Optimal. Leaf size=166 \[ \frac{4 b^2 c^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x}-\frac{2 b^2 \left (a+b \text{csch}^{-1}(c x)\right )}{9 x^3}-\frac{2}{3} b c^3 \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )^2+\frac{b c \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )^2}{3 x^2}-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{3 x^3}+\frac{2}{27} b^3 c^3 \left (\frac{1}{c^2 x^2}+1\right )^{3/2}-\frac{14}{9} b^3 c^3 \sqrt{\frac{1}{c^2 x^2}+1} \]
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Rubi [A] time = 0.170505, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6286, 5446, 3311, 3296, 2638, 2633} \[ \frac{4 b^2 c^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x}-\frac{2 b^2 \left (a+b \text{csch}^{-1}(c x)\right )}{9 x^3}-\frac{2}{3} b c^3 \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )^2+\frac{b c \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )^2}{3 x^2}-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{3 x^3}+\frac{2}{27} b^3 c^3 \left (\frac{1}{c^2 x^2}+1\right )^{3/2}-\frac{14}{9} b^3 c^3 \sqrt{\frac{1}{c^2 x^2}+1} \]
Antiderivative was successfully verified.
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Rule 6286
Rule 5446
Rule 3311
Rule 3296
Rule 2638
Rule 2633
Rubi steps
\begin{align*} \int \frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{x^4} \, dx &=-\left (c^3 \operatorname{Subst}\left (\int (a+b x)^3 \cosh (x) \sinh ^2(x) \, dx,x,\text{csch}^{-1}(c x)\right )\right )\\ &=-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{3 x^3}+\left (b c^3\right ) \operatorname{Subst}\left (\int (a+b x)^2 \sinh ^3(x) \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=-\frac{2 b^2 \left (a+b \text{csch}^{-1}(c x)\right )}{9 x^3}+\frac{b c \sqrt{1+\frac{1}{c^2 x^2}} \left (a+b \text{csch}^{-1}(c x)\right )^2}{3 x^2}-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{3 x^3}-\frac{1}{3} \left (2 b c^3\right ) \operatorname{Subst}\left (\int (a+b x)^2 \sinh (x) \, dx,x,\text{csch}^{-1}(c x)\right )+\frac{1}{9} \left (2 b^3 c^3\right ) \operatorname{Subst}\left (\int \sinh ^3(x) \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=-\frac{2 b^2 \left (a+b \text{csch}^{-1}(c x)\right )}{9 x^3}-\frac{2}{3} b c^3 \sqrt{1+\frac{1}{c^2 x^2}} \left (a+b \text{csch}^{-1}(c x)\right )^2+\frac{b c \sqrt{1+\frac{1}{c^2 x^2}} \left (a+b \text{csch}^{-1}(c x)\right )^2}{3 x^2}-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{3 x^3}+\frac{1}{3} \left (4 b^2 c^3\right ) \operatorname{Subst}\left (\int (a+b x) \cosh (x) \, dx,x,\text{csch}^{-1}(c x)\right )-\frac{1}{9} \left (2 b^3 c^3\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\sqrt{1+\frac{1}{c^2 x^2}}\right )\\ &=-\frac{2}{9} b^3 c^3 \sqrt{1+\frac{1}{c^2 x^2}}+\frac{2}{27} b^3 c^3 \left (1+\frac{1}{c^2 x^2}\right )^{3/2}-\frac{2 b^2 \left (a+b \text{csch}^{-1}(c x)\right )}{9 x^3}+\frac{4 b^2 c^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x}-\frac{2}{3} b c^3 \sqrt{1+\frac{1}{c^2 x^2}} \left (a+b \text{csch}^{-1}(c x)\right )^2+\frac{b c \sqrt{1+\frac{1}{c^2 x^2}} \left (a+b \text{csch}^{-1}(c x)\right )^2}{3 x^2}-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{3 x^3}-\frac{1}{3} \left (4 b^3 c^3\right ) \operatorname{Subst}\left (\int \sinh (x) \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=-\frac{14}{9} b^3 c^3 \sqrt{1+\frac{1}{c^2 x^2}}+\frac{2}{27} b^3 c^3 \left (1+\frac{1}{c^2 x^2}\right )^{3/2}-\frac{2 b^2 \left (a+b \text{csch}^{-1}(c x)\right )}{9 x^3}+\frac{4 b^2 c^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x}-\frac{2}{3} b c^3 \sqrt{1+\frac{1}{c^2 x^2}} \left (a+b \text{csch}^{-1}(c x)\right )^2+\frac{b c \sqrt{1+\frac{1}{c^2 x^2}} \left (a+b \text{csch}^{-1}(c x)\right )^2}{3 x^2}-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.300845, size = 200, normalized size = 1.2 \[ \frac{3 b \text{csch}^{-1}(c x) \left (-9 a^2+6 a b c x \sqrt{\frac{1}{c^2 x^2}+1} \left (1-2 c^2 x^2\right )+2 b^2 \left (6 c^2 x^2-1\right )\right )+9 a^2 b c x \sqrt{\frac{1}{c^2 x^2}+1} \left (1-2 c^2 x^2\right )-9 a^3+6 a b^2 \left (6 c^2 x^2-1\right )-9 b^2 \text{csch}^{-1}(c x)^2 \left (3 a+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left (2 c^2 x^2-1\right )\right )+2 b^3 c x \sqrt{\frac{1}{c^2 x^2}+1} \left (1-20 c^2 x^2\right )-9 b^3 \text{csch}^{-1}(c x)^3}{27 x^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.184, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccsch} \left (cx\right ) \right ) ^{3}}{{x}^{4}}}\, dx \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*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.56577, size = 640, normalized size = 3.86 \begin{align*} \frac{36 \, a b^{2} c^{2} x^{2} - 9 \, b^{3} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} - 9 \, a^{3} - 6 \, a b^{2} - 9 \,{\left (3 \, a b^{2} +{\left (2 \, b^{3} c^{3} x^{3} - b^{3} c x\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + 3 \,{\left (12 \, b^{3} c^{2} x^{2} - 9 \, a^{2} b - 2 \, b^{3} - 6 \,{\left (2 \, a b^{2} c^{3} x^{3} - a b^{2} c x\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) -{\left (2 \,{\left (9 \, a^{2} b + 20 \, b^{3}\right )} c^{3} x^{3} -{\left (9 \, a^{2} b + 2 \, b^{3}\right )} c x\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{27 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acsch}{\left (c x \right )}\right )^{3}}{x^{4}}\, dx \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{{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )}^{3}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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