Optimal. Leaf size=204 \[ \frac {1}{6} d x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b x \left (-c^2 x^2-1\right )^{5/2} \left (4 c^2 d-9 e\right )}{120 c^7 \sqrt {-c^2 x^2}}+\frac {b x \left (-c^2 x^2-1\right )^{3/2} \left (8 c^2 d-9 e\right )}{72 c^7 \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-c^2 x^2-1} \left (4 c^2 d-3 e\right )}{24 c^7 \sqrt {-c^2 x^2}}-\frac {b e x \left (-c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt {-c^2 x^2}} \]
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Rubi [A] time = 0.16, antiderivative size = 204, 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} \[ \frac {1}{6} d x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b x \left (-c^2 x^2-1\right )^{5/2} \left (4 c^2 d-9 e\right )}{120 c^7 \sqrt {-c^2 x^2}}+\frac {b x \left (-c^2 x^2-1\right )^{3/2} \left (8 c^2 d-9 e\right )}{72 c^7 \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-c^2 x^2-1} \left (4 c^2 d-3 e\right )}{24 c^7 \sqrt {-c^2 x^2}}-\frac {b e x \left (-c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt {-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 77
Rule 446
Rule 6302
Rubi steps
\begin {align*} \int x^5 \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=\frac {1}{6} d x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^5 \left (4 d+3 e x^2\right )}{24 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=\frac {1}{6} d x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^5 \left (4 d+3 e x^2\right )}{\sqrt {-1-c^2 x^2}} \, dx}{24 \sqrt {-c^2 x^2}}\\ &=\frac {1}{6} d x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \operatorname {Subst}\left (\int \frac {x^2 (4 d+3 e x)}{\sqrt {-1-c^2 x}} \, dx,x,x^2\right )}{48 \sqrt {-c^2 x^2}}\\ &=\frac {1}{6} d x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \operatorname {Subst}\left (\int \left (\frac {4 c^2 d-3 e}{c^6 \sqrt {-1-c^2 x}}+\frac {\left (8 c^2 d-9 e\right ) \sqrt {-1-c^2 x}}{c^6}+\frac {\left (4 c^2 d-9 e\right ) \left (-1-c^2 x\right )^{3/2}}{c^6}-\frac {3 e \left (-1-c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )}{48 \sqrt {-c^2 x^2}}\\ &=\frac {b \left (4 c^2 d-3 e\right ) x \sqrt {-1-c^2 x^2}}{24 c^7 \sqrt {-c^2 x^2}}+\frac {b \left (8 c^2 d-9 e\right ) x \left (-1-c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {-c^2 x^2}}+\frac {b \left (4 c^2 d-9 e\right ) x \left (-1-c^2 x^2\right )^{5/2}}{120 c^7 \sqrt {-c^2 x^2}}-\frac {b e x \left (-1-c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {-c^2 x^2}}+\frac {1}{6} d x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {csch}^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.25, size = 114, normalized size = 0.56 \[ \frac {x \left (105 a x^5 \left (4 d+3 e x^2\right )+\frac {b \sqrt {\frac {1}{c^2 x^2}+1} \left (c^6 \left (84 d x^4+45 e x^6\right )-2 c^4 \left (56 d x^2+27 e x^4\right )+8 c^2 \left (28 d+9 e x^2\right )-144 e\right )}{c^7}+105 b x^5 \text {csch}^{-1}(c x) \left (4 d+3 e x^2\right )\right )}{2520} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 165, normalized size = 0.81 \[ \frac {315 \, a c^{7} e x^{8} + 420 \, a c^{7} d x^{6} + 105 \, {\left (3 \, b c^{7} e x^{8} + 4 \, b c^{7} d x^{6}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (45 \, b c^{6} e x^{7} + 6 \, {\left (14 \, b c^{6} d - 9 \, b c^{4} e\right )} x^{5} - 8 \, {\left (14 \, b c^{4} d - 9 \, b c^{2} e\right )} x^{3} + 16 \, {\left (14 \, b c^{2} d - 9 \, b e\right )} x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{2520 \, c^{7}} \]
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 \[ \int {\left (e x^{2} + d\right )} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 152, normalized size = 0.75 \[ \frac {\frac {a \left (\frac {1}{8} e \,c^{8} x^{8}+\frac {1}{6} c^{8} d \,x^{6}\right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arccsch}\left (c x \right ) e \,c^{8} x^{8}}{8}+\frac {\mathrm {arccsch}\left (c x \right ) c^{8} x^{6} d}{6}+\frac {\left (c^{2} x^{2}+1\right ) \left (45 c^{6} e \,x^{6}+84 c^{6} d \,x^{4}-54 c^{4} e \,x^{4}-112 c^{4} d \,x^{2}+72 c^{2} x^{2} e +224 c^{2} d -144 e \right )}{2520 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 176, normalized size = 0.86 \[ \frac {1}{8} \, a e x^{8} + \frac {1}{6} \, a d x^{6} + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arcsch}\left (c x\right ) + \frac {3 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b d + \frac {1}{280} \, {\left (35 \, x^{8} \operatorname {arcsch}\left (c x\right ) + \frac {5 \, c^{6} x^{7} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} - 21 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 35 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{7}}\right )} b e \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^5\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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 \[ \int x^{5} \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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