3.16.52 \(\int \frac {\sqrt [3]{1+x^4} (3+x^4)}{x^{17}} \, dx\)

Optimal. Leaf size=107 \[ -\frac {5}{324} \log \left (\sqrt [3]{x^4+1}-1\right )+\frac {5}{648} \log \left (\left (x^4+1\right )^{2/3}+\sqrt [3]{x^4+1}+1\right )+\frac {5 \tan ^{-1}\left (\frac {2 \sqrt [3]{x^4+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{108 \sqrt {3}}+\frac {\sqrt [3]{x^4+1} \left (-10 x^{12}+6 x^8-45 x^4-81\right )}{432 x^{16}} \]

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Rubi [A]  time = 0.07, antiderivative size = 118, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {446, 78, 47, 51, 57, 618, 204, 31} \begin {gather*} -\frac {5 \sqrt [3]{x^4+1}}{216 x^4}-\frac {5}{216} \log \left (1-\sqrt [3]{x^4+1}\right )+\frac {5 \tan ^{-1}\left (\frac {2 \sqrt [3]{x^4+1}+1}{\sqrt {3}}\right )}{108 \sqrt {3}}-\frac {3 \left (x^4+1\right )^{4/3}}{16 x^{16}}+\frac {\sqrt [3]{x^4+1}}{12 x^{12}}+\frac {\sqrt [3]{x^4+1}}{72 x^8}+\frac {5 \log (x)}{162} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 + x^4)^(1/3)*(3 + x^4))/x^17,x]

[Out]

(1 + x^4)^(1/3)/(12*x^12) + (1 + x^4)^(1/3)/(72*x^8) - (5*(1 + x^4)^(1/3))/(216*x^4) - (3*(1 + x^4)^(4/3))/(16
*x^16) + (5*ArcTan[(1 + 2*(1 + x^4)^(1/3))/Sqrt[3]])/(108*Sqrt[3]) + (5*Log[x])/162 - (5*Log[1 - (1 + x^4)^(1/
3)])/216

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x]
&& PosQ[(b*c - a*d)/b]

Rule 78

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

Rule 204

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

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{17}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x} (3+x)}{x^5} \, dx,x,x^4\right )\\ &=-\frac {3 \left (1+x^4\right )^{4/3}}{16 x^{16}}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x}}{x^4} \, dx,x,x^4\right )\\ &=\frac {\sqrt [3]{1+x^4}}{12 x^{12}}-\frac {3 \left (1+x^4\right )^{4/3}}{16 x^{16}}-\frac {1}{36} \operatorname {Subst}\left (\int \frac {1}{x^3 (1+x)^{2/3}} \, dx,x,x^4\right )\\ &=\frac {\sqrt [3]{1+x^4}}{12 x^{12}}+\frac {\sqrt [3]{1+x^4}}{72 x^8}-\frac {3 \left (1+x^4\right )^{4/3}}{16 x^{16}}+\frac {5}{216} \operatorname {Subst}\left (\int \frac {1}{x^2 (1+x)^{2/3}} \, dx,x,x^4\right )\\ &=\frac {\sqrt [3]{1+x^4}}{12 x^{12}}+\frac {\sqrt [3]{1+x^4}}{72 x^8}-\frac {5 \sqrt [3]{1+x^4}}{216 x^4}-\frac {3 \left (1+x^4\right )^{4/3}}{16 x^{16}}-\frac {5}{324} \operatorname {Subst}\left (\int \frac {1}{x (1+x)^{2/3}} \, dx,x,x^4\right )\\ &=\frac {\sqrt [3]{1+x^4}}{12 x^{12}}+\frac {\sqrt [3]{1+x^4}}{72 x^8}-\frac {5 \sqrt [3]{1+x^4}}{216 x^4}-\frac {3 \left (1+x^4\right )^{4/3}}{16 x^{16}}+\frac {5 \log (x)}{162}+\frac {5}{216} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^4}\right )+\frac {5}{216} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^4}\right )\\ &=\frac {\sqrt [3]{1+x^4}}{12 x^{12}}+\frac {\sqrt [3]{1+x^4}}{72 x^8}-\frac {5 \sqrt [3]{1+x^4}}{216 x^4}-\frac {3 \left (1+x^4\right )^{4/3}}{16 x^{16}}+\frac {5 \log (x)}{162}-\frac {5}{216} \log \left (1-\sqrt [3]{1+x^4}\right )-\frac {5}{108} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^4}\right )\\ &=\frac {\sqrt [3]{1+x^4}}{12 x^{12}}+\frac {\sqrt [3]{1+x^4}}{72 x^8}-\frac {5 \sqrt [3]{1+x^4}}{216 x^4}-\frac {3 \left (1+x^4\right )^{4/3}}{16 x^{16}}+\frac {5 \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )}{108 \sqrt {3}}+\frac {5 \log (x)}{162}-\frac {5}{216} \log \left (1-\sqrt [3]{1+x^4}\right )\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 35, normalized size = 0.33 \begin {gather*} -\frac {3 \left (x^4+1\right )^{4/3} \left (x^{16} \, _2F_1\left (\frac {4}{3},4;\frac {7}{3};x^4+1\right )+1\right )}{16 x^{16}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x^4)^(1/3)*(3 + x^4))/x^17,x]

