Optimal. Leaf size=580 \[ -\frac {15 \sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{16384 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {15 \sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {15 \sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{16384 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {15 \sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^{5/2}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {a x-1} \sqrt {a x+1}}+\frac {45 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {15}{256} c x (1-a x) (a x+1) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)} \]
[Out]
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Rubi [A] time = 1.53, antiderivative size = 592, normalized size of antiderivative = 1.02, number of steps used = 40, number of rules used = 15, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5713, 5685, 5683, 5676, 5664, 5759, 5670, 5448, 12, 3308, 2180, 2204, 2205, 5716, 5780} \[ -\frac {15 \sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {Erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{16384 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {15 \sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {15 \sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {Erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{16384 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {15 \sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} c x (1-a x) (a x+1) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {a x-1} \sqrt {a x+1}}+\frac {45 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {15}{256} c x (1-a x) (a x+1) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5448
Rule 5664
Rule 5670
Rule 5676
Rule 5683
Rule 5685
Rule 5713
Rule 5716
Rule 5759
Rule 5780
Rubi steps
\begin {align*} \int \left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^{5/2} \, dx &=-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^{5/2} \, 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} \cosh ^{-1}(a x)^{5/2}+\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{5/2} \, dx}{4 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (5 a c \sqrt {c-a^2 c x^2}\right ) \int x \left (-1+a^2 x^2\right ) \cosh ^{-1}(a x)^{3/2} \, dx}{8 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\left (15 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}{64 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\cosh ^{-1}(a x)^{5/2}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{8 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 a c \sqrt {c-a^2 c x^2}\right ) \int x \cosh ^{-1}(a x)^{3/2} \, dx}{16 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \int \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)} \, dx}{256 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 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}{512 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (45 a^2 c \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}{64 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 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}{512 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (45 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}{128 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 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 )}{512 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\cosh ^{-1}(a x)}} \, dx}{1024 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\cosh ^{-1}(a x)}} \, dx}{256 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 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 )}{512 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 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 )}{1024 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 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 )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 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 )}{4096 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 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 )}{2048 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 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 )}{1024 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 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 )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 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 )}{8192 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 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 )}{8192 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 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 )}{4096 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 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 )}{4096 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 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 )}{2048 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 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 )}{512 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 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 )}{4096 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 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 )}{4096 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 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 )}{2048 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 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 )}{2048 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (45 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 )}{4096 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 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 )}{4096 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (45 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 )}{1024 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 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 )}{1024 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{16384 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{4096 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{16384 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{4096 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (45 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 )}{2048 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 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 )}{2048 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (45 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 )}{512 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (45 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 )}{512 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{16384 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 c \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 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{16384 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 c \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 = 0.51, size = 213, normalized size = 0.37 \[ \frac {c \sqrt {c-a^2 c x^2} \left (420 \sqrt {2 \pi } \sqrt {\cosh ^{-1}(a x)} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )-420 \sqrt {2 \pi } \sqrt {\cosh ^{-1}(a x)} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )-1536 \cosh ^{-1}(a x)^4-4480 \cosh \left (2 \cosh ^{-1}(a x)\right ) \cosh ^{-1}(a x)^2+3584 \cosh ^{-1}(a x)^3 \sinh \left (2 \cosh ^{-1}(a x)\right )+3360 \cosh ^{-1}(a x) \sinh \left (2 \cosh ^{-1}(a x)\right )+7 \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {7}{2},4 \cosh ^{-1}(a x)\right )+7 \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {7}{2},-4 \cosh ^{-1}(a x)\right )\right )}{14336 a \sqrt {\frac {a x-1}{a x+1}} (a x+1) \sqrt {\cosh ^{-1}(a x)}} \]
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 = 0.63, size = 0, normalized size = 0.00 \[ \int \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}}\, dx \]
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 \[ \int {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\mathrm {acosh}\left (a\,x\right )}^{5/2}\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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