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

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {\sqrt [3]{a \cos (c+d x)+a}}{x},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 1.51, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{a+a \cos (c+d x)}}{x} \, dx \]

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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}{x}\,{d x} \]

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)

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maple [A]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +a \cos \left (d x +c \right )\right )^{\frac {1}{3}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}{x}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{1/3}}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*cos(c + d*x))^(1/3)/x,x)

[Out]

int((a + a*cos(c + d*x))^(1/3)/x, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{a \left (\cos {\left (c + d x \right )} + 1\right )}}{x}\, dx \]

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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