Optimal. Leaf size=79 \[ -\frac {16 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-4}{2}} \, _2F_1\left (4,2-\frac {n}{2};3-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (4-n)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6191, 6195, 131} \[ -\frac {16 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-4}{2}} \, _2F_1\left (4,2-\frac {n}{2};3-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (4-n)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 131
Rule 6191
Rule 6195
Rubi steps
\begin {align*} \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx &=-\left (\left (a^2 c\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right ) x^2 \, dx\right )\\ &=\left (a^2 c\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{1-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{1+\frac {n}{2}}}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {16 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-4+n)} \, _2F_1\left (4,2-\frac {n}{2};3-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (4-n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.77, size = 111, normalized size = 1.41 \[ -\frac {c e^{n \coth ^{-1}(a x)} \left (2 a^3 x^3+a^2 n x^2+\left (n^2-4\right ) \, _2F_1\left (1,\frac {n}{2};\frac {n}{2}+1;e^{2 \coth ^{-1}(a x)}\right )+(n-2) n e^{2 \coth ^{-1}(a x)} \, _2F_1\left (1,\frac {n}{2}+1;\frac {n}{2}+2;e^{2 \coth ^{-1}(a x)}\right )+a n^2 x-6 a x-n\right )}{6 a} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c x^{2} - c\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -{\left (a^{2} c x^{2} - c\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int {\left (a^{2} c x^{2} - c\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,\left (c-a^2\,c\,x^2\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - c \left (\int a^{2} x^{2} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx + \int \left (- e^{n \operatorname {acoth}{\left (a x \right )}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________