Optimal. Leaf size=164 \[ -\frac{d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{2 d e \left (a+b \text{csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b c d^2 \sqrt{-c^2 x^2-1}}{9 x^2 \sqrt{-c^2 x^2}}-\frac{2 b c d \sqrt{-c^2 x^2-1} \left (c^2 d-9 e\right )}{9 \sqrt{-c^2 x^2}}-\frac{b e^2 x \tan ^{-1}\left (\frac{c x}{\sqrt{-c^2 x^2-1}}\right )}{\sqrt{-c^2 x^2}} \]
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Rubi [A] time = 0.142695, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {270, 6302, 12, 1265, 451, 217, 203} \[ -\frac{d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{2 d e \left (a+b \text{csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b c d^2 \sqrt{-c^2 x^2-1}}{9 x^2 \sqrt{-c^2 x^2}}-\frac{2 b c d \sqrt{-c^2 x^2-1} \left (c^2 d-9 e\right )}{9 \sqrt{-c^2 x^2}}-\frac{b e^2 x \tan ^{-1}\left (\frac{c x}{\sqrt{-c^2 x^2-1}}\right )}{\sqrt{-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 270
Rule 6302
Rule 12
Rule 1265
Rule 451
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^2 \left (a+b \text{csch}^{-1}(c x)\right )}{x^4} \, dx &=-\frac{d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{2 d e \left (a+b \text{csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(b c x) \int \frac{-d^2-6 d e x^2+3 e^2 x^4}{3 x^4 \sqrt{-1-c^2 x^2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=-\frac{d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{2 d e \left (a+b \text{csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(b c x) \int \frac{-d^2-6 d e x^2+3 e^2 x^4}{x^4 \sqrt{-1-c^2 x^2}} \, dx}{3 \sqrt{-c^2 x^2}}\\ &=\frac{b c d^2 \sqrt{-1-c^2 x^2}}{9 x^2 \sqrt{-c^2 x^2}}-\frac{d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{2 d e \left (a+b \text{csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(b c x) \int \frac{2 d \left (c^2 d-9 e\right )+9 e^2 x^2}{x^2 \sqrt{-1-c^2 x^2}} \, dx}{9 \sqrt{-c^2 x^2}}\\ &=-\frac{2 b c d \left (c^2 d-9 e\right ) \sqrt{-1-c^2 x^2}}{9 \sqrt{-c^2 x^2}}+\frac{b c d^2 \sqrt{-1-c^2 x^2}}{9 x^2 \sqrt{-c^2 x^2}}-\frac{d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{2 d e \left (a+b \text{csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text{csch}^{-1}(c x)\right )-\frac{\left (b c e^2 x\right ) \int \frac{1}{\sqrt{-1-c^2 x^2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=-\frac{2 b c d \left (c^2 d-9 e\right ) \sqrt{-1-c^2 x^2}}{9 \sqrt{-c^2 x^2}}+\frac{b c d^2 \sqrt{-1-c^2 x^2}}{9 x^2 \sqrt{-c^2 x^2}}-\frac{d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{2 d e \left (a+b \text{csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text{csch}^{-1}(c x)\right )-\frac{\left (b c e^2 x\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x^2} \, dx,x,\frac{x}{\sqrt{-1-c^2 x^2}}\right )}{\sqrt{-c^2 x^2}}\\ &=-\frac{2 b c d \left (c^2 d-9 e\right ) \sqrt{-1-c^2 x^2}}{9 \sqrt{-c^2 x^2}}+\frac{b c d^2 \sqrt{-1-c^2 x^2}}{9 x^2 \sqrt{-c^2 x^2}}-\frac{d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{2 d e \left (a+b \text{csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text{csch}^{-1}(c x)\right )-\frac{b e^2 x \tan ^{-1}\left (\frac{c x}{\sqrt{-1-c^2 x^2}}\right )}{\sqrt{-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.262792, size = 123, normalized size = 0.75 \[ \frac{b c d x \sqrt{\frac{1}{c^2 x^2}+1} \left (-2 c^2 d x^2+d+18 e x^2\right )-3 a \left (d^2+6 d e x^2-3 e^2 x^4\right )}{9 x^3}+\frac{b e^2 \log \left (x \left (\sqrt{\frac{1}{c^2 x^2}+1}+1\right )\right )}{c}-\frac{b \text{csch}^{-1}(c x) \left (d^2+6 d e x^2-3 e^2 x^4\right )}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.191, size = 191, normalized size = 1.2 \begin{align*}{c}^{3} \left ({\frac{a}{{c}^{4}} \left ( cx{e}^{2}-2\,{\frac{cde}{x}}-{\frac{c{d}^{2}}{3\,{x}^{3}}} \right ) }+{\frac{b}{{c}^{4}} \left ({\rm arccsch} \left (cx\right )cx{e}^{2}-2\,{\frac{{\rm arccsch} \left (cx\right )cde}{x}}-{\frac{{\rm arccsch} \left (cx\right ){d}^{2}c}{3\,{x}^{3}}}-{\frac{1}{9\,{c}^{4}{x}^{4}}\sqrt{{c}^{2}{x}^{2}+1} \left ( 2\,\sqrt{{c}^{2}{x}^{2}+1}{c}^{6}{x}^{2}{d}^{2}-18\,{c}^{4}de\sqrt{{c}^{2}{x}^{2}+1}{x}^{2}-{d}^{2}{c}^{4}\sqrt{{c}^{2}{x}^{2}+1}-9\,{e}^{2}{\it Arcsinh} \left ( cx \right ){c}^{3}{x}^{3} \right ){\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 = 1.02882, size = 205, normalized size = 1.25 \begin{align*} 2 \,{\left (c \sqrt{\frac{1}{c^{2} x^{2}} + 1} - \frac{\operatorname{arcsch}\left (c x\right )}{x}\right )} b d e + a e^{2} x + \frac{1}{9} \, b d^{2}{\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{{\left (2 \, c x \operatorname{arcsch}\left (c x\right ) + \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )\right )} b e^{2}}{2 \, c} - \frac{2 \, a d e}{x} - \frac{a d^{2}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.01253, size = 743, normalized size = 4.53 \begin{align*} \frac{9 \, a c e^{2} x^{4} - 9 \, b e^{2} x^{3} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 18 \, a c d e x^{2} - 3 \,{\left (b c d^{2} + 6 \, b c d e - 3 \, b c e^{2}\right )} x^{3} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) + 3 \,{\left (b c d^{2} + 6 \, b c d e - 3 \, b c e^{2}\right )} x^{3} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) - 3 \, a c d^{2} - 2 \,{\left (b c^{4} d^{2} - 9 \, b c^{2} d e\right )} x^{3} + 3 \,{\left (3 \, b c e^{2} x^{4} - 6 \, b c d e x^{2} - b c d^{2} +{\left (b c d^{2} + 6 \, b c d e - 3 \, b c e^{2}\right )} x^{3}\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) +{\left (b c^{2} d^{2} x - 2 \,{\left (b c^{4} d^{2} - 9 \, b c^{2} d e\right )} x^{3}\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{9 \, c 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 )^{2}}{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 )}^{2}{\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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