3.3.96 \(\int \frac {(1-2 \sqrt [3]{x})^{3/4}}{x} \, dx\) [296]

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 = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, 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 = {272, 52, 65, 304, 209, 212} \begin {gather*} 6 \text {ArcTan}\left (\sqrt [4]{1-2 \sqrt [3]{x}}\right )+4 \left (1-2 \sqrt [3]{x}\right )^{3/4}-6 \tanh ^{-1}\left (\sqrt [4]{1-2 \sqrt [3]{x}}\right ) \end {gather*}

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 52

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 65

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 209

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 272

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 304

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]

Rubi steps

\begin {align*} \int \frac {\left (1-2 \sqrt [3]{x}\right )^{3/4}}{x} \, dx &=3 \text {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 \text {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 \text {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 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1-2 \sqrt [3]{x}}\right )+6 \text {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*}

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Mathematica [A]
time = 0.03, size = 48, normalized size = 1.00 \begin {gather*} 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 {gather*}

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 = 0.07, size = 53, normalized size = 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	\(-\frac {9 \sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) \left (-\frac {4 \left (\frac {4}{3}-2 \ln \left (2\right )-\frac {\pi }{2}+\frac {\ln \left (x \right )}{3}+i \pi \right ) \pi \sqrt {2}}{3 \Gamma \left (\frac {3}{4}\right )}+\frac {2 \pi \sqrt {2}\, x^{\frac {1}{3}} \hypergeom \left (\left [\frac {1}{4}, 1, 1\right ], \left [2, 2\right ], 2 x^{\frac {1}{3}}\right )}{\Gamma \left (\frac {3}{4}\right )}\right )}{8 \pi }\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))

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Maxima [A]
time = 3.74, size = 52, normalized size = 1.08 \begin {gather*} 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 {gather*}

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 = 1.26, size = 52, normalized size = 1.08 \begin {gather*} 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 {gather*}

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] Result contains complex when optimal does not.
time = 1.13, size = 51, normalized size = 1.06 \begin {gather*} - \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 {gather*}

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 [A]
time = 2.83, size = 53, normalized size = 1.10 \begin {gather*} 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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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(abs((-
2*x^(1/3) + 1)^(1/4) - 1))

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Mupad [B]
time = 0.87, size = 36, normalized size = 0.75 \begin {gather*} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x^(1/3))^(3/4)/x,x)

[Out]

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)

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