Optimal. Leaf size=672 \[ -\frac {\sqrt {\pi } \sqrt {b} e^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^5}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^2 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^5}-\frac {\sqrt {\frac {\pi }{5}} \sqrt {b} e^2 e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{320 c^5}-\frac {\sqrt {\pi } \sqrt {b} e^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^5}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^2 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^5}-\frac {\sqrt {\frac {\pi }{5}} \sqrt {b} e^2 e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{320 c^5}-\frac {\sqrt {\pi } \sqrt {b} d e e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} d e e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {\sqrt {\pi } \sqrt {b} d e e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} d e e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {\sqrt {\pi } \sqrt {b} d^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {\pi } \sqrt {b} d^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}+d^2 x \sqrt {a+b \cosh ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \cosh ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \cosh ^{-1}(c x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.40, antiderivative size = 672, normalized size of antiderivative = 1.00, number of steps used = 42, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {5707, 5654, 5781, 3307, 2180, 2204, 2205, 5664, 3312} \[ -\frac {\sqrt {\pi } \sqrt {b} d e e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} d e e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {\sqrt {\pi } \sqrt {b} d e e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} d e e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {\sqrt {\pi } \sqrt {b} e^2 e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^5}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^2 e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^5}-\frac {\sqrt {\frac {\pi }{5}} \sqrt {b} e^2 e^{\frac {5 a}{b}} \text {Erf}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{320 c^5}-\frac {\sqrt {\pi } \sqrt {b} e^2 e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^5}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^2 e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^5}-\frac {\sqrt {\frac {\pi }{5}} \sqrt {b} e^2 e^{-\frac {5 a}{b}} \text {Erfi}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{320 c^5}-\frac {\sqrt {\pi } \sqrt {b} d^2 e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {\pi } \sqrt {b} d^2 e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}+d^2 x \sqrt {a+b \cosh ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \cosh ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \cosh ^{-1}(c x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 3312
Rule 5654
Rule 5664
Rule 5707
Rule 5781
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^2 \sqrt {a+b \cosh ^{-1}(c x)} \, dx &=\int \left (d^2 \sqrt {a+b \cosh ^{-1}(c x)}+2 d e x^2 \sqrt {a+b \cosh ^{-1}(c x)}+e^2 x^4 \sqrt {a+b \cosh ^{-1}(c x)}\right ) \, dx\\ &=d^2 \int \sqrt {a+b \cosh ^{-1}(c x)} \, dx+(2 d e) \int x^2 \sqrt {a+b \cosh ^{-1}(c x)} \, dx+e^2 \int x^4 \sqrt {a+b \cosh ^{-1}(c x)} \, dx\\ &=d^2 x \sqrt {a+b \cosh ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \cosh ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {1}{2} \left (b c d^2\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}} \, dx-\frac {1}{3} (b c d e) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}} \, dx-\frac {1}{10} \left (b c e^2\right ) \int \frac {x^5}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}} \, dx\\ &=d^2 x \sqrt {a+b \cosh ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \cosh ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {\left (b d^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c}-\frac {(b d e) \operatorname {Subst}\left (\int \frac {\cosh ^3(x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{3 c^3}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {\cosh ^5(x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{10 c^5}\\ &=d^2 x \sqrt {a+b \cosh ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \cosh ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {\left (b d^2\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c}-\frac {\left (b d^2\right ) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c}-\frac {(b d e) \operatorname {Subst}\left (\int \left (\frac {3 \cosh (x)}{4 \sqrt {a+b x}}+\frac {\cosh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 c^3}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \left (\frac {5 \cosh (x)}{8 \sqrt {a+b x}}+\frac {5 \cosh (3 x)}{16 \sqrt {a+b x}}+\frac {\cosh (5 x)}{16 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{10 c^5}\\ &=d^2 x \sqrt {a+b \cosh ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \cosh ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {d^2 \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{2 c}-\frac {d^2 \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{2 c}-\frac {(b d e) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{12 c^3}-\frac {(b d e) \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^3}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (5 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{160 c^5}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^5}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^5}\\ &=d^2 x \sqrt {a+b \cosh ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \cosh ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {\sqrt {b} d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {(b d e) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{24 c^3}-\frac {(b d e) \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{24 c^3}-\frac {(b d e) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^3}-\frac {(b d e) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^3}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{320 c^5}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {e^{5 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{320 c^5}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^5}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^5}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^5}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^5}\\ &=d^2 x \sqrt {a+b \cosh ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \cosh ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {\sqrt {b} d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {(d e) \operatorname {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{12 c^3}-\frac {(d e) \operatorname {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{12 c^3}-\frac {(d e) \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{4 c^3}-\frac {(d e) \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{4 c^3}-\frac {e^2 \operatorname {Subst}\left (\int e^{\frac {5 a}{b}-\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{160 c^5}-\frac {e^2 \operatorname {Subst}\left (\int e^{-\frac {5 a}{b}+\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{160 c^5}-\frac {e^2 \operatorname {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{32 c^5}-\frac {e^2 \operatorname {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{32 c^5}-\frac {e^2 \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{16 c^5}-\frac {e^2 \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{16 c^5}\\ &=d^2 x \sqrt {a+b \cosh ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \cosh ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {\sqrt {b} d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} d e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}-\frac {\sqrt {b} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^5}-\frac {\sqrt {b} d e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {\sqrt {b} e^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^5}-\frac {\sqrt {b} e^2 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{320 c^5}-\frac {\sqrt {b} d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} d e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}-\frac {\sqrt {b} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^5}-\frac {\sqrt {b} d e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {\sqrt {b} e^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^5}-\frac {\sqrt {b} e^2 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{320 c^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 6.69, size = 536, normalized size = 0.80 \[ \frac {b e^{-\frac {5 a}{b}} \left (450 e^{\frac {6 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\cosh ^{-1}(c x)\right ) \left (-b e \left (4 c^2 d+e\right ) \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \sqrt {-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}}+8 a c^4 d^2 \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)}+8 b c^4 d^2 \cosh ^{-1}(c x) \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)}\right )-e^{\frac {2 a}{b}} \left (25 \sqrt {3} b e \left (8 c^2 d+3 e\right ) \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \sqrt {-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}} \Gamma \left (\frac {3}{2},-\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+b e e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \sqrt {-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}} \left (25 \sqrt {3} \left (8 c^2 d+3 e\right ) \Gamma \left (\frac {3}{2},\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+9 \sqrt {5} e e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )+450 e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},-\frac {a+b \cosh ^{-1}(c x)}{b}\right ) \left (b e \left (4 c^2 d+e\right ) \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \sqrt {-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}}+8 a c^4 d^2 \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}}+8 b c^4 d^2 \cosh ^{-1}(c x) \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}}\right )\right )-9 \sqrt {5} b e^2 \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \sqrt {-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}} \Gamma \left (\frac {3}{2},-\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )}{7200 c^5 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (e \,x^{2}+d \right )^{2} \sqrt {a +b \,\mathrm {arccosh}\left (c x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x^{2} + d\right )}^{2} \sqrt {b \operatorname {arcosh}\left (c x\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {a+b\,\mathrm {acosh}\left (c\,x\right )}\,{\left (e\,x^2+d\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \left (d + e x^{2}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________