3.621 \(\int e^{n \tanh ^{-1}(a x)} (c-\frac{c}{a x}) \, dx\)

Optimal. Leaf size=187 \[ -\frac{2 c (a x+1)^{\frac{n-2}{2}} (1-a x)^{1-\frac{n}{2}} \text{Hypergeometric2F1}\left (1,\frac{n-2}{2},\frac{n}{2},\frac{a x+1}{1-a x}\right )}{a (2-n)}+\frac{c 2^{n/2} (1-n) (1-a x)^{2-\frac{n}{2}} \text{Hypergeometric2F1}\left (\frac{2-n}{2},2-\frac{n}{2},3-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a (2-n) (4-n)}+\frac{c (a x+1)^{\frac{n-2}{2}} (1-a x)^{2-\frac{n}{2}}}{a (2-n)} \]

[Out]

(c*(1 - a*x)^(2 - n/2)*(1 + a*x)^((-2 + n)/2))/(a*(2 - n)) - (2*c*(1 - a*x)^(1 - n/2)*(1 + a*x)^((-2 + n)/2)*H
ypergeometric2F1[1, (-2 + n)/2, n/2, (1 + a*x)/(1 - a*x)])/(a*(2 - n)) + (2^(n/2)*c*(1 - n)*(1 - a*x)^(2 - n/2
)*Hypergeometric2F1[(2 - n)/2, 2 - n/2, 3 - n/2, (1 - a*x)/2])/(a*(2 - n)*(4 - n))

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Rubi [A]  time = 0.132947, antiderivative size = 184, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6131, 6129, 105, 69, 131} \[ -\frac{c 2^{\frac{n}{2}+1} (1-a x)^{1-\frac{n}{2}} \, _2F_1\left (1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a (2-n)}-\frac{2 c (a x+1)^{n/2} (1-a x)^{-n/2} \, _2F_1\left (1,-\frac{n}{2};1-\frac{n}{2};\frac{1-a x}{a x+1}\right )}{a n}+\frac{c 2^{\frac{n}{2}+1} (1-a x)^{-n/2} \, _2F_1\left (-\frac{n}{2},-\frac{n}{2};1-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a n} \]

Warning: Unable to verify antiderivative.

[In]

Int[E^(n*ArcTanh[a*x])*(c - c/(a*x)),x]

[Out]

(-2*c*(1 + a*x)^(n/2)*Hypergeometric2F1[1, -n/2, 1 - n/2, (1 - a*x)/(1 + a*x)])/(a*n*(1 - a*x)^(n/2)) - (2^(1
+ n/2)*c*(1 - a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -n/2, 2 - n/2, (1 - a*x)/2])/(a*(2 - n)) + (2^(1 + n/2
)*c*Hypergeometric2F1[-n/2, -n/2, 1 - n/2, (1 - a*x)/2])/(a*n*(1 - a*x)^(n/2))

Rule 6131

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> Dist[d^p, Int[(u*(1 + (c*x)/d)
^p*E^(n*ArcTanh[a*x]))/x^p, x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]

Rule 6129

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^p, Int[(u*(1 + (d*x)/c)
^p*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ
[p] || GtQ[c, 0])

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rule 131

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*c -
a*d)^n*(a + b*x)^(m + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rubi steps

\begin{align*} \int e^{n \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right ) \, dx &=-\frac{c \int \frac{e^{n \tanh ^{-1}(a x)} (1-a x)}{x} \, dx}{a}\\ &=-\frac{c \int \frac{(1-a x)^{1-\frac{n}{2}} (1+a x)^{n/2}}{x} \, dx}{a}\\ &=c \int (1-a x)^{-n/2} (1+a x)^{n/2} \, dx-\frac{c \int \frac{(1-a x)^{-n/2} (1+a x)^{n/2}}{x} \, dx}{a}\\ &=-\frac{2^{1+\frac{n}{2}} c (1-a x)^{1-\frac{n}{2}} \, _2F_1\left (1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a (2-n)}+c \int (1-a x)^{-1-\frac{n}{2}} (1+a x)^{n/2} \, dx-\frac{c \int \frac{(1-a x)^{-1-\frac{n}{2}} (1+a x)^{n/2}}{x} \, dx}{a}\\ &=-\frac{2 c (1-a x)^{-n/2} (1+a x)^{n/2} \, _2F_1\left (1,-\frac{n}{2};1-\frac{n}{2};\frac{1-a x}{1+a x}\right )}{a n}-\frac{2^{1+\frac{n}{2}} c (1-a x)^{1-\frac{n}{2}} \, _2F_1\left (1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a (2-n)}+\frac{2^{1+\frac{n}{2}} c (1-a x)^{-n/2} \, _2F_1\left (-\frac{n}{2},-\frac{n}{2};1-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a n}\\ \end{align*}

Mathematica [A]  time = 0.2376, size = 180, normalized size = 0.96 \[ \frac{c e^{n \tanh ^{-1}(a x)} \left (e^{2 \tanh ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,-e^{2 \tanh ^{-1}(a x)}\right )+e^{2 \tanh ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,e^{2 \tanh ^{-1}(a x)}\right )-\frac{(n+2) \left (\text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,-e^{2 \tanh ^{-1}(a x)}\right )-\text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,e^{2 \tanh ^{-1}(a x)}\right )\right )}{n}+4 e^{2 \tanh ^{-1}(a x)} \text{Hypergeometric2F1}\left (2,\frac{n}{2}+1,\frac{n}{2}+2,-e^{2 \tanh ^{-1}(a x)}\right )\right )}{a (n+2)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(n*ArcTanh[a*x])*(c - c/(a*x)),x]

[Out]

(c*E^(n*ArcTanh[a*x])*(E^(2*ArcTanh[a*x])*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, -E^(2*ArcTanh[a*x])] + E^(2*A
rcTanh[a*x])*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcTanh[a*x])] - ((2 + n)*(Hypergeometric2F1[1, n/2,
1 + n/2, -E^(2*ArcTanh[a*x])] - Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcTanh[a*x])]))/n + 4*E^(2*ArcTanh[a*
x])*Hypergeometric2F1[2, 1 + n/2, 2 + n/2, -E^(2*ArcTanh[a*x])]))/(a*(2 + n))

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( c-{\frac{c}{ax}} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(a*x))*(c-c/a/x),x)

[Out]

int(exp(n*arctanh(a*x))*(c-c/a/x),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c - \frac{c}{a x}\right )} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*(c-c/a/x),x, algorithm="maxima")

[Out]

integrate((c - c/(a*x))*((a*x + 1)/(a*x - 1))^(1/2*n), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a c x - c\right )} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*(c-c/a/x),x, algorithm="fricas")

[Out]

integral((a*c*x - c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{c \left (\int a e^{n \operatorname{atanh}{\left (a x \right )}}\, dx + \int - \frac{e^{n \operatorname{atanh}{\left (a x \right )}}}{x}\, dx\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*atanh(a*x))*(c-c/a/x),x)

[Out]

c*(Integral(a*exp(n*atanh(a*x)), x) + Integral(-exp(n*atanh(a*x))/x, x))/a

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c - \frac{c}{a x}\right )} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*(c-c/a/x),x, algorithm="giac")

[Out]

integrate((c - c/(a*x))*((a*x + 1)/(a*x - 1))^(1/2*n), x)