Optimal. Leaf size=264 \[ -\frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^4}-\frac {b d e^2 x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{2 c^2}-\frac {b e^3 x^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{12 c^2}+\frac {(d+e x)^4 \left (a+b \text {sech}^{-1}(c x)\right )}{4 e}+\frac {b d \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {ArcSin}(c x)}{2 c^3}-\frac {b d^4 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{4 e} \]
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Rubi [A]
time = 0.24, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6423, 1823,
858, 222, 272, 65, 214} \begin {gather*} \frac {(d+e x)^4 \left (a+b \text {sech}^{-1}(c x)\right )}{4 e}+\frac {b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {ArcSin}(c x) \left (2 c^2 d^2+e^2\right )}{2 c^3}-\frac {b d^4 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{4 e}-\frac {b d e^2 x \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{2 c^2}-\frac {b e^3 x^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{12 c^2}-\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (9 c^2 d^2+e^2\right )}{6 c^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 858
Rule 1823
Rule 6423
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^4 \left (a+b \text {sech}^{-1}(c x)\right )}{4 e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {(d+e x)^4}{x \sqrt {1-c^2 x^2}} \, dx}{4 e}\\ &=-\frac {b e^3 x^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{12 c^2}+\frac {(d+e x)^4 \left (a+b \text {sech}^{-1}(c x)\right )}{4 e}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-3 c^2 d^4-12 c^2 d^3 e x-2 e^2 \left (9 c^2 d^2+e^2\right ) x^2-12 c^2 d e^3 x^3}{x \sqrt {1-c^2 x^2}} \, dx}{12 c^2 e}\\ &=-\frac {b d e^2 x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{2 c^2}-\frac {b e^3 x^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{12 c^2}+\frac {(d+e x)^4 \left (a+b \text {sech}^{-1}(c x)\right )}{4 e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {6 c^4 d^4+12 c^2 d e \left (2 c^2 d^2+e^2\right ) x+4 c^2 e^2 \left (9 c^2 d^2+e^2\right ) x^2}{x \sqrt {1-c^2 x^2}} \, dx}{24 c^4 e}\\ &=-\frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^4}-\frac {b d e^2 x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{2 c^2}-\frac {b e^3 x^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{12 c^2}+\frac {(d+e x)^4 \left (a+b \text {sech}^{-1}(c x)\right )}{4 e}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-6 c^6 d^4-12 c^4 d e \left (2 c^2 d^2+e^2\right ) x}{x \sqrt {1-c^2 x^2}} \, dx}{24 c^6 e}\\ &=-\frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^4}-\frac {b d e^2 x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{2 c^2}-\frac {b e^3 x^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{12 c^2}+\frac {(d+e x)^4 \left (a+b \text {sech}^{-1}(c x)\right )}{4 e}+\frac {\left (b d^4 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {1-c^2 x^2}} \, dx}{4 e}+\frac {\left (b d \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{2 c^2}\\ &=-\frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^4}-\frac {b d e^2 x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{2 c^2}-\frac {b e^3 x^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{12 c^2}+\frac {(d+e x)^4 \left (a+b \text {sech}^{-1}(c x)\right )}{4 e}+\frac {b d \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{2 c^3}+\frac {\left (b d^4 \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 )}{8 e}\\ &=-\frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^4}-\frac {b d e^2 x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{2 c^2}-\frac {b e^3 x^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{12 c^2}+\frac {(d+e x)^4 \left (a+b \text {sech}^{-1}(c x)\right )}{4 e}+\frac {b d \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{2 c^3}-\frac {\left (b d^4 \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 )}{4 c^2 e}\\ &=-\frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^4}-\frac {b d e^2 x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{2 c^2}-\frac {b e^3 x^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{12 c^2}+\frac {(d+e x)^4 \left (a+b \text {sech}^{-1}(c x)\right )}{4 e}+\frac {b d \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{2 c^3}-\frac {b d^4 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{4 e}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.23, size = 190, normalized size = 0.72 \begin {gather*} \frac {1}{4} \left (4 a d^3 x+6 a d^2 e x^2+4 a d e^2 x^3+a e^3 x^4-\frac {b e \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (2 e^2+c^2 \left (18 d^2+6 d e x+e^2 x^2\right )\right )}{3 c^4}+b x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right ) \text {sech}^{-1}(c x)+\frac {2 i b d \left (2 c^2 d^2+e^2\right ) \log \left (-2 i c x+2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )}{c^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 283, normalized size = 1.07
method | result | size |
