Optimal. Leaf size=62 \[ -\frac {b c \sqrt {\pi }}{6 x^2}-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x^3}+\frac {1}{3} b c^3 \sqrt {\pi } \log (x) \]
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Rubi [A]
time = 0.06, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {5800, 14}
\begin {gather*} -\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x^3}+\frac {1}{3} \sqrt {\pi } b c^3 \log (x)-\frac {\sqrt {\pi } b c}{6 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 5800
Rubi steps
\begin {align*} \int \frac {\sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x^3}+\frac {\left (b c \sqrt {\pi +c^2 \pi x^2}\right ) \int \frac {1+c^2 x^2}{x^3} \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x^3}+\frac {\left (b c \sqrt {\pi +c^2 \pi x^2}\right ) \int \left (\frac {1}{x^3}+\frac {c^2}{x}\right ) \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c \sqrt {\pi +c^2 \pi x^2}}{6 x^2 \sqrt {1+c^2 x^2}}-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x^3}+\frac {b c^3 \sqrt {\pi +c^2 \pi x^2} \log (x)}{3 \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 78, normalized size = 1.26 \begin {gather*} -\frac {\sqrt {\pi } \left (b c x+3 b c^3 x^3+2 a \left (1+c^2 x^2\right )^{3/2}+2 b \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)\right )}{6 x^3}+\frac {1}{3} b c^3 \sqrt {\pi } \log (x) \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. \(500\) vs.
\(2(50)=100\).
time = 6.54, size = 501, normalized size = 8.08
method | result | size |
default | \(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3 \pi \,x^{3}}-\frac {2 b \,c^{3} \sqrt {\pi }\, \arcsinh \left (c x \right )}{3}+\frac {b \sqrt {\pi }\, x^{4} \arcsinh \left (c x \right ) c^{7}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}-\frac {b \sqrt {\pi }\, x^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{6}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}+\frac {b \sqrt {\pi }\, x^{4} c^{7}}{18 c^{4} x^{4}+18 c^{2} x^{2}+6}-\frac {b \sqrt {\pi }\, x^{2} \left (c^{2} x^{2}+1\right ) c^{5}}{6 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right )}+\frac {b \sqrt {\pi }\, x^{2} \arcsinh \left (c x \right ) c^{5}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}-\frac {2 b \sqrt {\pi }\, x \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{4}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}-\frac {b \sqrt {\pi }\, \left (c^{2} x^{2}+1\right ) c^{3}}{3 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right )}+\frac {b \,c^{3} \sqrt {\pi }\, \arcsinh \left (c x \right )}{9 c^{4} x^{4}+9 c^{2} x^{2}+3}-\frac {4 b \sqrt {\pi }\, \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2}}{3 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) x}-\frac {b \sqrt {\pi }\, \left (c^{2} x^{2}+1\right ) c}{6 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) x^{2}}-\frac {b \sqrt {\pi }\, \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) x^{3}}+\frac {b \,c^{3} \sqrt {\pi }\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{3}\) | \(501\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 133 vs.
\(2 (50) = 100\).
time = 0.28, size = 133, normalized size = 2.15 \begin {gather*} -\frac {{\left (\pi ^{\frac {3}{2}} \left (-1\right )^{2 \, \pi + 2 \, \pi c^{2} x^{2}} c^{2} \log \left (2 \, \pi c^{2} + \frac {2 \, \pi }{x^{2}}\right ) - \pi ^{\frac {3}{2}} c^{2} \log \left (x^{2} + \frac {1}{c^{2}}\right ) + \frac {\pi \sqrt {\pi + \pi c^{4} x^{4} + 2 \, \pi c^{2} x^{2}}}{x^{2}}\right )} b c}{6 \, \pi } - \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} b \operatorname {arsinh}\left (c x\right )}{3 \, \pi x^{3}} - \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} a}{3 \, \pi x^{3}} \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. 217 vs.
\(2 (50) = 100\).
time = 0.38, size = 217, normalized size = 3.50 \begin {gather*} -\frac {2 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - \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 (2 \, a c^{4} x^{4} + 4 \, 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 (c^{2} x^{5} + 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*} \sqrt {\pi } \left (\int \frac {a \sqrt {c^{2} x^{2} + 1}}{x^{4}}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{4}}\, dx\right ) \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.02 \begin {gather*} \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {\Pi \,c^2\,x^2+\Pi }}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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