[Out]

(-3*(1 + x^4)^(4/3)*(1 + x^16*Hypergeometric2F1[4/3, 4, 7/3, 1 + x^4]))/(16*x^16)

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IntegrateAlgebraic [A]  time = 0.23, size = 107, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{1+x^4} \left (-81-45 x^4+6 x^8-10 x^{12}\right )}{432 x^{16}}+\frac {5 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )}{108 \sqrt {3}}-\frac {5}{324} \log \left (-1+\sqrt [3]{1+x^4}\right )+\frac {5}{648} \log \left (1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^4)^(1/3)*(3 + x^4))/x^17,x]

[Out]

((1 + x^4)^(1/3)*(-81 - 45*x^4 + 6*x^8 - 10*x^12))/(432*x^16) + (5*ArcTan[1/Sqrt[3] + (2*(1 + x^4)^(1/3))/Sqrt
[3]])/(108*Sqrt[3]) - (5*Log[-1 + (1 + x^4)^(1/3)])/324 + (5*Log[1 + (1 + x^4)^(1/3) + (1 + x^4)^(2/3)])/648

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fricas [A]  time = 0.64, size = 96, normalized size = 0.90 \begin {gather*} \frac {20 \, \sqrt {3} x^{16} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 10 \, x^{16} \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) - 20 \, x^{16} \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (10 \, x^{12} - 6 \, x^{8} + 45 \, x^{4} + 81\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{1296 \, x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)^(1/3)*(x^4+3)/x^17,x, algorithm="fricas")

[Out]

1/1296*(20*sqrt(3)*x^16*arctan(2/3*sqrt(3)*(x^4 + 1)^(1/3) + 1/3*sqrt(3)) + 10*x^16*log((x^4 + 1)^(2/3) + (x^4
+ 1)^(1/3) + 1) - 20*x^16*log((x^4 + 1)^(1/3) - 1) - 3*(10*x^12 - 6*x^8 + 45*x^4 + 81)*(x^4 + 1)^(1/3))/x^16

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giac [A]  time = 0.13, size = 96, normalized size = 0.90 \begin {gather*} \frac {5}{324} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {10 \, {\left (x^{4} + 1\right )}^{\frac {10}{3}} - 36 \, {\left (x^{4} + 1\right )}^{\frac {7}{3}} + 87 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}} + 20 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{432 \, x^{16}} + \frac {5}{648} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {5}{324} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)^(1/3)*(x^4+3)/x^17,x, algorithm="giac")

[Out]

5/324*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^4 + 1)^(1/3) + 1)) - 1/432*(10*(x^4 + 1)^(10/3) - 36*(x^4 + 1)^(7/3) +
87*(x^4 + 1)^(4/3) + 20*(x^4 + 1)^(1/3))/x^16 + 5/648*log((x^4 + 1)^(2/3) + (x^4 + 1)^(1/3) + 1) - 5/324*log((
x^4 + 1)^(1/3) - 1)

