Optimal. Leaf size=318 \[ \frac {b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2 \text {ArcTan}\left (e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \text {ArcTan}(c x)}{6 c^3 d^3}-\frac {i b \left (a+b \sinh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac {i b \left (a+b \sinh ^{-1}(c x)\right ) \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac {i b^2 \text {PolyLog}\left (3,-i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac {i b^2 \text {PolyLog}\left (3,i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3} \]
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Rubi [A]
time = 0.30, antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5810, 5788,
5789, 4265, 2611, 2320, 6724, 5798, 209, 205} \begin {gather*} \frac {\text {ArcTan}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^3 d^3}-\frac {i b \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3}+\frac {i b \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (c^2 x^2+1\right )}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt {c^2 x^2+1}}-\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{6 c^3 d^3 \left (c^2 x^2+1\right )^{3/2}}-\frac {b^2 \text {ArcTan}(c x)}{6 c^3 d^3}+\frac {i b^2 \text {Li}_3\left (-i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac {i b^2 \text {Li}_3\left (i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac {b^2 x}{12 c^2 d^3 \left (c^2 x^2+1\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 209
Rule 2320
Rule 2611
Rule 4265
Rule 5788
Rule 5789
Rule 5798
Rule 5810
Rule 6724
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^3} \, dx &=-\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {b \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{2 c d^3}+\frac {\int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^2} \, dx}{4 c^2 d}\\ &=-\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {b^2 \int \frac {1}{\left (1+c^2 x^2\right )^2} \, dx}{6 c^2 d^3}-\frac {b \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{4 c d^3}+\frac {\int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{d+c^2 d x^2} \, dx}{8 c^2 d^2}\\ &=\frac {b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {\text {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^3 d^3}+\frac {b^2 \int \frac {1}{1+c^2 x^2} \, dx}{12 c^2 d^3}-\frac {b^2 \int \frac {1}{1+c^2 x^2} \, dx}{4 c^2 d^3}\\ &=\frac {b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \tan ^{-1}(c x)}{6 c^3 d^3}-\frac {(i b) \text {Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^3 d^3}+\frac {(i b) \text {Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^3 d^3}\\ &=\frac {b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \tan ^{-1}(c x)}{6 c^3 d^3}-\frac {i b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac {i b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^3 d^3}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^3 d^3}\\ &=\frac {b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \tan ^{-1}(c x)}{6 c^3 d^3}-\frac {i b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac {i b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}\\ &=\frac {b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \tan ^{-1}(c x)}{6 c^3 d^3}-\frac {i b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac {i b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac {i b^2 \text {Li}_3\left (-i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac {i b^2 \text {Li}_3\left (i e^{\sinh ^{-1}(c x)}\right )}{4 c^3 d^3}\\ \end {align*}
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Mathematica [A]
time = 1.71, size = 550, normalized size = 1.73 \begin {gather*} \frac {-\frac {6 a^2 c x}{\left (1+c^2 x^2\right )^2}+\frac {3 a^2 c x}{1+c^2 x^2}+\frac {a b \left ((2+i c x) \sqrt {1+c^2 x^2}+3 i \sinh ^{-1}(c x)\right )}{(-i+c x)^2}+\frac {3 a b \left (-i \sqrt {1+c^2 x^2}+\sinh ^{-1}(c x)\right )}{-i+c x}+\frac {3 a b \left (i \sqrt {1+c^2 x^2}+\sinh ^{-1}(c x)\right )}{i+c x}-\frac {i a b \left ((2 i+c x) \sqrt {1+c^2 x^2}+3 \sinh ^{-1}(c x)\right )}{(i+c x)^2}+3 a^2 \text {ArcTan}(c x)+\frac {3}{2} i 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 {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )\right )-\frac {3}{2} i 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 {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )\right )+b^2 \left (\frac {2 c x}{1+c^2 x^2}-\frac {4 \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^{3/2}}+\frac {6 \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}-\frac {6 c x \sinh ^{-1}(c x)^2}{\left (1+c^2 x^2\right )^2}+\frac {3 c x \sinh ^{-1}(c x)^2}{1+c^2 x^2}-8 \text {ArcTan}\left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )-3 i \sinh ^{-1}(c x)^2 \log \left (1-i e^{-\sinh ^{-1}(c x)}\right )+3 i \sinh ^{-1}(c x)^2 \log \left (1+i e^{-\sinh ^{-1}(c x)}\right )-6 i \sinh ^{-1}(c x) \text {PolyLog}\left (2,-i e^{-\sinh ^{-1}(c x)}\right )+6 i \sinh ^{-1}(c x) \text {PolyLog}\left (2,i e^{-\sinh ^{-1}(c x)}\right )-6 i \text {PolyLog}\left (3,-i e^{-\sinh ^{-1}(c x)}\right )+6 i \text {PolyLog}\left (3,i e^{-\sinh ^{-1}(c x)}\right )\right )}{24 c^3 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (a +b \arcsinh \left (c x \right )\right )^{2}}{\left (c^{2} d \,x^{2}+d \right )^{3}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \frac {\int \frac {a^{2} x^{2}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x^{2} \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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