Integrand size = 24, antiderivative size = 128 \[ \int \frac {(f x)^m \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {(f x)^{1+m} \text {arccosh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{f (1+m)}+\frac {a (f x)^{2+m} \sqrt {-1+a x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};a^2 x^2\right )}{f^2 (1+m) (2+m) \sqrt {1-a x}} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {5948} \[ \int \frac {(f x)^m \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {a \sqrt {a x-1} (f x)^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;a^2 x^2\right )}{f^2 (m+1) (m+2) \sqrt {1-a x}}+\frac {\text {arccosh}(a x) (f x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},a^2 x^2\right )}{f (m+1)} \]
[In]
[Out]
Rule 5948
Rubi steps \begin{align*} \text {integral}& = \frac {(f x)^{1+m} \text {arccosh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{f (1+m)}+\frac {a (f x)^{2+m} \sqrt {-1+a x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};a^2 x^2\right )}{f^2 (1+m) (2+m) \sqrt {1-a x}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.97 \[ \int \frac {(f x)^m \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {x (f x)^m \left (\text {arccosh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )+\frac {a x \sqrt {-1+a x} \sqrt {1+a x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};a^2 x^2\right )}{(2+m) \sqrt {1-a^2 x^2}}\right )}{1+m} \]
[In]
[Out]
\[\int \frac {\left (f x \right )^{m} \operatorname {arccosh}\left (a x \right )}{\sqrt {-a^{2} x^{2}+1}}d x\]
[In]
[Out]
\[ \int \frac {(f x)^m \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {\left (f x\right )^{m} \operatorname {arcosh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]
[In]
[Out]
\[ \int \frac {(f x)^m \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\left (f x\right )^{m} \operatorname {acosh}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
[In]
[Out]
\[ \int \frac {(f x)^m \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {\left (f x\right )^{m} \operatorname {arcosh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]
[In]
[Out]
\[ \int \frac {(f x)^m \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {\left (f x\right )^{m} \operatorname {arcosh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(f x)^m \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathrm {acosh}\left (a\,x\right )\,{\left (f\,x\right )}^m}{\sqrt {1-a^2\,x^2}} \,d x \]
[In]
[Out]