\(\int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx\) [231]

Optimal result

Integrand size = 21, antiderivative size = 58 \[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=-\frac {3 (b \cos (c+d x))^{7/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {13}{6},\cos ^2(c+d x)\right ) \sin (c+d x)}{7 b^3 d \sqrt {\sin ^2(c+d x)}} \]

[Out]

-3/7*(b*cos(d*x+c))^(7/3)*hypergeom([1/2, 7/6],[13/6],cos(d*x+c)^2)*sin(d*x+c)/b^3/d/(sin(d*x+c)^2)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {16, 2722} \[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=-\frac {3 \sin (c+d x) (b \cos (c+d x))^{7/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {13}{6},\cos ^2(c+d x)\right )}{7 b^3 d \sqrt {\sin ^2(c+d x)}} \]

[In]

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

[Out]

(-3*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*b^3*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*} \text {integral}& = \frac {\int (b \cos (c+d x))^{4/3} \, dx}{b^2} \\ & = -\frac {3 (b \cos (c+d x))^{7/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {13}{6},\cos ^2(c+d x)\right ) \sin (c+d x)}{7 b^3 d \sqrt {\sin ^2(c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=-\frac {3 \cos ^2(c+d x) \cot (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {13}{6},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{7 d (b \cos (c+d x))^{2/3}} \]

[In]

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

[Out]

(-3*Cos[c + d*x]^2*Cot[c + d*x]*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sqrt[Sin[c + d*x]^2])/(7*d*(
b*Cos[c + d*x])^(2/3))

Maple [F]

\[\int \frac {\cos ^{2}\left (d x +c \right )}{\left (\cos \left (d x +c \right ) b \right )^{\frac {2}{3}}}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3}} \,d x \]

[In]

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

[Out]

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