Optimal. Leaf size=153 \[ \frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi \sqrt{\pi c^2 x^2+\pi }}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi x \sqrt{\pi c^2 x^2+\pi }}-\frac{a+b \sinh ^{-1}(c x)}{3 \pi x^3 \sqrt{\pi c^2 x^2+\pi }}-\frac{b c^3 \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2}}-\frac{5 b c^3 \log (x)}{3 \pi ^{3/2}}-\frac{b c}{6 \pi ^{3/2} x^2} \]
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Rubi [A] time = 0.175009, 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 271
Rule 191
Rule 5732
Rule 12
Rule 1251
Rule 893
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*}
Mathematica [A] time = 0.217, size = 127, normalized size = 0.83 \[ \frac{2 a \left (8 c^4 x^4+4 c^2 x^2-1\right )-b c x \sqrt{c^2 x^2+1}+2 b \left (8 c^4 x^4+4 c^2 x^2-1\right ) \sinh ^{-1}(c x)}{6 \pi ^{3/2} x^3 \sqrt{c^2 x^2+1}}+\frac{-\frac{3}{2} b c^3 \log \left (c^2 x^2+1\right )-5 b c^3 \log (x)}{3 \pi ^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.19, size = 601, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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