Optimal. Leaf size=248 \[ -\frac {b^2 c^2 d^2}{3 x}+2 b^2 c^4 d^2 x-\frac {5}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {8}{3} c^4 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {4 c^2 d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac {22}{3} b c^3 d^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\frac {11}{3} b^2 c^3 d^2 \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+\frac {11}{3} b^2 c^3 d^2 \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \]
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Rubi [A]
time = 0.47, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5807, 5772,
5798, 8, 5806, 5816, 4267, 2317, 2438, 14} \begin {gather*} \frac {8}{3} c^4 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {22}{3} b c^3 d^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac {4 c^2 d^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {b c d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {d^2 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac {5}{3} b c^3 d^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+2 b^2 c^4 d^2 x-\frac {11}{3} b^2 c^3 d^2 \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+\frac {11}{3} b^2 c^3 d^2 \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )-\frac {b^2 c^2 d^2}{3 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 14
Rule 2317
Rule 2438
Rule 4267
Rule 5772
Rule 5798
Rule 5806
Rule 5807
Rule 5816
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac {d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} \left (4 c^2 d\right ) \int \frac {\left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{x^2} \, dx+\frac {1}{3} \left (2 b c d^2\right ) \int \frac {\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x^3} \, dx\\ &=-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {4 c^2 d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} \left (b^2 c^2 d^2\right ) \int \frac {1+c^2 x^2}{x^2} \, dx+\left (b c^3 d^2\right ) \int \frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\frac {1}{3} \left (8 b c^3 d^2\right ) \int \frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\frac {1}{3} \left (8 c^4 d^2\right ) \int \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx\\ &=\frac {11}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {8}{3} c^4 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {4 c^2 d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} \left (b^2 c^2 d^2\right ) \int \left (c^2+\frac {1}{x^2}\right ) \, dx+\left (b c^3 d^2\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \sqrt {1+c^2 x^2}} \, dx+\frac {1}{3} \left (8 b c^3 d^2\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \sqrt {1+c^2 x^2}} \, dx-\left (b^2 c^4 d^2\right ) \int 1 \, dx-\frac {1}{3} \left (8 b^2 c^4 d^2\right ) \int 1 \, dx-\frac {1}{3} \left (16 b c^5 d^2\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {b^2 c^2 d^2}{3 x}-\frac {10}{3} b^2 c^4 d^2 x-\frac {5}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {8}{3} c^4 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {4 c^2 d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\left (b c^3 d^2\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )+\frac {1}{3} \left (8 b c^3 d^2\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )+\frac {1}{3} \left (16 b^2 c^4 d^2\right ) \int 1 \, dx\\ &=-\frac {b^2 c^2 d^2}{3 x}+2 b^2 c^4 d^2 x-\frac {5}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {8}{3} c^4 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {4 c^2 d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac {22}{3} b c^3 d^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\left (b^2 c^3 d^2\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )+\left (b^2 c^3 d^2\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )-\frac {1}{3} \left (8 b^2 c^3 d^2\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )+\frac {1}{3} \left (8 b^2 c^3 d^2\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=-\frac {b^2 c^2 d^2}{3 x}+2 b^2 c^4 d^2 x-\frac {5}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {8}{3} c^4 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {4 c^2 d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac {22}{3} b c^3 d^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\left (b^2 c^3 d^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )+\left (b^2 c^3 d^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )-\frac {1}{3} \left (8 b^2 c^3 d^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )+\frac {1}{3} \left (8 b^2 c^3 d^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )\\ &=-\frac {b^2 c^2 d^2}{3 x}+2 b^2 c^4 d^2 x-\frac {5}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {8}{3} c^4 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {4 c^2 d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac {22}{3} b c^3 d^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\frac {11}{3} b^2 c^3 d^2 \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+\frac {11}{3} b^2 c^3 d^2 \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.57, size = 357, normalized size = 1.44 \begin {gather*} \frac {d^2 \left (-a^2-6 a^2 c^2 x^2-b^2 c^2 x^2+3 a^2 c^4 x^4+6 b^2 c^4 x^4-a b c x \sqrt {1+c^2 x^2}-6 a b c^3 x^3 \sqrt {1+c^2 x^2}-2 a b \sinh ^{-1}(c x)-12 a b c^2 x^2 \sinh ^{-1}(c x)+6 a b c^4 x^4 \sinh ^{-1}(c x)-b^2 c x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)-6 b^2 c^3 x^3 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)-b^2 \sinh ^{-1}(c x)^2-6 b^2 c^2 x^2 \sinh ^{-1}(c x)^2+3 b^2 c^4 x^4 \sinh ^{-1}(c x)^2-11 a b c^3 x^3 \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )+11 b^2 c^3 x^3 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-11 b^2 c^3 x^3 \sinh ^{-1}(c x) \log \left (1+e^{-\sinh ^{-1}(c x)}\right )+11 b^2 c^3 x^3 \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-11 b^2 c^3 x^3 \text {PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )\right )}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 6.10, size = 363, normalized size = 1.46
method | result | size |
derivativedivides | \(c^{3} \left (a^{2} d^{2} \left (c x -\frac {1}{3 c^{3} x^{3}}-\frac {2}{c x}\right )+b^{2} d^{2} \arcsinh \left (c x \right )^{2} c x -2 b^{2} d^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 b^{2} d^{2} c x -\frac {2 b^{2} d^{2} \arcsinh \left (c x \right )^{2}}{c x}-\frac {b^{2} d^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3 c^{2} x^{2}}-\frac {b^{2} d^{2} \arcsinh \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {b^{2} d^{2}}{3 c x}+\frac {11 b^{2} d^{2} \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {11 b^{2} d^{2} \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {11 b^{2} d^{2} \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {11 b^{2} d^{2} \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+2 a \,d^{2} b \left (\arcsinh \left (c x \right ) c x -\frac {\arcsinh \left (c x \right )}{3 c^{3} x^{3}}-\frac {2 \arcsinh \left (c x \right )}{c x}-\sqrt {c^{2} x^{2}+1}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {11 \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\) | \(363\) |
default | \(c^{3} \left (a^{2} d^{2} \left (c x -\frac {1}{3 c^{3} x^{3}}-\frac {2}{c x}\right )+b^{2} d^{2} \arcsinh \left (c x \right )^{2} c x -2 b^{2} d^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 b^{2} d^{2} c x -\frac {2 b^{2} d^{2} \arcsinh \left (c x \right )^{2}}{c x}-\frac {b^{2} d^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3 c^{2} x^{2}}-\frac {b^{2} d^{2} \arcsinh \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {b^{2} d^{2}}{3 c x}+\frac {11 b^{2} d^{2} \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {11 b^{2} d^{2} \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {11 b^{2} d^{2} \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {11 b^{2} d^{2} \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+2 a \,d^{2} b \left (\arcsinh \left (c x \right ) c x -\frac {\arcsinh \left (c x \right )}{3 c^{3} x^{3}}-\frac {2 \arcsinh \left (c x \right )}{c x}-\sqrt {c^{2} x^{2}+1}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {11 \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\) | \(363\) |
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*} d^{2} \left (\int a^{2} c^{4}\, dx + \int \frac {a^{2}}{x^{4}}\, dx + \int \frac {2 a^{2} c^{2}}{x^{2}}\, dx + \int b^{2} c^{4} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int 2 a b c^{4} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {2 b^{2} c^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {4 a b c^{2} \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx\right ) \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 = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^2}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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