Optimal. Leaf size=93 \[ -\frac {2 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {a+b \sinh ^{-1}(c x)}{\pi x \sqrt {\pi c^2 x^2+\pi }}+\frac {b c \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2}}+\frac {b c \log (x)}{\pi ^{3/2}} \]
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Rubi [A] time = 0.14, antiderivative size = 95, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {271, 191, 5732, 446, 72} \[ -\frac {2 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{3/2} \sqrt {c^2 x^2+1}}-\frac {a+b \sinh ^{-1}(c x)}{\pi ^{3/2} x \sqrt {c^2 x^2+1}}+\frac {b c \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2}}+\frac {b c \log (x)}{\pi ^{3/2}} \]
Antiderivative was successfully verified.
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Rule 72
Rule 191
Rule 271
Rule 446
Rule 5732
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{\pi ^{3/2} x \sqrt {1+c^2 x^2}}-\frac {2 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{3/2} \sqrt {1+c^2 x^2}}-\frac {(b c) \int \frac {-1-2 c^2 x^2}{x \left (1+c^2 x^2\right )} \, dx}{\pi ^{3/2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{\pi ^{3/2} x \sqrt {1+c^2 x^2}}-\frac {2 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{3/2} \sqrt {1+c^2 x^2}}-\frac {(b c) \operatorname {Subst}\left (\int \frac {-1-2 c^2 x}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 \pi ^{3/2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{\pi ^{3/2} x \sqrt {1+c^2 x^2}}-\frac {2 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{3/2} \sqrt {1+c^2 x^2}}-\frac {(b c) \operatorname {Subst}\left (\int \left (-\frac {1}{x}-\frac {c^2}{1+c^2 x}\right ) \, dx,x,x^2\right )}{2 \pi ^{3/2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{\pi ^{3/2} x \sqrt {1+c^2 x^2}}-\frac {2 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{3/2} \sqrt {1+c^2 x^2}}+\frac {b c \log (x)}{\pi ^{3/2}}+\frac {b c \log \left (1+c^2 x^2\right )}{2 \pi ^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 69, normalized size = 0.74 \[ \frac {b \left (\frac {1}{2} c \log \left (c^2 x^2+1\right )+c \log (x)\right )}{\pi ^{3/2}}-\frac {\left (2 c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{3/2} x \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\pi + \pi c^{2} x^{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\pi ^{2} c^{4} x^{6} + 2 \, \pi ^{2} c^{2} x^{4} + \pi ^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 180, normalized size = 1.94 \[ -\frac {a}{\pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {2 a \,c^{2} x}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {4 b c \arcsinh \left (c x \right )}{\pi ^{\frac {3}{2}}}+\frac {2 b \arcsinh \left (c x \right ) x^{2} c^{3}}{\pi ^{\frac {3}{2}} \left (c^{2} x^{2}+1\right )}-\frac {2 b \arcsinh \left (c x \right ) x \,c^{2}}{\pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}}+\frac {2 b \arcsinh \left (c x \right ) c}{\pi ^{\frac {3}{2}} \left (c^{2} x^{2}+1\right )}-\frac {b \arcsinh \left (c x \right )}{\pi ^{\frac {3}{2}} x \sqrt {c^{2} x^{2}+1}}+\frac {b c \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{4}-1\right )}{\pi ^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 119, normalized size = 1.28 \[ \frac {1}{2} \, b c {\left (\frac {\log \left (c^{2} x^{2} + 1\right )}{\pi ^{\frac {3}{2}}} + \frac {2 \, \log \relax (x)}{\pi ^{\frac {3}{2}}}\right )} - {\left (\frac {2 \, c^{2} x}{\pi \sqrt {\pi + \pi c^{2} x^{2}}} + \frac {1}{\pi \sqrt {\pi + \pi c^{2} x^{2}} x}\right )} b \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, c^{2} x}{\pi \sqrt {\pi + \pi c^{2} x^{2}}} + \frac {1}{\pi \sqrt {\pi + \pi c^{2} x^{2}} x}\right )} a \]
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 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^2\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \]
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 \[ \frac {\int \frac {a}{c^{2} x^{4} \sqrt {c^{2} x^{2} + 1} + x^{2} \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{4} \sqrt {c^{2} x^{2} + 1} + x^{2} \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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