3.198 \(\int \frac{\cos ^2(c+d x)}{\sqrt [3]{b \sec (c+d x)}} \, dx\)

Optimal. Leaf size=58 \[ -\frac{3 b^3 \sin (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{5}{3},\frac{8}{3},\cos ^2(c+d x)\right )}{10 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{10/3}} \]

[Out]

(-3*b^3*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*d*(b*Sec[c + d*x])^(10/3)*Sqrt[Sin[
c + d*x]^2])

________________________________________________________________________________________

Rubi [A]  time = 0.04462, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {16, 3772, 2643} \[ -\frac{3 b^3 \sin (c+d x) \, _2F_1\left (\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right )}{10 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{10/3}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2/(b*Sec[c + d*x])^(1/3),x]

[Out]

(-3*b^3*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*d*(b*Sec[c + d*x])^(10/3)*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 \frac{\cos ^2(c+d x)}{\sqrt [3]{b \sec (c+d x)}} \, dx &=b^2 \int \frac{1}{(b \sec (c+d x))^{7/3}} \, dx\\ &=\left (b^2 \left (\frac{\cos (c+d x)}{b}\right )^{2/3} (b \sec (c+d x))^{2/3}\right ) \int \left (\frac{\cos (c+d x)}{b}\right )^{7/3} \, dx\\ &=-\frac{3 \cos ^4(c+d x) \, _2F_1\left (\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{2/3} \sin (c+d x)}{10 b d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0697015, size = 60, normalized size = 1.03 \[ -\frac{3 b^2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) \text{Hypergeometric2F1}\left (-\frac{7}{6},\frac{1}{2},-\frac{1}{6},\sec ^2(c+d x)\right )}{7 d (b \sec (c+d x))^{7/3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[c + d*x]^2/(b*Sec[c + d*x])^(1/3),x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.142, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}{\frac{1}{\sqrt [3]{b\sec \left ( dx+c \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(c + d*x)**2/(b*sec(c + d*x))**(1/3), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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