3.29 \(\int \frac{1}{\cos ^{\frac{2}{3}}(a+b x)} \, dx\)

Optimal. Leaf size=51 \[ -\frac{3 \sin (a+b x) \sqrt [3]{\cos (a+b x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(a+b x)\right )}{b \sqrt{\sin ^2(a+b x)}} \]

[Out]

(-3*Cos[a + b*x]^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[a + b*x]^2]*Sin[a + b*x])/(b*Sqrt[Sin[a + b*x]^2])

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Rubi [A]  time = 0.0118621, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2643} \[ -\frac{3 \sin (a+b x) \sqrt [3]{\cos (a+b x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(a+b x)\right )}{b \sqrt{\sin ^2(a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^(-2/3),x]

[Out]

(-3*Cos[a + b*x]^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[a + b*x]^2]*Sin[a + b*x])/(b*Sqrt[Sin[a + b*x]^2])

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin{align*} \int \frac{1}{\cos ^{\frac{2}{3}}(a+b x)} \, dx &=-\frac{3 \sqrt [3]{\cos (a+b x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(a+b x)\right ) \sin (a+b x)}{b \sqrt{\sin ^2(a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.0247452, size = 51, normalized size = 1. \[ -\frac{3 \sqrt{\sin ^2(a+b x)} \sqrt [3]{\cos (a+b x)} \csc (a+b x) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(a+b x)\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^(-2/3),x]

[Out]

(-3*Cos[a + b*x]^(1/3)*Csc[a + b*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[a + b*x]^2]*Sqrt[Sin[a + b*x]^2])/b

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Maple [F]  time = 0.139, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( bx+a \right ) \right ) ^{-{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\cos \left (b x + a\right )^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cos(b*x + a)^(-2/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\cos \left (b x + a\right )^{\frac{2}{3}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cos(b*x + a)^(-2/3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\cos ^{\frac{2}{3}}{\left (a + b x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(a + b*x)**(-2/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\cos \left (b x + a\right )^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cos(b*x + a)^(-2/3), x)