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maple [C]  time = 4.46, size = 81, normalized size = 0.76

method	result	size
risch	\(-\frac {10 x^{16}+4 x^{12}+39 x^{8}+126 x^{4}+81}{432 x^{16} \left (x^{4}+1\right )^{\frac {2}{3}}}-\frac {5 \left (-\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{4} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], -x^{4}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+4 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )\right )}{324 \Gamma \left (\frac {2}{3}\right )}\)	\(81\)
meijerg	\(-\frac {\frac {10 \Gamma \left (\frac {2}{3}\right ) x^{4} \hypergeom \left (\left [1, 1, \frac {11}{3}\right ], \left [2, 5\right ], -x^{4}\right )}{81}-\frac {5 \left (\frac {4}{15}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+4 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )}{27}+\frac {\Gamma \left (\frac {2}{3}\right )}{x^{12}}+\frac {\Gamma \left (\frac {2}{3}\right )}{2 x^{8}}-\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{4}}}{12 \Gamma \left (\frac {2}{3}\right )}-\frac {-\frac {22 \Gamma \left (\frac {2}{3}\right ) x^{4} \hypergeom \left (\left [1, 1, \frac {14}{3}\right ], \left [2, 6\right ], -x^{4}\right )}{243}+\frac {10 \left (\frac {47}{120}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+4 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )}{81}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{4 x^{16}}+\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{12}}-\frac {\Gamma \left (\frac {2}{3}\right )}{6 x^{8}}+\frac {5 \Gamma \left (\frac {2}{3}\right )}{27 x^{4}}}{4 \Gamma \left (\frac {2}{3}\right )}\)	\(144\)
trager	\(-\frac {\left (10 x^{12}-6 x^{8}+45 x^{4}+81\right ) \left (x^{4}+1\right )^{\frac {1}{3}}}{432 x^{16}}-\frac {5 \ln \left (\frac {1801935 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}+2398401 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+798074 x^{4}-7791669 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}}-1801935 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+6586200 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}}+401823 \left (x^{4}+1\right )^{\frac {2}{3}}+1806114 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-2597223 \left (x^{4}+1\right )^{\frac {1}{3}}+1995185}{x^{4}}\right )}{324}+\frac {5 \ln \left (\frac {153 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-264 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+111 x^{4}-351 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}}-153 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+495 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}}-48 \left (x^{4}+1\right )^{\frac {2}{3}}-93 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-117 \left (x^{4}+1\right )^{\frac {1}{3}}+148}{x^{4}}\right )}{324}-\frac {5 \ln \left (\frac {153 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-264 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+111 x^{4}-351 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}}-153 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+495 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}}-48 \left (x^{4}+1\right )^{\frac {2}{3}}-93 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-117 \left (x^{4}+1\right )^{\frac {1}{3}}+148}{x^{4}}\right ) \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )}{108}\)	\(444\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+1)^(1/3)*(x^4+3)/x^17,x,method=_RETURNVERBOSE)

[Out]

-1/432*(10*x^16+4*x^12+39*x^8+126*x^4+81)/x^16/(x^4+1)^(2/3)-5/324/GAMMA(2/3)*(-2/3*GAMMA(2/3)*x^4*hypergeom([
1,1,5/3],[2,2],-x^4)+(1/6*Pi*3^(1/2)-3/2*ln(3)+4*ln(x))*GAMMA(2/3))

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maxima [B]  time = 0.43, size = 182, normalized size = 1.70 \begin {gather*} \frac {5}{324} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {20 \, {\left (x^{4} + 1\right )}^{\frac {10}{3}} - 72 \, {\left (x^{4} + 1\right )}^{\frac {7}{3}} + 93 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}} + 40 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{432 \, {\left ({\left (x^{4} + 1\right )}^{4} - 4 \, x^{4} - 4 \, {\left (x^{4} + 1\right )}^{3} + 6 \, {\left (x^{4} + 1\right )}^{2} - 3\right )}} + \frac {5 \, {\left (x^{4} + 1\right )}^{\frac {7}{3}} - 13 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}} - 10 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{216 \, {\left (3 \, x^{4} + {\left (x^{4} + 1\right )}^{3} - 3 \, {\left (x^{4} + 1\right )}^{2} + 2\right )}} + \frac {5}{648} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {5}{324} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)^(1/3)*(x^4+3)/x^17,x, algorithm="maxima")

[Out]

5/324*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^4 + 1)^(1/3) + 1)) - 1/432*(20*(x^4 + 1)^(10/3) - 72*(x^4 + 1)^(7/3) +
93*(x^4 + 1)^(4/3) + 40*(x^4 + 1)^(1/3))/((x^4 + 1)^4 - 4*x^4 - 4*(x^4 + 1)^3 + 6*(x^4 + 1)^2 - 3) + 1/216*(5*
(x^4 + 1)^(7/3) - 13*(x^4 + 1)^(4/3) - 10*(x^4 + 1)^(1/3))/(3*x^4 + (x^4 + 1)^3 - 3*(x^4 + 1)^2 + 2) + 5/648*l
og((x^4 + 1)^(2/3) + (x^4 + 1)^(1/3) + 1) - 5/324*log((x^4 + 1)^(1/3) - 1)

