3.3.32 \(\int \frac {\cos (c+d x)}{(b \cos (c+d x))^{2/3}} \, dx\) [232]

Optimal. Leaf size=58 \[ -\frac {3 (b \cos (c+d x))^{4/3} \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{4 b^2 d \sqrt {\sin ^2(c+d x)}} \]

[Out]

-3/4*(b*cos(d*x+c))^(4/3)*hypergeom([1/2, 2/3],[5/3],cos(d*x+c)^2)*sin(d*x+c)/b^2/d/(sin(d*x+c)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {16, 2722} \begin {gather*} -\frac {3 \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(c+d x)\right )}{4 b^2 d \sqrt {\sin ^2(c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*b^2*d*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 2722

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 58, normalized size = 1.00 \begin {gather*} -\frac {3 \sqrt [3]{b \cos (c+d x)} \cot (c+d x) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{4 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(b*Cos[c + d*x])^(1/3)*Cot[c + d*x]*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sqrt[Sin[c + d*x]^2])
/(4*b*d)

________________________________________________________________________________________

Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\cos \left (d x +c \right )}{\left (b \cos \left (d x +c \right )\right )^{\frac {2}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*cos(d*x + c))^(1/3)/b, x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos {\left (c + d x \right )}}{\left (b \cos {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\cos \left (c+d\,x\right )}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________