Integrand size = 17, antiderivative size = 48 \[ \int \frac {\left (1-2 \sqrt [3]{x}\right )^{3/4}}{x} \, dx=4 \left (1-2 \sqrt [3]{x}\right )^{3/4}+6 \arctan \left (\sqrt [4]{1-2 \sqrt [3]{x}}\right )-6 \text {arctanh}\left (\sqrt [4]{1-2 \sqrt [3]{x}}\right ) \] Output:
4*(1-2*x^(1/3))^(3/4)+6*arctan((1-2*x^(1/3))^(1/4))-6*arctanh((1-2*x^(1/3) )^(1/4))
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-2 \sqrt [3]{x}\right )^{3/4}}{x} \, dx=4 \left (1-2 \sqrt [3]{x}\right )^{3/4}+6 \arctan \left (\sqrt [4]{1-2 \sqrt [3]{x}}\right )-6 \text {arctanh}\left (\sqrt [4]{1-2 \sqrt [3]{x}}\right ) \] Input:
Integrate[(1 - 2*x^(1/3))^(3/4)/x,x]
Output:
4*(1 - 2*x^(1/3))^(3/4) + 6*ArcTan[(1 - 2*x^(1/3))^(1/4)] - 6*ArcTanh[(1 - 2*x^(1/3))^(1/4)]
Time = 0.17 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.23, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {798, 60, 73, 27, 827, 216, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (1-2 \sqrt [3]{x}\right )^{3/4}}{x} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 3 \int \frac {\left (1-2 \sqrt [3]{x}\right )^{3/4}}{\sqrt [3]{x}}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle 3 \left (\int \frac {1}{\sqrt [4]{1-2 \sqrt [3]{x}} \sqrt [3]{x}}d\sqrt [3]{x}+\frac {4}{3} \left (1-2 \sqrt [3]{x}\right )^{3/4}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle 3 \left (\frac {4}{3} \left (1-2 \sqrt [3]{x}\right )^{3/4}-2 \int \frac {2 x^{2/3}}{1-x^{4/3}}d\sqrt [4]{1-2 \sqrt [3]{x}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 3 \left (\frac {4}{3} \left (1-2 \sqrt [3]{x}\right )^{3/4}-4 \int \frac {x^{2/3}}{1-x^{4/3}}d\sqrt [4]{1-2 \sqrt [3]{x}}\right )\) |
\(\Big \downarrow \) 827 |
\(\displaystyle 3 \left (\frac {4}{3} \left (1-2 \sqrt [3]{x}\right )^{3/4}-4 \left (\frac {1}{2} \int \frac {1}{1-x^{2/3}}d\sqrt [4]{1-2 \sqrt [3]{x}}-\frac {1}{2} \int \frac {1}{x^{2/3}+1}d\sqrt [4]{1-2 \sqrt [3]{x}}\right )\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle 3 \left (\frac {4}{3} \left (1-2 \sqrt [3]{x}\right )^{3/4}-4 \left (\frac {1}{2} \int \frac {1}{1-x^{2/3}}d\sqrt [4]{1-2 \sqrt [3]{x}}-\frac {1}{2} \arctan \left (\sqrt [4]{1-2 \sqrt [3]{x}}\right )\right )\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 3 \left (\frac {4}{3} \left (1-2 \sqrt [3]{x}\right )^{3/4}-4 \left (\frac {1}{2} \text {arctanh}\left (\sqrt [4]{1-2 \sqrt [3]{x}}\right )-\frac {1}{2} \arctan \left (\sqrt [4]{1-2 \sqrt [3]{x}}\right )\right )\right )\) |
