Optimal. Leaf size=351 \[ -\frac {\sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {\sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{16 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {\sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{16 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\cosh ^{-1}(a x)}-\frac {c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{4 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)} \]
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Rubi [A] time = 0.72, antiderivative size = 363, normalized size of antiderivative = 1.03, number of steps used = 25, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5713, 5685, 5683, 5676, 5670, 5448, 12, 3308, 2180, 2204, 2205, 5780} \[ -\frac {\sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {Erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {\sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{16 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {\sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {Erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{16 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{4 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {1}{4} c x (1-a x) (a x+1) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5448
Rule 5670
Rule 5676
Rule 5683
Rule 5685
Rule 5713
Rule 5780
Rubi steps
\begin {align*} \int \left (c-a^2 c x^2\right )^{3/2} \sqrt {\cosh ^{-1}(a x)} \, dx &=-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int (-1+a x)^{3/2} (1+a x)^{3/2} \sqrt {\cosh ^{-1}(a x)} \, dx}{\sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)} \, dx}{4 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x \left (-1+a^2 x^2\right )}{\sqrt {\cosh ^{-1}(a x)}} \, dx}{8 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt {\cosh ^{-1}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^3(x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\cosh ^{-1}(a x)}} \, dx}{16 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}-\frac {c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{4 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {\sinh (2 x)}{4 \sqrt {x}}+\frac {\sinh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}-\frac {c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{4 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh (4 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}-\frac {c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{4 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{128 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{128 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}-\frac {c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{4 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{32 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}-\frac {c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{4 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{32 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{32 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}-\frac {c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{4 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 154, normalized size = 0.44 \[ -\frac {c \sqrt {c-a^2 c x^2} \left (-\sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-4 \cosh ^{-1}(a x)\right )+8 \sqrt {2} \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-2 \cosh ^{-1}(a x)\right )+\sqrt {\cosh ^{-1}(a x)} \left (32 \cosh ^{-1}(a x)^{3/2}+8 \sqrt {2} \Gamma \left (\frac {3}{2},2 \cosh ^{-1}(a x)\right )-\Gamma \left (\frac {3}{2},4 \cosh ^{-1}(a x)\right )\right )\right )}{128 a \sqrt {\frac {a x-1}{a x+1}} (a x+1) \sqrt {\cosh ^{-1}(a x)}} \]
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(-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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maple [F] time = 0.80, size = 0, normalized size = 0.00 \[ \int \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {\mathrm {arccosh}\left (a x \right )}\, dx \]
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 \[ \int {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {\operatorname {arcosh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {\mathrm {acosh}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \]
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 \[ \int \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \sqrt {\operatorname {acosh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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