Optimal. Leaf size=44 \[ -\frac {(c-a c x)^{2+p} \, _2F_1\left (1,2+p;3+p;\frac {1}{2} (1-a x)\right )}{2 a c^2 (2+p)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6265, 21, 70}
\begin {gather*} -\frac {(c-a c x)^{p+2} \, _2F_1\left (1,p+2;p+3;\frac {1}{2} (1-a x)\right )}{2 a c^2 (p+2)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 70
Rule 6265
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} (c-a c x)^p \, dx &=\int \frac {(1-a x) (c-a c x)^p}{1+a x} \, dx\\ &=\frac {\int \frac {(c-a c x)^{1+p}}{1+a x} \, dx}{c}\\ &=-\frac {(c-a c x)^{2+p} \, _2F_1\left (1,2+p;3+p;\frac {1}{2} (1-a x)\right )}{2 a c^2 (2+p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 43, normalized size = 0.98 \begin {gather*} \frac {(-1+a x) (c-a c x)^p \left (-1+\, _2F_1\left (1,1+p;2+p;\frac {1}{2} (1-a x)\right )\right )}{a (1+p)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.74, size = 0, normalized size = 0.00 \[\int \frac {\left (-c x a +c \right )^{p} \left (-a^{2} x^{2}+1\right )}{\left (a x +1\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {\left (- a c x + c\right )^{p}}{a x + 1}\right )\, dx - \int \frac {a x \left (- a c x + c\right )^{p}}{a x + 1}\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int \frac {\left (a^2\,x^2-1\right )\,{\left (c-a\,c\,x\right )}^p}{{\left (a\,x+1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________