Input:
Int[(1 - 2*x^(1/3))^(3/4)/x,x]
Output:
3*((4*(1 - 2*x^(1/3))^(3/4))/3 - 4*(-1/2*ArcTan[(1 - 2*x^(1/3))^(1/4)] + A rcTanh[(1 - 2*x^(1/3))^(1/4)]/2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.10
method | result | size |
derivativedivides | \(4 \left (1-2 x^{\frac {1}{3}}\right )^{\frac {3}{4}}+3 \ln \left (\left (1-2 x^{\frac {1}{3}}\right )^{\frac {1}{4}}-1\right )-3 \ln \left (\left (1-2 x^{\frac {1}{3}}\right )^{\frac {1}{4}}+1\right )+6 \arctan \left (\left (1-2 x^{\frac {1}{3}}\right )^{\frac {1}{4}}\right )\) | \(53\) |
default | \(4 \left (1-2 x^{\frac {1}{3}}\right )^{\frac {3}{4}}+3 \ln \left (\left (1-2 x^{\frac {1}{3}}\right )^{\frac {1}{4}}-1\right )-3 \ln \left (\left (1-2 x^{\frac {1}{3}}\right )^{\frac {1}{4}}+1\right )+6 \arctan \left (\left (1-2 x^{\frac {1}{3}}\right )^{\frac {1}{4}}\right )\) | \(53\) |
meijerg | \(-\frac {9 \sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) \left (-\frac {4 \left (\frac {4}{3}-2 \ln \left (2\right )-\frac {\pi }{2}+\frac {\ln \left (x \right )}{3}+i \pi \right ) \pi \sqrt {2}}{3 \Gamma \left (\frac {3}{4}\right )}+\frac {2 \pi \sqrt {2}\, x^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{4}, 1, 1\right ], \left [2, 2\right ], 2 x^{\frac {1}{3}}\right )}{\Gamma \left (\frac {3}{4}\right )}\right )}{8 \pi }\) | \(65\) |
Input:
int((1-2*x^(1/3))^(3/4)/x,x,method=_RETURNVERBOSE)
Output:
4*(1-2*x^(1/3))^(3/4)+3*ln((1-2*x^(1/3))^(1/4)-1)-3*ln((1-2*x^(1/3))^(1/4) +1)+6*arctan((1-2*x^(1/3))^(1/4))
Time = 0.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.08 \[ \int \frac {\left (1-2 \sqrt [3]{x}\right )^{3/4}}{x} \, dx=4 \, {\left (-2 \, x^{\frac {1}{3}} + 1\right )}^{\frac {3}{4}} + 6 \, \arctan \left ({\left (-2 \, x^{\frac {1}{3}} + 1\right )}^{\frac {1}{4}}\right ) - 3 \, \log \left ({\left (-2 \, x^{\frac {1}{3}} + 1\right )}^{\frac {1}{4}} + 1\right ) + 3 \, \log \left ({\left (-2 \, x^{\frac {1}{3}} + 1\right )}^{\frac {1}{4}} - 1\right ) \] Input:
integrate((1-2*x^(1/3))^(3/4)/x,x, algorithm="fricas")
Output:
4*(-2*x^(1/3) + 1)^(3/4) + 6*arctan((-2*x^(1/3) + 1)^(1/4)) - 3*log((-2*x^ (1/3) + 1)^(1/4) + 1) + 3*log((-2*x^(1/3) + 1)^(1/4) - 1)
Result contains complex when optimal does not.
