Optimal. Leaf size=97 \[ -\frac {b c}{6 \sqrt {\pi } x^2}-\frac {\sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x^3}+\frac {2 c^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x}-\frac {2 b c^3 \log (x)}{3 \sqrt {\pi }} \]
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Rubi [A]
time = 0.12, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5809, 5800, 29,
30} \begin {gather*} \frac {2 c^2 \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x}-\frac {\sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x^3}-\frac {2 b c^3 \log (x)}{3 \sqrt {\pi }}-\frac {b c}{6 \sqrt {\pi } x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 30
Rule 5800
Rule 5809
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{x^4 \sqrt {\pi +c^2 \pi x^2}} \, dx &=-\frac {\sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x^3}-\frac {1}{3} \left (2 c^2\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x^2 \sqrt {\pi +c^2 \pi x^2}} \, dx+\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x^3} \, dx}{3 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{6 x^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {\sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x^3}+\frac {2 c^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x}-\frac {\left (2 b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x} \, dx}{3 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{6 x^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {\sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x^3}+\frac {2 c^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x}-\frac {2 b c^3 \sqrt {1+c^2 x^2} \log (x)}{3 \sqrt {\pi +c^2 \pi x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 99, normalized size = 1.02 \begin {gather*} \frac {2 a \sqrt {1+c^2 x^2} \left (-1+2 c^2 x^2\right )+b c x \left (-1+6 c^2 x^2\right )+2 b \sqrt {1+c^2 x^2} \left (-1+2 c^2 x^2\right ) \sinh ^{-1}(c x)}{6 \sqrt {\pi } x^3}-\frac {2 b c^3 \log (x)}{3 \sqrt {\pi }} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(372\) vs.
\(2(81)=162\).
time = 3.88, size = 373, normalized size = 3.85
method | result | size |
default | \(a \left (-\frac {\sqrt {\pi \,c^{2} x^{2}+\pi }}{3 \pi \,x^{3}}+\frac {2 c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}{3 \pi x}\right )+\frac {4 b \,c^{3} \arcsinh \left (c x \right )}{3 \sqrt {\pi }}-\frac {2 b \,x^{4} c^{7}}{3 \sqrt {\pi }\, \left (3 c^{2} x^{2}-1\right )}+\frac {2 b \,x^{2} \left (c^{2} x^{2}+1\right ) c^{5}}{3 \sqrt {\pi }\, \left (3 c^{2} x^{2}-1\right )}-\frac {2 b \,x^{2} \arcsinh \left (c x \right ) c^{5}}{\sqrt {\pi }\, \left (3 c^{2} x^{2}-1\right )}+\frac {2 b x \sqrt {c^{2} x^{2}+1}\, \arcsinh \left (c x \right ) c^{4}}{\sqrt {\pi }\, \left (3 c^{2} x^{2}-1\right )}-\frac {2 b \left (c^{2} x^{2}+1\right ) c^{3}}{3 \sqrt {\pi }\, \left (3 c^{2} x^{2}-1\right )}+\frac {2 b \arcsinh \left (c x \right ) c^{3}}{3 \sqrt {\pi }\, \left (3 c^{2} x^{2}-1\right )}-\frac {5 b \sqrt {c^{2} x^{2}+1}\, \arcsinh \left (c x \right ) c^{2}}{3 \sqrt {\pi }\, \left (3 c^{2} x^{2}-1\right ) x}+\frac {b \left (c^{2} x^{2}+1\right ) c}{6 \sqrt {\pi }\, \left (3 c^{2} x^{2}-1\right ) x^{2}}+\frac {b \sqrt {c^{2} x^{2}+1}\, \arcsinh \left (c x \right )}{3 \sqrt {\pi }\, \left (3 c^{2} x^{2}-1\right ) x^{3}}-\frac {2 b \,c^{3} \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{3 \sqrt {\pi }}\) | \(373\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 121, normalized size = 1.25 \begin {gather*} -\frac {1}{6} \, {\left (\frac {4 \, c^{2} \log \left (x\right )}{\sqrt {\pi }} + \frac {1}{\sqrt {\pi } x^{2}}\right )} b c + \frac {1}{3} \, b {\left (\frac {2 \, \sqrt {\pi + \pi c^{2} x^{2}} c^{2}}{\pi x} - \frac {\sqrt {\pi + \pi c^{2} x^{2}}}{\pi x^{3}}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{3} \, a {\left (\frac {2 \, \sqrt {\pi + \pi c^{2} x^{2}} c^{2}}{\pi x} - \frac {\sqrt {\pi + \pi c^{2} x^{2}}}{\pi x^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 222 vs.
\(2 (81) = 162\).
time = 0.40, size = 222, normalized size = 2.29 \begin {gather*} \frac {2 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (2 \, b c^{4} x^{4} + b c^{2} x^{2} - b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, \sqrt {\pi } {\left (b c^{5} x^{5} + b c^{3} x^{3}\right )} \log \left (\frac {\pi + \pi c^{2} x^{6} + \pi c^{2} x^{2} + \pi x^{4} - \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} {\left (x^{4} - 1\right )}}{c^{2} x^{4} + x^{2}}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (4 \, a c^{4} x^{4} + 2 \, a c^{2} x^{2} + {\left (b c x^{3} - b c x\right )} \sqrt {c^{2} x^{2} + 1} - 2 \, a\right )}}{6 \, {\left (\pi c^{2} x^{5} + \pi x^{3}\right )}} \end {gather*}
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 \begin {gather*} \frac {\int \frac {a}{x^{4} \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x^{4} \sqrt {c^{2} x^{2} + 1}}\, dx}{\sqrt {\pi }} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^4\,\sqrt {\Pi \,c^2\,x^2+\Pi }} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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