Optimal. Leaf size=160 \[ \frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {b x \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{e^{3/2} \sqrt {-c^2 x^2}}+\frac {2 b c \sqrt {d} x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{e^2 \sqrt {-c^2 x^2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 160, 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, 6437,
12, 587, 163, 65, 223, 209, 95, 210} \begin {gather*} \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {b x \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{e^{3/2} \sqrt {-c^2 x^2}}+\frac {2 b c \sqrt {d} x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{e^2 \sqrt {-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 65
Rule 95
Rule 163
Rule 209
Rule 210
Rule 223
Rule 272
Rule 587
Rule 6437
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}-\frac {(b c x) \int \frac {2 d+e x^2}{e^2 x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}-\frac {(b c x) \int \frac {2 d+e x^2}{x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{e^2 \sqrt {-c^2 x^2}}\\ &=\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}-\frac {(b c x) \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 \sqrt {-c^2 x^2}}\\ &=\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}-\frac {(b c d x) \text {Subst}\left (\int \frac {1}{x \sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{e^2 \sqrt {-c^2 x^2}}-\frac {(b c x) \text {Subst}\left (\int \frac {1}{\sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{2 e \sqrt {-c^2 x^2}}\\ &=\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}-\frac {(2 b c d x) \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 \sqrt {-c^2 x^2}}+\frac {(b x) \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 e \sqrt {-c^2 x^2}}\\ &=\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 b c \sqrt {d} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{e^2 \sqrt {-c^2 x^2}}+\frac {(b x) \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 e \sqrt {-c^2 x^2}}\\ &=\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {b x \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{e^{3/2} \sqrt {-c^2 x^2}}+\frac {2 b c \sqrt {d} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{e^2 \sqrt {-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.95, size = 147, normalized size = 0.92 \begin {gather*} \frac {\left (2 d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}-\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x \left (2 c \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {1+c^2 x^2}}{\sqrt {d+e x^2}}\right )-\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )\right )}{e^2 \sqrt {1+c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]
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. 553 vs.
\(2 (135) = 270\).
time = 0.45, size = 1146, normalized size = 7.16 \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 )} \log \left (c^{4} d^{2} + {\left (8 \, c^{4} x^{4} + 8 \, c^{2} x^{2} + 1\right )} \cosh \left (1\right )^{2} + {\left (8 \, c^{4} x^{4} + 8 \, c^{2} x^{2} + 1\right )} \sinh \left (1\right )^{2} + 4 \, {\left (c^{4} d x + {\left (2 \, c^{4} x^{3} + c^{2} x\right )} \cosh \left (1\right ) + {\left (2 \, c^{4} x^{3} + c^{2} 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 (4 \, c^{4} d x^{2} + 3 \, c^{2} d\right )} \cosh \left (1\right ) + 2 \, {\left (4 \, c^{4} d x^{2} + 3 \, c^{2} d + {\left (8 \, c^{4} x^{4} + 8 \, c^{2} x^{2} + 1\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right ) + 4 \, {\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 (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 ) + 4 \, {\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}}{4 \, {\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 {4 \, {\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 )} \log \left (c^{4} d^{2} + {\left (8 \, c^{4} x^{4} + 8 \, c^{2} x^{2} + 1\right )} \cosh \left (1\right )^{2} + {\left (8 \, c^{4} x^{4} + 8 \, c^{2} x^{2} + 1\right )} \sinh \left (1\right )^{2} + 4 \, {\left (c^{4} d x + {\left (2 \, c^{4} x^{3} + c^{2} x\right )} \cosh \left (1\right ) + {\left (2 \, c^{4} x^{3} + c^{2} 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 (4 \, c^{4} d x^{2} + 3 \, c^{2} d\right )} \cosh \left (1\right ) + 2 \, {\left (4 \, c^{4} d x^{2} + 3 \, c^{2} d + {\left (8 \, c^{4} x^{4} + 8 \, c^{2} x^{2} + 1\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right ) + 4 \, {\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 ) + 4 \, {\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}}{4 \, {\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 {acsch}{\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 {asinh}\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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