3.65 \(\int e^{2 \text {sech}^{-1}(a x)} x^4 \, dx\)

Optimal. Leaf size=203 \[ \frac {(1-a x) (a x+1)^4}{5 a^5}+\frac {\sqrt {\frac {1-a x}{a x+1}} \left (5-6 \sqrt {\frac {1-a x}{a x+1}}\right ) (a x+1)^4}{10 a^5}-\frac {\left (45 \sqrt {\frac {1-a x}{a x+1}}+4\right ) (a x+1)^3}{30 a^5}+\frac {5 \sqrt {\frac {1-a x}{a x+1}} (a x+1)^2}{4 a^5}+\frac {\left (4-\sqrt {\frac {1-a x}{a x+1}}\right ) (a x+1)}{4 a^5}-\frac {\tan ^{-1}\left (\sqrt {\frac {1-a x}{a x+1}}\right )}{2 a^5} \]

[Out]

1/5*(-a*x+1)*(a*x+1)^4/a^5-1/2*arctan(((-a*x+1)/(a*x+1))^(1/2))/a^5+1/4*(a*x+1)*(4-((-a*x+1)/(a*x+1))^(1/2))/a
^5+5/4*(a*x+1)^2*((-a*x+1)/(a*x+1))^(1/2)/a^5+1/10*(a*x+1)^4*(5-6*((-a*x+1)/(a*x+1))^(1/2))*((-a*x+1)/(a*x+1))
^(1/2)/a^5-1/30*(a*x+1)^3*(4+45*((-a*x+1)/(a*x+1))^(1/2))/a^5

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Rubi [A]  time = 0.70, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6337, 1804, 1811, 1814, 639, 203} \[ \frac {(1-a x) (a x+1)^4}{5 a^5}+\frac {\sqrt {\frac {1-a x}{a x+1}} \left (5-6 \sqrt {\frac {1-a x}{a x+1}}\right ) (a x+1)^4}{10 a^5}-\frac {\left (45 \sqrt {\frac {1-a x}{a x+1}}+4\right ) (a x+1)^3}{30 a^5}+\frac {5 \sqrt {\frac {1-a x}{a x+1}} (a x+1)^2}{4 a^5}+\frac {\left (4-\sqrt {\frac {1-a x}{a x+1}}\right ) (a x+1)}{4 a^5}-\frac {\tan ^{-1}\left (\sqrt {\frac {1-a x}{a x+1}}\right )}{2 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)^2)/(4*a^5) + ((1 - a*x)*(1 + a*x)^4)/(5*a^5) + (Sqrt[(1 - a*x)/(1 + a*x
)]*(1 + a*x)^4*(5 - 6*Sqrt[(1 - a*x)/(1 + a*x)]))/(10*a^5) + ((1 + a*x)*(4 - Sqrt[(1 - a*x)/(1 + a*x)]))/(4*a^
5) - ((1 + a*x)^3*(4 + 45*Sqrt[(1 - a*x)/(1 + a*x)]))/(30*a^5) - ArcTan[Sqrt[(1 - a*x)/(1 + a*x)]]/(2*a^5)

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 639

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a
*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rule 1804

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x
^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x
], x, 1]}, Simp[((c*x)^m*(a + b*x^2)^(p + 1)*(a*g - b*f*x))/(2*a*b*(p + 1)), x] + Dist[c/(2*a*b*(p + 1)), Int[
(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]

Rule 1811

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[x*PolynomialQuotient[Pq, x, x]*(a + b*x^2)^p, x] /; Fre
eQ[{a, b, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] &&  !MatchQ[Pq, x^(m_.)*(u_.) /; IntegerQ[m]]

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rule 6337

Int[E^(ArcSech[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(1/u + Sqrt[(1 - u)/(1 + u)] + (1*Sqrt[(1 - u)/(1 +
u)])/u)^n, x] /; FreeQ[m, x] && IntegerQ[n]

