Optimal. Leaf size=433 \[ \frac {3 \sqrt {\pi } c^2 \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{8 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {15 \sqrt {\frac {\pi }{2}} c^2 \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{16 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {\frac {3 \pi }{2}} c^2 \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {6} \sqrt {\cosh ^{-1}(a x)}\right )}{16 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {3 \sqrt {\pi } c^2 \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {15 \sqrt {\frac {\pi }{2}} c^2 \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{16 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {\sqrt {\frac {3 \pi }{2}} c^2 \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {6} \sqrt {\cosh ^{-1}(a x)}\right )}{16 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\cosh ^{-1}(a x)}} \]
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Rubi [A] time = 0.46, antiderivative size = 444, normalized size of antiderivative = 1.03, number of steps used = 20, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5713, 5697, 5780, 5448, 3308, 2180, 2204, 2205} \[ \frac {3 \sqrt {\pi } c^2 \sqrt {c-a^2 c x^2} \text {Erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{8 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {15 \sqrt {\frac {\pi }{2}} c^2 \sqrt {c-a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{16 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {\frac {3 \pi }{2}} c^2 \sqrt {c-a^2 c x^2} \text {Erf}\left (\sqrt {6} \sqrt {\cosh ^{-1}(a x)}\right )}{16 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {3 \sqrt {\pi } c^2 \sqrt {c-a^2 c x^2} \text {Erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {15 \sqrt {\frac {\pi }{2}} c^2 \sqrt {c-a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{16 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {\sqrt {\frac {3 \pi }{2}} c^2 \sqrt {c-a^2 c x^2} \text {Erfi}\left (\sqrt {6} \sqrt {\cosh ^{-1}(a x)}\right )}{16 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 c^2 (a x+1)^{5/2} (1-a x)^3 \sqrt {c-a^2 c x^2}}{a \sqrt {a x-1} \sqrt {\cosh ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5448
Rule 5697
Rule 5713
Rule 5780
Rubi steps
\begin {align*} \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\cosh ^{-1}(a x)^{3/2}} \, dx &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {(-1+a x)^{5/2} (1+a x)^{5/2}}{\cosh ^{-1}(a x)^{3/2}} \, dx}{\sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 c^2 (1-a x)^3 (1+a x)^{5/2} \sqrt {c-a^2 c x^2}}{a \sqrt {-1+a x} \sqrt {\cosh ^{-1}(a x)}}+\frac {\left (12 a c^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {x \left (-1+a^2 x^2\right )^2}{\sqrt {\cosh ^{-1}(a x)}} \, dx}{\sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 c^2 (1-a x)^3 (1+a x)^{5/2} \sqrt {c-a^2 c x^2}}{a \sqrt {-1+a x} \sqrt {\cosh ^{-1}(a x)}}+\frac {\left (12 c^2 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^5(x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 c^2 (1-a x)^3 (1+a x)^{5/2} \sqrt {c-a^2 c x^2}}{a \sqrt {-1+a x} \sqrt {\cosh ^{-1}(a x)}}+\frac {\left (12 c^2 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {5 \sinh (2 x)}{32 \sqrt {x}}-\frac {\sinh (4 x)}{8 \sqrt {x}}+\frac {\sinh (6 x)}{32 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 c^2 (1-a x)^3 (1+a x)^{5/2} \sqrt {c-a^2 c x^2}}{a \sqrt {-1+a x} \sqrt {\cosh ^{-1}(a x)}}+\frac {\left (3 c^2 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh (6 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 c^2 \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 )}{2 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 c^2 \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 )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 c^2 (1-a x)^3 (1+a x)^{5/2} \sqrt {c-a^2 c x^2}}{a \sqrt {-1+a x} \sqrt {\cosh ^{-1}(a x)}}-\frac {\left (3 c^2 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{-6 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 c^2 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{6 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 c^2 \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 )}{4 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 c^2 \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 )}{4 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 c^2 \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 )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 c^2 \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 )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 c^2 (1-a x)^3 (1+a x)^{5/2} \sqrt {c-a^2 c x^2}}{a \sqrt {-1+a x} \sqrt {\cosh ^{-1}(a x)}}-\frac {\left (3 c^2 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{-6 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 c^2 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{6 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 c^2 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{2 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 c^2 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{2 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 c^2 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 c^2 \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {2 c^2 (1-a x)^3 (1+a x)^{5/2} \sqrt {c-a^2 c x^2}}{a \sqrt {-1+a x} \sqrt {\cosh ^{-1}(a x)}}+\frac {3 c^2 \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 c^2 \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^2 \sqrt {\frac {3 \pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {6} \sqrt {\cosh ^{-1}(a x)}\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 c^2 \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 c^2 \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}}+\frac {c^2 \sqrt {\frac {3 \pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {6} \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 = 1.27, size = 411, normalized size = 0.95 \[ \frac {c^2 \sqrt {c-a^2 c x^2} e^{-6 \cosh ^{-1}(a x)} \left (-64 a^2 x^2 e^{6 \cosh ^{-1}(a x)}-16 \sqrt {2 \pi } e^{6 \cosh ^{-1}(a x)} \sqrt {\cosh ^{-1}(a x)} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )+16 \sqrt {2 \pi } e^{6 \cosh ^{-1}(a x)} \sqrt {\cosh ^{-1}(a x)} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )+6 e^{2 \cosh ^{-1}(a x)}+e^{4 \cosh ^{-1}(a x)}+52 e^{6 \cosh ^{-1}(a x)}+e^{8 \cosh ^{-1}(a x)}+6 e^{10 \cosh ^{-1}(a x)}-e^{12 \cosh ^{-1}(a x)}+\sqrt {6} e^{6 \cosh ^{-1}(a x)} \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-6 \cosh ^{-1}(a x)\right )-12 e^{6 \cosh ^{-1}(a x)} \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 \cosh ^{-1}(a x)\right )-\sqrt {2} e^{6 \cosh ^{-1}(a x)} \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 \cosh ^{-1}(a x)\right )-\sqrt {2} e^{6 \cosh ^{-1}(a x)} \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 \cosh ^{-1}(a x)\right )-12 e^{6 \cosh ^{-1}(a x)} \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 \cosh ^{-1}(a x)\right )+\sqrt {6} e^{6 \cosh ^{-1}(a x)} \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},6 \cosh ^{-1}(a x)\right )-1\right )}{32 a \sqrt {\frac {a x-1}{a x+1}} (a x+1) \sqrt {\cosh ^{-1}(a x)}} \]
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] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{\mathrm {arccosh}\left (a x \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^{5/2}}{{\mathrm {acosh}\left (a\,x\right )}^{3/2}} \,d x \]
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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