3.32 \(\int (c \sec (a+b x))^{2/3} \, dx\)

Optimal. Leaf size=54 \[ -\frac{3 c \sin (a+b x) \text{Hypergeometric2F1}\left (\frac{1}{6},\frac{1}{2},\frac{7}{6},\cos ^2(a+b x)\right )}{b \sqrt{\sin ^2(a+b x)} \sqrt [3]{c \sec (a+b x)}} \]

[Out]

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

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Rubi [A]  time = 0.0295325, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3772, 2643} \[ -\frac{3 c \sin (a+b x) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(a+b x)\right )}{b \sqrt{\sin ^2(a+b x)} \sqrt [3]{c \sec (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sec[a + b*x])^(2/3),x]

[Out]

(-3*c*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[a + b*x]^2]*Sin[a + b*x])/(b*(c*Sec[a + b*x])^(1/3)*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 (c \sec (a+b x))^{2/3} \, dx &=\left (\frac{\cos (a+b x)}{c}\right )^{2/3} (c \sec (a+b x))^{2/3} \int \frac{1}{\left (\frac{\cos (a+b x)}{c}\right )^{2/3}} \, dx\\ &=-\frac{3 \cos (a+b x) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(a+b x)\right ) (c \sec (a+b x))^{2/3} \sin (a+b x)}{b \sqrt{\sin ^2(a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.0410417, size = 57, normalized size = 1.06 \[ \frac{3 \sqrt{-\tan ^2(a+b x)} \cot (a+b x) (c \sec (a+b x))^{2/3} \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{1}{2},\frac{4}{3},\sec ^2(a+b x)\right )}{2 b} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c*Sec[a + b*x])^(2/3),x]

[Out]

(3*Cot[a + b*x]*Hypergeometric2F1[1/3, 1/2, 4/3, Sec[a + b*x]^2]*(c*Sec[a + b*x])^(2/3)*Sqrt[-Tan[a + b*x]^2])
/(2*b)

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Maple [F]  time = 0.084, size = 0, normalized size = 0. \begin{align*} \int \left ( c\sec \left ( bx+a \right ) \right ) ^{{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*sec(b*x + a))^(2/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c*sec(b*x + a))^(2/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c*sec(a + b*x))**(2/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*sec(b*x + a))^(2/3), x)