3.86 \(\int e^{-\frac{1}{2} \coth ^{-1}(a x)} x^4 \, dx\)

Optimal. Leaf size=253 \[ \frac{11 x^3 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{48 a^2}-\frac{269 x^2 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{960 a^3}+\frac{611 x \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{1920 a^4}+\frac{31 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}-\frac{31 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}+\frac{1}{5} x^5 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}-\frac{9 x^4 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{40 a} \]

[Out]

(611*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(1920*a^4) - (269*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^2)
/(960*a^3) + (11*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^3)/(48*a^2) - (9*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^
(3/4)*x^4)/(40*a) + ((1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^5)/5 + (31*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a
*x))^(1/4)])/(128*a^5) - (31*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5)

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Rubi [A]  time = 0.139715, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ \frac{11 x^3 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{48 a^2}-\frac{269 x^2 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{960 a^3}+\frac{611 x \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{1920 a^4}+\frac{31 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}-\frac{31 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}+\frac{1}{5} x^5 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}-\frac{9 x^4 \sqrt [4]{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/4}}{40 a} \]

Antiderivative was successfully verified.

[In]

Int[x^4/E^(ArcCoth[a*x]/2),x]

[Out]

(611*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(1920*a^4) - (269*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^2)
/(960*a^3) + (11*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^3)/(48*a^2) - (9*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^
(3/4)*x^4)/(40*a) + ((1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^5)/5 + (31*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a
*x))^(1/4)])/(128*a^5) - (31*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5)

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{1}{2} \coth ^{-1}(a x)} x^4 \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [4]{1-\frac{x}{a}}}{x^6 \sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{5} \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^5-\frac{1}{5} \operatorname{Subst}\left (\int \frac{-\frac{9}{2 a}+\frac{4 x}{a^2}}{x^5 \left (1-\frac{x}{a}\right )^{3/4} \sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{9 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^4}{40 a}+\frac{1}{5} \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^5+\frac{1}{20} \operatorname{Subst}\left (\int \frac{-\frac{55}{4 a^2}+\frac{27 x}{2 a^3}}{x^4 \left (1-\frac{x}{a}\right )^{3/4} \sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{11 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^3}{48 a^2}-\frac{9 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^4}{40 a}+\frac{1}{5} \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^5-\frac{1}{60} \operatorname{Subst}\left (\int \frac{-\frac{269}{8 a^3}+\frac{55 x}{2 a^4}}{x^3 \left (1-\frac{x}{a}\right )^{3/4} \sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{269 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^2}{960 a^3}+\frac{11 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^3}{48 a^2}-\frac{9 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^4}{40 a}+\frac{1}{5} \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^5+\frac{1}{120} \operatorname{Subst}\left (\int \frac{-\frac{611}{16 a^4}+\frac{269 x}{8 a^5}}{x^2 \left (1-\frac{x}{a}\right )^{3/4} \sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{611 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x}{1920 a^4}-\frac{269 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^2}{960 a^3}+\frac{11 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^3}{48 a^2}-\frac{9 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^4}{40 a}+\frac{1}{5} \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^5-\frac{1}{120} \operatorname{Subst}\left (\int -\frac{465}{32 a^5 x \left (1-\frac{x}{a}\right )^{3/4} \sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{611 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x}{1920 a^4}-\frac{269 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^2}{960 a^3}+\frac{11 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^3}{48 a^2}-\frac{9 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^4}{40 a}+\frac{1}{5} \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^5+\frac{31 \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 )}{256 a^5}\\ &=\frac{611 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x}{1920 a^4}-\frac{269 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^2}{960 a^3}+\frac{11 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^3}{48 a^2}-\frac{9 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^4}{40 a}+\frac{1}{5} \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^5+\frac{31 \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 )}{64 a^5}\\ &=\frac{611 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x}{1920 a^4}-\frac{269 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^2}{960 a^3}+\frac{11 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^3}{48 a^2}-\frac{9 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^4}{40 a}+\frac{1}{5} \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^5-\frac{31 \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 )}{128 a^5}+\frac{31 \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 )}{128 a^5}\\ &=\frac{611 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x}{1920 a^4}-\frac{269 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^2}{960 a^3}+\frac{11 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^3}{48 a^2}-\frac{9 \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^4}{40 a}+\frac{1}{5} \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4} x^5+\frac{31 \tan ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}-\frac{31 \tanh ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}\\ \end{align*}

Mathematica [A]  time = 5.28007, size = 173, normalized size = 0.68 \[ \frac{\frac{9620 e^{\frac{3}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}-\frac{34000 e^{\frac{7}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^2}+\frac{64640 e^{\frac{11}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^3}-\frac{62976 e^{\frac{15}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^4}+\frac{24576 e^{\frac{19}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^5}+465 \log \left (1-e^{-\frac{1}{2} \coth ^{-1}(a x)}\right )-465 \log \left (e^{-\frac{1}{2} \coth ^{-1}(a x)}+1\right )-930 \tan ^{-1}\left (e^{-\frac{1}{2} \coth ^{-1}(a x)}\right )}{3840 a^5} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4/E^(ArcCoth[a*x]/2),x]

