Optimal. Leaf size=177 \[ \frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{c e^{3/2}}-\frac {2 b \sqrt {d} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{e^2} \]
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Rubi [A]
time = 0.18, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {272, 45, 6436,
12, 587, 163, 65, 223, 209, 95, 213} \begin {gather*} \frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{c e^{3/2}}-\frac {2 b \sqrt {d} \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 65
Rule 95
Rule 163
Rule 209
Rule 213
Rule 223
Rule 272
Rule 587
Rule 6436
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {2 d+e x^2}{e^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \, dx\\ &=\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {2 d+e x^2}{x \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{e^2}\\ &=\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {2 d+e x}{x \sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{2 e^2}\\ &=\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{e^2}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{2 e}\\ &=\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (2 b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{-d+x^2} \, dx,x,\frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}}\right )}{e^2}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d+\frac {e}{c^2}-\frac {e x^2}{c^2}}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{c^2 e}\\ &=\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}-\frac {2 b \sqrt {d} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{e^2}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{1+\frac {e x^2}{c^2}} \, dx,x,\frac {\sqrt {1-c^2 x^2}}{\sqrt {d+e x^2}}\right )}{c^2 e}\\ &=\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{c e^{3/2}}-\frac {2 b \sqrt {d} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{e^2}\\ \end {align*}
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Mathematica [A]
time = 21.06, size = 249, normalized size = 1.41 \begin {gather*} \frac {\left (2 d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {b \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^2 x^2} \left (\sqrt {-c^2} \sqrt {-c^2 d-e} \sqrt {e} \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}} \text {ArcSin}\left (\frac {c \sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {-c^2} \sqrt {-c^2 d-e}}\right )+2 c^3 \sqrt {d} \sqrt {-d-e x^2} \text {ArcTan}\left (\frac {\sqrt {d} \sqrt {1-c^2 x^2}}{\sqrt {-d-e x^2}}\right )\right )}{c^3 e^2 (-1+c x) \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.29, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 529 vs.
\(2 (116) = 232\).
time = 0.49, size = 1095, normalized size = 6.19 \begin {gather*} \left [-\frac {{\left (b x^{2} \cosh \left (1\right ) + b x^{2} \sinh \left (1\right ) + b d\right )} \sqrt {\cosh \left (1\right ) + \sinh \left (1\right )} \arctan \left (\frac {{\left (c^{2} d x + {\left (2 \, c^{2} x^{3} - x\right )} \cosh \left (1\right ) + {\left (2 \, c^{2} x^{3} - x\right )} \sinh \left (1\right )\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} \sqrt {\cosh \left (1\right ) + \sinh \left (1\right )}}{2 \, {\left ({\left (c^{2} x^{4} - x^{2}\right )} \cosh \left (1\right )^{2} + {\left (c^{2} x^{4} - x^{2}\right )} \sinh \left (1\right )^{2} + {\left (c^{2} d x^{2} - d\right )} \cosh \left (1\right ) + {\left (c^{2} d x^{2} + 2 \, {\left (c^{2} x^{4} - x^{2}\right )} \cosh \left (1\right ) - d\right )} \sinh \left (1\right )\right )}}\right ) - 2 \, {\left (b c x^{2} \cosh \left (1\right ) + b c x^{2} \sinh \left (1\right ) + 2 \, b c d\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (b c x^{2} \cosh \left (1\right ) + b c x^{2} \sinh \left (1\right ) + b c d\right )} \sqrt {d} \log \left (\frac {c^{4} d^{2} x^{4} - 8 \, c^{2} d^{2} x^{2} + x^{4} \cosh \left (1\right )^{2} + x^{4} \sinh \left (1\right )^{2} + 4 \, {\left (c^{3} d x^{3} - c x^{3} \cosh \left (1\right ) - c x^{3} \sinh \left (1\right ) - 2 \, c d x\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \sqrt {d} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 8 \, d^{2} - 2 \, {\left (3 \, c^{2} d x^{4} - 4 \, d x^{2}\right )} \cosh \left (1\right ) - 2 \, {\left (3 \, c^{2} d x^{4} - x^{4} \cosh \left (1\right ) - 4 \, d x^{2}\right )} \sinh \left (1\right )}{x^{4}}\right ) - 2 \, {\left (a c x^{2} \cosh \left (1\right ) + a c x^{2} \sinh \left (1\right ) + 2 \, a c d\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d}}{2 \, {\left (c x^{2} \cosh \left (1\right )^{3} + c x^{2} \sinh \left (1\right )^{3} + c d \cosh \left (1\right )^{2} + {\left (3 \, c x^{2} \cosh \left (1\right ) + c d\right )} \sinh \left (1\right )^{2} + {\left (3 \, c x^{2} \cosh \left (1\right )^{2} + 2 \, c d \cosh \left (1\right )\right )} \sinh \left (1\right )\right )}}, -\frac {2 \, {\left (b c x^{2} \cosh \left (1\right ) + b c x^{2} \sinh \left (1\right ) + b c d\right )} \sqrt {-d} \arctan \left (-\frac {{\left (c^{3} d x^{3} - c x^{3} \cosh \left (1\right ) - c x^{3} \sinh \left (1\right ) - 2 \, c d x\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \sqrt {-d} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{2 \, {\left (c^{2} d^{2} x^{2} - d^{2} + {\left (c^{2} d x^{4} - d x^{2}\right )} \cosh \left (1\right ) + {\left (c^{2} d x^{4} - d x^{2}\right )} \sinh \left (1\right )\right )}}\right ) + {\left (b x^{2} \cosh \left (1\right ) + b x^{2} \sinh \left (1\right ) + b d\right )} \sqrt {\cosh \left (1\right ) + \sinh \left (1\right )} \arctan \left (\frac {{\left (c^{2} d x + {\left (2 \, c^{2} x^{3} - x\right )} \cosh \left (1\right ) + {\left (2 \, c^{2} x^{3} - x\right )} \sinh \left (1\right )\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} \sqrt {\cosh \left (1\right ) + \sinh \left (1\right )}}{2 \, {\left ({\left (c^{2} x^{4} - x^{2}\right )} \cosh \left (1\right )^{2} + {\left (c^{2} x^{4} - x^{2}\right )} \sinh \left (1\right )^{2} + {\left (c^{2} d x^{2} - d\right )} \cosh \left (1\right ) + {\left (c^{2} d x^{2} + 2 \, {\left (c^{2} x^{4} - x^{2}\right )} \cosh \left (1\right ) - d\right )} \sinh \left (1\right )\right )}}\right ) - 2 \, {\left (b c x^{2} \cosh \left (1\right ) + b c x^{2} \sinh \left (1\right ) + 2 \, b c d\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 2 \, {\left (a c x^{2} \cosh \left (1\right ) + a c x^{2} \sinh \left (1\right ) + 2 \, a c d\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d}}{2 \, {\left (c x^{2} \cosh \left (1\right )^{3} + c x^{2} \sinh \left (1\right )^{3} + c d \cosh \left (1\right )^{2} + {\left (3 \, c x^{2} \cosh \left (1\right ) + c d\right )} \sinh \left (1\right )^{2} + {\left (3 \, c x^{2} \cosh \left (1\right )^{2} + 2 \, c d \cosh \left (1\right )\right )} \sinh \left (1\right )\right )}}\right ] \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 \frac {x^{3} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, 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 = -0.01 \begin {gather*} \int \frac {x^3\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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