3.7.81 \(\int \frac {\cos ^{\frac {5}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx\) [681]

Optimal. Leaf size=28 \[ \text {Int}\left (\frac {\cos ^{\frac {5}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}},x\right ) \]

[Out]

Unintegrable(cos(d*x+c)^(5/3)/(a+b*cos(d*x+c))^(1/2),x)

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Rubi [A]
time = 0.04, 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 = {} \begin {gather*} \int \frac {\cos ^{\frac {5}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Cos[c + d*x]^(5/3)/Sqrt[a + b*Cos[c + d*x]],x]

[Out]

Defer[Int][Cos[c + d*x]^(5/3)/Sqrt[a + b*Cos[c + d*x]], x]

Rubi steps

\begin {align*} \int \frac {\cos ^{\frac {5}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx &=\int \frac {\cos ^{\frac {5}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx\\ \end {align*}

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Mathematica [A]
time = 86.63, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^{\frac {5}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Cos[c + d*x]^(5/3)/Sqrt[a + b*Cos[c + d*x]],x]

[Out]

Integrate[Cos[c + d*x]^(5/3)/Sqrt[a + b*Cos[c + d*x]], x]

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Maple [A]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\cos ^{\frac {5}{3}}\left (d x +c \right )}{\sqrt {a +b \cos \left (d x +c \right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(5/3)/(a+b*cos(d*x+c))^(1/2),x)

[Out]

int(cos(d*x+c)^(5/3)/(a+b*cos(d*x+c))^(1/2),x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(5/3)/(a+b*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)^(5/3)/sqrt(b*cos(d*x + c) + a), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(5/3)/(a+b*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(cos(d*x + c)^(5/3)/sqrt(b*cos(d*x + c) + a), x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**(5/3)/(a+b*cos(d*x+c))**(1/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4369 deep

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(5/3)/(a+b*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(cos(d*x + c)^(5/3)/sqrt(b*cos(d*x + c) + a), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^{5/3}}{\sqrt {a+b\,\cos \left (c+d\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^(5/3)/(a + b*cos(c + d*x))^(1/2),x)

[Out]

int(cos(c + d*x)^(5/3)/(a + b*cos(c + d*x))^(1/2), x)

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