Optimal. Leaf size=203 \[ \frac {\left (d+e x^2\right )^3 \left (a+b \text {csch}^{-1}(c x)\right )}{6 e}-\frac {b c d^3 x \tan ^{-1}\left (\sqrt {-c^2 x^2-1}\right )}{6 e \sqrt {-c^2 x^2}}-\frac {b e x \left (-c^2 x^2-1\right )^{3/2} \left (3 c^2 d-2 e\right )}{18 c^5 \sqrt {-c^2 x^2}}+\frac {b e^2 x \left (-c^2 x^2-1\right )^{5/2}}{30 c^5 \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-c^2 x^2-1} \left (3 c^4 d^2-3 c^2 d e+e^2\right )}{6 c^5 \sqrt {-c^2 x^2}} \]
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Rubi [A] time = 0.16, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6300, 446, 88, 63, 205} \[ \frac {\left (d+e x^2\right )^3 \left (a+b \text {csch}^{-1}(c x)\right )}{6 e}+\frac {b x \sqrt {-c^2 x^2-1} \left (3 c^4 d^2-3 c^2 d e+e^2\right )}{6 c^5 \sqrt {-c^2 x^2}}-\frac {b c d^3 x \tan ^{-1}\left (\sqrt {-c^2 x^2-1}\right )}{6 e \sqrt {-c^2 x^2}}-\frac {b e x \left (-c^2 x^2-1\right )^{3/2} \left (3 c^2 d-2 e\right )}{18 c^5 \sqrt {-c^2 x^2}}+\frac {b e^2 x \left (-c^2 x^2-1\right )^{5/2}}{30 c^5 \sqrt {-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 88
Rule 205
Rule 446
Rule 6300
Rubi steps
\begin {align*} \int x \left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=\frac {\left (d+e x^2\right )^3 \left (a+b \text {csch}^{-1}(c x)\right )}{6 e}-\frac {(b c x) \int \frac {\left (d+e x^2\right )^3}{x \sqrt {-1-c^2 x^2}} \, dx}{6 e \sqrt {-c^2 x^2}}\\ &=\frac {\left (d+e x^2\right )^3 \left (a+b \text {csch}^{-1}(c x)\right )}{6 e}-\frac {(b c x) \operatorname {Subst}\left (\int \frac {(d+e x)^3}{x \sqrt {-1-c^2 x}} \, dx,x,x^2\right )}{12 e \sqrt {-c^2 x^2}}\\ &=\frac {\left (d+e x^2\right )^3 \left (a+b \text {csch}^{-1}(c x)\right )}{6 e}-\frac {(b c x) \operatorname {Subst}\left (\int \left (\frac {e \left (3 c^4 d^2-3 c^2 d e+e^2\right )}{c^4 \sqrt {-1-c^2 x}}+\frac {d^3}{x \sqrt {-1-c^2 x}}-\frac {\left (3 c^2 d-2 e\right ) e^2 \sqrt {-1-c^2 x}}{c^4}+\frac {e^3 \left (-1-c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{12 e \sqrt {-c^2 x^2}}\\ &=\frac {b \left (3 c^4 d^2-3 c^2 d e+e^2\right ) x \sqrt {-1-c^2 x^2}}{6 c^5 \sqrt {-c^2 x^2}}-\frac {b \left (3 c^2 d-2 e\right ) e x \left (-1-c^2 x^2\right )^{3/2}}{18 c^5 \sqrt {-c^2 x^2}}+\frac {b e^2 x \left (-1-c^2 x^2\right )^{5/2}}{30 c^5 \sqrt {-c^2 x^2}}+\frac {\left (d+e x^2\right )^3 \left (a+b \text {csch}^{-1}(c x)\right )}{6 e}-\frac {\left (b c d^3 x\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-1-c^2 x}} \, dx,x,x^2\right )}{12 e \sqrt {-c^2 x^2}}\\ &=\frac {b \left (3 c^4 d^2-3 c^2 d e+e^2\right ) x \sqrt {-1-c^2 x^2}}{6 c^5 \sqrt {-c^2 x^2}}-\frac {b \left (3 c^2 d-2 e\right ) e x \left (-1-c^2 x^2\right )^{3/2}}{18 c^5 \sqrt {-c^2 x^2}}+\frac {b e^2 x \left (-1-c^2 x^2\right )^{5/2}}{30 c^5 \sqrt {-c^2 x^2}}+\frac {\left (d+e x^2\right )^3 \left (a+b \text {csch}^{-1}(c x)\right )}{6 e}+\frac {\left (b d^3 x\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {-1-c^2 