Optimal. Leaf size=214 \[ \frac{1}{5} d x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b x^4 \sqrt{-c^2 x^2-1} \left (42 c^2 d-25 e\right )}{840 c^3 \sqrt{-c^2 x^2}}-\frac{b x^2 \sqrt{-c^2 x^2-1} \left (42 c^2 d-25 e\right )}{560 c^5 \sqrt{-c^2 x^2}}-\frac{b x \left (42 c^2 d-25 e\right ) \tan ^{-1}\left (\frac{c x}{\sqrt{-c^2 x^2-1}}\right )}{560 c^6 \sqrt{-c^2 x^2}}+\frac{b e x^6 \sqrt{-c^2 x^2-1}}{42 c \sqrt{-c^2 x^2}} \]
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Rubi [A] time = 0.132, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {14, 6302, 12, 459, 321, 217, 203} \[ \frac{1}{5} d x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b x^4 \sqrt{-c^2 x^2-1} \left (42 c^2 d-25 e\right )}{840 c^3 \sqrt{-c^2 x^2}}-\frac{b x^2 \sqrt{-c^2 x^2-1} \left (42 c^2 d-25 e\right )}{560 c^5 \sqrt{-c^2 x^2}}-\frac{b x \left (42 c^2 d-25 e\right ) \tan ^{-1}\left (\frac{c x}{\sqrt{-c^2 x^2-1}}\right )}{560 c^6 \sqrt{-c^2 x^2}}+\frac{b e x^6 \sqrt{-c^2 x^2-1}}{42 c \sqrt{-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 6302
Rule 12
Rule 459
Rule 321
Rule 217
Rule 203
Rubi steps
\begin{align*} \int x^4 \left (d+e x^2\right ) \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=\frac{1}{5} d x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(b c x) \int \frac{x^4 \left (7 d+5 e x^2\right )}{35 \sqrt{-1-c^2 x^2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=\frac{1}{5} d x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(b c x) \int \frac{x^4 \left (7 d+5 e x^2\right )}{\sqrt{-1-c^2 x^2}} \, dx}{35 \sqrt{-c^2 x^2}}\\ &=\frac{b e x^6 \sqrt{-1-c^2 x^2}}{42 c \sqrt{-c^2 x^2}}+\frac{1}{5} d x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{\left (b c \left (42 d-\frac{25 e}{c^2}\right ) x\right ) \int \frac{x^4}{\sqrt{-1-c^2 x^2}} \, dx}{210 \sqrt{-c^2 x^2}}\\ &=\frac{b \left (42 c^2 d-25 e\right ) x^4 \sqrt{-1-c^2 x^2}}{840 c^3 \sqrt{-c^2 x^2}}+\frac{b e x^6 \sqrt{-1-c^2 x^2}}{42 c \sqrt{-c^2 x^2}}+\frac{1}{5} d x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{\left (b \left (42 d-\frac{25 e}{c^2}\right ) x\right ) \int \frac{x^2}{\sqrt{-1-c^2 x^2}} \, dx}{280 c \sqrt{-c^2 x^2}}\\ &=-\frac{b \left (42 c^2 d-25 e\right ) x^2 \sqrt{-1-c^2 x^2}}{560 c^5 \sqrt{-c^2 x^2}}+\frac{b \left (42 c^2 d-25 e\right ) x^4 \sqrt{-1-c^2 x^2}}{840 c^3 \sqrt{-c^2 x^2}}+\frac{b e x^6 \sqrt{-1-c^2 x^2}}{42 c \sqrt{-c^2 x^2}}+\frac{1}{5} d x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{\left (b \left (42 d-\frac{25 e}{c^2}\right ) x\right ) \int \frac{1}{\sqrt{-1-c^2 x^2}} \, dx}{560 c^3 \sqrt{-c^2 x^2}}\\ &=-\frac{b \left (42 c^2 d-25 e\right ) x^2 \sqrt{-1-c^2 x^2}}{560 c^5 \sqrt{-c^2 x^2}}+\frac{b \left (42 c^2 d-25 e\right ) x^4 \sqrt{-1-c^2 x^2}}{840 c^3 \sqrt{-c^2 x^2}}+\frac{b e x^6 \sqrt{-1-c^2 x^2}}{42 c \sqrt{-c^2 x^2}}+\frac{1}{5} d x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{\left (b \left (42 d-\frac{25 e}{c^2}\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x^2} \, dx,x,\frac{x}{\sqrt{-1-c^2 x^2}}\right )}{560 c^3 \sqrt{-c^2 x^2}}\\ &=-\frac{b \left (42 c^2 d-25 e\right ) x^2 \sqrt{-1-c^2 x^2}}{560 c^5 \sqrt{-c^2 x^2}}+\frac{b \left (42 c^2 d-25 e\right ) x^4 \sqrt{-1-c^2 x^2}}{840 c^3 \sqrt{-c^2 x^2}}+\frac{b e x^6 \sqrt{-1-c^2 x^2}}{42 c \sqrt{-c^2 x^2}}+\frac{1}{5} d x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{b \left (42 c^2 d-25 e\right ) x \tan ^{-1}\left (\frac{c x}{\sqrt{-1-c^2 x^2}}\right )}{560 c^6 \sqrt{-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.232764, size = 138, normalized size = 0.64 \[ \frac{48 a c^7 x^5 \left (7 d+5 e x^2\right )+b c^2 x^2 \sqrt{\frac{1}{c^2 x^2}+1} \left (c^4 \left (84 d x^2+40 e x^4\right )-2 c^2 \left (63 d+25 e x^2\right )+75 e\right )+3 b \left (42 c^2 d-25 e\right ) \log \left (x \left (\sqrt{\frac{1}{c^2 x^2}+1}+1\right )\right )+48 b c^7 x^5 \text{csch}^{-1}(c x) \left (7 d+5 e x^2\right )}{1680 c^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.196, size = 211, normalized size = 1. \begin{align*}{\frac{1}{{c}^{5}} \left ({\frac{a}{{c}^{2}} \left ({\frac{e{c}^{7}{x}^{7}}{7}}+{\frac{{c}^{7}{x}^{5}d}{5}} \right ) }+{\frac{b}{{c}^{2}} \left ({\frac{{\rm arccsch} \left (cx\right )e{c}^{7}{x}^{7}}{7}}+{\frac{{\rm arccsch} \left (cx\right ){c}^{7}{x}^{5}d}{5}}+{\frac{1}{1680\,cx}\sqrt{{c}^{2}{x}^{2}+1} \left ( 40\,e{c}^{5}{x}^{5}\sqrt{{c}^{2}{x}^{2}+1}+84\,{c}^{5}d{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}-50\,e{c}^{3}{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}-126\,{c}^{3}dx\sqrt{{c}^{2}{x}^{2}+1}+126\,{c}^{2}d{\it Arcsinh} \left ( cx \right ) +75\,ecx\sqrt{{c}^{2}{x}^{2}+1}-75\,e{\it Arcsinh} \left ( cx \right ) \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 = 1.0222, size = 390, normalized size = 1.82 \begin{align*} \frac{1}{7} \, a e x^{7} + \frac{1}{5} \, a d x^{5} + \frac{1}{80} \,{\left (16 \, x^{5} \operatorname{arcsch}\left (c x\right ) - \frac{\frac{2 \,{\left (3 \,{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{\frac{1}{c^{2} x^{2}} + 1}\right )}}{c^{4}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{2} - 2 \, c^{4}{\left (\frac{1}{c^{2} x^{2}} + 1\right )} + c^{4}} - \frac{3 \, \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac{3 \, \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b d + \frac{1}{672} \,{\left (96 \, x^{7} \operatorname{arcsch}\left (c x\right ) + \frac{\frac{2 \,{\left (15 \,{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} - 40 \,{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 33 \, \sqrt{\frac{1}{c^{2} x^{2}} + 1}\right )}}{c^{6}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{3} - 3 \, c^{6}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{2} + 3 \, c^{6}{\left (\frac{1}{c^{2} x^{2}} + 1\right )} - c^{6}} - \frac{15 \, \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} + \frac{15 \, \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{6}}}{c}\right )} b e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.32047, size = 680, normalized size = 3.18 \begin{align*} \frac{240 \, a c^{7} e x^{7} + 336 \, a c^{7} d x^{5} + 48 \,{\left (7 \, b c^{7} d + 5 \, b c^{7} e\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - 3 \,{\left (42 \, b c^{2} d - 25 \, b e\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 48 \,{\left (7 \, b c^{7} d + 5 \, b c^{7} e\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 48 \,{\left (5 \, b c^{7} e x^{7} + 7 \, b c^{7} d x^{5} - 7 \, b c^{7} d - 5 \, b c^{7} e\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) +{\left (40 \, b c^{6} e x^{6} + 2 \,{\left (42 \, b c^{6} d - 25 \, b c^{4} e\right )} x^{4} - 3 \,{\left (42 \, b c^{4} d - 25 \, b c^{2} e\right )} x^{2}\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{1680 \, c^{7}} \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 x^{4} \left (a + b \operatorname{acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, 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{\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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