\(\int \frac {\cos ^m(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx\) [230]

Optimal result

Integrand size = 21, antiderivative size = 82 \[ \int \frac {\cos ^m(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=-\frac {3 \cos ^{1+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (1+3 m),\frac {1}{6} (7+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+3 m) (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^m(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=-\frac {\cos ^{1+m}(c+d x) \csc (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} \left (\frac {1}{3}+m\right ),\frac {1}{2} \left (\frac {7}{3}+m\right ),\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{d \left (\frac {1}{3}+m\right ) (b \cos (c+d x))^{2/3}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2034, 3042, 3122}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\cos ^m(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx\)

\(\Big \downarrow \) 2034

\(\displaystyle \frac {\cos ^{\frac {2}{3}}(c+d x) \int \cos ^{m-\frac {2}{3}}(c+d x)dx}{(b \cos (c+d x))^{2/3}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cos ^{\frac {2}{3}}(c+d x) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{m-\frac {2}{3}}dx}{(b \cos (c+d x))^{2/3}}\)

\(\Big \downarrow \) 3122

\(\displaystyle -\frac {3 \sin (c+d x) \cos ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (3 m+1),\frac {1}{6} (3 m+7),\cos ^2(c+d x)\right )}{d (3 m+1) \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2034

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3122

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]))*Hypergeometric2 
F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n}, x] 
 &&  !IntegerQ[2*n]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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