Optimal. Leaf size=93 \[ -\frac {c (5-p) \left (c-\frac {c}{a x}\right )^{-1+p}}{a (1-p)}+c \left (c-\frac {c}{a x}\right )^{-1+p} x+\frac {(4-p) \left (c-\frac {c}{a x}\right )^p \, _2F_1\left (1,p;1+p;1-\frac {1}{a x}\right )}{a p} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6268, 25, 528,
382, 91, 80, 67} \begin {gather*} \frac {(4-p) \left (c-\frac {c}{a x}\right )^p \, _2F_1\left (1,p;p+1;1-\frac {1}{a x}\right )}{a p}-\frac {c (5-p) \left (c-\frac {c}{a x}\right )^{p-1}}{a (1-p)}+c x \left (c-\frac {c}{a x}\right )^{p-1} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 25
Rule 67
Rule 80
Rule 91
Rule 382
Rule 528
Rule 6268
Rubi steps
\begin {align*} \int e^{4 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx &=\int \frac {\left (c-\frac {c}{a x}\right )^p (1+a x)^2}{(1-a x)^2} \, dx\\ &=\frac {c^2 \int \frac {\left (c-\frac {c}{a x}\right )^{-2+p} (1+a x)^2}{x^2} \, dx}{a^2}\\ &=\frac {c^2 \int \left (a+\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{-2+p} \, dx}{a^2}\\ &=-\frac {c^2 \text {Subst}\left (\int \frac {(a+x)^2 \left (c-\frac {c x}{a}\right )^{-2+p}}{x^2} \, dx,x,\frac {1}{x}\right )}{a^2}\\ &=c \left (c-\frac {c}{a x}\right )^{-1+p} x-\frac {c \text {Subst}\left (\int \frac {(a c (4-p)+c x) \left (c-\frac {c x}{a}\right )^{-2+p}}{x} \, dx,x,\frac {1}{x}\right )}{a^2}\\ &=-\frac {c (5-p) \left (c-\frac {c}{a x}\right )^{-1+p}}{a (1-p)}+c \left (c-\frac {c}{a x}\right )^{-1+p} x-\frac {(c (4-p)) \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{-1+p}}{x} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {c (5-p) \left (c-\frac {c}{a x}\right )^{-1+p}}{a (1-p)}+c \left (c-\frac {c}{a x}\right )^{-1+p} x+\frac {(4-p) \left (c-\frac {c}{a x}\right )^p \, _2F_1\left (1,p;1+p;1-\frac {1}{a x}\right )}{a p}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 81, normalized size = 0.87 \begin {gather*} \frac {\left (c-\frac {c}{a x}\right )^p \left (a p x (5-a x+p (-1+a x))-\left (4-5 p+p^2\right ) (-1+a x) \, _2F_1\left (1,p;1+p;1-\frac {1}{a x}\right )\right )}{a (-1+p) p (-1+a x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.97, size = 0, normalized size = 0.00 \[\int \frac {\left (a x +1\right )^{4} \left (c -\frac {c}{a x}\right )^{p}}{\left (-a^{2} x^{2}+1\right )^{2}}\, dx\]
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 \frac {\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{p} \left (a x + 1\right )^{2}}{\left (a x - 1\right )^{2}}\, 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 = -0.01 \begin {gather*} \int \frac {{\left (c-\frac {c}{a\,x}\right )}^p\,{\left (a\,x+1\right )}^4}{{\left (a^2\,x^2-1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________