3.81 \(\int e^{\frac{5}{2} \coth ^{-1}(a x)} \, dx\)

Optimal. Leaf size=130 \[ \frac{x \left (\frac{1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{10 \sqrt [4]{\frac{1}{a x}+1}}{a \sqrt [4]{1-\frac{1}{a x}}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a}+\frac{5 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a} \]

[Out]

(-10*(1 + 1/(a*x))^(1/4))/(a*(1 - 1/(a*x))^(1/4)) + ((1 + 1/(a*x))^(5/4)*x)/(1 - 1/(a*x))^(1/4) + (5*ArcTan[(1
+ 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/a + (5*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/a

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Rubi [A]  time = 0.0440145, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6170, 94, 93, 212, 206, 203} \[ \frac{x \left (\frac{1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{10 \sqrt [4]{\frac{1}{a x}+1}}{a \sqrt [4]{1-\frac{1}{a x}}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a}+\frac{5 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[E^((5*ArcCoth[a*x])/2),x]

[Out]

(-10*(1 + 1/(a*x))^(1/4))/(a*(1 - 1/(a*x))^(1/4)) + ((1 + 1/(a*x))^(5/4)*x)/(1 - 1/(a*x))^(1/4) + (5*ArcTan[(1
+ 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/a + (5*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/a

Rule 6170

Int[E^(ArcCoth[(a_.)*(x_)]*(n_)), x_Symbol] :> -Subst[Int[(1 + x/a)^(n/2)/(x^2*(1 - x/a)^(n/2)), x], x, 1/x] /
; FreeQ[{a, n}, x] &&  !IntegerQ[n]

Rule 94

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int e^{\frac{5}{2} \coth ^{-1}(a x)} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{5/4}}{x^2 \left (1-\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\left (1+\frac{1}{a x}\right )^{5/4} x}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{5 \operatorname{Subst}\left (\int \frac{\sqrt [4]{1+\frac{x}{a}}}{x \left (1-\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\frac{10 \sqrt [4]{1+\frac{1}{a x}}}{a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\left (1+\frac{1}{a x}\right )^{5/4} x}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{x \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\frac{10 \sqrt [4]{1+\frac{1}{a x}}}{a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\left (1+\frac{1}{a x}\right )^{5/4} x}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{10 \operatorname{Subst}\left (\int \frac{1}{-1+x^4} \, dx,x,\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a}\\ &=-\frac{10 \sqrt [4]{1+\frac{1}{a x}}}{a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\left (1+\frac{1}{a x}\right )^{5/4} x}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a}\\ &=-\frac{10 \sqrt [4]{1+\frac{1}{a x}}}{a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\left (1+\frac{1}{a x}\right )^{5/4} x}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a}+\frac{5 \tanh ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.133485, size = 67, normalized size = 0.52 \[ \frac{-\frac{2 e^{\frac{1}{2} \coth ^{-1}(a x)} \left (4 e^{2 \coth ^{-1}(a x)}-5\right )}{e^{2 \coth ^{-1}(a x)}-1}+5 \tan ^{-1}\left (e^{\frac{1}{2} \coth ^{-1}(a x)}\right )+5 \tanh ^{-1}\left (e^{\frac{1}{2} \coth ^{-1}(a x)}\right )}{a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^((5*ArcCoth[a*x])/2),x]

[Out]

((-2*E^(ArcCoth[a*x]/2)*(-5 + 4*E^(2*ArcCoth[a*x])))/(-1 + E^(2*ArcCoth[a*x])) + 5*ArcTan[E^(ArcCoth[a*x]/2)]
+ 5*ArcTanh[E^(ArcCoth[a*x]/2)])/a

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Maple [F]  time = 0.322, size = 0, normalized size = 0. \begin{align*} \int \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{5}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(5/4),x)

[Out]

int(1/((a*x-1)/(a*x+1))^(5/4),x)

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Maxima [A]  time = 1.48489, size = 177, normalized size = 1.36 \begin{align*} -\frac{1}{2} \, a{\left (\frac{4 \,{\left (\frac{5 \,{\left (a x - 1\right )}}{a x + 1} - 4\right )}}{a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}} - a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}} + \frac{10 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{2}} - \frac{5 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{2}} + \frac{5 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{a^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(5/4),x, algorithm="maxima")

[Out]

-1/2*a*(4*(5*(a*x - 1)/(a*x + 1) - 4)/(a^2*((a*x - 1)/(a*x + 1))^(5/4) - a^2*((a*x - 1)/(a*x + 1))^(1/4)) + 10
*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^2 - 5*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^2 + 5*log(((a*x - 1)/(a*x
+ 1))^(1/4) - 1)/a^2)

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Fricas [A]  time = 1.60109, size = 304, normalized size = 2.34 \begin{align*} -\frac{10 \,{\left (a x - 1\right )} \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - 5 \,{\left (a x - 1\right )} \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) + 5 \,{\left (a x - 1\right )} \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right ) - 2 \,{\left (a^{2} x^{2} - 8 \, a x - 9\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{2 \,{\left (a^{2} x - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(5/4),x, algorithm="fricas")

[Out]

-1/2*(10*(a*x - 1)*arctan(((a*x - 1)/(a*x + 1))^(1/4)) - 5*(a*x - 1)*log(((a*x - 1)/(a*x + 1))^(1/4) + 1) + 5*
(a*x - 1)*log(((a*x - 1)/(a*x + 1))^(1/4) - 1) - 2*(a^2*x^2 - 8*a*x - 9)*((a*x - 1)/(a*x + 1))^(3/4))/(a^2*x -
a)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))**(5/4),x)

[Out]

Integral(((a*x - 1)/(a*x + 1))**(-5/4), x)

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Giac [A]  time = 1.20936, size = 190, normalized size = 1.46 \begin{align*} -\frac{1}{2} \, a{\left (\frac{10 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{2}} - \frac{5 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{2}} + \frac{5 \, \log \left ({\left | \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1 \right |}\right )}{a^{2}} + \frac{4 \,{\left (\frac{5 \,{\left (a x - 1\right )}}{a x + 1} - 4\right )}}{a^{2}{\left (\frac{{\left (a x - 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a x + 1} - \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(5/4),x, algorithm="giac")

[Out]

-1/2*a*(10*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^2 - 5*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^2 + 5*log(abs(((
a*x - 1)/(a*x + 1))^(1/4) - 1))/a^2 + 4*(5*(a*x - 1)/(a*x + 1) - 4)/(a^2*((a*x - 1)*((a*x - 1)/(a*x + 1))^(1/4
)/(a*x + 1) - ((a*x - 1)/(a*x + 1))^(1/4))))