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

Optimal. Leaf size=28 \[ \text {Int}\left (\frac {\sqrt [3]{\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}},x\right ) \]

[Out]

Unintegrable(cos(d*x+c)^(1/3)/(a+b*cos(d*x+c))^(1/2),x)

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Rubi [A]  time = 0.05, 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 = {} \[ \int \frac {\sqrt [3]{\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 2.74, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (d x + c\right )^{\frac {1}{3}}}{\sqrt {b \cos \left (d x + c\right ) + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(1/3)/(a+b*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(cos(d*x + c)^(1/3)/sqrt(b*cos(d*x + c) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(1/3)/(a+b*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^{\frac {1}{3}}\left (d x +c \right )}{\sqrt {a +b \cos \left (d x +c \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(1/3)/(a+b*cos(d*x+c))^(1/2),x)

[Out]

int(cos(d*x+c)^(1/3)/(a+b*cos(d*x+c))^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(1/3)/(a+b*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\cos \left (c+d\,x\right )}^{1/3}}{\sqrt {a+b\,\cos \left (c+d\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^(1/3)/(a + b*cos(c + d*x))^(1/2),x)

[Out]

int(cos(c + d*x)^(1/3)/(a + b*cos(c + d*x))^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**(1/3)/(a+b*cos(d*x+c))**(1/2),x)

[Out]

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

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