Integrand size = 23, antiderivative size = 155 \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{b d \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d} \]
[Out]
Time = 0.16 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5996, 12, 5885, 3388, 2211, 2236, 2235} \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=\frac {\sqrt {\frac {\pi }{2}} e e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {\sqrt {\frac {\pi }{2}} e e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}-\frac {2 e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b d \sqrt {a+b \text {arccosh}(c+d x)}} \]
[In]
[Out]
Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5885
Rule 5996
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e x}{(a+b \text {arccosh}(x))^{3/2}} \, dx,x,c+d x\right )}{d} \\ & = \frac {e \text {Subst}\left (\int \frac {x}{(a+b \text {arccosh}(x))^{3/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{b d \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {(2 e) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{b^2 d} \\ & = -\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{b d \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {e \text {Subst}\left (\int \frac {e^{-i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{b^2 d}+\frac {e \text {Subst}\left (\int \frac {e^{i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{b^2 d} \\ & = -\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{b d \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {(2 e) \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c+d x)}\right )}{b^2 d}+\frac {(2 e) \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c+d x)}\right )}{b^2 d} \\ & = -\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{b d \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(314\) vs. \(2(155)=310\).
Time = 4.19 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.03 \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=\frac {e \left (-2 c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-2 c \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )+\sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )-\frac {2 \sqrt {b} e^{-\frac {a}{b}} \left (c e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )-c \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )+e^{a/b} \sinh (2 \text {arccosh}(c+d x))\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}\right )}{2 b^{3/2} d} \]
[In]
[Out]
\[\int \frac {d e x +c e}{\left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{\frac {3}{2}}}d x\]
[In]
[Out]
Exception generated. \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=e \left (\int \frac {c}{a \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )}}\, dx + \int \frac {d x}{a \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )}}\, dx\right ) \]
[In]
[Out]
\[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=\int { \frac {d e x + c e}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=\int { \frac {d e x + c e}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=\int \frac {c\,e+d\,e\,x}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]
[In]
[Out]