Optimal. Leaf size=108 \[ -\frac {b \sqrt {1+c^2 x^2}}{c^3 d}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \text {ArcTan}\left (e^{\sinh ^{-1}(c x)}\right )}{c^3 d}+\frac {i b \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{c^3 d}-\frac {i b \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{c^3 d} \]
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Rubi [A]
time = 0.10, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5812, 5789,
4265, 2317, 2438, 267} \begin {gather*} -\frac {2 \text {ArcTan}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^3 d}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d}+\frac {i b \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{c^3 d}-\frac {i b \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{c^3 d}-\frac {b \sqrt {c^2 x^2+1}}{c^3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 2317
Rule 2438
Rule 4265
Rule 5789
Rule 5812
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{d+c^2 d x^2} \, dx &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d}-\frac {\int \frac {a+b \sinh ^{-1}(c x)}{d+c^2 d x^2} \, dx}{c^2}-\frac {b \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{c d}\\ &=-\frac {b \sqrt {1+c^2 x^2}}{c^3 d}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d}-\frac {\text {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^3 d}\\ &=-\frac {b \sqrt {1+c^2 x^2}}{c^3 d}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^3 d}+\frac {(i b) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^3 d}-\frac {(i b) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^3 d}\\ &=-\frac {b \sqrt {1+c^2 x^2}}{c^3 d}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^3 d}+\frac {(i b) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c^3 d}-\frac {(i b) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c^3 d}\\ &=-\frac {b \sqrt {1+c^2 x^2}}{c^3 d}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^3 d}+\frac {i b \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{c^3 d}-\frac {i b \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{c^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 121, normalized size = 1.12 \begin {gather*} \frac {a c x-b \sqrt {1+c^2 x^2}+b c x \sinh ^{-1}(c x)-a \text {ArcTan}(c x)-i b \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )+i b \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )+i b \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )-i b \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{c^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.56, size = 194, normalized size = 1.80
method | result | size |
derivativedivides | \(\frac {\frac {a c x}{d}-\frac {a \arctan \left (c x \right )}{d}+\frac {i b \dilog \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{d}-\frac {b \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{d}-\frac {i b \dilog \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{d}-\frac {b \arcsinh \left (c x \right ) \arctan \left (c x \right )}{d}+\frac {b \arcsinh \left (c x \right ) c x}{d}+\frac {b \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{d}-\frac {b \sqrt {c^{2} x^{2}+1}}{d}}{c^{3}}\) | \(194\) |
default | \(\frac {\frac {a c x}{d}-\frac {a \arctan \left (c x \right )}{d}+\frac {i b \dilog \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{d}-\frac {b \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{d}-\frac {i b \dilog \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{d}-\frac {b \arcsinh \left (c x \right ) \arctan \left (c x \right )}{d}+\frac {b \arcsinh \left (c x \right ) c x}{d}+\frac {b \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{d}-\frac {b \sqrt {c^{2} x^{2}+1}}{d}}{c^{3}}\) | \(194\) |
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 x^{2}}{c^{2} x^{2} + 1}\, dx + \int \frac {b x^{2} \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} + 1}\, dx}{d} \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 = -0.01 \begin {gather*} \int \frac {x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{d\,c^2\,x^2+d} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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