3.184 \(\int \frac{\sqrt [3]{a+a \cos (c+d x)}}{x} \, dx\)

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{\sqrt [3]{a \cos (c+d x)+a}}{x},x\right ) \]

[Out]

Unintegrable[(a + a*Cos[c + d*x])^(1/3)/x, x]

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Rubi [A]  time = 0.0648875, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sqrt [3]{a+a \cos (c+d x)}}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + a*Cos[c + d*x])^(1/3)/x,x]

[Out]

Defer[Int][(a + a*Cos[c + d*x])^(1/3)/x, x]

Rubi steps

\begin{align*} \int \frac{\sqrt [3]{a+a \cos (c+d x)}}{x} \, dx &=\int \frac{\sqrt [3]{a+a \cos (c+d x)}}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 1.0715, size = 0, normalized size = 0. \[ \int \frac{\sqrt [3]{a+a \cos (c+d x)}}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + a*Cos[c + d*x])^(1/3)/x,x]

[Out]

Integrate[(a + a*Cos[c + d*x])^(1/3)/x, x]

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Maple [A]  time = 0.164, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\sqrt [3]{a+\cos \left ( dx+c \right ) a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+cos(d*x+c)*a)^(1/3)/x,x)

[Out]

int((a+cos(d*x+c)*a)^(1/3)/x,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{1}{3}}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^(1/3)/x,x, algorithm="maxima")

[Out]

integrate((a*cos(d*x + c) + a)^(1/3)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^(1/3)/x,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [3]{a \left (\cos{\left (c + d x \right )} + 1\right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))**(1/3)/x,x)

[Out]

Integral((a*(cos(c + d*x) + 1))**(1/3)/x, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{1}{3}}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^(1/3)/x,x, algorithm="giac")

[Out]

integrate((a*cos(d*x + c) + a)^(1/3)/x, x)