3.685 \(\int \frac{1}{\sqrt [3]{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx\)

Optimal. Leaf size=27 \[ \text{Unintegrable}\left (\frac{1}{\sqrt [3]{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0530685, 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 = {} \[ \int \frac{1}{\sqrt [3]{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [3]{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx &=\int \frac{1}{\sqrt [3]{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx\\ \end{align*}

Mathematica [A]  time = 1.74168, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [3]{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.315, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [3]{\cos \left ( dx+c \right ) }}}{\frac{1}{\sqrt{a+b\cos \left ( dx+c \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac{2}{3}}}{b \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*cos(d*x + c) + a)*cos(d*x + c)^(2/3)/(b*cos(d*x + c)^2 + a*cos(d*x + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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