3.296 \(\int \frac{(1-2 \sqrt [3]{x})^{3/4}}{x} \, dx\)

Optimal. Leaf size=48 \[ 4 \left (1-2 \sqrt [3]{x}\right )^{3/4}+6 \tan ^{-1}\left (\sqrt [4]{1-2 \sqrt [3]{x}}\right )-6 \tanh ^{-1}\left (\sqrt [4]{1-2 \sqrt [3]{x}}\right ) \]

[Out]

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)]

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Rubi [A]  time = 0.0176575, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {266, 50, 63, 298, 203, 206} \[ 4 \left (1-2 \sqrt [3]{x}\right )^{3/4}+6 \tan ^{-1}\left (\sqrt [4]{1-2 \sqrt [3]{x}}\right )-6 \tanh ^{-1}\left (\sqrt [4]{1-2 \sqrt [3]{x}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x^(1/3))^(3/4)/x,x]

[Out]

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)]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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]]

Rule 50

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] + Dist[(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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, 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 \frac{\left (1-2 \sqrt [3]{x}\right )^{3/4}}{x} \, dx &=3 \operatorname{Subst}\left (\int \frac{(1-2 x)^{3/4}}{x} \, dx,x,\sqrt [3]{x}\right )\\ &=4 \left (1-2 \sqrt [3]{x}\right )^{3/4}+3 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-2 x} x} \, dx,x,\sqrt [3]{x}\right )\\ &=4 \left (1-2 \sqrt [3]{x}\right )^{3/4}-6 \operatorname{Subst}\left (\int \frac{x^2}{\frac{1}{2}-\frac{x^4}{2}} \, dx,x,\sqrt [4]{1-2 \sqrt [3]{x}}\right )\\ &=4 \left (1-2 \sqrt [3]{x}\right )^{3/4}-6 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt [4]{1-2 \sqrt [3]{x}}\right )+6 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt [4]{1-2 \sqrt [3]{x}}\right )\\ &=4 \left (1-2 \sqrt [3]{x}\right )^{3/4}+6 \tan ^{-1}\left (\sqrt [4]{1-2 \sqrt [3]{x}}\right )-6 \tanh ^{-1}\left (\sqrt [4]{1-2 \sqrt [3]{x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0123142, size = 48, normalized size = 1. \[ 4 \left (1-2 \sqrt [3]{x}\right )^{3/4}+6 \tan ^{-1}\left (\sqrt [4]{1-2 \sqrt [3]{x}}\right )-6 \tanh ^{-1}\left (\sqrt [4]{1-2 \sqrt [3]{x}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x^(1/3))^(3/4)/x,x]

[Out]

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)]

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Maple [A]  time = 0.007, size = 53, normalized size = 1.1 \begin{align*} 4\, \left ( 1-2\,\sqrt [3]{x} \right ) ^{3/4}+3\,\ln \left ( -1+\sqrt [4]{1-2\,\sqrt [3]{x}} \right ) -3\,\ln \left ( 1+\sqrt [4]{1-2\,\sqrt [3]{x}} \right ) +6\,\arctan \left ( \sqrt [4]{1-2\,\sqrt [3]{x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x^(1/3))^(3/4)/x,x)

[Out]

4*(1-2*x^(1/3))^(3/4)+3*ln(-1+(1-2*x^(1/3))^(1/4))-3*ln(1+(1-2*x^(1/3))^(1/4))+6*arctan((1-2*x^(1/3))^(1/4))

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Maxima [A]  time = 1.46042, size = 70, normalized size = 1.46 \begin{align*} 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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x^(1/3))^(3/4)/x,x, algorithm="maxima")

[Out]

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)

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Fricas [A]  time = 1.83267, size = 180, normalized size = 3.75 \begin{align*} 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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x^(1/3))^(3/4)/x,x, algorithm="fricas")

[Out]

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)

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Sympy [C]  time = 3.48925, size = 51, normalized size = 1.06 \begin{align*} - \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x**(1/3))**(3/4)/x,x)

[Out]

-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)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x^(1/3))^(3/4)/x,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError