Optimal. Leaf size=229 \[ \frac{1}{5} d x^5 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b x^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (42 c^2 d+25 e\right )}{840 c^4}-\frac{b x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (42 c^2 d+25 e\right )}{560 c^6}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (42 c^2 d+25 e\right ) \sin ^{-1}(c x)}{560 c^7}-\frac{b e x^5 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{42 c^2} \]
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Rubi [A] time = 0.127182, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {14, 6301, 12, 459, 321, 216} \[ \frac{1}{5} d x^5 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b x^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (42 c^2 d+25 e\right )}{840 c^4}-\frac{b x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (42 c^2 d+25 e\right )}{560 c^6}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (42 c^2 d+25 e\right ) \sin ^{-1}(c x)}{560 c^7}-\frac{b e x^5 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{42 c^2} \]
Antiderivative was successfully verified.
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Rule 14
Rule 6301
Rule 12
Rule 459
Rule 321
Rule 216
Rubi steps
\begin{align*} \int x^4 \left (d+e x^2\right ) \left (a+b \text{sech}^{-1}(c x)\right ) \, dx &=\frac{1}{5} d x^5 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^4 \left (7 d+5 e x^2\right )}{35 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{5} d x^5 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{35} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^4 \left (7 d+5 e x^2\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{b e x^5 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{42 c^2}+\frac{1}{5} d x^5 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{210} \left (b \left (42 d+\frac{25 e}{c^2}\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{b \left (42 c^2 d+25 e\right ) x^3 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{840 c^4}-\frac{b e x^5 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{42 c^2}+\frac{1}{5} d x^5 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{\left (b \left (42 d+\frac{25 e}{c^2}\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{280 c^2}\\ &=-\frac{b \left (42 c^2 d+25 e\right ) x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{560 c^6}-\frac{b \left (42 c^2 d+25 e\right ) x^3 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{840 c^4}-\frac{b e x^5 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{42 c^2}+\frac{1}{5} d x^5 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{\left (b \left (42 d+\frac{25 e}{c^2}\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{560 c^4}\\ &=-\frac{b \left (42 c^2 d+25 e\right ) x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{560 c^6}-\frac{b \left (42 c^2 d+25 e\right ) x^3 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{840 c^4}-\frac{b e x^5 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{42 c^2}+\frac{1}{5} d x^5 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{b \left (42 c^2 d+25 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sin ^{-1}(c x)}{560 c^7}\\ \end{align*}
Mathematica [C] time = 0.317618, size = 162, normalized size = 0.71 \[ \frac{48 a c^7 x^5 \left (7 d+5 e x^2\right )-b c x \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (c^4 \left (84 d x^2+40 e x^4\right )+2 c^2 \left (63 d+25 e x^2\right )+75 e\right )+48 b c^7 x^5 \text{sech}^{-1}(c x) \left (7 d+5 e x^2\right )+3 i b \left (42 c^2 d+25 e\right ) \log \left (2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right )}{1680 c^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.185, size = 224, 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 arcsech} \left (cx\right )e{c}^{7}{x}^{7}}{7}}+{\frac{{\rm arcsech} \left (cx\right ){c}^{7}{x}^{5}d}{5}}+{\frac{cx}{1680}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}} \left ( -40\,{c}^{5}{x}^{5}e\sqrt{-{c}^{2}{x}^{2}+1}-84\,{c}^{5}{x}^{3}d\sqrt{-{c}^{2}{x}^{2}+1}-50\,e{c}^{3}{x}^{3}\sqrt{-{c}^{2}{x}^{2}+1}-126\,\sqrt{-{c}^{2}{x}^{2}+1}{c}^{3}xd+126\,\arcsin \left ( cx \right ){c}^{2}d-75\,ecx\sqrt{-{c}^{2}{x}^{2}+1}+75\,e\arcsin \left ( cx \right ) \right ){\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47971, size = 329, normalized size = 1.44 \begin{align*} \frac{1}{7} \, a e x^{7} + \frac{1}{5} \, a d x^{5} + \frac{1}{40} \,{\left (8 \, x^{5} \operatorname{arsech}\left (c x\right ) - \frac{\frac{3 \,{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{3}{2}} + 5 \, \sqrt{\frac{1}{c^{2} x^{2}} - 1}}{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 \, \arctan \left (\sqrt{\frac{1}{c^{2} x^{2}} - 1}\right )}{c^{4}}}{c}\right )} b d + \frac{1}{336} \,{\left (48 \, x^{7} \operatorname{arsech}\left (c x\right ) - \frac{\frac{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}}{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 \, \arctan \left (\sqrt{\frac{1}{c^{2} x^{2}} - 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 = 2.98418, size = 590, normalized size = 2.58 \begin{align*} \frac{240 \, a c^{7} e x^{7} + 336 \, a c^{7} d x^{5} - 6 \,{\left (42 \, b c^{2} d + 25 \, b e\right )} \arctan \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - 48 \,{\left (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}{x}\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{asech}{\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{arsech}\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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