Integrand size = 20, antiderivative size = 287 \[ \int \frac {d+e x^2}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=-\frac {d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}-\frac {e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}-\frac {e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}+\frac {d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}+\frac {e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}+\frac {e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3} \]
[Out]
Time = 0.36 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5909, 5881, 3389, 2211, 2236, 2235, 5887, 5556} \[ \int \frac {d+e x^2}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=-\frac {\sqrt {\pi } e e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}-\frac {\sqrt {\frac {\pi }{3}} e e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}+\frac {\sqrt {\pi } e e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}+\frac {\sqrt {\frac {\pi }{3}} e e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}-\frac {\sqrt {\pi } d e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}+\frac {\sqrt {\pi } d e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c} \]
[In]
[Out]
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5881
Rule 5887
Rule 5909
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d}{\sqrt {a+b \text {arccosh}(c x)}}+\frac {e x^2}{\sqrt {a+b \text {arccosh}(c x)}}\right ) \, dx \\ & = d \int \frac {1}{\sqrt {a+b \text {arccosh}(c x)}} \, dx+e \int \frac {x^2}{\sqrt {a+b \text {arccosh}(c x)}} \, dx \\ & = -\frac {d \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{b c}-\frac {e \text {Subst}\left (\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{b c^3} \\ & = -\frac {d \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b c}+\frac {d \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b c}-\frac {e \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 \sqrt {x}}+\frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b c^3} \\ & = -\frac {d \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{b c}+\frac {d \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{b c}-\frac {e \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b c^3}-\frac {e \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b c^3} \\ & = -\frac {d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}+\frac {d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}-\frac {e \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b c^3}+\frac {e \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b c^3}-\frac {e \text {Subst}\left (\int \frac {e^{-i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b c^3}+\frac {e \text {Subst}\left (\int \frac {e^{i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b c^3} \\ & = -\frac {d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}+\frac {d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}-\frac {e \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{4 b c^3}-\frac {e \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{4 b c^3}+\frac {e \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{4 b c^3}+\frac {e \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{4 b c^3} \\ & = -\frac {d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}-\frac {e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}-\frac {e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}+\frac {d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}+\frac {e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}+\frac {e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.74 \[ \int \frac {d+e x^2}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=\frac {e^{-\frac {3 a}{b}} \left (3 \left (4 c^2 d+e\right ) e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c x)\right )+\sqrt {3} e \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+3 \left (4 c^2 d+e\right ) e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )+\sqrt {3} e e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )}{24 c^3 \sqrt {a+b \text {arccosh}(c x)}} \]
[In]
[Out]
\[\int \frac {e \,x^{2}+d}{\sqrt {a +b \,\operatorname {arccosh}\left (c x \right )}}d x\]
[In]
[Out]
Exception generated. \[ \int \frac {d+e x^2}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \frac {d+e x^2}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=\int \frac {d + e x^{2}}{\sqrt {a + b \operatorname {acosh}{\left (c x \right )}}}\, dx \]
[In]
[Out]
\[ \int \frac {d+e x^2}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=\int { \frac {e x^{2} + d}{\sqrt {b \operatorname {arcosh}\left (c x\right ) + a}} \,d x } \]
[In]
[Out]
\[ \int \frac {d+e x^2}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=\int { \frac {e x^{2} + d}{\sqrt {b \operatorname {arcosh}\left (c x\right ) + a}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {d+e x^2}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=\int \frac {e\,x^2+d}{\sqrt {a+b\,\mathrm {acosh}\left (c\,x\right )}} \,d x \]
[In]
[Out]