Rubi steps

\begin {align*} \int e^{2 \text {sech}^{-1}(a x)} x^4 \, dx &=\int x^4 \left (\frac {1}{a x}+\sqrt {\frac {1-a x}{1+a x}}+\frac {\sqrt {\frac {1-a x}{1+a x}}}{a x}\right )^2 \, dx\\ &=-\frac {4 \operatorname {Subst}\left (\int \frac {(-1+x)^2 x (1+x)^6}{\left (1+x^2\right )^6} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^5}\\ &=\frac {(1-a x) (1+a x)^4}{5 a^5}+\frac {2 \operatorname {Subst}\left (\int \frac {-42 x-40 x^2+130 x^3+80 x^4-30 x^5-40 x^6-10 x^7}{\left (1+x^2\right )^5} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{5 a^5}\\ &=\frac {(1-a x) (1+a x)^4}{5 a^5}+\frac {2 \operatorname {Subst}\left (\int \frac {x \left (-42-40 x+130 x^2+80 x^3-30 x^4-40 x^5-10 x^6\right )}{\left (1+x^2\right )^5} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{5 a^5}\\ &=\frac {(1-a x) (1+a x)^4}{5 a^5}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^4 \left (5-6 \sqrt {\frac {1-a x}{1+a x}}\right )}{10 a^5}-\frac {\operatorname {Subst}\left (\int \frac {160-48 x-960 x^2+160 x^3+320 x^4+80 x^5}{\left (1+x^2\right )^4} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{20 a^5}\\ &=\frac {(1-a x) (1+a x)^4}{5 a^5}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^4 \left (5-6 \sqrt {\frac {1-a x}{1+a x}}\right )}{10 a^5}-\frac {(1+a x)^3 \left (4+45 \sqrt {\frac {1-a x}{1+a x}}\right )}{30 a^5}+\frac {\operatorname {Subst}\left (\int \frac {480-480 x-1920 x^2-480 x^3}{\left (1+x^2\right )^3} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{120 a^5}\\ &=\frac {5 \sqrt {\frac {1-a x}{1+a x}} (1+a x)^2}{4 a^5}+\frac {(1-a x) (1+a x)^4}{5 a^5}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^4 \left (5-6 \sqrt {\frac {1-a x}{1+a x}}\right )}{10 a^5}-\frac {(1+a x)^3 \left (4+45 \sqrt {\frac {1-a x}{1+a x}}\right )}{30 a^5}-\frac {\operatorname {Subst}\left (\int \frac {480+1920 x}{\left (1+x^2\right )^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{480 a^5}\\ &=\frac {5 \sqrt {\frac {1-a x}{1+a x}} (1+a x)^2}{4 a^5}+\frac {(1-a x) (1+a x)^4}{5 a^5}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^4 \left (5-6 \sqrt {\frac {1-a x}{1+a x}}\right )}{10 a^5}+\frac {(1+a x) \left (4-\sqrt {\frac {1-a x}{1+a x}}\right )}{4 a^5}-\frac {(1+a x)^3 \left (4+45 \sqrt {\frac {1-a x}{1+a x}}\right )}{30 a^5}-\frac {\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{2 a^5}\\ &=\frac {5 \sqrt {\frac {1-a x}{1+a x}} (1+a x)^2}{4 a^5}+\frac {(1-a x) (1+a x)^4}{5 a^5}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^4 \left (5-6 \sqrt {\frac {1-a x}{1+a x}}\right )}{10 a^5}+\frac {(1+a x) \left (4-\sqrt {\frac {1-a x}{1+a x}}\right )}{4 a^5}-\frac {(1+a x)^3 \left (4+45 \sqrt {\frac {1-a x}{1+a x}}\right )}{30 a^5}-\frac {\tan ^{-1}\left (\sqrt {\frac {1-a x}{1+a x}}\right )}{2 a^5}\\ \end {align*}

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Mathematica [C]  time = 0.16, size = 105, normalized size = 0.52 \[ \frac {-12 a^5 x^5+40 a^3 x^3-15 a \sqrt {\frac {1-a x}{a x+1}} \left (-2 a^3 x^4-2 a^2 x^3+a x^2+x\right )+15 i \log \left (2 \sqrt {\frac {1-a x}{a x+1}} (a x+1)-2 i a x\right )}{60 a^5} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(40*a^3*x^3 - 12*a^5*x^5 - 15*a*Sqrt[(1 - a*x)/(1 + a*x)]*(x + a*x^2 - 2*a^2*x^3 - 2*a^3*x^4) + (15*I)*Log[(-2
*I)*a*x + 2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)])/(60*a^5)

