Optimal. Leaf size=115 \[ -\frac {\sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d}-\frac {\sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d}+\frac {(c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}}{d} \]
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Rubi [A] time = 0.35, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5864, 5654, 5781, 3307, 2180, 2204, 2205} \[ -\frac {\sqrt {\pi } \sqrt {b} e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d}-\frac {\sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d}+\frac {(c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5654
Rule 5781
Rule 5864
Rubi steps
\begin {align*} \int \sqrt {a+b \cosh ^{-1}(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}}{d}-\frac {b \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x} \sqrt {a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{2 d}\\ &=\frac {(c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}}{d}-\frac {b \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 d}\\ &=\frac {(c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}}{d}-\frac {b \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{4 d}-\frac {b \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{4 d}\\ &=\frac {(c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}}{d}-\frac {\operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{2 d}-\frac {\operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{2 d}\\ &=\frac {(c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}}{d}-\frac {\sqrt {b} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d}-\frac {\sqrt {b} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 110, normalized size = 0.96 \[ \frac {e^{-\frac {a}{b}} \sqrt {a+b \cosh ^{-1}(c+d x)} \left (\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{\sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{\sqrt {-\frac {a+b \cosh ^{-1}(c+d x)}{b}}}\right )}{2 d} \]
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 \[ \int \sqrt {b \operatorname {arcosh}\left (d x + c\right ) + a}\,{d x} \]
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 \[ \int \sqrt {a +b \,\mathrm {arccosh}\left (d x +c \right )}\, dx \]
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 \[ \int \sqrt {b \operatorname {arcosh}\left (d x + c\right ) + a}\,{d x} \]
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 \[ \int \sqrt {a+b\,\mathrm {acosh}\left (c+d\,x\right )} \,d x \]
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 \[ \int \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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