Time = 0.88 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.06 \[ \int \frac {\left (1-2 \sqrt [3]{x}\right )^{3/4}}{x} \, dx=- \frac {3 \cdot 2^{\frac {3}{4}} \sqrt [4]{x} e^{\frac {3 i \pi }{4}} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {1}{2 \sqrt [3]{x}}} \right )}}{\Gamma \left (\frac {1}{4}\right )} \] Input:
integrate((1-2*x**(1/3))**(3/4)/x,x)
Output:
-3*2**(3/4)*x**(1/4)*exp(3*I*pi/4)*gamma(-3/4)*hyper((-3/4, -3/4), (1/4,), 1/(2*x**(1/3)))/gamma(1/4)
Time = 0.10 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.08 \[ \int \frac {\left (1-2 \sqrt [3]{x}\right )^{3/4}}{x} \, dx=4 \, {\left (-2 \, x^{\frac {1}{3}} + 1\right )}^{\frac {3}{4}} + 6 \, \arctan \left ({\left (-2 \, x^{\frac {1}{3}} + 1\right )}^{\frac {1}{4}}\right ) - 3 \, \log \left ({\left (-2 \, x^{\frac {1}{3}} + 1\right )}^{\frac {1}{4}} + 1\right ) + 3 \, \log \left ({\left (-2 \, x^{\frac {1}{3}} + 1\right )}^{\frac {1}{4}} - 1\right ) \] Input:
integrate((1-2*x^(1/3))^(3/4)/x,x, algorithm="maxima")
Output:
4*(-2*x^(1/3) + 1)^(3/4) + 6*arctan((-2*x^(1/3) + 1)^(1/4)) - 3*log((-2*x^ (1/3) + 1)^(1/4) + 1) + 3*log((-2*x^(1/3) + 1)^(1/4) - 1)
Time = 2.13 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.10 \[ \int \frac {\left (1-2 \sqrt [3]{x}\right )^{3/4}}{x} \, dx=4 \, {\left (-2 \, x^{\frac {1}{3}} + 1\right )}^{\frac {3}{4}} + 6 \, \arctan \left ({\left (-2 \, x^{\frac {1}{3}} + 1\right )}^{\frac {1}{4}}\right ) - 3 \, \log \left ({\left (-2 \, x^{\frac {1}{3}} + 1\right )}^{\frac {1}{4}} + 1\right ) + 3 \, \log \left ({\left | {\left (-2 \, x^{\frac {1}{3}} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \] Input:
integrate((1-2*x^(1/3))^(3/4)/x,x, algorithm="giac")
Output:
4*(-2*x^(1/3) + 1)^(3/4) + 6*arctan((-2*x^(1/3) + 1)^(1/4)) - 3*log((-2*x^ (1/3) + 1)^(1/4) + 1) + 3*log(abs((-2*x^(1/3) + 1)^(1/4) - 1))
Time = 0.48 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.75 \[ \int \frac {\left (1-2 \sqrt [3]{x}\right )^{3/4}}{x} \, dx=6\,\mathrm {atan}\left ({\left (1-2\,x^{1/3}\right )}^{1/4}\right )-6\,\mathrm {atanh}\left ({\left (1-2\,x^{1/3}\right )}^{1/4}\right )+4\,{\left (1-2\,x^{1/3}\right )}^{3/4} \] Input:
int((1 - 2*x^(1/3))^(3/4)/x,x)
Output:
6*atan((1 - 2*x^(1/3))^(1/4)) - 6*atanh((1 - 2*x^(1/3))^(1/4)) + 4*(1 - 2* x^(1/3))^(3/4)
Time = 0.15 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.08 \[ \int \frac {\left (1-2 \sqrt [3]{x}\right )^{3/4}}{x} \, dx=6 \mathit {atan} \left (\left (-2 x^{\frac {1}{3}}+1\right )^{\frac {1}{4}}\right )+4 \left (-2 x^{\frac {1}{3}}+1\right )^{\frac {3}{4}}+3 \,\mathrm {log}\left (\left (-2 x^{\frac {1}{3}}+1\right )^{\frac {1}{4}}-1\right )-3 \,\mathrm {log}\left (\left (-2 x^{\frac {1}{3}}+1\right )^{\frac {1}{4}}+1\right ) \] Input:
int((1-2*x^(1/3))^(3/4)/x,x)
Output:
6*atan(( - 2*x**(1/3) + 1)**(1/4)) + 4*( - 2*x**(1/3) + 1)**(3/4) + 3*log( ( - 2*x**(1/3) + 1)**(1/4) - 1) - 3*log(( - 2*x**(1/3) + 1)**(1/4) + 1)