Optimal. Leaf size=330 \[ \frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)} \]
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Rubi [A] time = 0.71, antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5713, 5683, 5676, 5664, 5759, 5670, 5448, 12, 3308, 2180, 2204, 2205} \[ \frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5448
Rule 5664
Rule 5670
Rule 5676
Rule 5683
Rule 5713
Rule 5759
Rubi steps
\begin {align*} \int \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{5/2} \, dx}{\sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \int \frac {\cosh ^{-1}(a x)^{5/2}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (5 a \sqrt {c-a^2 c x^2}\right ) \int x \cosh ^{-1}(a x)^{3/2} \, dx}{4 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 a^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2 \sqrt {\cosh ^{-1}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{16 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt {\cosh ^{-1}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{32 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 a \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\cosh ^{-1}(a x)}} \, dx}{64 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 \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 )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 \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 )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 \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 )}{128 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 \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 )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 \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 )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{128 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{128 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 148, normalized size = 0.45 \[ -\frac {\sqrt {-c (a x-1) (a x+1)} \left (-105 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )+105 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )+8 \sqrt {\cosh ^{-1}(a x)} \left (64 \cosh ^{-1}(a x)^3+140 \cosh \left (2 \cosh ^{-1}(a x)\right ) \cosh ^{-1}(a x)-7 \left (16 \cosh ^{-1}(a x)^2+15\right ) \sinh \left (2 \cosh ^{-1}(a x)\right )\right )\right )}{3584 a \sqrt {\frac {a x-1}{a x+1}} (a x+1)} \]
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 = 0.69, size = 0, normalized size = 0.00 \[ \int \sqrt {-a^{2} c \,x^{2}+c}\, \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}}\, dx \]
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 \[ \int \sqrt {-a^{2} c x^{2} + c} \operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\mathrm {acosh}\left (a\,x\right )}^{5/2}\,\sqrt {c-a^2\,c\,x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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