Optimal. Leaf size=287 \[ -\frac {b c \sqrt {1+c^2 x^2}}{2 d x \sqrt {d+c^2 d x^2}}-\frac {3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d \sqrt {d+c^2 d x^2}}-\frac {a+b \sinh ^{-1}(c x)}{2 d x^2 \sqrt {d+c^2 d x^2}}+\frac {b c^2 \sqrt {1+c^2 x^2} \text {ArcTan}(c x)}{d \sqrt {d+c^2 d x^2}}+\frac {3 c^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {3 b c^2 \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{2 d \sqrt {d+c^2 d x^2}}-\frac {3 b c^2 \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{2 d \sqrt {d+c^2 d x^2}} \]
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Rubi [A]
time = 0.26, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5809, 5811,
5816, 4267, 2317, 2438, 209, 331} \begin {gather*} -\frac {3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d \sqrt {c^2 d x^2+d}}-\frac {a+b \sinh ^{-1}(c x)}{2 d x^2 \sqrt {c^2 d x^2+d}}+\frac {3 c^2 \sqrt {c^2 x^2+1} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {c^2 d x^2+d}}+\frac {b c^2 \sqrt {c^2 x^2+1} \text {ArcTan}(c x)}{d \sqrt {c^2 d x^2+d}}+\frac {3 b c^2 \sqrt {c^2 x^2+1} \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{2 d \sqrt {c^2 d x^2+d}}-\frac {3 b c^2 \sqrt {c^2 x^2+1} \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{2 d \sqrt {c^2 d x^2+d}}-\frac {b c \sqrt {c^2 x^2+1}}{2 d x \sqrt {c^2 d x^2+d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 331
Rule 2317
Rule 2438
Rule 4267
Rule 5809
Rule 5811
Rule 5816
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{x^3 \left (d+c^2 d x^2\right )^{3/2}} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{2 d x^2 \sqrt {d+c^2 d x^2}}-\frac {1}{2} \left (3 c^2\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \left (d+c^2 d x^2\right )^{3/2}} \, dx+\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx}{2 d \sqrt {d+c^2 d x^2}}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{2 d x \sqrt {d+c^2 d x^2}}-\frac {3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d \sqrt {d+c^2 d x^2}}-\frac {a+b \sinh ^{-1}(c x)}{2 d x^2 \sqrt {d+c^2 d x^2}}-\frac {\left (3 c^2\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \sqrt {d+c^2 d x^2}} \, dx}{2 d}-\frac {\left (b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 d \sqrt {d+c^2 d x^2}}+\frac {\left (3 b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 d \sqrt {d+c^2 d x^2}}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{2 d x \sqrt {d+c^2 d x^2}}-\frac {3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d \sqrt {d+c^2 d x^2}}-\frac {a+b \sinh ^{-1}(c x)}{2 d x^2 \sqrt {d+c^2 d x^2}}+\frac {b c^2 \sqrt {1+c^2 x^2} \tan ^{-1}(c x)}{d \sqrt {d+c^2 d x^2}}-\frac {\left (3 c^2 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \sqrt {1+c^2 x^2}} \, dx}{2 d \sqrt {d+c^2 d x^2}}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{2 d x \sqrt {d+c^2 d x^2}}-\frac {3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d \sqrt {d+c^2 d x^2}}-\frac {a+b \sinh ^{-1}(c x)}{2 d x^2 \sqrt {d+c^2 d x^2}}+\frac {b c^2 \sqrt {1+c^2 x^2} \tan ^{-1}(c x)}{d \sqrt {d+c^2 d x^2}}-\frac {\left (3 c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 d \sqrt {d+c^2 d x^2}}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{2 d x \sqrt {d+c^2 d x^2}}-\frac {3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d \sqrt {d+c^2 d x^2}}-\frac {a+b \sinh ^{-1}(c x)}{2 d x^2 \sqrt {d+c^2 d x^2}}+\frac {b c^2 \sqrt {1+c^2 x^2} \tan ^{-1}(c x)}{d \sqrt {d+c^2 d x^2}}+\frac {3 c^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {\left (3 b c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 d \sqrt {d+c^2 d x^2}}-\frac {\left (3 b c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 d \sqrt {d+c^2 d x^2}}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{2 d x \sqrt {d+c^2 d x^2}}-\frac {3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d \sqrt {d+c^2 d x^2}}-\frac {a+b \sinh ^{-1}(c x)}{2 d x^2 \sqrt {d+c^2 d x^2}}+\frac {b c^2 \sqrt {1+c^2 x^2} \tan ^{-1}(c x)}{d \sqrt {d+c^2 d x^2}}+\frac {3 c^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {\left (3 b c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 d \sqrt {d+c^2 d x^2}}-\frac {\left (3 b c^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 d \sqrt {d+c^2 d x^2}}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{2 d x \sqrt {d+c^2 d x^2}}-\frac {3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d \sqrt {d+c^2 d x^2}}-\frac {a+b \sinh ^{-1}(c x)}{2 d x^2 \sqrt {d+c^2 d x^2}}+\frac {b c^2 \sqrt {1+c^2 x^2} \tan ^{-1}(c x)}{d \sqrt {d+c^2 d x^2}}+\frac {3 c^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {3 b c^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{2 d \sqrt {d+c^2 d x^2}}-\frac {3 b c^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{2 d \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 5.66, size = 369, normalized size = 1.29 \begin {gather*} \frac {-\frac {4 a \left (1+3 c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{x^2+c^2 x^4}-12 a c^2 \sqrt {d} \log (x)+12 a c^2 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {b c^2 d \left (-8 \sinh ^{-1}(c x)+16 \sqrt {1+c^2 x^2} \text {ArcTan}\left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )-2 \sqrt {1+c^2 x^2} \coth \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-\sqrt {1+c^2 x^2} \sinh ^{-1}(c x) \text {csch}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )-12 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )+12 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x) \log \left (1+e^{-\sinh ^{-1}(c x)}\right )-12 \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )+12 \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )-\sqrt {1+c^2 x^2} \sinh ^{-1}(c x) \text {sech}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )+2 \sqrt {1+c^2 x^2} \tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )}{\sqrt {d+c^2 d x^2}}}{8 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.23, size = 389, normalized size = 1.36
method | result | size |
default | \(-\frac {a}{2 d \,x^{2} \sqrt {c^{2} d \,x^{2}+d}}-\frac {3 a \,c^{2}}{2 d \sqrt {c^{2} d \,x^{2}+d}}+\frac {3 a \,c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )}{2 d^{\frac {3}{2}}}-\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) c^{2}}{2 d^{2} \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c}{2 d^{2} \sqrt {c^{2} x^{2}+1}\, x}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{2 d^{2} \left (c^{2} x^{2}+1\right ) x^{2}}+\frac {2 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arctan \left (c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}\, d^{2}}+\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \dilog \left (c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d^{2}}+\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \dilog \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d^{2}}+\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d^{2}}\) | \(389\) |
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*} \int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{x^{3} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \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 {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^3\,{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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