Optimal. Leaf size=140 \[ \frac {\sqrt {\frac {\pi }{2}} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}+\frac {\sqrt {\frac {\pi }{2}} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \cosh ^{-1}(c x)}} \]
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Rubi [A] time = 0.15, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5666, 3307, 2180, 2204, 2205} \[ \frac {\sqrt {\frac {\pi }{2}} e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}+\frac {\sqrt {\frac {\pi }{2}} e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \cosh ^{-1}(c x)}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5666
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac {2 x \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {2 \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 x \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {\operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2}+\frac {\operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 x \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {2 \operatorname {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{b^2 c^2}+\frac {2 \operatorname {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{b^2 c^2}\\ &=-\frac {2 x \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}+\frac {e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}\\ \end {align*}
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Mathematica [A] time = 1.29, size = 135, normalized size = 0.96 \[ \frac {\sqrt {2 \pi } \left (\sinh \left (\frac {2 a}{b}\right )+\cosh \left (\frac {2 a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )-\frac {2 \sqrt {b} \sinh \left (2 \cosh ^{-1}(c x)\right )}{\sqrt {a+b \cosh ^{-1}(c x)}}}{2 b^{3/2} c^2} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {x}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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