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mupad [B]  time = 1.82, size = 268, normalized size = 2.50 \begin {gather*} \frac {5\,\ln \left (\frac {25\,{\left (x^4+1\right )}^{1/3}}{11664}-\frac {25}{11664}\right )}{324}-\frac {5\,\ln \left (\frac {25\,{\left (x^4+1\right )}^{1/3}}{2916}-\frac {25}{2916}\right )}{162}-\frac {\frac {5\,{\left (x^4+1\right )}^{1/3}}{108}+\frac {13\,{\left (x^4+1\right )}^{4/3}}{216}-\frac {5\,{\left (x^4+1\right )}^{7/3}}{216}}{{\left (x^4+1\right )}^3-3\,{\left (x^4+1\right )}^2+3\,x^4+2}+\frac {\frac {5\,{\left (x^4+1\right )}^{1/3}}{54}+\frac {31\,{\left (x^4+1\right )}^{4/3}}{144}-\frac {{\left (x^4+1\right )}^{7/3}}{6}+\frac {5\,{\left (x^4+1\right )}^{10/3}}{108}}{4\,{\left (x^4+1\right )}^3-6\,{\left (x^4+1\right )}^2-{\left (x^4+1\right )}^4+4\,x^4+3}-\ln \left (\frac {5\,{\left (x^4+1\right )}^{1/3}}{18}+\frac {5}{36}-\frac {\sqrt {3}\,5{}\mathrm {i}}{36}\right )\,\left (-\frac {5}{324}+\frac {\sqrt {3}\,5{}\mathrm {i}}{324}\right )+\ln \left (\frac {5\,{\left (x^4+1\right )}^{1/3}}{18}+\frac {5}{36}+\frac {\sqrt {3}\,5{}\mathrm {i}}{36}\right )\,\left (\frac {5}{324}+\frac {\sqrt {3}\,5{}\mathrm {i}}{324}\right )+\ln \left (\frac {5\,{\left (x^4+1\right )}^{1/3}}{36}+\frac {5}{72}-\frac {\sqrt {3}\,5{}\mathrm {i}}{72}\right )\,\left (-\frac {5}{648}+\frac {\sqrt {3}\,5{}\mathrm {i}}{648}\right )-\ln \left (\frac {5\,{\left (x^4+1\right )}^{1/3}}{36}+\frac {5}{72}+\frac {\sqrt {3}\,5{}\mathrm {i}}{72}\right )\,\left (\frac {5}{648}+\frac {\sqrt {3}\,5{}\mathrm {i}}{648}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 + 1)^(1/3)*(x^4 + 3))/x^17,x)

[Out]

(5*log((25*(x^4 + 1)^(1/3))/11664 - 25/11664))/324 - (5*log((25*(x^4 + 1)^(1/3))/2916 - 25/2916))/162 - ((5*(x
^4 + 1)^(1/3))/108 + (13*(x^4 + 1)^(4/3))/216 - (5*(x^4 + 1)^(7/3))/216)/((x^4 + 1)^3 - 3*(x^4 + 1)^2 + 3*x^4
+ 2) + ((5*(x^4 + 1)^(1/3))/54 + (31*(x^4 + 1)^(4/3))/144 - (x^4 + 1)^(7/3)/6 + (5*(x^4 + 1)^(10/3))/108)/(4*(
x^4 + 1)^3 - 6*(x^4 + 1)^2 - (x^4 + 1)^4 + 4*x^4 + 3) - log((5*(x^4 + 1)^(1/3))/18 - (3^(1/2)*5i)/36 + 5/36)*(
(3^(1/2)*5i)/324 - 5/324) + log((3^(1/2)*5i)/36 + (5*(x^4 + 1)^(1/3))/18 + 5/36)*((3^(1/2)*5i)/324 + 5/324) +
log((5*(x^4 + 1)^(1/3))/36 - (3^(1/2)*5i)/72 + 5/72)*((3^(1/2)*5i)/648 - 5/648) - log((3^(1/2)*5i)/72 + (5*(x^
4 + 1)^(1/3))/36 + 5/72)*((3^(1/2)*5i)/648 + 5/648)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+1)**(1/3)*(x**4+3)/x**17,x)

[Out]

Timed out

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