Optimal. Leaf size=180 \[ \frac {1}{4} d x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2} \left (3 c^2 d+4 e\right )}{36 c^6}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (3 c^2 d+2 e\right )}{12 c^6}-\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{30 c^6} \]
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Rubi [A] time = 0.13, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {14, 6301, 12, 446, 77} \[ \frac {1}{4} d x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2} \left (3 c^2 d+4 e\right )}{36 c^6}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (3 c^2 d+2 e\right )}{12 c^6}-\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{30 c^6} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 77
Rule 446
Rule 6301
Rubi steps
\begin {align*} \int x^3 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {1}{4} d x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} e x^6 \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^3 \left (3 d+2 e x^2\right )}{12 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{4} d x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{12} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^3 \left (3 d+2 e x^2\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{4} d x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{24} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {x (3 d+2 e x)}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{4} d x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{24} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \left (\frac {3 c^2 d+2 e}{c^4 \sqrt {1-c^2 x}}+\frac {\left (-3 c^2 d-4 e\right ) \sqrt {1-c^2 x}}{c^4}+\frac {2 e \left (1-c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )\\ &=-\frac {b \left (3 c^2 d+2 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{12 c^6}+\frac {b \left (3 c^2 d+4 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{36 c^6}-\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{30 c^6}+\frac {1}{4} d x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \text {sech}^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.19, size = 106, normalized size = 0.59 \[ \frac {1}{180} \left (15 a x^4 \left (3 d+2 e x^2\right )-\frac {b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (3 c^4 \left (5 d x^2+2 e x^4\right )+c^2 \left (30 d+8 e x^2\right )+16 e\right )}{c^6}+15 b x^4 \text {sech}^{-1}(c x) \left (3 d+2 e x^2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 147, normalized size = 0.82 \[ \frac {30 \, a c^{5} e x^{6} + 45 \, a c^{5} d x^{4} + 15 \, {\left (2 \, b c^{5} e x^{6} + 3 \, b c^{5} d x^{4}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (6 \, b c^{4} e x^{5} + {\left (15 \, b c^{4} d + 8 \, b c^{2} e\right )} x^{3} + 2 \, {\left (15 \, b c^{2} d + 8 \, b e\right )} x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{180 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x^{2} + d\right )} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 132, normalized size = 0.73 \[ \frac {\frac {a \left (\frac {1}{6} c^{6} e \,x^{6}+\frac {1}{4} c^{6} d \,x^{4}\right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) c^{6} x^{6} e}{6}+\frac {\mathrm {arcsech}\left (c x \right ) c^{6} x^{4} d}{4}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (6 c^{4} e \,x^{4}+15 c^{4} d \,x^{2}+8 c^{2} x^{2} e +30 c^{2} d +16 e \right )}{180}\right )}{c^{2}}}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 138, normalized size = 0.77 \[ \frac {1}{6} \, a e x^{6} + \frac {1}{4} \, a d x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arsech}\left (c x\right ) + \frac {c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} - 3 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{3}}\right )} b d + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arsech}\left (c x\right ) - \frac {3 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} - 10 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{5}}\right )} b e \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.81, size = 177, normalized size = 0.98 \[ \begin {cases} \frac {a d x^{4}}{4} + \frac {a e x^{6}}{6} + \frac {b d x^{4} \operatorname {asech}{\left (c x \right )}}{4} + \frac {b e x^{6} \operatorname {asech}{\left (c x \right )}}{6} - \frac {b d x^{2} \sqrt {- c^{2} x^{2} + 1}}{12 c^{2}} - \frac {b e x^{4} \sqrt {- c^{2} x^{2} + 1}}{30 c^{2}} - \frac {b d \sqrt {- c^{2} x^{2} + 1}}{6 c^{4}} - \frac {2 b e x^{2} \sqrt {- c^{2} x^{2} + 1}}{45 c^{4}} - \frac {4 b e \sqrt {- c^{2} x^{2} + 1}}{45 c^{6}} & \text {for}\: c \neq 0 \\\left (a + \infty b\right ) \left (\frac {d x^{4}}{4} + \frac {e x^{6}}{6}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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