x^2}\right )}{6 c e \sqrt {-c^2 x^2}}\\ &=\frac {b \left (3 c^4 d^2-3 c^2 d e+e^2\right ) x \sqrt {-1-c^2 x^2}}{6 c^5 \sqrt {-c^2 x^2}}-\frac {b \left (3 c^2 d-2 e\right ) e x \left (-1-c^2 x^2\right )^{3/2}}{18 c^5 \sqrt {-c^2 x^2}}+\frac {b e^2 x \left (-1-c^2 x^2\right )^{5/2}}{30 c^5 \sqrt {-c^2 x^2}}+\frac {\left (d+e x^2\right )^3 \left (a+b \text {csch}^{-1}(c x)\right )}{6 e}-\frac {b c d^3 x \tan ^{-1}\left (\sqrt {-1-c^2 x^2}\right )}{6 e \sqrt {-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 123, normalized size = 0.61 \[ \frac {1}{90} x \left (15 a x \left (3 d^2+3 d e x^2+e^2 x^4\right )+\frac {b \sqrt {\frac {1}{c^2 x^2}+1} \left (3 c^4 \left (15 d^2+5 d e x^2+e^2 x^4\right )-2 c^2 e \left (15 d+2 e x^2\right )+8 e^2\right )}{c^5}+15 b x \text {csch}^{-1}(c x) \left (3 d^2+3 d e x^2+e^2 x^4\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.16, size = 189, normalized size = 0.93 \[ \frac {15 \, a c^{5} e^{2} x^{6} + 45 \, a c^{5} d e x^{4} + 45 \, a c^{5} d^{2} x^{2} + 15 \, {\left (b c^{5} e^{2} x^{6} + 3 \, b c^{5} d e x^{4} + 3 \, b c^{5} d^{2} x^{2}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (3 \, b c^{4} e^{2} x^{5} + {\left (15 \, b c^{4} d e - 4 \, b c^{2} e^{2}\right )} x^{3} + {\left (45 \, b c^{4} d^{2} - 30 \, b c^{2} d e + 8 \, b e^{2}\right )} x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{90 \, c^{5}} \]
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 \[ \int {\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 182, normalized size = 0.90 \[ \frac {\frac {a \left (\frac {1}{6} c^{6} e^{2} x^{6}+\frac {1}{2} c^{6} d e \,x^{4}+\frac {1}{2} c^{6} d^{2} x^{2}\right )}{c^{4}}+\frac {b \left (\frac {\mathrm {arccsch}\left (c x \right ) e^{2} c^{6} x^{6}}{6}+\frac {\mathrm {arccsch}\left (c x \right ) c^{6} d e \,x^{4}}{2}+\frac {\mathrm {arccsch}\left (c x \right ) c^{6} x^{2} d^{2}}{2}+\frac {\left (c^{2} x^{2}+1\right ) \left (3 c^{4} e^{2} x^{4}+15 c^{4} d e \,x^{2}+45 d^{2} c^{4}-4 c^{2} e^{2} x^{2}-30 c^{2} d e +8 e^{2}\right )}{90 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 183, normalized size = 0.90 \[ \frac {1}{6} \, a e^{2} x^{6} + \frac {1}{2} \, a d e x^{4} + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{2} \, {\left (x^{2} \operatorname {arcsch}\left (c x\right ) + \frac {x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b d^{2} + \frac {1}{6} \, {\left (3 \, x^{4} \operatorname {arcsch}\left (c x\right ) + \frac {c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b d e + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arcsch}\left (c x\right ) + \frac {3 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b e^{2} \]
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 \[ \int x\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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 \[ \int x \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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