3.192 \(\int \cos (c+d x) (b \sec (c+d x))^{4/3} \, dx\)

Optimal. Leaf size=58 \[ -\frac{3 b^2 \sin (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{1}{2},\frac{4}{3},\cos ^2(c+d x)\right )}{2 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}} \]

[Out]

(-3*b^2*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c
+ d*x]^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0381933, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {16, 3772, 2643} \[ -\frac{3 b^2 \sin (c+d x) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right )}{2 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]*(b*Sec[c + d*x])^(4/3),x]

[Out]

(-3*b^2*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c
+ d*x]^2])

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

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 \cos (c+d x) (b \sec (c+d x))^{4/3} \, dx &=b \int \sqrt [3]{b \sec (c+d x)} \, dx\\ &=\left (b \sqrt [3]{\frac{\cos (c+d x)}{b}} \sqrt [3]{b \sec (c+d x)}\right ) \int \frac{1}{\sqrt [3]{\frac{\cos (c+d x)}{b}}} \, dx\\ &=-\frac{3 b \cos (c+d x) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right ) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{2 d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.009011, size = 56, normalized size = 0.97 \[ \frac{3 b \sqrt{-\tan ^2(c+d x)} \cot (c+d x) \sqrt [3]{b \sec (c+d x)} \text{Hypergeometric2F1}\left (\frac{1}{6},\frac{1}{2},\frac{7}{6},\sec ^2(c+d x)\right )}{d} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[c + d*x]*(b*Sec[c + d*x])^(4/3),x]

[Out]

(3*b*Cot[c + d*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Sec[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sqrt[-Tan[c + d*x]^2
])/d

________________________________________________________________________________________

Maple [F]  time = 0.106, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( dx+c \right ) \left ( b\sec \left ( dx+c \right ) \right ) ^{{\frac{4}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*(b*sec(d*x+c))^(4/3),x)

[Out]

int(cos(d*x+c)*(b*sec(d*x+c))^(4/3),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec \left (d x + c\right )\right )^{\frac{4}{3}} \cos \left (d x + c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(b*sec(d*x+c))^(4/3),x, algorithm="maxima")

[Out]

integrate((b*sec(d*x + c))^(4/3)*cos(d*x + c), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(b*sec(d*x+c))^(4/3),x, algorithm="fricas")

[Out]

integral((b*sec(d*x + c))^(1/3)*b*cos(d*x + c)*sec(d*x + c), x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(b*sec(d*x+c))**(4/3),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec \left (d x + c\right )\right )^{\frac{4}{3}} \cos \left (d x + c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(b*sec(d*x+c))^(4/3),x, algorithm="giac")

[Out]

integrate((b*sec(d*x + c))^(4/3)*cos(d*x + c), x)