Optimal. Leaf size=61 \[ -\frac {\sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {\sqrt {\pi } c \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+\sqrt {\pi } b c \log (x) \]
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Rubi [A] time = 0.11, antiderivative size = 105, normalized size of antiderivative = 1.72, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {5737, 29, 5675} \[ \frac {c \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b \sqrt {c^2 x^2+1}}-\frac {\sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {b c \sqrt {\pi c^2 x^2+\pi } \log (x)}{\sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 29
Rule 5675
Rule 5737
Rubi steps
\begin {align*} \int \frac {\sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {\sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {\left (b c \sqrt {\pi +c^2 \pi x^2}\right ) \int \frac {1}{x} \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (c^2 \sqrt {\pi +c^2 \pi x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {\sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {c \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b \sqrt {1+c^2 x^2}}+\frac {b c \sqrt {\pi +c^2 \pi x^2} \log (x)}{\sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 75, normalized size = 1.23 \[ \frac {\sqrt {\pi } \left (2 \sinh ^{-1}(c x) \left (a c x-b \sqrt {c^2 x^2+1}\right )-2 a \sqrt {c^2 x^2+1}+2 b c x \log (c x)+b c x \sinh ^{-1}(c x)^2\right )}{2 x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\pi + \pi c^{2} x^{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{2}}, x\right ) \]
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 [B] time = 0.16, size = 155, normalized size = 2.54 \[ -\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{\pi x}+a \,c^{2} x \sqrt {\pi \,c^{2} x^{2}+\pi }+\frac {a \,c^{2} \pi \ln \left (\frac {\pi x \,c^{2}}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\sqrt {\pi \,c^{2}}}+\frac {b c \sqrt {\pi }\, \arcsinh \left (c x \right )^{2}}{2}-b c \sqrt {\pi }\, \arcsinh \left (c x \right )-\frac {b \sqrt {\pi }\, \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{x}+b c \sqrt {\pi }\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) \]
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 \[ {\left (\sqrt {\pi } c \operatorname {arsinh}\left (c x\right ) - \frac {\sqrt {\pi + \pi c^{2} x^{2}}}{x}\right )} a + b \int \frac {\sqrt {\pi + \pi c^{2} x^{2}} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {\Pi \,c^2\,x^2+\Pi }}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.01, size = 110, normalized size = 1.80 \[ - \frac {\sqrt {\pi } a c^{2} x}{\sqrt {c^{2} x^{2} + 1}} + \sqrt {\pi } a c \operatorname {asinh}{\left (c x \right )} - \frac {\sqrt {\pi } a}{x \sqrt {c^{2} x^{2} + 1}} + \sqrt {\pi } b c \log {\relax (x )} + \frac {\sqrt {\pi } b c \operatorname {asinh}^{2}{\left (c x \right )}}{2} - \frac {\sqrt {\pi } b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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