derivativedivides | \(\frac {\frac {\left (c e x +c d \right )^{4} a}{4 c^{3} e}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) c^{4} d^{4}}{4 e}+\mathrm {arcsech}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\mathrm {arcsech}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \mathrm {arcsech}\left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \mathrm {arcsech}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-3 c^{4} d^{4} \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )+12 c^{3} d^{3} e \arcsin \left (c x \right )-18 c^{2} d^{2} e^{2} \sqrt {-c^{2} x^{2}+1}-6 c^{2} d \,e^{3} x \sqrt {-c^{2} x^{2}+1}-e^{4} \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+6 c d \,e^{3} \arcsin \left (c x \right )-2 e^{4} \sqrt {-c^{2} x^{2}+1}\right )}{12 e \sqrt {-c^{2} x^{2}+1}}\right )}{c^{3}}}{c}\) | \(283\) |
default | \(\frac {\frac {\left (c e x +c d \right )^{4} a}{4 c^{3} e}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) c^{4} d^{4}}{4 e}+\mathrm {arcsech}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\mathrm {arcsech}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \mathrm {arcsech}\left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \mathrm {arcsech}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-3 c^{4} d^{4} \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )+12 c^{3} d^{3} e \arcsin \left (c x \right )-18 c^{2} d^{2} e^{2} \sqrt {-c^{2} x^{2}+1}-6 c^{2} d \,e^{3} x \sqrt {-c^{2} x^{2}+1}-e^{4} \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+6 c d \,e^{3} \arcsin \left (c x \right )-2 e^{4} \sqrt {-c^{2} x^{2}+1}\right )}{12 e \sqrt {-c^{2} x^{2}+1}}\right )}{c^{3}}}{c}\) | \(283\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 219, normalized size = 0.83 \begin {gather*} \frac {1}{4} \, a x^{4} e^{3} + a d x^{3} e^{2} + \frac {3}{2} \, a d^{2} x^{2} e + a d^{3} x + \frac {3}{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} e + \frac {{\left (c x \operatorname {arsech}\left (c x\right ) - \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )\right )} b d^{3}}{c} + \frac {1}{2} \, {\left (2 \, x^{3} \operatorname {arsech}\left (c x\right ) - \frac {\frac {\sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{2}}}{c}\right )} b d e^{2} + \frac {1}{12} \, {\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 e^{3} \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. 753 vs.
\(2 (140) = 280\).
time = 0.45, size = 753, normalized size = 2.85 \begin {gather*} \frac {3 \, a c^{3} x^{4} \cosh \left (1\right )^{3} + 3 \, a c^{3} x^{4} \sinh \left (1\right )^{3} + 12 \, a c^{3} d x^{3} \cosh \left (1\right )^{2} + 18 \, a c^{3} d^{2} x^{2} \cosh \left (1\right ) + 12 \, a c^{3} d^{3} x + 3 \, {\left (3 \, a c^{3} x^{4} \cosh \left (1\right ) + 4 \, a c^{3} d x^{3}\right )} \sinh \left (1\right )^{2} - 12 \, {\left (2 \, b c^{2} d^{3} + b d \cosh \left (1\right )^{2} + 2 \, b d \cosh \left (1\right ) \sinh \left (1\right ) + b d \sinh \left (1\right )^{2}\right )} \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - 3 \, {\left (4 \, b c^{3} d^{3} + 6 \, b c^{3} d^{2} \cosh \left (1\right ) + 4 \, b c^{3} d \cosh \left (1\right )^{2} + b c^{3} \cosh \left (1\right )^{3} + b c^{3} \sinh \left (1\right )^{3} + {\left (4 \, b c^{3} d + 3 \, b c^{3} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + {\left (6 \, b c^{3} d^{2} + 8 \, b c^{3} d \cosh \left (1\right ) + 3 \, b c^{3} \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 3 \, {\left (4 \, b c^{3} d^{3} x - 4 \, b c^{3} d^{3} + {\left (b c^{3} x^{4} - b c^{3}\right )} \cosh \left (1\right )^{3} + {\left (b c^{3} x^{4} - b c^{3}\right )} \sinh \left (1\right )^{3} + 4 \, {\left (b c^{3} d x^{3} - b c^{3} d\right )} \cosh \left (1\right )^{2} + {\left (4 \, b c^{3} d x^{3} - 4 \, b c^{3} d + 3 \, {\left (b c^{3} x^{4} - b c^{3}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 6 \, {\left (b c^{3} d^{2} x^{2} - b c^{3} d^{2}\right )} \cosh \left (1\right ) + {\left (6 \, b c^{3} d^{2} x^{2} - 6 \, b c^{3} d^{2} + 3 \, {\left (b c^{3} x^{4} - b c^{3}\right )} \cosh \left (1\right )^{2} + 8 \, {\left (b c^{3} d x^{3} - b c^{3} d\right )} \cosh \left (1\right )\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 ) + 3 \, {\left (3 \, a c^{3} x^{4} \cosh \left (1\right )^{2} + 8 \, a c^{3} d x^{3} \cosh \left (1\right ) + 6 \, a c^{3} d^{2} x^{2}\right )} \sinh \left (1\right ) - {\left (6 \, b c^{2} d x^{2} \cosh \left (1\right )^{2} + 18 \, b c^{2} d^{2} x \cosh \left (1\right ) + {\left (b c^{2} x^{3} + 2 \, b x\right )} \cosh \left (1\right )^{3} + {\left (b c^{2} x^{3} + 2 \, b x\right )} \sinh \left (1\right )^{3} + 3 \, {\left (2 \, b c^{2} d x^{2} + {\left (b c^{2} x^{3} + 2 \, b x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 3 \, {\left (4 \, b c^{2} d x^{2} \cosh \left (1\right ) + 6 \, b c^{2} d^{2} x + {\left (b c^{2} x^{3} + 2 \, b x\right )} \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{12 \, c^{3}} \end {gather*}
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 \begin {gather*} \int \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x\right )^{3}\, dx \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 \left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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