3.70 \(\int e^{\frac{3}{2} \coth ^{-1}(a x)} x^2 \, dx\)

Optimal. Leaf size=179 \[ \frac{23 x \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{24 a^2}-\frac{17 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}+\frac{17 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}+\frac{1}{3} x^3 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}+\frac{7 x^2 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{12 a} \]

[Out]

(23*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(24*a^2) + (7*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^2)/(12*
a) + ((1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^3)/3 - (17*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(8
*a^3) + (17*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(8*a^3)

________________________________________________________________________________________

Rubi [A]  time = 0.0936718, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {6171, 99, 151, 12, 93, 298, 203, 206} \[ \frac{23 x \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{24 a^2}-\frac{17 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}+\frac{17 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}+\frac{1}{3} x^3 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}+\frac{7 x^2 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{12 a} \]

Antiderivative was successfully verified.

[In]

Int[E^((3*ArcCoth[a*x])/2)*x^2,x]

[Out]

(23*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(24*a^2) + (7*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^2)/(12*
a) + ((1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^3)/3 - (17*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(8
*a^3) + (17*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(8*a^3)

Rule 6171

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

Rule 99

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[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 298

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

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])

Rubi steps

\begin{align*} \int e^{\frac{3}{2} \coth ^{-1}(a x)} x^2 \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/4}}{x^4 \left (1-\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^3-\frac{1}{3} \operatorname{Subst}\left (\int \frac{\frac{7}{2 a}+\frac{2 x}{a^2}}{x^3 \left (1-\frac{x}{a}\right )^{3/4} \sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{7 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^2}{12 a}+\frac{1}{3} \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^3+\frac{1}{6} \operatorname{Subst}\left (\int \frac{-\frac{23}{4 a^2}-\frac{7 x}{2 a^3}}{x^2 \left (1-\frac{x}{a}\right )^{3/4} \sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{23 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x}{24 a^2}+\frac{7 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^2}{12 a}+\frac{1}{3} \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^3-\frac{1}{6} \operatorname{Subst}\left (\int \frac{51}{8 a^3 x \left (1-\frac{x}{a}\right )^{3/4} \sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{23 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x}{24 a^2}+\frac{7 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^2}{12 a}+\frac{1}{3} \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^3-\frac{17 \operatorname{Subst}\left (\int \frac{1}{x \left (1-\frac{x}{a}\right )^{3/4} \sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{16 a^3}\\ &=\frac{23 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x}{24 a^2}+\frac{7 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^2}{12 a}+\frac{1}{3} \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^3-\frac{17 \operatorname{Subst}\left (\int \frac{x^2}{-1+x^4} \, dx,x,\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{4 a^3}\\ &=\frac{23 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x}{24 a^2}+\frac{7 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^2}{12 a}+\frac{1}{3} \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^3+\frac{17 \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 )}{8 a^3}-\frac{17 \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 )}{8 a^3}\\ &=\frac{23 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x}{24 a^2}+\frac{7 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^2}{12 a}+\frac{1}{3} \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^3-\frac{17 \tan ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}+\frac{17 \tanh ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}\\ \end{align*}

Mathematica [A]  time = 5.15991, size = 125, normalized size = 0.7 \[ \frac{\frac{180 e^{\frac{3}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}+\frac{240 e^{\frac{3}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^2}+\frac{128 e^{\frac{3}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^3}-51 \log \left (1-e^{\frac{1}{2} \coth ^{-1}(a x)}\right )+51 \log \left (e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right )-102 \tan ^{-1}\left (e^{\frac{1}{2} \coth ^{-1}(a x)}\right )}{48 a^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^((3*ArcCoth[a*x])/2)*x^2,x]

[Out]

((128*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^3 + (240*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*
x]))^2 + (180*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x])) - 102*ArcTan[E^(ArcCoth[a*x]/2)] - 51*Log[1 -
E^(ArcCoth[a*x]/2)] + 51*Log[1 + E^(ArcCoth[a*x]/2)])/(48*a^3)

________________________________________________________________________________________

Maple [F]  time = 0.138, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(3/4)*x^2,x)

[Out]

int(1/((a*x-1)/(a*x+1))^(3/4)*x^2,x)

________________________________________________________________________________________

Maxima [A]  time = 1.50305, size = 252, normalized size = 1.41 \begin{align*} -\frac{1}{48} \, a{\left (\frac{4 \,{\left (17 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{4}} - 30 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}} + 45 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}}{\frac{3 \,{\left (a x - 1\right )} a^{4}}{a x + 1} - \frac{3 \,{\left (a x - 1\right )}^{2} a^{4}}{{\left (a x + 1\right )}^{2}} + \frac{{\left (a x - 1\right )}^{3} a^{4}}{{\left (a x + 1\right )}^{3}} - a^{4}} - \frac{102 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{4}} - \frac{51 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{4}} + \frac{51 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{a^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/4)*x^2,x, algorithm="maxima")

[Out]

-1/48*a*(4*(17*((a*x - 1)/(a*x + 1))^(9/4) - 30*((a*x - 1)/(a*x + 1))^(5/4) + 45*((a*x - 1)/(a*x + 1))^(1/4))/
(3*(a*x - 1)*a^4/(a*x + 1) - 3*(a*x - 1)^2*a^4/(a*x + 1)^2 + (a*x - 1)^3*a^4/(a*x + 1)^3 - a^4) - 102*arctan((
(a*x - 1)/(a*x + 1))^(1/4))/a^4 - 51*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^4 + 51*log(((a*x - 1)/(a*x + 1))^(
1/4) - 1)/a^4)

________________________________________________________________________________________

Fricas [A]  time = 1.70283, size = 279, normalized size = 1.56 \begin{align*} \frac{2 \,{\left (8 \, a^{3} x^{3} + 22 \, a^{2} x^{2} + 37 \, a x + 23\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 102 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) + 51 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) - 51 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{48 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/4)*x^2,x, algorithm="fricas")

[Out]

1/48*(2*(8*a^3*x^3 + 22*a^2*x^2 + 37*a*x + 23)*((a*x - 1)/(a*x + 1))^(1/4) + 102*arctan(((a*x - 1)/(a*x + 1))^
(1/4)) + 51*log(((a*x - 1)/(a*x + 1))^(1/4) + 1) - 51*log(((a*x - 1)/(a*x + 1))^(1/4) - 1))/a^3

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))**(3/4)*x**2,x)

[Out]

Integral(x**2/((a*x - 1)/(a*x + 1))**(3/4), x)

________________________________________________________________________________________

Giac [A]  time = 1.20814, size = 232, normalized size = 1.3 \begin{align*} \frac{1}{48} \, a{\left (\frac{102 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{4}} + \frac{51 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{4}} - \frac{51 \, \log \left ({\left | \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1 \right |}\right )}{a^{4}} + \frac{4 \,{\left (\frac{30 \,{\left (a x - 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a x + 1} - \frac{17 \,{\left (a x - 1\right )}^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{{\left (a x + 1\right )}^{2}} - 45 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}}{a^{4}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/4)*x^2,x, algorithm="giac")

[Out]

1/48*a*(102*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^4 + 51*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^4 - 51*log(abs
(((a*x - 1)/(a*x + 1))^(1/4) - 1))/a^4 + 4*(30*(a*x - 1)*((a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1) - 17*(a*x - 1)^
2*((a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1)^2 - 45*((a*x - 1)/(a*x + 1))^(1/4))/(a^4*((a*x - 1)/(a*x + 1) - 1)^3))