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fricas [A]  time = 0.89, size = 103, normalized size = 0.51 \[ -\frac {12 \, a^{5} x^{5} - 40 \, a^{3} x^{3} - 15 \, {\left (2 \, a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 15 \, \arctan \left (\sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}\right )}{60 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(12*a^5*x^5 - 40*a^3*x^3 - 15*(2*a^4*x^4 - a^2*x^2)*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 15*ar
ctan(sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x))))/a^5

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, choosing root of [1,0,%%%{-4,[1,1]%%%},0,%%%{4,[4,4]%%%}] at parameters values [86,-97]Warning, choosing
root of [1,0,%%%{4,[1,1]%%%},0,%%%{4,[4,4]%%%}] at parameters values [-82,7]Warning, integration of abs or sig
n assumes constant sign by intervals (correct if the argument is real):Check [abs(t_nostep)]Warning, choosing
root of [1,0,%%%{-4,[1,1]%%%},0,%%%{4,[4,4]%%%}] at parameters values [-27,26]Warning, choosing root of [1,0,%
%%{4,[1,1]%%%},0,%%%{4,[4,4]%%%}] at parameters values [-89,63]Warning, choosing root of [1,0,%%%{-4,[1,1]%%%}
,0,%%%{4,[4,4]%%%}] at parameters values [-49,-86]Warning, choosing root of [1,0,%%%{4,[1,1]%%%},0,%%%{4,[4,4]
%%%}] at parameters values [-64,-30]Warning, choosing root of [1,0,%%%{-4,[1,1]%%%},0,%%%{4,[4,4]%%%}] at para
meters values [70,22]Warning, choosing root of [1,0,%%%{4,[1,1]%%%},0,%%%{4,[4,4]%%%}] at parameters values [4
2,56]Warning, choosing root of [1,0,%%%{-4,[1,1]%%%},0,%%%{4,[4,4]%%%}] at parameters values [-9,-13]Warning,
choosing root of [1,0,%%%{4,[1,1]%%%},0,%%%{4,[4,4]%%%}] at parameters values [46,24]Warning, choosing root of
[1,0,%%%{-4,[1,1]%%%},0,%%%{4,[4,4]%%%}] at parameters values [49,-6]Unable to divide, perhaps due to roundin
g error%%%{-1,[4,2,0,0]%%%}+%%%{2,[3,1,1,1]%%%}+%%%{2,[2,0,0,0]%%%} / %%%{1,[0,2,0,0]%%%} Error: Bad Argument
Value

