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

Optimal. Leaf size=53 \[ -\frac{3 \sin (a+b x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{5}{6},\frac{11}{6},\cos ^2(a+b x)\right )}{5 b \sqrt{\sin ^2(a+b x)} \sec ^{\frac{5}{3}}(a+b x)} \]

[Out]

(-3*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[a + b*x]^2]*Sin[a + b*x])/(5*b*Sec[a + b*x]^(5/3)*Sqrt[Sin[a + b*x]^
2])

________________________________________________________________________________________

Rubi [A]  time = 0.0254632, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3772, 2643} \[ -\frac{3 \sin (a+b x) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(a+b x)\right )}{5 b \sqrt{\sin ^2(a+b x)} \sec ^{\frac{5}{3}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*x]^(-2/3),x]

[Out]

(-3*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[a + b*x]^2]*Sin[a + b*x])/(5*b*Sec[a + b*x]^(5/3)*Sqrt[Sin[a + b*x]^
2])

Rule 3772

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x])^(n - 1)*((Sin[c + d*x]/b)^(n - 1)
*Int[1/(Sin[c + d*x]/b)^n, x]), x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

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}{\sec ^{\frac{2}{3}}(a+b x)} \, dx &=\sqrt [3]{\cos (a+b x)} \sqrt [3]{\sec (a+b x)} \int \cos ^{\frac{2}{3}}(a+b x) \, dx\\ &=-\frac{3 \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(a+b x)\right ) \sin (a+b x)}{5 b \sec ^{\frac{5}{3}}(a+b x) \sqrt{\sin ^2(a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.0829981, size = 55, normalized size = 1.04 \[ -\frac{3 \sqrt{-\tan ^2(a+b x)} \csc (a+b x) \text{Hypergeometric2F1}\left (-\frac{1}{3},\frac{1}{2},\frac{2}{3},\sec ^2(a+b x)\right )}{2 b \sec ^{\frac{5}{3}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[a + b*x]^(-2/3),x]

[Out]

(-3*Csc[a + b*x]*Hypergeometric2F1[-1/3, 1/2, 2/3, Sec[a + b*x]^2]*Sqrt[-Tan[a + b*x]^2])/(2*b*Sec[a + b*x]^(5
/3))

________________________________________________________________________________________

Maple [F]  time = 0.093, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( bx+a \right ) \right ) ^{-{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sec \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/sec(b*x+a)^(2/3),x, algorithm="maxima")

[Out]

integrate(sec(b*x + a)^(-2/3), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sec \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/sec(b*x+a)^(2/3),x, algorithm="fricas")

[Out]

integral(sec(b*x + a)^(-2/3), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sec(a + b*x)**(-2/3), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sec \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/sec(b*x+a)^(2/3),x, algorithm="giac")

[Out]

integrate(sec(b*x + a)^(-2/3), x)