Integrand size = 24, antiderivative size = 580 \[ \int \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2} \, dx=\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{7/2}}{28 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{16384 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{16384 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}} \]
[Out]
Time = 1.00 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.00, number of steps used = 41, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {5897, 5895, 5893, 5884, 5939, 5887, 5556, 12, 3389, 2211, 2235, 2236, 5912, 5914, 5898, 5896, 5952} \[ \int \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2} \, dx=-\frac {15 \sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{16384 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {15 \sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {15 \sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{16384 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {15 \sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {3 c \text {arccosh}(a x)^{7/2} \sqrt {c-a^2 c x^2}}{28 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{4} x \text {arccosh}(a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{8} c x \text {arccosh}(a x)^{5/2} \sqrt {c-a^2 c x^2}+\frac {5 c \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^{3/2} \sqrt {c-a^2 c x^2}}{32 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {15 a c x^2 \text {arccosh}(a x)^{3/2} \sqrt {c-a^2 c x^2}}{32 \sqrt {a x-1} \sqrt {a x+1}}+\frac {45 c \text {arccosh}(a x)^{3/2} \sqrt {c-a^2 c x^2}}{256 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {225}{512} c x \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}+\frac {15}{256} c x (1-a x) (a x+1) \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2} \]
[In]
[Out]
Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5884
Rule 5887
Rule 5893
Rule 5895
Rule 5896
Rule 5897
Rule 5898
Rule 5912
Rule 5914
Rule 5939
Rule 5952
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2}+\frac {1}{4} (3 c) \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2} \, dx+\frac {\left (5 a c \sqrt {c-a^2 c x^2}\right ) \int x (-1+a x) (1+a x) \text {arccosh}(a x)^{3/2} \, dx}{8 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {3}{8} c x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\text {arccosh}(a x)^{5/2}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{8 \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 ) \text {arccosh}(a x)^{3/2} \, 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 \text {arccosh}(a x)^{3/2} \, dx}{16 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = -\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \text {arccosh}(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 ) \int (-1+a x)^{3/2} (1+a x)^{3/2} \sqrt {\text {arccosh}(a x)} \, dx}{64 \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 {\text {arccosh}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{64 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {45}{128} c x \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \text {arccosh}(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 {\text {arccosh}(a x)} \, dx}{256 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{128 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x (-1+a x) (1+a x)}{\sqrt {\text {arccosh}(a x)}} \, dx}{512 \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 {\text {arccosh}(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 {\text {arccosh}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {15 c \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{64 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \text {arccosh}(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 {\text {arccosh}(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 ) \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{256 a \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 {\text {arccosh}(a x)}} \, dx}{512 \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 {\text {arccosh}(a x)}} \, dx}{1024 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \text {arccosh}(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 ) \text {Subst}\left (\int \frac {\cosh (x) \sinh ^3(x)}{\sqrt {x}} \, dx,x,\text {arccosh}(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 ) \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\text {arccosh}(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 ) \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\text {arccosh}(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 {\text {arccosh}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \text {arccosh}(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 ) \text {Subst}\left (\int \left (-\frac {\sinh (2 x)}{4 \sqrt {x}}+\frac {\sinh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\text {arccosh}(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 ) \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\text {arccosh}(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 ) \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(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 {\text {arccosh}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \text {arccosh}(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 ) \text {Subst}\left (\int \frac {\sinh (4 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(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 ) \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(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 ) \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(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 ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(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 ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(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 {\text {arccosh}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \text {arccosh}(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 ) \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(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 ) \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(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 ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(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 ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(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 ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(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 ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(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 ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arccosh}(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 ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arccosh}(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 {\text {arccosh}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{7/2}}{28 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {45 c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{1024 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {45 c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{1024 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\text {arccosh}(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 ) \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\text {arccosh}(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 ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arccosh}(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 ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arccosh}(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 ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arccosh}(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 ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{2048 a \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {15}{256} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2}-\frac {3 c \sqrt {c-a^2 c x^2} \text {arccosh}(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 {\text {arccosh}(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 {\text {arccosh}(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 {\text {arccosh}(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 {\text {arccosh}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.37 \[ \int \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2} \, dx=\frac {c \sqrt {c-a^2 c x^2} \left (-1536 \text {arccosh}(a x)^4-4480 \text {arccosh}(a x)^2 \cosh (2 \text {arccosh}(a x))+420 \sqrt {2 \pi } \sqrt {\text {arccosh}(a x)} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-420 \sqrt {2 \pi } \sqrt {\text {arccosh}(a x)} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+7 \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {7}{2},-4 \text {arccosh}(a x)\right )+7 \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {7}{2},4 \text {arccosh}(a x)\right )+3360 \text {arccosh}(a x) \sinh (2 \text {arccosh}(a x))+3584 \text {arccosh}(a x)^3 \sinh (2 \text {arccosh}(a x))\right )}{14336 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \sqrt {\text {arccosh}(a x)}} \]
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\[\int \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \operatorname {arccosh}\left (a x \right )^{\frac {5}{2}}d x\]
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Exception generated. \[ \int \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2} \, dx=\text {Timed out} \]
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\[ \int \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2} \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \operatorname {arcosh}\left (a x\right )^{\frac {5}{2}} \,d x } \]
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Exception generated. \[ \int \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^{5/2} \, dx=\int {\mathrm {acosh}\left (a\,x\right )}^{5/2}\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \]
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