \text {Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}\\ \end {align*}
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Mathematica [C] time = 1.86, size = 428, normalized size = 2.22 \[ \frac {\frac {a^2}{c^2 x^2+1}-a^2 \log \left (c^2 x^2+1\right )+2 a^2 \log (c x)-\frac {a b \left (\sqrt {c^2 x^2+1}-i \sinh ^{-1}(c x)\right )}{c x+i}-\frac {a b \left (\sqrt {c^2 x^2+1}+i \sinh ^{-1}(c x)\right )}{c x-i}+2 a b \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )+a b \left (\sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)-4 \log \left (1+i e^{\sinh ^{-1}(c x)}\right )\right )-4 \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )\right )+a b \left (\sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)-4 \log \left (1-i e^{\sinh ^{-1}(c x)}\right )\right )-4 \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )\right )-2 a b \sinh ^{-1}(c x)^2+4 a b \sinh ^{-1}(c x) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+2 b^2 \left (\frac {1}{2} \log \left (c^2 x^2+1\right )+\frac {\sinh ^{-1}(c x)^2}{2 c^2 x^2+2}-\frac {c x \sinh ^{-1}(c x)}{\sqrt {c^2 x^2+1}}+\sinh ^{-1}(c x) \text {Li}_2\left (-e^{-2 \sinh ^{-1}(c x)}\right )+\sinh ^{-1}(c x) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )+\frac {1}{2} \text {Li}_3\left (-e^{-2 \sinh ^{-1}(c x)}\right )-\frac {1}{2} \text {Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )-\frac {2}{3} \sinh ^{-1}(c x)^3-\sinh ^{-1}(c x)^2 \log \left (e^{-2 \sinh ^{-1}(c x)}+1\right )+\sinh ^{-1}(c x)^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+\frac {i \pi ^3}{24}\right )}{2 d^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname {arsinh}\left (c x\right ) + a^{2}}{c^{4} d^{2} x^{5} + 2 \, c^{2} d^{2} x^{3} + d^{2} x}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 724, normalized size = 3.75 \[ \frac {a b \,c^{2} x^{2}}{d^{2} \left (c^{2} x^{2}+1\right )}-\frac {a b c x}{d^{2} \sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \arcsinh \left (c x \right ) c x}{d^{2} \sqrt {c^{2} x^{2}+1}}+\frac {b^{2} \arcsinh \left (c x \right ) c^{2} x^{2}}{d^{2} \left (c^{2} x^{2}+1\right )}+\frac {a b}{d^{2} \left (c^{2} x^{2}+1\right )}+\frac {2 a b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {2 a b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {2 b^{2} \arcsinh \left (c x \right ) \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {2 b^{2} \arcsinh \left (c x \right ) \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b^{2} \arcsinh \left (c x \right )^{2}}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \arcsinh \left (c x \right )}{d^{2} \left (c^{2} x^{2}+1\right )}-\frac {b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {b^{2} \arcsinh \left (c x \right ) \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {a b \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}+\frac {a b \arcsinh \left (c x \right )}{d^{2} \left (c^{2} x^{2}+1\right )}-\frac {2 a b \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}+\frac {2 a b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {2 a b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b^{2} \polylog \left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2 d^{2}}-\frac {2 b^{2} \polylog \left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}-\frac {2 b^{2} \polylog \left (3, c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {a^{2}}{2 d^{2} \left (c^{2} x^{2}+1\right )}-\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}+\frac {a^{2} \ln \left (c x \right )}{d^{2}}-\frac {2 b^{2} \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}} \]
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 \[ \frac {1}{2} \, a^{2} {\left (\frac {1}{c^{2} d^{2} x^{2} + d^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{d^{2}} + \frac {2 \, \log \relax (x)}{d^{2}}\right )} + \int \frac {b^{2} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{4} d^{2} x^{5} + 2 \, c^{2} d^{2} x^{3} + d^{2} x} + \frac {2 \, a b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{c^{4} d^{2} x^{5} + 2 \, c^{2} d^{2} x^{3} + d^{2} x}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x\,{\left (d\,c^2\,x^2+d\right )}^2} \,d x \]
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 \[ \frac {\int \frac {a^{2}}{c^{4} x^{5} + 2 c^{2} x^{3} + x}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{4} x^{5} + 2 c^{2} x^{3} + x}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{5} + 2 c^{2} x^{3} + x}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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