Optimal. Leaf size=66 \[ \frac{4 c (c-a c x)^{p-1}}{a (1-p)}+\frac{4 (c-a c x)^p}{a p}-\frac{(c-a c x)^{p+1}}{a c (p+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0810893, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6167, 6130, 21, 43} \[ \frac{4 c (c-a c x)^{p-1}}{a (1-p)}+\frac{4 (c-a c x)^p}{a p}-\frac{(c-a c x)^{p+1}}{a c (p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6167
Rule 6130
Rule 21
Rule 43
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} (c-a c x)^p \, dx &=\int e^{4 \tanh ^{-1}(a x)} (c-a c x)^p \, dx\\ &=\int \frac{(1+a x)^2 (c-a c x)^p}{(1-a x)^2} \, dx\\ &=c^2 \int (1+a x)^2 (c-a c x)^{-2+p} \, dx\\ &=c^2 \int \left (4 (c-a c x)^{-2+p}-\frac{4 (c-a c x)^{-1+p}}{c}+\frac{(c-a c x)^p}{c^2}\right ) \, dx\\ &=\frac{4 c (c-a c x)^{-1+p}}{a (1-p)}+\frac{4 (c-a c x)^p}{a p}-\frac{(c-a c x)^{1+p}}{a c (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0763295, size = 50, normalized size = 0.76 \[ \frac{\left (\frac{a x}{p+1}+\frac{4}{(p-1) (a x-1)}+\frac{3 p+4}{p (p+1)}\right ) (c-a c x)^p}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 74, normalized size = 1.1 \begin{align*}{\frac{ \left ( -acx+c \right ) ^{p} \left ({a}^{2}{p}^{2}{x}^{2}-{a}^{2}{x}^{2}p+2\,a{p}^{2}x+2\,apx-4\,ax+{p}^{2}+3\,p+4 \right ) }{ \left ({p}^{2}-1 \right ) ap \left ( ax-1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.07685, size = 207, normalized size = 3.14 \begin{align*} \frac{{\left ({\left (p^{2} - p\right )} a^{2} c^{p} x^{2} + 2 \, a c^{p}{\left (p - 1\right )} x + 2 \, c^{p}\right )}{\left (-a x + 1\right )}^{p} a^{2}}{{\left (p^{3} - p\right )} a^{4} x -{\left (p^{3} - p\right )} a^{3}} + \frac{2 \,{\left (a c^{p}{\left (p - 1\right )} x + c^{p}\right )}{\left (-a x + 1\right )}^{p} a}{{\left (p^{2} - p\right )} a^{3} x -{\left (p^{2} - p\right )} a^{2}} + \frac{{\left (-a x + 1\right )}^{p} c^{p}}{a^{2}{\left (p - 1\right )} x - a{\left (p - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.52886, size = 161, normalized size = 2.44 \begin{align*} -\frac{{\left ({\left (a^{2} p^{2} - a^{2} p\right )} x^{2} + p^{2} + 2 \,{\left (a p^{2} + a p - 2 \, a\right )} x + 3 \, p + 4\right )}{\left (-a c x + c\right )}^{p}}{a p^{3} - a p -{\left (a^{2} p^{3} - a^{2} p\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.96831, size = 512, normalized size = 7.76 \begin{align*} \begin{cases} c^{p} x & \text{for}\: a = 0 \\- \frac{a^{2} x^{2} \log{\left (x - \frac{1}{a} \right )}}{a^{3} c x^{2} - 2 a^{2} c x + a c} + \frac{2 a^{2} x^{2}}{a^{3} c x^{2} - 2 a^{2} c x + a c} + \frac{2 a x \log{\left (x - \frac{1}{a} \right )}}{a^{3} c x^{2} - 2 a^{2} c x + a c} - \frac{\log{\left (x - \frac{1}{a} \right )}}{a^{3} c x^{2} - 2 a^{2} c x + a c} & \text{for}\: p = -1 \\\frac{a^{2} x^{2}}{a^{2} x - a} + \frac{4 a x \log{\left (x - \frac{1}{a} \right )}}{a^{2} x - a} - \frac{4 \log{\left (x - \frac{1}{a} \right )}}{a^{2} x - a} - \frac{5}{a^{2} x - a} & \text{for}\: p = 0 \\- \frac{a c x^{2}}{2} - 3 c x - \frac{4 c \log{\left (x - \frac{1}{a} \right )}}{a} & \text{for}\: p = 1 \\\frac{a^{2} p^{2} x^{2} \left (- a c x + c\right )^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} - \frac{a^{2} p x^{2} \left (- a c x + c\right )^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} + \frac{2 a p^{2} x \left (- a c x + c\right )^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} + \frac{2 a p x \left (- a c x + c\right )^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} - \frac{4 a x \left (- a c x + c\right )^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} + \frac{p^{2} \left (- a c x + c\right )^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} + \frac{3 p \left (- a c x + c\right )^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} + \frac{4 \left (- a c x + c\right )^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{2}{\left (-a c x + c\right )}^{p}}{{\left (a x - 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]