[Out]

((24576*E^((19*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^5 - (62976*E^((15*ArcCoth[a*x])/2))/(-1 + E^(2*ArcC
oth[a*x]))^4 + (64640*E^((11*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^3 - (34000*E^((7*ArcCoth[a*x])/2))/(-
1 + E^(2*ArcCoth[a*x]))^2 + (9620*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x])) - 930*ArcTan[E^(-ArcCoth[a
*x]/2)] + 465*Log[1 - E^(-ArcCoth[a*x]/2)] - 465*Log[1 + E^(-ArcCoth[a*x]/2)])/(3840*a^5)

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Maple [F]  time = 0.137, size = 0, normalized size = 0. \begin{align*} \int{x}^{4}\sqrt [4]{{\frac{ax-1}{ax+1}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.50989, size = 350, normalized size = 1.38 \begin{align*} -\frac{1}{3840} \, a{\left (\frac{4 \,{\left (2405 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{17}{4}} - 1120 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{13}{4}} + 5090 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{4}} - 696 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}} + 465 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}}{\frac{5 \,{\left (a x - 1\right )} a^{6}}{a x + 1} - \frac{10 \,{\left (a x - 1\right )}^{2} a^{6}}{{\left (a x + 1\right )}^{2}} + \frac{10 \,{\left (a x - 1\right )}^{3} a^{6}}{{\left (a x + 1\right )}^{3}} - \frac{5 \,{\left (a x - 1\right )}^{4} a^{6}}{{\left (a x + 1\right )}^{4}} + \frac{{\left (a x - 1\right )}^{5} a^{6}}{{\left (a x + 1\right )}^{5}} - a^{6}} + \frac{930 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{6}} + \frac{465 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{6}} - \frac{465 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{a^{6}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3840*a*(4*(2405*((a*x - 1)/(a*x + 1))^(17/4) - 1120*((a*x - 1)/(a*x + 1))^(13/4) + 5090*((a*x - 1)/(a*x + 1
))^(9/4) - 696*((a*x - 1)/(a*x + 1))^(5/4) + 465*((a*x - 1)/(a*x + 1))^(1/4))/(5*(a*x - 1)*a^6/(a*x + 1) - 10*
(a*x - 1)^2*a^6/(a*x + 1)^2 + 10*(a*x - 1)^3*a^6/(a*x + 1)^3 - 5*(a*x - 1)^4*a^6/(a*x + 1)^4 + (a*x - 1)^5*a^6
/(a*x + 1)^5 - a^6) + 930*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^6 + 465*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a
^6 - 465*log(((a*x - 1)/(a*x + 1))^(1/4) - 1)/a^6)

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Fricas [A]  time = 1.60675, size = 323, normalized size = 1.28 \begin{align*} \frac{2 \,{\left (384 \, a^{5} x^{5} - 48 \, a^{4} x^{4} + 8 \, a^{3} x^{3} - 98 \, a^{2} x^{2} + 73 \, a x + 611\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 930 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - 465 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) + 465 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{3840 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3840*(2*(384*a^5*x^5 - 48*a^4*x^4 + 8*a^3*x^3 - 98*a^2*x^2 + 73*a*x + 611)*((a*x - 1)/(a*x + 1))^(1/4) - 930
*arctan(((a*x - 1)/(a*x + 1))^(1/4)) - 465*log(((a*x - 1)/(a*x + 1))^(1/4) + 1) + 465*log(((a*x - 1)/(a*x + 1)
)^(1/4) - 1))/a^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.23068, size = 316, normalized size = 1.25 \begin{align*} -\frac{1}{3840} \, a{\left (\frac{930 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{6}} + \frac{465 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{6}} - \frac{465 \, \log \left ({\left | \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1 \right |}\right )}{a^{6}} - \frac{4 \,{\left (\frac{696 \,{\left (a x - 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a x + 1} - \frac{5090 \,{\left (a x - 1\right )}^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{{\left (a x + 1\right )}^{2}} + \frac{1120 \,{\left (a x - 1\right )}^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{{\left (a x + 1\right )}^{3}} - \frac{2405 \,{\left (a x - 1\right )}^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{{\left (a x + 1\right )}^{4}} - 465 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}}{a^{6}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3840*a*(930*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^6 + 465*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^6 - 465*lo
g(abs(((a*x - 1)/(a*x + 1))^(1/4) - 1))/a^6 - 4*(696*(a*x - 1)*((a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1) - 5090*(a
*x - 1)^2*((a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1)^2 + 1120*(a*x - 1)^3*((a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1)^3 -
2405*(a*x - 1)^4*((a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1)^4 - 465*((a*x - 1)/(a*x + 1))^(1/4))/(a^6*((a*x - 1)/(
a*x + 1) - 1)^5))