\(\int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \, dx\) [209]

Optimal result

Integrand size = 21, antiderivative size = 82 \[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \, dx=-\frac {3 \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (5+3 m),\frac {1}{6} (11+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5+3 m) \sqrt {\sin ^2(c+d x)}} \]

[Out]

-3*cos(d*x+c)^(1+m)*(b*cos(d*x+c))^(2/3)*hypergeom([1/2, 5/6+1/2*m],[11/6+1/2*m],cos(d*x+c)^2)*sin(d*x+c)/d/(5
+3*m)/(sin(d*x+c)^2)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 82, 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 = {20, 2722} \[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \, dx=-\frac {3 \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (3 m+5),\frac {1}{6} (3 m+11),\cos ^2(c+d x)\right )}{d (3 m+5) \sqrt {\sin ^2(c+d x)}} \]

[In]

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

[Out]

(-3*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/2, (5 + 3*m)/6, (11 + 3*m)/6, Cos[c + d*x]
^2]*Sin[c + d*x])/(d*(5 + 3*m)*Sqrt[Sin[c + d*x]^2])

Rule 20

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

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 {(b \cos (c+d x))^{2/3} \int \cos ^{\frac {2}{3}+m}(c+d x) \, dx}{\cos ^{\frac {2}{3}}(c+d x)} \\ & = -\frac {3 \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (5+3 m),\frac {1}{6} (11+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5+3 m) \sqrt {\sin ^2(c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \, dx=-\frac {\cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \csc (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} \left (\frac {5}{3}+m\right ),\frac {1}{2} \left (\frac {11}{3}+m\right ),\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{d \left (\frac {5}{3}+m\right )} \]

[In]

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

[Out]

-((Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Csc[c + d*x]*Hypergeometric2F1[1/2, (5/3 + m)/2, (11/3 + m)/2,
Cos[c + d*x]^2]*Sqrt[Sin[c + d*x]^2])/(d*(5/3 + m)))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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