Optimal. Leaf size=120 \[ \frac {\sqrt {\pi } e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {\sqrt {\pi } e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}-\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \cosh ^{-1}(c x)}} \]
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Rubi [A] time = 0.41, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5656, 5781, 3307, 2180, 2204, 2205} \[ \frac {\sqrt {\pi } e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {\sqrt {\pi } e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}-\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \cosh ^{-1}(c x)}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5656
Rule 5781
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac {2 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {(2 c) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 \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 (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c}\\ &=-\frac {2 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {\operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c}+\frac {\operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c}\\ &=-\frac {2 \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 {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{b^2 c}+\frac {2 \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{b^2 c}\\ &=-\frac {2 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 132, normalized size = 1.10 \[ \frac {e^{-\frac {a}{b}} \left (-2 e^{a/b} \sqrt {\frac {c x-1}{c x+1}} (c x+1)-e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\cosh ^{-1}(c x)\right )+\sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \cosh ^{-1}(c x)}{b}\right )\right )}{b c \sqrt {a+b \cosh ^{-1}(c x)}} \]
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 \[ \int \frac {1}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {1}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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