Optimal. Leaf size=115 \[ -\frac {b c \pi ^{3/2}}{6 x^2}-\frac {c^2 \pi \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}+\frac {c^3 \pi ^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+\frac {4}{3} b c^3 \pi ^{3/2} \log (x) \]
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Rubi [A]
time = 0.15, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5807, 5805, 29,
5783, 14} \begin {gather*} \frac {\pi ^{3/2} c^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}-\frac {\pi c^2 \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}+\frac {4}{3} \pi ^{3/2} b c^3 \log (x)-\frac {\pi ^{3/2} b c}{6 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 29
Rule 5783
Rule 5805
Rule 5807
Rubi steps
\begin {align*} \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \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 x^3}+\left (c^2 \pi \right ) \int \frac {\sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x^2} \, dx+\frac {\left (b c \pi \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 {c^2 \pi \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}+\frac {\left (b c \pi \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 {\left (b c^3 \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \frac {1}{x} \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (c^4 \pi \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 {b c \pi \sqrt {\pi +c^2 \pi x^2}}{6 x^2 \sqrt {1+c^2 x^2}}-\frac {c^2 \pi \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}+\frac {c^3 \pi \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 {4 b c^3 \pi \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 = 0.14, size = 125, normalized size = 1.09 \begin {gather*} \frac {\pi ^{3/2} \left (-b c x-2 a \sqrt {1+c^2 x^2}-8 a c^2 x^2 \sqrt {1+c^2 x^2}+\left (6 a c^3 x^3-2 b \sqrt {1+c^2 x^2} \left (1+4 c^2 x^2\right )\right ) \sinh ^{-1}(c x)+3 b c^3 x^3 \sinh ^{-1}(c x)^2+8 b c^3 x^3 \log (c x)\right )}{6 x^3} \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. \(621\) vs.
\(2(97)=194\).
time = 6.62, size = 622, normalized size = 5.41
method | result | size |
default | \(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{3 \pi \,x^{3}}-\frac {2 a \,c^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{3 \pi x}+\frac {2 a \,c^{4} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3}+a \,c^{4} \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }+\frac {a \,c^{4} \pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\sqrt {\pi \,c^{2}}}+\frac {b \,c^{3} \pi ^{\frac {3}{2}} \arcsinh \left (c x \right )^{2}}{2}-\frac {8 b \,c^{3} \pi ^{\frac {3}{2}} \arcsinh \left (c x \right )}{3}+\frac {32 b \,\pi ^{\frac {3}{2}} x^{4} \arcsinh \left (c x \right ) c^{7}}{24 c^{4} x^{4}+9 c^{2} x^{2}+1}-\frac {32 b \,\pi ^{\frac {3}{2}} x^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{6}}{24 c^{4} x^{4}+9 c^{2} x^{2}+1}+\frac {8 b \,\pi ^{\frac {3}{2}} x^{4} c^{7}}{3 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right )}-\frac {8 b \,\pi ^{\frac {3}{2}} x^{2} \left (c^{2} x^{2}+1\right ) c^{5}}{3 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right )}+\frac {12 b \,\pi ^{\frac {3}{2}} x^{2} \arcsinh \left (c x \right ) c^{5}}{24 c^{4} x^{4}+9 c^{2} x^{2}+1}-\frac {20 b \,\pi ^{\frac {3}{2}} x \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{4}}{24 c^{4} x^{4}+9 c^{2} x^{2}+1}-\frac {4 b \,\pi ^{\frac {3}{2}} \left (c^{2} x^{2}+1\right ) c^{3}}{3 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right )}+\frac {4 b \,\pi ^{\frac {3}{2}} \arcsinh \left (c x \right ) c^{3}}{3 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right )}-\frac {13 b \,\pi ^{\frac {3}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2}}{3 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right ) x}-\frac {b \,\pi ^{\frac {3}{2}} \left (c^{2} x^{2}+1\right ) c}{6 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right ) x^{2}}-\frac {b \,\pi ^{\frac {3}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right ) x^{3}}+\frac {4 b \,c^{3} \pi ^{\frac {3}{2}} \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{3}\) | \(622\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \pi ^{\frac {3}{2}} \left (\int \frac {a \sqrt {c^{2} x^{2} + 1}}{x^{4}}\, dx + \int \frac {a c^{2} \sqrt {c^{2} x^{2} + 1}}{x^{2}}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {b c^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, 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 = -0.01 \begin {gather*} \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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