Optimal. Leaf size=150 \[ -\frac{8 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^2 \sqrt{\pi c^2 x^2+\pi }}-\frac{4 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}-\frac{a+b \sinh ^{-1}(c x)}{\pi x \left (\pi c^2 x^2+\pi \right )^{3/2}}-\frac{b c}{6 \pi ^{5/2} \left (c^2 x^2+1\right )}+\frac{5 b c \log \left (c^2 x^2+1\right )}{6 \pi ^{5/2}}+\frac{b c \log (x)}{\pi ^{5/2}} \]
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Rubi [A] time = 0.176281, antiderivative size = 153, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {271, 192, 191, 5732, 12, 1251, 893} \[ -\frac{8 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{5/2} \sqrt{c^2 x^2+1}}-\frac{4 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{5/2} \left (c^2 x^2+1\right )^{3/2}}-\frac{a+b \sinh ^{-1}(c x)}{\pi ^{5/2} x \left (c^2 x^2+1\right )^{3/2}}-\frac{b c}{6 \pi ^{5/2} \left (c^2 x^2+1\right )}+\frac{5 b c \log \left (c^2 x^2+1\right )}{6 \pi ^{5/2}}+\frac{b c \log (x)}{\pi ^{5/2}} \]
Antiderivative was successfully verified.
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Rule 271
Rule 192
Rule 191
Rule 5732
Rule 12
Rule 1251
Rule 893
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx &=-\frac{a+b \sinh ^{-1}(c x)}{\pi ^{5/2} x \left (1+c^2 x^2\right )^{3/2}}-\frac{4 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{5/2} \left (1+c^2 x^2\right )^{3/2}}-\frac{8 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{5/2} \sqrt{1+c^2 x^2}}-\frac{(b c) \int \frac{-3-12 c^2 x^2-8 c^4 x^4}{3 x \left (1+c^2 x^2\right )^2} \, dx}{\pi ^{5/2}}\\ &=-\frac{a+b \sinh ^{-1}(c x)}{\pi ^{5/2} x \left (1+c^2 x^2\right )^{3/2}}-\frac{4 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{5/2} \left (1+c^2 x^2\right )^{3/2}}-\frac{8 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{5/2} \sqrt{1+c^2 x^2}}-\frac{(b c) \int \frac{-3-12 c^2 x^2-8 c^4 x^4}{x \left (1+c^2 x^2\right )^2} \, dx}{3 \pi ^{5/2}}\\ &=-\frac{a+b \sinh ^{-1}(c x)}{\pi ^{5/2} x \left (1+c^2 x^2\right )^{3/2}}-\frac{4 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{5/2} \left (1+c^2 x^2\right )^{3/2}}-\frac{8 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{5/2} \sqrt{1+c^2 x^2}}-\frac{(b c) \operatorname{Subst}\left (\int \frac{-3-12 c^2 x-8 c^4 x^2}{x \left (1+c^2 x\right )^2} \, dx,x,x^2\right )}{6 \pi ^{5/2}}\\ &=-\frac{a+b \sinh ^{-1}(c x)}{\pi ^{5/2} x \left (1+c^2 x^2\right )^{3/2}}-\frac{4 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{5/2} \left (1+c^2 x^2\right )^{3/2}}-\frac{8 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{5/2} \sqrt{1+c^2 x^2}}-\frac{(b c) \operatorname{Subst}\left (\int \left (-\frac{3}{x}-\frac{c^2}{\left (1+c^2 x\right )^2}-\frac{5 c^2}{1+c^2 x}\right ) \, dx,x,x^2\right )}{6 \pi ^{5/2}}\\ &=-\frac{b c}{6 \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac{a+b \sinh ^{-1}(c x)}{\pi ^{5/2} x \left (1+c^2 x^2\right )^{3/2}}-\frac{4 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{5/2} \left (1+c^2 x^2\right )^{3/2}}-\frac{8 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{5/2} \sqrt{1+c^2 x^2}}+\frac{b c \log (x)}{\pi ^{5/2}}+\frac{5 b c \log \left (1+c^2 x^2\right )}{6 \pi ^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.207856, size = 123, normalized size = 0.82 \[ \frac{-2 a \left (8 c^4 x^4+12 c^2 x^2+3\right )-b c x \sqrt{c^2 x^2+1}-2 b \left (8 c^4 x^4+12 c^2 x^2+3\right ) \sinh ^{-1}(c x)}{6 \pi ^{5/2} x \left (c^2 x^2+1\right )^{3/2}}+\frac{\frac{5}{2} b c \log \left (c^2 x^2+1\right )+3 b c \log (x)}{3 \pi ^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.179, size = 778, normalized size = 5.2 \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} \, a{\left (\frac{4 \, c^{2} x}{\pi{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{3}{2}}} + \frac{8 \, c^{2} x}{\pi ^{2} \sqrt{\pi + \pi c^{2} x^{2}}} + \frac{3}{\pi{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{3}{2}} x}\right )} + b \int \frac{\log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{5}{2}} x^{2}}\,{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 ^{3} c^{6} x^{8} + 3 \, \pi ^{3} c^{4} x^{6} + 3 \, \pi ^{3} c^{2} x^{4} + \pi ^{3} x^{2}}, 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{5}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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