Optimal. Leaf size=159 \[ \frac {1}{4} d x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b x \left (-c^2 x^2-1\right )^{3/2} \left (3 c^2 d-4 e\right )}{36 c^5 \sqrt {-c^2 x^2}}-\frac {b x \sqrt {-c^2 x^2-1} \left (3 c^2 d-2 e\right )}{12 c^5 \sqrt {-c^2 x^2}}+\frac {b e x \left (-c^2 x^2-1\right )^{5/2}}{30 c^5 \sqrt {-c^2 x^2}} \]
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Rubi [A] time = 0.13, antiderivative size = 159, 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, 6302, 12, 446, 77} \[ \frac {1}{4} d x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b x \left (-c^2 x^2-1\right )^{3/2} \left (3 c^2 d-4 e\right )}{36 c^5 \sqrt {-c^2 x^2}}-\frac {b x \sqrt {-c^2 x^2-1} \left (3 c^2 d-2 e\right )}{12 c^5 \sqrt {-c^2 x^2}}+\frac {b e x \left (-c^2 x^2-1\right )^{5/2}}{30 c^5 \sqrt {-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 77
Rule 446
Rule 6302
Rubi steps
\begin {align*} \int x^3 \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=\frac {1}{4} d x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^3 \left (3 d+2 e x^2\right )}{12 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=\frac {1}{4} d x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^3 \left (3 d+2 e x^2\right )}{\sqrt {-1-c^2 x^2}} \, dx}{12 \sqrt {-c^2 x^2}}\\ &=\frac {1}{4} d x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \operatorname {Subst}\left (\int \frac {x (3 d+2 e x)}{\sqrt {-1-c^2 x}} \, dx,x,x^2\right )}{24 \sqrt {-c^2 x^2}}\\ &=\frac {1}{4} d x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \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 )}{24 \sqrt {-c^2 x^2}}\\ &=-\frac {b \left (3 c^2 d-2 e\right ) x \sqrt {-1-c^2 x^2}}{12 c^5 \sqrt {-c^2 x^2}}-\frac {b \left (3 c^2 d-4 e\right ) x \left (-1-c^2 x^2\right )^{3/2}}{36 c^5 \sqrt {-c^2 x^2}}+\frac {b e x \left (-1-c^2 x^2\right )^{5/2}}{30 c^5 \sqrt {-c^2 x^2}}+\frac {1}{4} d x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \text {csch}^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.21, size = 97, normalized size = 0.61 \[ \frac {1}{180} x \left (15 a x^3 \left (3 d+2 e x^2\right )+\frac {b \sqrt {\frac {1}{c^2 x^2}+1} \left (3 c^4 \left (5 d x^2+2 e x^4\right )-2 c^2 \left (15 d+4 e x^2\right )+16 e\right )}{c^5}+15 b x^3 \text {csch}^{-1}(c x) \left (3 d+2 e x^2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 144, normalized size = 0.91 \[ \frac {30 \, a c^{5} e x^{6} + 45 \, a c^{5} d x^{4} + 15 \, {\left (2 \, b c^{5} e x^{6} + 3 \, b c^{5} d x^{4}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (6 \, b c^{4} e x^{5} + {\left (15 \, b c^{4} d - 8 \, b c^{2} e\right )} x^{3} - 2 \, {\left (15 \, b c^{2} d - 8 \, b e\right )} x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{180 \, c^{5}} \]
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 \[ \int {\left (e x^{2} + d\right )} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 134, normalized size = 0.84 \[ \frac {\frac {a \left (\frac {1}{6} c^{6} e \,x^{6}+\frac {1}{4} c^{6} d \,x^{4}\right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arccsch}\left (c x \right ) c^{6} x^{6} e}{6}+\frac {\mathrm {arccsch}\left (c x \right ) c^{6} x^{4} d}{4}+\frac {\left (c^{2} x^{2}+1\right ) \left (6 c^{4} e \,x^{4}+15 c^{4} d \,x^{2}-8 c^{2} x^{2} e -30 c^{2} d +16 e \right )}{180 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 137, normalized size = 0.86 \[ \frac {1}{6} \, a e x^{6} + \frac {1}{4} \, a d x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arcsch}\left (c x\right ) + \frac {c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b d + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arcsch}\left (c x\right ) + \frac {3 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b e \]
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 \[ \int x^3\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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