Optimal. Leaf size=230 \[ -\frac {b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^6}+\frac {b e \left (3 c^2 d+2 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{18 c^6}-\frac {b e^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{30 c^6}+\frac {\left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right )}{6 e}-\frac {b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{6 e} \]
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Rubi [A]
time = 0.17, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6434, 531, 457,
90, 65, 214} \begin {gather*} \frac {\left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right )}{6 e}-\frac {b d^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{6 e}+\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2} \left (3 c^2 d+2 e\right )}{18 c^6}-\frac {b e^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{30 c^6}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (3 c^4 d^2+3 c^2 d e+e^2\right )}{6 c^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 90
Rule 214
Rule 457
Rule 531
Rule 6434
Rubi steps
\begin {align*} \int x \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {\left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right )}{6 e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (d+e x^2\right )^3}{x \sqrt {1-c x} \sqrt {1+c x}} \, dx}{6 e}\\ &=\frac {\left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right )}{6 e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (d+e x^2\right )^3}{x \sqrt {1-c^2 x^2}} \, dx}{6 e}\\ &=\frac {\left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right )}{6 e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {(d+e x)^3}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{12 e}\\ &=\frac {\left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right )}{6 e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {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 {e^2 \left (3 c^2 d+2 e\right ) \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}\\ &=-\frac {b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^6}+\frac {b e \left (3 c^2 d+2 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{18 c^6}-\frac {b e^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{30 c^6}+\frac {\left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right )}{6 e}+\frac {\left (b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{12 e}\\ &=-\frac {b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^6}+\frac {b e \left (3 c^2 d+2 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{18 c^6}-\frac {b e^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{30 c^6}+\frac {\left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right )}{6 e}-\frac {\left (b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {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^2 e}\\ &=-\frac {b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^6}+\frac {b e \left (3 c^2 d+2 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{18 c^6}-\frac {b e^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{30 c^6}+\frac {\left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right )}{6 e}-\frac {b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{6 e}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 139, normalized size = 0.60 \begin {gather*} \frac {1}{6} a x^2 \left (3 d^2+3 d e x^2+e^2 x^4\right )-\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (8 e^2+2 c^2 e \left (15 d+2 e x^2\right )+3 c^4 \left (15 d^2+5 d e x^2+e^2 x^4\right )\right )}{90 c^6}+\frac {1}{6} b x^2 \left (3 d^2+3 d e x^2+e^2 x^4\right ) \text {sech}^{-1}(c x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 295, normalized size = 1.28
method | result | size |
derivativedivides | \(\frac {\frac {\left (e \,c^{2} x^{2}+c^{2} d \right )^{3} a}{6 c^{4} e}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) c^{6} d^{3}}{6 e}+\frac {\mathrm {arcsech}\left (c x \right ) c^{6} d^{2} x^{2}}{2}+\frac {e \,\mathrm {arcsech}\left (c x \right ) c^{6} d \,x^{4}}{2}+\frac {e^{2} \mathrm {arcsech}\left (c x \right ) c^{6} x^{6}}{6}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (15 c^{6} d^{3} \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )+45 c^{4} d^{2} e \sqrt {-c^{2} x^{2}+1}+15 c^{4} d \,e^{2} \sqrt {-c^{2} x^{2}+1}\, x^{2}+3 e^{3} \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}+30 c^{2} d \,e^{2} \sqrt {-c^{2} x^{2}+1}+4 e^{3} c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}+8 e^{3} \sqrt {-c^{2} x^{2}+1}\right )}{90 e \sqrt {-c^{2} x^{2}+1}}\right )}{c^{4}}}{c^{2}}\) | \(295\) |
default | \(\frac {\frac {\left (e \,c^{2} x^{2}+c^{2} d \right )^{3} a}{6 c^{4} e}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) c^{6} d^{3}}{6 e}+\frac {\mathrm {arcsech}\left (c x \right ) c^{6} d^{2} x^{2}}{2}+\frac {e \,\mathrm {arcsech}\left (c x \right ) c^{6} d \,x^{4}}{2}+\frac {e^{2} \mathrm {arcsech}\left (c x \right ) c^{6} x^{6}}{6}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (15 c^{6} d^{3} \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )+45 c^{4} d^{2} e \sqrt {-c^{2} x^{2}+1}+15 c^{4} d \,e^{2} \sqrt {-c^{2} x^{2}+1}\, x^{2}+3 e^{3} \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}+30 c^{2} d \,e^{2} \sqrt {-c^{2} x^{2}+1}+4 e^{3} c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}+8 e^{3} \sqrt {-c^{2} x^{2}+1}\right )}{90 e \sqrt {-c^{2} x^{2}+1}}\right )}{c^{4}}}{c^{2}}\) | \(295\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 185, normalized size = 0.80 \begin {gather*} \frac {1}{6} \, a x^{6} e^{2} + \frac {1}{2} \, a d x^{4} e + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{2} \, {\left (x^{2} \operatorname {arsech}\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 {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 e + \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^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 349 vs.
