Optimal. Leaf size=94 \[ -\frac {e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a-b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a-b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d} \]
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Rubi [A]
time = 0.09, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5995, 5881,
3389, 2211, 2236, 2235} \begin {gather*} \frac {\sqrt {\pi } e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a-b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d}-\frac {\sqrt {\pi } e^{a/b} \text {Erf}\left (\frac {\sqrt {a-b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5881
Rule 5995
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a-b \cosh ^{-1}(c+d x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a-b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a-b \cosh ^{-1}(c+d x)\right )}{b d}\\ &=-\frac {\text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a-b \cosh ^{-1}(c+d x)\right )}{2 b d}+\frac {\text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a-b \cosh ^{-1}(c+d x)\right )}{2 b d}\\ &=-\frac {\text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a-b \cosh ^{-1}(c+d x)}\right )}{b d}+\frac {\text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a-b \cosh ^{-1}(c+d x)}\right )}{b d}\\ &=-\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a-b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a-b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 111, normalized size = 1.18 \begin {gather*} \frac {e^{-\frac {a}{b}} \left (e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}-\cosh ^{-1}(c+d x)} \Gamma \left (\frac {1}{2},\frac {a}{b}-\cosh ^{-1}(c+d x)\right )+\sqrt {-\frac {a}{b}+\cosh ^{-1}(c+d x)} \Gamma \left (\frac {1}{2},-\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )}{2 d \sqrt {a-b \cosh ^{-1}(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {a -b \,\mathrm {arccosh}\left (d x +c \right )}}\, dx\]
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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {1}{\sqrt {a - b \operatorname {acosh}{\left (c + d x \right )}}}\, 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 {1}{\sqrt {a-b\,\mathrm {acosh}\left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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