Optimal. Leaf size=101 \[ \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {b \sqrt {-1+c^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{d \sqrt {e} \sqrt {-1+c x} \sqrt {1+c x}} \]
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Rubi [A]
time = 0.14, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {197, 5908, 12,
533, 455, 65, 223, 212} \begin {gather*} \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {b \sqrt {c^2 x^2-1} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{d \sqrt {e} \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 197
Rule 212
Rule 223
Rule 455
Rule 533
Rule 5908
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-(b c) \int \frac {x}{d \sqrt {-1+c x} \sqrt {1+c x} \sqrt {d+e x^2}} \, dx\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {(b c) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {d+e x^2}} \, dx}{d}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {x}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{2 d \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {\left (b \sqrt {-1+c^2 x^2}\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 d \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {\left (b \sqrt {-1+c^2 x^2}\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 d \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {b \sqrt {-1+c^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{d \sqrt {e} \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 12.38, size = 556, normalized size = 5.50 \begin {gather*} \frac {a x+b x \cosh ^{-1}(c x)+\frac {2 b (-1+c x)^{3/2} \sqrt {\frac {\left (c \sqrt {d}-i \sqrt {e}\right ) (1+c x)}{\left (c \sqrt {d}+i \sqrt {e}\right ) (-1+c x)}} \left (\frac {c \left (-i c \sqrt {d}+\sqrt {e}\right ) \left (i \sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {1+\frac {i c \sqrt {d}}{\sqrt {e}}-c x+\frac {i \sqrt {e} x}{\sqrt {d}}}{1-c x}} F\left (\text {ArcSin}\left (\sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{2-2 c x}}\right )|\frac {4 i c \sqrt {d} \sqrt {e}}{\left (c \sqrt {d}+i \sqrt {e}\right )^2}\right )}{-1+c x}+c \sqrt {d} \left (-c \sqrt {d}+i \sqrt {e}\right ) \sqrt {\frac {\left (c^2 d+e\right ) \left (d+e x^2\right )}{d e (-1+c x)^2}} \sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{1-c x}} \Pi \left (\frac {2 c \sqrt {d}}{c \sqrt {d}+i \sqrt {e}};\text {ArcSin}\left (\sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{2-2 c x}}\right )|\frac {4 i c \sqrt {d} \sqrt {e}}{\left (c \sqrt {d}+i \sqrt {e}\right )^2}\right )\right )}{c \left (c^2 d+e\right ) \sqrt {1+c x} \sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{1-c x}}}}{d \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {a +b \,\mathrm {arccosh}\left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 333 vs.
\(2 (85) = 170\).
time = 0.39, size = 333, normalized size = 3.30 \begin {gather*} \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^{3} d + {\left (2 \, c^{3} x^{2} - c\right )} \cosh \left (1\right ) + {\left (2 \, c^{3} x^{2} - c\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \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 x \cosh \left (1\right ) + b x \sinh \left (1\right )\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 4 \, {\left (a x \cosh \left (1\right ) + a x \sinh \left (1\right )\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d}}{4 \, {\left (d x^{2} \cosh \left (1\right )^{2} + d x^{2} \sinh \left (1\right )^{2} + d^{2} \cosh \left (1\right ) + {\left (2 \, d x^{2} \cosh \left (1\right ) + d^{2}\right )} \sinh \left (1\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 {a + b \operatorname {acosh}{\left (c x \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 {a+b\,\mathrm {acosh}\left (c\,x\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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