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maple [C]  time = 0.06, size = 123, normalized size = 0.61 \[ -\frac {x^{5}}{5}+\frac {2 x^{3}}{3 a^{2}}+\frac {\sqrt {-\frac {a x -1}{a x}}\, x \sqrt {\frac {a x +1}{a x}}\, \left (2 \,\mathrm {csgn}\relax (a ) x^{3} a^{3} \sqrt {-a^{2} x^{2}+1}-x \sqrt {-a^{2} x^{2}+1}\, \mathrm {csgn}\relax (a ) a +\arctan \left (\frac {\mathrm {csgn}\relax (a ) a x}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) \mathrm {csgn}\relax (a )}{4 a^{4} \sqrt {-a^{2} x^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*x^5+2/3*x^3/a^2+1/4/a^4*(-(a*x-1)/a/x)^(1/2)*x*((a*x+1)/a/x)^(1/2)*(2*csgn(a)*x^3*a^3*(-a^2*x^2+1)^(1/2)-
x*(-a^2*x^2+1)^(1/2)*csgn(a)*a+arctan(csgn(a)*a*x/(-a^2*x^2+1)^(1/2)))*csgn(a)/(-a^2*x^2+1)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, x^{3}}{3 \, a^{2}} + \frac {2 \, {\left (-\frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{4 \, a^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} x}{8 \, a^{2}} + \frac {\arcsin \left (a x\right )}{8 \, a^{3}}\right )}}{a^{2}} - \int x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 19.73, size = 808, normalized size = 3.98 \[ -\frac {\frac {1{}\mathrm {i}}{512\,a^5}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,3{}\mathrm {i}}{64\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4\,53{}\mathrm {i}}{256\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6\,87{}\mathrm {i}}{128\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8\,657{}\mathrm {i}}{512\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{10}\,121{}\mathrm {i}}{128\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{10}}}{\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}+\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {6\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}+\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{10}}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{12}}}-\frac {\frac {1{}\mathrm {i}}{16\,a^5}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{8\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4\,15{}\mathrm {i}}{16\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}}{\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+\frac {2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}}-\frac {x^5\,\left (\frac {a^2}{5}-\frac {2}{3\,x^2}\right )}{a^2}-\frac {\ln \left (\frac {a\,\sqrt {\frac {1}{a\,x}+1}-\frac {1}{x}+a\,\sqrt {\frac {1}{a\,x}-1}\,1{}\mathrm {i}}{2\,a-2\,a\,\sqrt {\frac {1}{a\,x}+1}+\frac {1}{x}}\right )\,3{}\mathrm {i}}{4\,a^5}-\frac {\ln \left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )\,1{}\mathrm {i}}{4\,a^5}+\frac {\ln \left (\frac {2\,a\,\sqrt {\frac {a+\frac {1}{x}}{a}}-\frac {2}{x}+a\,\sqrt {-\frac {a-\frac {1}{x}}{a}}\,2{}\mathrm {i}}{2\,a+\frac {1}{x}-2\,a\,\sqrt {\frac {a+\frac {1}{x}}{a}}}\right )\,1{}\mathrm {i}}{a^5}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{128\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4\,1{}\mathrm {i}}{512\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log((a*(-(a - 1/x)/a)^(1/2)*2i - 2/x + 2*a*((a + 1/x)/a)^(1/2))/(2*a + 1/x - 2*a*((a + 1/x)/a)^(1/2)))*1i)/a^
5 - (1i/(512*a^5) - (((1/(a*x) - 1)^(1/2) - 1i)^2*3i)/(64*a^5*((1/(a*x) + 1)^(1/2) - 1)^2) - (((1/(a*x) - 1)^(
1/2) - 1i)^4*53i)/(256*a^5*((1/(a*x) + 1)^(1/2) - 1)^4) + (((1/(a*x) - 1)^(1/2) - 1i)^6*87i)/(128*a^5*((1/(a*x
) + 1)^(1/2) - 1)^6) + (((1/(a*x) - 1)^(1/2) - 1i)^8*657i)/(512*a^5*((1/(a*x) + 1)^(1/2) - 1)^8) + (((1/(a*x)
- 1)^(1/2) - 1i)^10*121i)/(128*a^5*((1/(a*x) + 1)^(1/2) - 1)^10))/(((1/(a*x) - 1)^(1/2) - 1i)^4/((1/(a*x) + 1)
^(1/2) - 1)^4 + (4*((1/(a*x) - 1)^(1/2) - 1i)^6)/((1/(a*x) + 1)^(1/2) - 1)^6 + (6*((1/(a*x) - 1)^(1/2) - 1i)^8
)/((1/(a*x) + 1)^(1/2) - 1)^8 + (4*((1/(a*x) - 1)^(1/2) - 1i)^10)/((1/(a*x) + 1)^(1/2) - 1)^10 + ((1/(a*x) - 1
)^(1/2) - 1i)^12/((1/(a*x) + 1)^(1/2) - 1)^12) - (log(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1))*1i
)/(4*a^5) - (1i/(16*a^5) + (((1/(a*x) - 1)^(1/2) - 1i)^2*1i)/(8*a^5*((1/(a*x) + 1)^(1/2) - 1)^2) - (((1/(a*x)
- 1)^(1/2) - 1i)^4*15i)/(16*a^5*((1/(a*x) + 1)^(1/2) - 1)^4))/(((1/(a*x) - 1)^(1/2) - 1i)^2/((1/(a*x) + 1)^(1/
2) - 1)^2 + (2*((1/(a*x) - 1)^(1/2) - 1i)^4)/((1/(a*x) + 1)^(1/2) - 1)^4 + ((1/(a*x) - 1)^(1/2) - 1i)^6/((1/(a
*x) + 1)^(1/2) - 1)^6) - (log((a*(1/(a*x) - 1)^(1/2)*1i + a*(1/(a*x) + 1)^(1/2) - 1/x)/(2*a - 2*a*(1/(a*x) + 1
)^(1/2) + 1/x))*3i)/(4*a^5) - (((1/(a*x) - 1)^(1/2) - 1i)^2*1i)/(128*a^5*((1/(a*x) + 1)^(1/2) - 1)^2) - (((1/(
a*x) - 1)^(1/2) - 1i)^4*1i)/(512*a^5*((1/(a*x) + 1)^(1/2) - 1)^4) - (x^5*(a^2/5 - 2/(3*x^2)))/a^2

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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