Optimal. Leaf size=175 \[ \frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{4 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{4 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{a \sqrt {a x-1} \sqrt {a x+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5713, 5701, 3312, 3307, 2180, 2204, 2205} \[ \frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{4 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{4 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{a \sqrt {a x-1} \sqrt {a x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 3312
Rule 5701
Rule 5713
Rubi steps
\begin {align*} \int \frac {\sqrt {c-a^2 c x^2}}{\sqrt {\cosh ^{-1}(a x)}} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int \frac {\sqrt {-1+a x} \sqrt {1+a x}}{\sqrt {\cosh ^{-1}(a x)}} \, dx}{\sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {\sqrt {c-a^2 c x^2} \operatorname {Subst}\left (\int \frac {\sinh ^2(x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {\sqrt {c-a^2 c x^2} \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cosh (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {\sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\sqrt {c-a^2 c x^2} \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {\sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\sqrt {c-a^2 c x^2} \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\sqrt {c-a^2 c x^2} \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {\sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\sqrt {c-a^2 c x^2} \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{2 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\sqrt {c-a^2 c x^2} \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{2 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {\sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}{a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{4 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{4 a \sqrt {-1+a x} \sqrt {1+a x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 114, normalized size = 0.65 \[ -\frac {\sqrt {-c (a x-1) (a x+1)} \left (8 \cosh ^{-1}(a x)+\sqrt {2} \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 \cosh ^{-1}(a x)\right )-\sqrt {2} \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 \cosh ^{-1}(a x)\right )\right )}{8 a \sqrt {\frac {a x-1}{a x+1}} (a x+1) \sqrt {\cosh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c}}{\sqrt {\operatorname {arcosh}\left (a x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.71, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a^{2} c \,x^{2}+c}}{\sqrt {\mathrm {arccosh}\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c}}{\sqrt {\operatorname {arcosh}\left (a x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c-a^2\,c\,x^2}}{\sqrt {\mathrm {acosh}\left (a\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{\sqrt {\operatorname {acosh}{\left (a x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________