Integrand size = 22, antiderivative size = 57 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\left (c-\frac {c}{a x}\right )^p x+\frac {(2-p) \left (c-\frac {c}{a x}\right )^p \operatorname {Hypergeometric2F1}\left (1,p,1+p,1-\frac {1}{a x}\right )}{a p} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6302, 6268, 25, 528, 382, 79, 67} \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\frac {(2-p) \left (c-\frac {c}{a x}\right )^p \operatorname {Hypergeometric2F1}\left (1,p,p+1,1-\frac {1}{a x}\right )}{a p}+x \left (c-\frac {c}{a x}\right )^p \]
[In]
[Out]
Rule 25
Rule 67
Rule 79
Rule 382
Rule 528
Rule 6268
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx \\ & = -\int \frac {\left (c-\frac {c}{a x}\right )^p (1+a x)}{1-a x} \, dx \\ & = \frac {c \int \frac {\left (c-\frac {c}{a x}\right )^{-1+p} (1+a x)}{x} \, dx}{a} \\ & = \frac {c \int \left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{-1+p} \, dx}{a} \\ & = -\frac {c \text {Subst}\left (\int \frac {(a+x) \left (c-\frac {c x}{a}\right )^{-1+p}}{x^2} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \left (c-\frac {c}{a x}\right )^p x-\frac {(c (2-p)) \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{-1+p}}{x} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \left (c-\frac {c}{a x}\right )^p x+\frac {(2-p) \left (c-\frac {c}{a x}\right )^p \operatorname {Hypergeometric2F1}\left (1,p,1+p,1-\frac {1}{a x}\right )}{a p} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\frac {\left (c-\frac {c}{a x}\right )^p \left (a p x-(-2+p) \operatorname {Hypergeometric2F1}\left (1,p,1+p,1-\frac {1}{a x}\right )\right )}{a p} \]
[In]
[Out]
\[\int \frac {\left (a x +1\right ) \left (c -\frac {c}{a x}\right )^{p}}{a x -1}d x\]
[In]
[Out]
\[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int { \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{p}}{a x - 1} \,d x } \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 4.90 (sec) , antiderivative size = 269, normalized size of antiderivative = 4.72 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=a \left (\begin {cases} \frac {0^{p} x}{a} + \frac {0^{p} \log {\left (a x - 1 \right )}}{a^{2}} - \frac {a^{- p} c^{p} p x^{2 - p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (2 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 2 - p \\ 3 - p \end {matrix}\middle | {a x} \right )}}{\Gamma \left (3 - p\right ) \Gamma \left (p + 1\right )} & \text {for}\: \left |{a x}\right | > 1 \\\frac {0^{p} x}{a} + \frac {0^{p} \log {\left (- a x + 1 \right )}}{a^{2}} - \frac {a^{- p} c^{p} p x^{2 - p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (2 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 2 - p \\ 3 - p \end {matrix}\middle | {a x} \right )}}{\Gamma \left (3 - p\right ) \Gamma \left (p + 1\right )} & \text {otherwise} \end {cases}\right ) + \begin {cases} \frac {0^{p} \log {\left (a x - 1 \right )}}{a} - \frac {a^{- p} c^{p} p x^{1 - p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 1 - p \\ 2 - p \end {matrix}\middle | {a x} \right )}}{\Gamma \left (2 - p\right ) \Gamma \left (p + 1\right )} & \text {for}\: \left |{a x}\right | > 1 \\\frac {0^{p} \log {\left (- a x + 1 \right )}}{a} - \frac {a^{- p} c^{p} p x^{1 - p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 1 - p \\ 2 - p \end {matrix}\middle | {a x} \right )}}{\Gamma \left (2 - p\right ) \Gamma \left (p + 1\right )} & \text {otherwise} \end {cases} \]
[In]
[Out]
\[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int { \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{p}}{a x - 1} \,d x } \]
[In]
[Out]
\[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int { \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{p}}{a x - 1} \,d x } \]
[In]
[Out]
Timed out. \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^p\,\left (a\,x+1\right )}{a\,x-1} \,d x \]
[In]
[Out]