Optimal. Leaf size=199 \[ \frac {b^2 x^2}{4 c^2 d}-\frac {b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^4 d}+\frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^4 d}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^4 d}-\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{c^4 d}+\frac {b^2 \text {PolyLog}\left (3,-e^{2 \sinh ^{-1}(c x)}\right )}{2 c^4 d} \]
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Rubi [A]
time = 0.30, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5812, 5797,
3799, 2221, 2611, 2320, 6724, 5783, 30} \begin {gather*} -\frac {b \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^4 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^4 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^4 d}-\frac {\log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c^4 d}+\frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}-\frac {b x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d}+\frac {b^2 \text {Li}_3\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 c^4 d}+\frac {b^2 x^2}{4 c^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2221
Rule 2320
Rule 2611
Rule 3799
Rule 5783
Rule 5797
Rule 5812
Rule 6724
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{d+c^2 d x^2} \, dx &=\frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}-\frac {\int \frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{d+c^2 d x^2} \, dx}{c^2}-\frac {b \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{c d}\\ &=-\frac {b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d}+\frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}-\frac {\text {Subst}\left (\int (a+b x)^2 \tanh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^4 d}+\frac {b \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 c^3 d}+\frac {b^2 \int x \, dx}{2 c^2 d}\\ &=\frac {b^2 x^2}{4 c^2 d}-\frac {b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^4 d}+\frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^4 d}-\frac {2 \text {Subst}\left (\int \frac {e^{2 x} (a+b x)^2}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{c^4 d}\\ &=\frac {b^2 x^2}{4 c^2 d}-\frac {b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^4 d}+\frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^4 d}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^4 d}+\frac {(2 b) \text {Subst}\left (\int (a+b x) \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^4 d}\\ &=\frac {b^2 x^2}{4 c^2 d}-\frac {b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^4 d}+\frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^4 d}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^4 d}-\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{c^4 d}+\frac {b^2 \text {Subst}\left (\int \text {Li}_2\left (-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^4 d}\\ &=\frac {b^2 x^2}{4 c^2 d}-\frac {b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^4 d}+\frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^4 d}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^4 d}-\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{c^4 d}+\frac {b^2 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 c^4 d}\\ &=\frac {b^2 x^2}{4 c^2 d}-\frac {b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^4 d}+\frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^4 d}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^4 d}-\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{c^4 d}+\frac {b^2 \text {Li}_3\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 c^4 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.32, size = 292, normalized size = 1.47 \begin {gather*} \frac {12 a^2 c^2 x^2-12 a b c x \sqrt {1+c^2 x^2}+24 a b c^2 x^2 \sinh ^{-1}(c x)+24 a b \sinh ^{-1}(c x)^2-8 b^2 \sinh ^{-1}(c x)^3+12 a b \tanh ^{-1}\left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )+3 b^2 \cosh \left (2 \sinh ^{-1}(c x)\right )+6 b^2 \sinh ^{-1}(c x)^2 \cosh \left (2 \sinh ^{-1}(c x)\right )-24 b^2 \sinh ^{-1}(c x)^2 \log \left (1+e^{-2 \sinh ^{-1}(c x)}\right )-48 a b \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )-48 a b \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )-12 a^2 \log \left (1+c^2 x^2\right )+24 b^2 \sinh ^{-1}(c x) \text {PolyLog}\left (2,-e^{-2 \sinh ^{-1}(c x)}\right )-48 a b \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )-48 a b \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )+12 b^2 \text {PolyLog}\left (3,-e^{-2 \sinh ^{-1}(c x)}\right )-6 b^2 \sinh ^{-1}(c x) \sinh \left (2 \sinh ^{-1}(c x)\right )}{24 c^4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.80, size = 347, normalized size = 1.74
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} c^{2} x^{2}}{2 d}-\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {b^{2} \arcsinh \left (c x \right )^{3}}{3 d}+\frac {b^{2} \arcsinh \left (c x \right )^{2} c^{2} x^{2}}{2 d}-\frac {b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{2 d}+\frac {b^{2} \arcsinh \left (c x \right )^{2}}{4 d}+\frac {b^{2} c^{2} x^{2}}{4 d}+\frac {b^{2}}{8 d}-\frac {b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {b^{2} \arcsinh \left (c x \right ) \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {b^{2} \polylog \left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2 d}+\frac {a b \arcsinh \left (c x \right )^{2}}{d}+\frac {a b \arcsinh \left (c x \right ) c^{2} x^{2}}{d}-\frac {a b c x \sqrt {c^{2} x^{2}+1}}{2 d}+\frac {a b \arcsinh \left (c x \right )}{2 d}-\frac {2 a b \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {a b \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}}{c^{4}}\) | \(347\) |
default | \(\frac {\frac {a^{2} c^{2} x^{2}}{2 d}-\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {b^{2} \arcsinh \left (c x \right )^{3}}{3 d}+\frac {b^{2} \arcsinh \left (c x \right )^{2} c^{2} x^{2}}{2 d}-\frac {b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{2 d}+\frac {b^{2} \arcsinh \left (c x \right )^{2}}{4 d}+\frac {b^{2} c^{2} x^{2}}{4 d}+\frac {b^{2}}{8 d}-\frac {b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {b^{2} \arcsinh \left (c x \right ) \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {b^{2} \polylog \left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2 d}+\frac {a b \arcsinh \left (c x \right )^{2}}{d}+\frac {a b \arcsinh \left (c x \right ) c^{2} x^{2}}{d}-\frac {a b c x \sqrt {c^{2} x^{2}+1}}{2 d}+\frac {a b \arcsinh \left (c x \right )}{2 d}-\frac {2 a b \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {a b \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}}{c^{4}}\) | \(347\) |
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^{3}}{c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x^{3} \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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^3\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{d\,c^2\,x^2+d} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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