Optimal. Leaf size=250 \[ \frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b e x \left (-c^2 x^2-1\right )^{5/2} \left (8 c^2 d-9 e\right )}{120 c^7 \sqrt {-c^2 x^2}}-\frac {b e^2 x \left (-c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt {-c^2 x^2}}-\frac {b x \left (-c^2 x^2-1\right )^{3/2} \left (6 c^4 d^2-16 c^2 d e+9 e^2\right )}{72 c^7 \sqrt {-c^2 x^2}}-\frac {b x \sqrt {-c^2 x^2-1} \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )}{24 c^7 \sqrt {-c^2 x^2}} \]
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Rubi [A] time = 0.23, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {266, 43, 6302, 12, 1251, 771} \[ \frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b x \left (-c^2 x^2-1\right )^{3/2} \left (6 c^4 d^2-16 c^2 d e+9 e^2\right )}{72 c^7 \sqrt {-c^2 x^2}}-\frac {b x \sqrt {-c^2 x^2-1} \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )}{24 c^7 \sqrt {-c^2 x^2}}+\frac {b e x \left (-c^2 x^2-1\right )^{5/2} \left (8 c^2 d-9 e\right )}{120 c^7 \sqrt {-c^2 x^2}}-\frac {b e^2 x \left (-c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt {-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 266
Rule 771
Rule 1251
Rule 6302
Rubi steps
\begin {align*} \int x^3 \left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=\frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{24 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=\frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{\sqrt {-1-c^2 x^2}} \, dx}{24 \sqrt {-c^2 x^2}}\\ &=\frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \operatorname {Subst}\left (\int \frac {x \left (6 d^2+8 d e x+3 e^2 x^2\right )}{\sqrt {-1-c^2 x}} \, dx,x,x^2\right )}{48 \sqrt {-c^2 x^2}}\\ &=\frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \operatorname {Subst}\left (\int \left (\frac {-6 c^4 d^2+8 c^2 d e-3 e^2}{c^6 \sqrt {-1-c^2 x}}+\frac {\left (-6 c^4 d^2+16 c^2 d e-9 e^2\right ) \sqrt {-1-c^2 x}}{c^6}+\frac {\left (8 c^2 d-9 e\right ) e \left (-1-c^2 x\right )^{3/2}}{c^6}-\frac {3 e^2 \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 (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x \sqrt {-1-c^2 x^2}}{24 c^7 \sqrt {-c^2 x^2}}-\frac {b \left (6 c^4 d^2-16 c^2 d e+9 e^2\right ) x \left (-1-c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {-c^2 x^2}}+\frac {b \left (8 c^2 d-9 e\right ) e x \left (-1-c^2 x^2\right )^{5/2}}{120 c^7 \sqrt {-c^2 x^2}}-\frac {b e^2 x \left (-1-c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {-c^2 x^2}}+\frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.39, size = 159, normalized size = 0.64 \[ \frac {x \left (105 a x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )+\frac {b \sqrt {\frac {1}{c^2 x^2}+1} \left (3 c^6 \left (70 d^2 x^2+56 d e x^4+15 e^2 x^6\right )-2 c^4 \left (210 d^2+112 d e x^2+27 e^2 x^4\right )+8 c^2 e \left (56 d+9 e x^2\right )-144 e^2\right )}{c^7}+105 b x^3 \text {csch}^{-1}(c x) \left (6 d^2+8 d e x^2+3 e^2 x^4\right )\right )}{2520} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.25, size = 224, normalized size = 0.90 \[ \frac {315 \, a c^{7} e^{2} x^{8} + 840 \, a c^{7} d e x^{6} + 630 \, a c^{7} d^{2} x^{4} + 105 \, {\left (3 \, b c^{7} e^{2} x^{8} + 8 \, b c^{7} d e x^{6} + 6 \, b c^{7} d^{2} x^{4}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (45 \, b c^{6} e^{2} x^{7} + 6 \, {\left (28 \, b c^{6} d e - 9 \, b c^{4} e^{2}\right )} x^{5} + 2 \, {\left (105 \, b c^{6} d^{2} - 112 \, b c^{4} d e + 36 \, b c^{2} e^{2}\right )} x^{3} - 4 \, {\left (105 \, b c^{4} d^{2} - 112 \, b c^{2} d e + 36 \, b e^{2}\right )} x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{2520 \, c^{7}} \]
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 \[ \int {\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 214, normalized size = 0.86 \[ \frac {\frac {a \left (\frac {1}{8} e^{2} c^{8} x^{8}+\frac {1}{3} c^{8} d e \,x^{6}+\frac {1}{4} c^{8} d^{2} x^{4}\right )}{c^{4}}+\frac {b \left (\frac {\mathrm {arccsch}\left (c x \right ) e^{2} c^{8} x^{8}}{8}+\frac {\mathrm {arccsch}\left (c x \right ) c^{8} d e \,x^{6}}{3}+\frac {\mathrm {arccsch}\left (c x \right ) c^{8} x^{4} d^{2}}{4}+\frac {\left (c^{2} x^{2}+1\right ) \left (45 c^{6} e^{2} x^{6}+168 c^{6} d e \,x^{4}+210 c^{6} d^{2} x^{2}-54 c^{4} e^{2} x^{4}-224 c^{4} d e \,x^{2}-420 d^{2} c^{4}+72 c^{2} e^{2} x^{2}+448 c^{2} d e -144 e^{2}\right )}{2520 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 244, normalized size = 0.98 \[ \frac {1}{8} \, a e^{2} x^{8} + \frac {1}{3} \, a d e x^{6} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arcsch}\left (c x\right ) + \frac {c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b d^{2} + \frac {1}{45} \, {\left (15 \, x^{6} \operatorname {arcsch}\left (c x\right ) + \frac {3 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b d e + \frac {1}{280} \, {\left (35 \, x^{8} \operatorname {arcsch}\left (c x\right ) + \frac {5 \, c^{6} x^{7} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} - 21 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 35 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{7}}\right )} b e^{2} \]
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 \[ \int x^3\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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 \[ \int x^{3} \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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