Optimal. Leaf size=194 \[ \frac {a+b \sinh ^{-1}(c x)}{d \sqrt {c^2 d x^2+d}}-\frac {2 \sqrt {c^2 x^2+1} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {c^2 d x^2+d}}-\frac {b \sqrt {c^2 x^2+1} \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{d \sqrt {c^2 d x^2+d}}+\frac {b \sqrt {c^2 x^2+1} \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt {c^2 d x^2+d}}-\frac {b \sqrt {c^2 x^2+1} \tan ^{-1}(c x)}{d \sqrt {c^2 d x^2+d}} \]
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Rubi [A] time = 0.30, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {5755, 5764, 5760, 4182, 2279, 2391, 203} \[ -\frac {b \sqrt {c^2 x^2+1} \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{d \sqrt {c^2 d x^2+d}}+\frac {b \sqrt {c^2 x^2+1} \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{d \sqrt {c^2 d x^2+d}}+\frac {a+b \sinh ^{-1}(c x)}{d \sqrt {c^2 d x^2+d}}-\frac {2 \sqrt {c^2 x^2+1} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {c^2 d x^2+d}}-\frac {b \sqrt {c^2 x^2+1} \tan ^{-1}(c x)}{d \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 2279
Rule 2391
Rule 4182
Rule 5755
Rule 5760
Rule 5764
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{x \left (d+c^2 d x^2\right )^{3/2}} \, dx &=\frac {a+b \sinh ^{-1}(c x)}{d \sqrt {d+c^2 d x^2}}+\frac {\int \frac {a+b \sinh ^{-1}(c x)}{x \sqrt {d+c^2 d x^2}} \, dx}{d}-\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \int \frac {1}{1+c^2 x^2} \, dx}{d \sqrt {d+c^2 d x^2}}\\ &=\frac {a+b \sinh ^{-1}(c x)}{d \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \tan ^{-1}(c x)}{d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \int \frac {a+b \sinh ^{-1}(c x)}{x \sqrt {1+c^2 x^2}} \, dx}{d \sqrt {d+c^2 d x^2}}\\ &=\frac {a+b \sinh ^{-1}(c x)}{d \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \tan ^{-1}(c x)}{d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d \sqrt {d+c^2 d x^2}}\\ &=\frac {a+b \sinh ^{-1}(c x)}{d \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \tan ^{-1}(c x)}{d \sqrt {d+c^2 d x^2}}-\frac {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 (b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d \sqrt {d+c^2 d x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d \sqrt {d+c^2 d x^2}}\\ &=\frac {a+b \sinh ^{-1}(c x)}{d \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \tan ^{-1}(c x)}{d \sqrt {d+c^2 d x^2}}-\frac {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 (b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}\\ &=\frac {a+b \sinh ^{-1}(c x)}{d \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \tan ^{-1}(c x)}{d \sqrt {d+c^2 d x^2}}-\frac {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 {b \sqrt {1+c^2 x^2} \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {b \sqrt {1+c^2 x^2} \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.85, size = 231, normalized size = 1.19 \[ \frac {\frac {a \sqrt {c^2 d x^2+d}}{c^2 x^2+1}-a \sqrt {d} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+d\right )+a \sqrt {d} \log (x)+\frac {b d \left (\sqrt {c^2 x^2+1} \text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-\sqrt {c^2 x^2+1} \text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )+\sqrt {c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\sqrt {c^2 x^2+1} \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )-2 \sqrt {c^2 x^2+1} \tan ^{-1}\left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )+\sinh ^{-1}(c x)\right )}{\sqrt {c^2 d x^2+d}}}{d^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{c^{4} d^{2} x^{5} + 2 \, c^{2} d^{2} x^{3} + d^{2} x}, x\right ) \]
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 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 274, normalized size = 1.41 \[ \frac {a}{d \sqrt {c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{d^{2} \left (c^{2} x^{2}+1\right )}-\frac {2 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arctan \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \dilog \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \dilog \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2}} \]
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 \[ -a {\left (\frac {\operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right )}{d^{\frac {3}{2}}} - \frac {1}{\sqrt {c^{2} d x^{2} + d} d}\right )} + b \int \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x\,{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{x \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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