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

Optimal. Leaf size=216 \[ \frac{15 x^2 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}}{32 a^2}-\frac{63 x \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}}{64 a^3}+\frac{123 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^4}+\frac{123 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^4}+\frac{1}{4} x^4 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}-\frac{3 x^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}}{8 a} \]

[Out]

(-63*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x)/(64*a^3) + (15*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^2)/(3
2*a^2) - (3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^3)/(8*a) + ((1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^4)
/4 + (123*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(64*a^4) + (123*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/
(a*x))^(1/4)])/(64*a^4)

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Rubi [A]  time = 0.116205, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 10, 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, 212, 206, 203} \[ \frac{15 x^2 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}}{32 a^2}-\frac{63 x \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}}{64 a^3}+\frac{123 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^4}+\frac{123 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^4}+\frac{1}{4} x^4 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}-\frac{3 x^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}}{8 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-63*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x)/(64*a^3) + (15*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^2)/(3
2*a^2) - (3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^3)/(8*a) + ((1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^4)
/4 + (123*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(64*a^4) + (123*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/
(a*x))^(1/4)])/(64*a^4)

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 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{3}{2} \coth ^{-1}(a x)} x^3 \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{3/4}}{x^5 \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{4} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^4-\frac{1}{4} \operatorname{Subst}\left (\int \frac{-\frac{9}{2 a}+\frac{3 x}{a^2}}{x^4 \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3}{8 a}+\frac{1}{4} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^4+\frac{1}{12} \operatorname{Subst}\left (\int \frac{-\frac{45}{4 a^2}+\frac{9 x}{a^3}}{x^3 \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{15 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^2}{32 a^2}-\frac{3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3}{8 a}+\frac{1}{4} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^4-\frac{1}{24} \operatorname{Subst}\left (\int \frac{-\frac{189}{8 a^3}+\frac{45 x}{4 a^4}}{x^2 \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{63 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x}{64 a^3}+\frac{15 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^2}{32 a^2}-\frac{3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3}{8 a}+\frac{1}{4} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^4+\frac{1}{24} \operatorname{Subst}\left (\int -\frac{369}{16 a^4 x \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{63 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x}{64 a^3}+\frac{15 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^2}{32 a^2}-\frac{3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3}{8 a}+\frac{1}{4} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^4-\frac{123 \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 )}{128 a^4}\\ &=-\frac{63 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x}{64 a^3}+\frac{15 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^2}{32 a^2}-\frac{3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3}{8 a}+\frac{1}{4} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^4-\frac{123 \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 )}{32 a^4}\\ &=-\frac{63 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x}{64 a^3}+\frac{15 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^2}{32 a^2}-\frac{3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3}{8 a}+\frac{1}{4} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^4+\frac{123 \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 )}{64 a^4}+\frac{123 \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 )}{64 a^4}\\ &=-\frac{63 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x}{64 a^3}+\frac{15 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^2}{32 a^2}-\frac{3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3}{8 a}+\frac{1}{4} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^4+\frac{123 \tan ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^4}+\frac{123 \tanh ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^4}\\ \end{align*}

Mathematica [A]  time = 5.25472, size = 149, normalized size = 0.69 \[ \frac{-\frac{532 e^{\frac{1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}+\frac{1008 e^{\frac{5}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^2}-\frac{1152 e^{\frac{9}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^3}+\frac{512 e^{\frac{13}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^4}-123 \log \left (1-e^{-\frac{1}{2} \coth ^{-1}(a x)}\right )+123 \log \left (e^{-\frac{1}{2} \coth ^{-1}(a x)}+1\right )-246 \tan ^{-1}\left (e^{-\frac{1}{2} \coth ^{-1}(a x)}\right )}{128 a^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((512*E^((13*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^4 - (1152*E^((9*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[
a*x]))^3 + (1008*E^((5*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^2 - (532*E^(ArcCoth[a*x]/2))/(-1 + E^(2*Arc
Coth[a*x])) - 246*ArcTan[E^(-ArcCoth[a*x]/2)] - 123*Log[1 - E^(-ArcCoth[a*x]/2)] + 123*Log[1 + E^(-ArcCoth[a*x
]/2)])/(128*a^4)

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Maple [F]  time = 0.135, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \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(x^3*((a*x-1)/(a*x+1))^(3/4),x)

[Out]

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

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Maxima [A]  time = 1.51981, size = 302, normalized size = 1.4 \begin{align*} -\frac{1}{128} \, a{\left (\frac{4 \,{\left (133 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{15}{4}} - 147 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{11}{4}} + 183 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{4}} - 41 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}\right )}}{\frac{4 \,{\left (a x - 1\right )} a^{5}}{a x + 1} - \frac{6 \,{\left (a x - 1\right )}^{2} a^{5}}{{\left (a x + 1\right )}^{2}} + \frac{4 \,{\left (a x - 1\right )}^{3} a^{5}}{{\left (a x + 1\right )}^{3}} - \frac{{\left (a x - 1\right )}^{4} a^{5}}{{\left (a x + 1\right )}^{4}} - a^{5}} + \frac{246 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{5}} - \frac{123 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{5}} + \frac{123 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{a^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128*a*(4*(133*((a*x - 1)/(a*x + 1))^(15/4) - 147*((a*x - 1)/(a*x + 1))^(11/4) + 183*((a*x - 1)/(a*x + 1))^(
7/4) - 41*((a*x - 1)/(a*x + 1))^(3/4))/(4*(a*x - 1)*a^5/(a*x + 1) - 6*(a*x - 1)^2*a^5/(a*x + 1)^2 + 4*(a*x - 1
)^3*a^5/(a*x + 1)^3 - (a*x - 1)^4*a^5/(a*x + 1)^4 - a^5) + 246*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^5 - 123*l
og(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^5 + 123*log(((a*x - 1)/(a*x + 1))^(1/4) - 1)/a^5)

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Fricas [A]  time = 1.64639, size = 300, normalized size = 1.39 \begin{align*} \frac{2 \,{\left (16 \, a^{4} x^{4} - 8 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 33 \, a x - 63\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}} - 246 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) + 123 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) - 123 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{128 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*(2*(16*a^4*x^4 - 8*a^3*x^3 + 6*a^2*x^2 - 33*a*x - 63)*((a*x - 1)/(a*x + 1))^(3/4) - 246*arctan(((a*x - 1
)/(a*x + 1))^(1/4)) + 123*log(((a*x - 1)/(a*x + 1))^(1/4) + 1) - 123*log(((a*x - 1)/(a*x + 1))^(1/4) - 1))/a^4

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.22975, size = 274, normalized size = 1.27 \begin{align*} -\frac{1}{128} \, a{\left (\frac{246 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{5}} - \frac{123 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{5}} + \frac{123 \, \log \left ({\left | \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1 \right |}\right )}{a^{5}} - \frac{4 \,{\left (\frac{183 \,{\left (a x - 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{a x + 1} - \frac{147 \,{\left (a x - 1\right )}^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{{\left (a x + 1\right )}^{2}} + \frac{133 \,{\left (a x - 1\right )}^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{{\left (a x + 1\right )}^{3}} - 41 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}\right )}}{a^{5}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128*a*(246*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^5 - 123*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^5 + 123*log
(abs(((a*x - 1)/(a*x + 1))^(1/4) - 1))/a^5 - 4*(183*(a*x - 1)*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1) - 147*(a*x
- 1)^2*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1)^2 + 133*(a*x - 1)^3*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1)^3 - 41
*((a*x - 1)/(a*x + 1))^(3/4))/(a^5*((a*x - 1)/(a*x + 1) - 1)^4))