Optimal. Leaf size=109 \[ -\frac{d \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{x}-\frac{b c \sqrt{-c^2 x^2-1} \left (2 c^2 d-9 e\right )}{9 \sqrt{-c^2 x^2}}+\frac{b c d \sqrt{-c^2 x^2-1}}{9 x^2 \sqrt{-c^2 x^2}} \]
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Rubi [A] time = 0.079277, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {14, 6302, 12, 453, 264} \[ -\frac{d \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{x}-\frac{b c \sqrt{-c^2 x^2-1} \left (2 c^2 d-9 e\right )}{9 \sqrt{-c^2 x^2}}+\frac{b c d \sqrt{-c^2 x^2-1}}{9 x^2 \sqrt{-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 6302
Rule 12
Rule 453
Rule 264
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{x^4} \, dx &=-\frac{d \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{x}-\frac{(b c x) \int \frac{-d-3 e x^2}{3 x^4 \sqrt{-1-c^2 x^2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=-\frac{d \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{x}-\frac{(b c x) \int \frac{-d-3 e x^2}{x^4 \sqrt{-1-c^2 x^2}} \, dx}{3 \sqrt{-c^2 x^2}}\\ &=\frac{b c d \sqrt{-1-c^2 x^2}}{9 x^2 \sqrt{-c^2 x^2}}-\frac{d \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{x}-\frac{\left (b c \left (2 c^2 d-9 e\right ) x\right ) \int \frac{1}{x^2 \sqrt{-1-c^2 x^2}} \, dx}{9 \sqrt{-c^2 x^2}}\\ &=-\frac{b c \left (2 c^2 d-9 e\right ) \sqrt{-1-c^2 x^2}}{9 \sqrt{-c^2 x^2}}+\frac{b c d \sqrt{-1-c^2 x^2}}{9 x^2 \sqrt{-c^2 x^2}}-\frac{d \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{x}\\ \end{align*}
Mathematica [A] time = 0.0851028, size = 68, normalized size = 0.62 \[ \frac{-3 a \left (d+3 e x^2\right )+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left (-2 c^2 d x^2+d+9 e x^2\right )-3 b \text{csch}^{-1}(c x) \left (d+3 e x^2\right )}{9 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.2, size = 122, normalized size = 1.1 \begin{align*}{c}^{3} \left ({\frac{a}{{c}^{2}} \left ( -{\frac{e}{cx}}-{\frac{d}{3\,c{x}^{3}}} \right ) }+{\frac{b}{{c}^{2}} \left ( -{\frac{{\rm arccsch} \left (cx\right )e}{cx}}-{\frac{{\rm arccsch} \left (cx\right )d}{3\,c{x}^{3}}}-{\frac{ \left ({c}^{2}{x}^{2}+1 \right ) \left ( 2\,{c}^{4}d{x}^{2}-9\,{c}^{2}{x}^{2}e-{c}^{2}d \right ) }{9\,{c}^{4}{x}^{4}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}}}}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994452, size = 123, normalized size = 1.13 \begin{align*}{\left (c \sqrt{\frac{1}{c^{2} x^{2}} + 1} - \frac{\operatorname{arcsch}\left (c x\right )}{x}\right )} b e + \frac{1}{9} \, b d{\left (\frac{c^{4}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 3 \, c^{4} \sqrt{\frac{1}{c^{2} x^{2}} + 1}}{c} - \frac{3 \, \operatorname{arcsch}\left (c x\right )}{x^{3}}\right )} - \frac{a e}{x} - \frac{a d}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.35906, size = 234, normalized size = 2.15 \begin{align*} -\frac{9 \, a e x^{2} + 3 \, a d + 3 \,{\left (3 \, b e x^{2} + b d\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) -{\left (b c d x -{\left (2 \, b c^{3} d - 9 \, b c e\right )} x^{3}\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{9 \, 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 ) \left (d + e x^{2}\right )}{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 (e x^{2} + d\right )}{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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