Optimal. Leaf size=66 \[ \frac {4 c (c-a c x)^{-1+p}}{a (1-p)}+\frac {4 (c-a c x)^p}{a p}-\frac {(c-a c x)^{1+p}}{a c (1+p)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6265, 21, 45}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 45
Rule 6265
Rubi steps
\begin {align*} \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 = 0.06, size = 50, normalized size = 0.76 \begin {gather*} \frac {(c-a c x)^p \left (\frac {4+3 p}{p (1+p)}+\frac {a x}{1+p}+\frac {4}{(-1+p) (-1+a x)}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.13, size = 74, normalized size = 1.12
method | result | size |
gosper | \(\frac {\left (a^{2} p^{2} x^{2}-p \,x^{2} a^{2}+2 a \,p^{2} x +2 p a x -4 a x +p^{2}+3 p +4\right ) \left (-c x a +c \right )^{p}}{\left (a x -1\right ) a p \left (p^{2}-1\right )}\) | \(74\) |
risch | \(\frac {\left (a^{2} p^{2} x^{2}-p \,x^{2} a^{2}+2 a \,p^{2} x +2 p a x -4 a x +p^{2}+3 p +4\right ) \left (-c x a +c \right )^{p}}{a p \left (p +1\right ) \left (-1+p \right ) \left (a x -1\right )}\) | \(77\) |
norman | \(\frac {\frac {a^{2} x^{3} {\mathrm e}^{p \ln \left (-c x a +c \right )}}{p +1}+\frac {\left (3 p +5\right ) x \,{\mathrm e}^{p \ln \left (-c x a +c \right )}}{p^{2}-1}+\frac {\left (p^{2}+3 p +4\right ) {\mathrm e}^{p \ln \left (-c x a +c \right )}}{a p \left (p^{2}-1\right )}+\frac {\left (3 p +4\right ) a \,x^{2} {\mathrm e}^{p \ln \left (-c x a +c \right )}}{p \left (p +1\right )}}{a^{2} x^{2}-1}\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 77, normalized size = 1.17 \begin {gather*} \frac {{\left ({\left (p^{2} - p\right )} a^{2} c^{p} x^{2} + 2 \, {\left (p^{2} + p - 2\right )} a c^{p} x + {\left (p^{2} + 3 \, p + 4\right )} c^{p}\right )} {\left (-a x + 1\right )}^{p}}{{\left (p^{3} - p\right )} a^{2} x - {\left (p^{3} - p\right )} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 81, normalized size = 1.23 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 541 vs.
\(2 (48) = 96\).
time = 1.36, size = 541, normalized size = 8.20 \begin {gather*} \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 x \log {\left (x - \frac {1}{a} \right )}}{a^{3} c x^{2} - 2 a^{2} c x + a c} + \frac {4 a x}{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} - \frac {2}{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 {a x}{a^{2} x - a} - \frac {4 \log {\left (x - \frac {1}{a} \right )}}{a^{2} x - a} - \frac {4}{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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.94, size = 57, normalized size = 0.86 \begin {gather*} \frac {4\,{\left (c-a\,c\,x\right )}^p}{a\,\left (a\,x-1\right )\,\left (p-1\right )}+\frac {{\left (c-a\,c\,x\right )}^p\,\left (3\,p+a\,p\,x+4\right )}{a\,p\,\left (p+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________