Optimal. Leaf size=153 \[ \frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x \sqrt {\pi c^2 x^2+\pi }}-\frac {a+b \sinh ^{-1}(c x)}{3 \pi x^3 \sqrt {\pi c^2 x^2+\pi }}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi \sqrt {\pi c^2 x^2+\pi }}-\frac {5 b c^3 \log (x)}{3 \pi ^{3/2}}-\frac {b c^3 \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2}}-\frac {b c}{6 \pi ^{3/2} x^2} \]
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Rubi [A] time = 0.18, antiderivative size = 156, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {271, 191, 5732, 12, 1251, 893} \[ \frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{3/2} \sqrt {c^2 x^2+1}}+\frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{3/2} x \sqrt {c^2 x^2+1}}-\frac {a+b \sinh ^{-1}(c x)}{3 \pi ^{3/2} x^3 \sqrt {c^2 x^2+1}}-\frac {b c^3 \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2}}-\frac {5 b c^3 \log (x)}{3 \pi ^{3/2}}-\frac {b c}{6 \pi ^{3/2} x^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 191
Rule 271
Rule 893
Rule 1251
Rule 5732
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{x^4 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{3 \pi ^{3/2} x^3 \sqrt {1+c^2 x^2}}+\frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{3/2} x \sqrt {1+c^2 x^2}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{3/2} \sqrt {1+c^2 x^2}}-\frac {(b c) \int \frac {-1+4 c^2 x^2+8 c^4 x^4}{3 x^3 \left (1+c^2 x^2\right )} \, dx}{\pi ^{3/2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{3 \pi ^{3/2} x^3 \sqrt {1+c^2 x^2}}+\frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{3/2} x \sqrt {1+c^2 x^2}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{3/2} \sqrt {1+c^2 x^2}}-\frac {(b c) \int \frac {-1+4 c^2 x^2+8 c^4 x^4}{x^3 \left (1+c^2 x^2\right )} \, dx}{3 \pi ^{3/2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{3 \pi ^{3/2} x^3 \sqrt {1+c^2 x^2}}+\frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{3/2} x \sqrt {1+c^2 x^2}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{3/2} \sqrt {1+c^2 x^2}}-\frac {(b c) \operatorname {Subst}\left (\int \frac {-1+4 c^2 x+8 c^4 x^2}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right )}{6 \pi ^{3/2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{3 \pi ^{3/2} x^3 \sqrt {1+c^2 x^2}}+\frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{3/2} x \sqrt {1+c^2 x^2}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{3/2} \sqrt {1+c^2 x^2}}-\frac {(b c) \operatorname {Subst}\left (\int \left (-\frac {1}{x^2}+\frac {5 c^2}{x}+\frac {3 c^4}{1+c^2 x}\right ) \, dx,x,x^2\right )}{6 \pi ^{3/2}}\\ &=-\frac {b c}{6 \pi ^{3/2} x^2}-\frac {a+b \sinh ^{-1}(c x)}{3 \pi ^{3/2} x^3 \sqrt {1+c^2 x^2}}+\frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{3/2} x \sqrt {1+c^2 x^2}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{3/2} \sqrt {1+c^2 x^2}}-\frac {5 b c^3 \log (x)}{3 \pi ^{3/2}}-\frac {b c^3 \log \left (1+c^2 x^2\right )}{2 \pi ^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 127, normalized size = 0.83 \[ \frac {2 a \left (8 c^4 x^4+4 c^2 x^2-1\right )-b c x \sqrt {c^2 x^2+1}+2 b \left (8 c^4 x^4+4 c^2 x^2-1\right ) \sinh ^{-1}(c x)}{6 \pi ^{3/2} x^3 \sqrt {c^2 x^2+1}}+\frac {-5 b c^3 \log (x)-\frac {3}{2} b c^3 \log \left (c^2 x^2+1\right )}{3 \pi ^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\pi + \pi c^{2} x^{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\pi ^{2} c^{4} x^{8} + 2 \, \pi ^{2} c^{2} x^{6} + \pi ^{2} x^{4}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.32, size = 601, normalized size = 3.93 \[ -\frac {a}{3 \pi \,x^{3} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {4 a \,c^{2}}{3 \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {8 a \,c^{4} x}{3 \pi \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {16 b \,c^{3} \arcsinh \left (c x \right )}{3 \pi ^{\frac {3}{2}}}-\frac {32 b \,x^{8} c^{11}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+1\right )}+\frac {32 b \,x^{6} c^{9}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right )}-\frac {64 b \,x^{6} c^{9}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+1\right )}+\frac {32 b \,x^{4} c^{7}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right )}-\frac {64 b \,x^{4} \arcsinh \left (c x \right ) c^{7}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+1\right )}+\frac {64 b \,x^{3} \arcsinh \left (c x \right ) c^{6}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \sqrt {c^{2} x^{2}+1}}-\frac {32 b \,x^{4} c^{7}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+1\right )}-\frac {56 b \,x^{2} \arcsinh \left (c x \right ) c^{5}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+1\right )}+\frac {8 b x \arcsinh \left (c x \right ) c^{4}}{\pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \sqrt {c^{2} x^{2}+1}}-\frac {4 b \,c^{3}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right )}+\frac {8 b \arcsinh \left (c x \right ) c^{3}}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+1\right )}-\frac {4 b \arcsinh \left (c x \right ) c^{2}}{\pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) x \sqrt {c^{2} x^{2}+1}}+\frac {b c}{6 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) x^{2}}+\frac {b \arcsinh \left (c x \right )}{3 \pi ^{\frac {3}{2}} \left (8 c^{2} x^{2}-1\right ) x^{3} \sqrt {c^{2} x^{2}+1}}-\frac {b \,c^{3} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\pi ^{\frac {3}{2}}}-\frac {5 b \,c^{3} \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{3 \pi ^{\frac {3}{2}}} \]
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 \[ \frac {1}{3} \, {\left (\frac {8 \, c^{4} x}{\pi \sqrt {\pi + \pi c^{2} x^{2}}} + \frac {4 \, c^{2}}{\pi \sqrt {\pi + \pi c^{2} x^{2}} x} - \frac {1}{\pi \sqrt {\pi + \pi c^{2} x^{2}} x^{3}}\right )} a + b \int \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^4\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \]
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 \[ \frac {\int \frac {a}{c^{2} x^{6} \sqrt {c^{2} x^{2} + 1} + x^{4} \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{6} \sqrt {c^{2} x^{2} + 1} + x^{4} \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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