Optimal. Leaf size=145 \[ -\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^2}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^2}-\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \cosh ^{-1}(c x)} \]
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Rubi [A] time = 0.65, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5664, 5781, 3312, 3307, 2180, 2204, 2205} \[ -\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^2}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^2}-\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \cosh ^{-1}(c x)} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 3312
Rule 5664
Rule 5781
Rubi steps
\begin {align*} \int x \sqrt {a+b \cosh ^{-1}(c x)} \, dx &=\frac {1}{2} x^2 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {1}{4} (b c) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}} \, dx\\ &=\frac {1}{2} x^2 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {b \operatorname {Subst}\left (\int \frac {\cosh ^2(x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^2}\\ &=\frac {1}{2} x^2 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {b \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {a+b x}}+\frac {\cosh (2 x)}{2 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^2}\\ &=-\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {b \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^2}\\ &=-\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {b \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^2}-\frac {b \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^2}\\ &=-\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {\operatorname {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 c^2}-\frac {\operatorname {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 c^2}\\ &=-\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{4 c^2}+\frac {1}{2} x^2 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {\sqrt {b} 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 )}{16 c^2}-\frac {\sqrt {b} 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 )}{16 c^2}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 136, normalized size = 0.94 \[ \frac {-\sqrt {2 \pi } \sqrt {b} \left (\sinh \left (\frac {2 a}{b}\right )+\cosh \left (\frac {2 a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \sqrt {b} \left (\sinh \left (\frac {2 a}{b}\right )-\cosh \left (\frac {2 a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )+8 \cosh \left (2 \cosh ^{-1}(c x)\right ) \sqrt {a+b \cosh ^{-1}(c x)}}{32 c^2} \]
Antiderivative was successfully verified.
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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 \[ \int \sqrt {b \operatorname {arcosh}\left (c x\right ) + a} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.17, size = 0, normalized size = 0.00 \[ \int x \sqrt {a +b \,\mathrm {arccosh}\left (c x \right )}\, dx \]
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 \[ \int \sqrt {b \operatorname {arcosh}\left (c x\right ) + a} x\,{d x} \]
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 \[ \int x\,\sqrt {a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]
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 \[ \int x \sqrt {a + b \operatorname {acosh}{\left (c x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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