\(2 (132) = 264\).
time = 0.49, size = 349, normalized size = 1.52 \begin {gather*} \frac {15 \, a c^{5} x^{6} \cosh \left (1\right )^{2} + 15 \, a c^{5} x^{6} \sinh \left (1\right )^{2} + 45 \, a c^{5} d x^{4} \cosh \left (1\right ) + 45 \, a c^{5} d^{2} x^{2} + 15 \, {\left (b c^{5} x^{6} \cosh \left (1\right )^{2} + b c^{5} x^{6} \sinh \left (1\right )^{2} + 3 \, b c^{5} d x^{4} \cosh \left (1\right ) + 3 \, b c^{5} d^{2} x^{2} + {\left (2 \, b c^{5} x^{6} \cosh \left (1\right ) + 3 \, b c^{5} d x^{4}\right )} \sinh \left (1\right )\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 15 \, {\left (2 \, a c^{5} x^{6} \cosh \left (1\right ) + 3 \, a c^{5} d x^{4}\right )} \sinh \left (1\right ) - {\left (45 \, b c^{4} d^{2} x + {\left (3 \, b c^{4} x^{5} + 4 \, b c^{2} x^{3} + 8 \, b x\right )} \cosh \left (1\right )^{2} + {\left (3 \, b c^{4} x^{5} + 4 \, b c^{2} x^{3} + 8 \, b x\right )} \sinh \left (1\right )^{2} + 15 \, {\left (b c^{4} d x^{3} + 2 \, b c^{2} d x\right )} \cosh \left (1\right ) + {\left (15 \, b c^{4} d x^{3} + 30 \, b c^{2} d x + 2 \, {\left (3 \, b c^{4} x^{5} + 4 \, b c^{2} x^{3} + 8 \, b x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{90 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.95, size = 252, normalized size = 1.10 \begin {gather*} \begin {cases} \frac {a d^{2} x^{2}}{2} + \frac {a d e x^{4}}{2} + \frac {a e^{2} x^{6}}{6} + \frac {b d^{2} x^{2} \operatorname {asech}{\left (c x \right )}}{2} + \frac {b d e x^{4} \operatorname {asech}{\left (c x \right )}}{2} + \frac {b e^{2} x^{6} \operatorname {asech}{\left (c x \right )}}{6} - \frac {b d^{2} \sqrt {- c^{2} x^{2} + 1}}{2 c^{2}} - \frac {b d e x^{2} \sqrt {- c^{2} x^{2} + 1}}{6 c^{2}} - \frac {b e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{30 c^{2}} - \frac {b d e \sqrt {- c^{2} x^{2} + 1}}{3 c^{4}} - \frac {2 b e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{45 c^{4}} - \frac {4 b e^{2} \sqrt {- c^{2} x^{2} + 1}}{45 c^{6}} & \text {for}\: c \neq 0 \\\left (a + \infty b\right ) \left (\frac {d^{2} x^{2}}{2} + \frac {d e x^{4}}{2} + \frac {e^{2} x^{6}}{6}\right ) & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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