Optimal. Leaf size=89 \[ \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )-2 \sqrt {\pi } \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\sqrt {\pi } b \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+\sqrt {\pi } b \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )+\sqrt {\pi } (-b) c x \]
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Rubi [A] time = 0.19, antiderivative size = 177, normalized size of antiderivative = 1.99, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5742, 5760, 4182, 2279, 2391, 8} \[ -\frac {b \sqrt {\pi c^2 x^2+\pi } \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}+\frac {b \sqrt {\pi c^2 x^2+\pi } \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}+\sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )-\frac {2 \sqrt {\pi c^2 x^2+\pi } \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}-\frac {b c x \sqrt {\pi c^2 x^2+\pi }}{\sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2279
Rule 2391
Rule 4182
Rule 5742
Rule 5760
Rubi steps
\begin {align*} \int \frac {\sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx &=\sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {\sqrt {\pi +c^2 \pi x^2} \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 \sqrt {\pi +c^2 \pi x^2}\right ) \int 1 \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {b c x \sqrt {\pi +c^2 \pi x^2}}{\sqrt {1+c^2 x^2}}+\sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {\sqrt {\pi +c^2 \pi x^2} \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}\\ &=-\frac {b c x \sqrt {\pi +c^2 \pi x^2}}{\sqrt {1+c^2 x^2}}+\sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2 \sqrt {\pi +c^2 \pi 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 \sqrt {\pi +c^2 \pi 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 \sqrt {\pi +c^2 \pi 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 {b c x \sqrt {\pi +c^2 \pi x^2}}{\sqrt {1+c^2 x^2}}+\sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2 \sqrt {\pi +c^2 \pi 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 \sqrt {\pi +c^2 \pi 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 \sqrt {\pi +c^2 \pi 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 {b c x \sqrt {\pi +c^2 \pi x^2}}{\sqrt {1+c^2 x^2}}+\sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2 \sqrt {\pi +c^2 \pi 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 \sqrt {\pi +c^2 \pi x^2} \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {b \sqrt {\pi +c^2 \pi 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 = 0.20, size = 131, normalized size = 1.47 \[ \sqrt {\pi } \left (a \sqrt {c^2 x^2+1}-a \log \left (\pi \left (\sqrt {c^2 x^2+1}+1\right )\right )+a \log (x)+b \left (\sqrt {c^2 x^2+1} \sinh ^{-1}(c x)+\text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-\text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )-c x+\sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\pi + \pi c^{2} x^{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x}, x\right ) \]
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 = 0.22, size = 171, normalized size = 1.92 \[ -\sqrt {\pi }\, \arctanh \left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right ) a +a \sqrt {\pi \,c^{2} x^{2}+\pi }+\sqrt {\pi }\, \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) b +\sqrt {\pi }\, \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, b -\sqrt {\pi }\, \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) b -b c x \sqrt {\pi }+b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) \sqrt {\pi }-b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) \sqrt {\pi } \]
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 \[ -{\left (\sqrt {\pi } \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) - \sqrt {\pi + \pi c^{2} x^{2}}\right )} a + b \int \frac {\sqrt {\pi + \pi c^{2} x^{2}} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {\Pi \,c^2\,x^2+\Pi }}{x} \,d x \]
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 \[ \sqrt {\pi } \left (\int \frac {a \sqrt {c^{2} x^{2} + 1}}{x}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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