Optimal. Leaf size=297 \[ -\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac {2 c^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \text {ArcTan}\left (e^{\sinh ^{-1}(c x)}\right )}{d}+\frac {14 b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d}+\frac {7 b^2 c^3 \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{3 d}-\frac {2 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac {2 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{d}-\frac {7 b^2 c^3 \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{3 d}+\frac {2 i b^2 c^3 \text {PolyLog}\left (3,-i e^{\sinh ^{-1}(c x)}\right )}{d}-\frac {2 i b^2 c^3 \text {PolyLog}\left (3,i e^{\sinh ^{-1}(c x)}\right )}{d} \]
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Rubi [A]
time = 0.46, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {5809, 5789,
4265, 2611, 2320, 6724, 5816, 4267, 2317, 2438, 30} \begin {gather*} \frac {2 c^3 \text {ArcTan}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d}-\frac {2 i b c^3 \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}+\frac {2 i b c^3 \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}+\frac {14 b c^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d}-\frac {b c \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2}+\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac {7 b^2 c^3 \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{3 d}-\frac {7 b^2 c^3 \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{3 d}+\frac {2 i b^2 c^3 \text {Li}_3\left (-i e^{\sinh ^{-1}(c x)}\right )}{d}-\frac {2 i b^2 c^3 \text {Li}_3\left (i e^{\sinh ^{-1}(c x)}\right )}{d}-\frac {b^2 c^2}{3 d x} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 4265
Rule 4267
Rule 5789
Rule 5809
Rule 5816
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x^4 \left (d+c^2 d x^2\right )} \, dx &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}-c^2 \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x^2 \left (d+c^2 d x^2\right )} \, dx+\frac {(2 b c) \int \frac {a+b \sinh ^{-1}(c x)}{x^3 \sqrt {1+c^2 x^2}} \, dx}{3 d}\\ &=-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+c^4 \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{d+c^2 d x^2} \, dx+\frac {\left (b^2 c^2\right ) \int \frac {1}{x^2} \, dx}{3 d}-\frac {\left (b c^3\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \sqrt {1+c^2 x^2}} \, dx}{3 d}-\frac {\left (2 b c^3\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \sqrt {1+c^2 x^2}} \, dx}{d}\\ &=-\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac {c^3 \text {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d}-\frac {\left (b c^3\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d}-\frac {\left (2 b c^3\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=-\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac {2 c^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}+\frac {14 b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d}-\frac {\left (2 i b c^3\right ) \text {Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}+\frac {\left (2 i b c^3\right ) \text {Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}+\frac {\left (b^2 c^3\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d}-\frac {\left (b^2 c^3\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d}+\frac {\left (2 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}-\frac {\left (2 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=-\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac {2 c^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}+\frac {14 b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d}-\frac {2 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac {2 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac {\left (2 i b^2 c^3\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}-\frac {\left (2 i b^2 c^3\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}+\frac {\left (b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{3 d}-\frac {\left (b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{3 d}+\frac {\left (2 b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d}-\frac {\left (2 b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d}\\ &=-\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac {2 c^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}+\frac {14 b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d}+\frac {7 b^2 c^3 \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{3 d}-\frac {2 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac {2 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{d}-\frac {7 b^2 c^3 \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{3 d}+\frac {\left (2 i b^2 c^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d}-\frac {\left (2 i b^2 c^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d}\\ &=-\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac {2 c^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}+\frac {14 b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d}+\frac {7 b^2 c^3 \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{3 d}-\frac {2 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac {2 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{d}-\frac {7 b^2 c^3 \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{3 d}+\frac {2 i b^2 c^3 \text {Li}_3\left (-i e^{\sinh ^{-1}(c x)}\right )}{d}-\frac {2 i b^2 c^3 \text {Li}_3\left (i e^{\sinh ^{-1}(c x)}\right )}{d}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(602\) vs. \(2(297)=594\).
time = 7.29, size = 602, normalized size = 2.03 \begin {gather*} -\frac {a^2}{3 d x^3}+\frac {a^2 c^2}{d x}+\frac {a^2 c^3 \text {ArcTan}(c x)}{d}+\frac {2 a b \left (-\frac {c \sqrt {1+c^2 x^2}}{6 x^2}-\frac {\sinh ^{-1}(c x)}{3 x^3}+\frac {1}{6} c^3 \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )-c^2 \left (-\frac {\sinh ^{-1}(c x)}{x}-c \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )\right )-\frac {1}{2} i c^4 \left (-\frac {\sinh ^{-1}(c x)^2}{2 c}+\frac {2 \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )}{c}+\frac {2 \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{c}\right )+\frac {1}{2} i c^4 \left (-\frac {\sinh ^{-1}(c x)^2}{2 c}+\frac {2 \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )}{c}+\frac {2 \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{c}\right )\right )}{d}+\frac {b^2 c^3 \left (-4 \coth \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+14 \sinh ^{-1}(c x)^2 \coth \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-2 \sinh ^{-1}(c x) \text {csch}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )-\frac {1}{2} c x \sinh ^{-1}(c x)^2 \text {csch}^4\left (\frac {1}{2} \sinh ^{-1}(c x)\right )-56 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-24 i \sinh ^{-1}(c x)^2 \log \left (1-i e^{-\sinh ^{-1}(c x)}\right )+24 i \sinh ^{-1}(c x)^2 \log \left (1+i e^{-\sinh ^{-1}(c x)}\right )+56 \sinh ^{-1}(c x) \log \left (1+e^{-\sinh ^{-1}(c x)}\right )-56 \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-48 i \sinh ^{-1}(c x) \text {PolyLog}\left (2,-i e^{-\sinh ^{-1}(c x)}\right )+48 i \sinh ^{-1}(c x) \text {PolyLog}\left (2,i e^{-\sinh ^{-1}(c x)}\right )+56 \text {PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )-48 i \text {PolyLog}\left (3,-i e^{-\sinh ^{-1}(c x)}\right )+48 i \text {PolyLog}\left (3,i e^{-\sinh ^{-1}(c x)}\right )-2 \sinh ^{-1}(c x) \text {sech}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )-\frac {8 \sinh ^{-1}(c x)^2 \sinh ^4\left (\frac {1}{2} \sinh ^{-1}(c x)\right )}{c^3 x^3}+4 \tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-14 \sinh ^{-1}(c x)^2 \tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )}{24 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsinh \left (c x \right )\right )^{2}}{x^{4} \left (c^{2} d \,x^{2}+d \right )}\, dx\]
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}}{c^{2} x^{6} + x^{4}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{2} x^{6} + x^{4}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{6} + x^{4}}\, 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 = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^4\,\left (d\,c^2\,x^2+d\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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