Optimal. Leaf size=249 \[ \frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2 d \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}-\frac {b d \sqrt {c^2 d x^2+d} \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}+\frac {b d \sqrt {c^2 d x^2+d} \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {4 b c d x \sqrt {c^2 d x^2+d}}{3 \sqrt {c^2 x^2+1}}-\frac {b c^3 d x^3 \sqrt {c^2 d x^2+d}}{9 \sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.30, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {5744, 5742, 5760, 4182, 2279, 2391, 8} \[ -\frac {b d \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}+\frac {b d \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2 d \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}-\frac {b c^3 d x^3 \sqrt {c^2 d x^2+d}}{9 \sqrt {c^2 x^2+1}}-\frac {4 b c d x \sqrt {c^2 d x^2+d}}{3 \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2279
Rule 2391
Rule 4182
Rule 5742
Rule 5744
Rule 5760
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx &=\frac {1}{3} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+d \int \frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx-\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right ) \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c d x \sqrt {d+c^2 d x^2}}{3 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int 1 \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {4 b c d x \sqrt {d+c^2 d x^2}}{3 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}\\ &=-\frac {4 b c d x \sqrt {d+c^2 d x^2}}{3 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (b d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (b d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}\\ &=-\frac {4 b c d x \sqrt {d+c^2 d x^2}}{3 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (b d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (b d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}\\ &=-\frac {4 b c d x \sqrt {d+c^2 d x^2}}{3 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b d \sqrt {d+c^2 d x^2} \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {b d \sqrt {d+c^2 d x^2} \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.77, size = 248, normalized size = 1.00 \[ -a d^{3/2} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+d\right )+\frac {1}{3} a d \left (c^2 x^2+4\right ) \sqrt {c^2 d x^2+d}+a d^{3/2} \log (x)+\frac {b d \sqrt {c^2 d x^2+d} \left (\sqrt {c^2 x^2+1} \sinh ^{-1}(c x)+\text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-\text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )-c x+\sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )\right )}{\sqrt {c^2 x^2+1}}+\frac {b d \sqrt {c^2 d x^2+d} \left (3 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)-c x \left (c^2 x^2+3\right )\right )}{9 \sqrt {c^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a c^{2} d x^{2} + a d + {\left (b c^{2} d x^{2} + b d\right )} \operatorname {arsinh}\left (c x\right )\right )} \sqrt {c^{2} d x^{2} + d}}{x}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 428, normalized size = 1.72 \[ \frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a}{3}-a \,d^{\frac {3}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )+a \sqrt {c^{2} d \,x^{2}+d}\, d +\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \arcsinh \left (c x \right )}{3 \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) d}{\sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) d}{\sqrt {c^{2} x^{2}+1}}+\frac {5 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \arcsinh \left (c x \right ) x^{2} c^{2}}{3 \left (c^{2} x^{2}+1\right )}-\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d c x}{3 \sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \arcsinh \left (c x \right ) x^{4} c^{4}}{3 c^{2} x^{2}+3}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{3} x^{3}}{9 \sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) d}{\sqrt {c^{2} x^{2}+1}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) d}{\sqrt {c^{2} x^{2}+1}} \]
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 \[ -\frac {1}{3} \, {\left (3 \, d^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) - {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {c^{2} d x^{2} + d} d\right )} a + b \int \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{3/2}}{x} \,d x \]
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 \[ \int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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