3.3.14 \(\int \frac {\cos ^m(c+d x) (B \cos (c+d x)+C \cos ^2(c+d x))}{(b \cos (c+d x))^{2/3}} \, dx\) [214]

3.3.14.1 Optimal result

Integrand size = 40, antiderivative size = 167 \[ \int \frac {\cos ^m(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{2/3}} \, dx=-\frac {3 B \cos ^{2+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (4+3 m),\frac {1}{6} (10+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (4+3 m) (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}}-\frac {3 C \cos ^{3+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (7+3 m),\frac {1}{6} (13+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (7+3 m) (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}} \]

output

-3*B*cos(d*x+c)^(2+m)*hypergeom([1/2, 2/3+1/2*m],[5/3+1/2*m],cos(d*x+c)^2) 
*sin(d*x+c)/d/(4+3*m)/(b*cos(d*x+c))^(2/3)/(sin(d*x+c)^2)^(1/2)-3*C*cos(d* 
x+c)^(3+m)*hypergeom([1/2, 7/6+1/2*m],[13/6+1/2*m],cos(d*x+c)^2)*sin(d*x+c 
)/d/(7+3*m)/(b*cos(d*x+c))^(2/3)/(sin(d*x+c)^2)^(1/2)
 

3.3.14.2 Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.84 \[ \int \frac {\cos ^m(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{2/3}} \, dx=-\frac {3 \cos ^{2+m}(c+d x) \csc (c+d x) \left (B (7+3 m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (4+3 m),\frac {5}{3}+\frac {m}{2},\cos ^2(c+d x)\right )+C (4+3 m) \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (7+3 m),\frac {1}{6} (13+3 m),\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (4+3 m) (7+3 m) (b \cos (c+d x))^{2/3}} \]

input

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

output

(-3*Cos[c + d*x]^(2 + m)*Csc[c + d*x]*(B*(7 + 3*m)*Hypergeometric2F1[1/2, 
(4 + 3*m)/6, 5/3 + m/2, Cos[c + d*x]^2] + C*(4 + 3*m)*Cos[c + d*x]*Hyperge 
ometric2F1[1/2, (7 + 3*m)/6, (13 + 3*m)/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d 
*x]^2])/(d*(4 + 3*m)*(7 + 3*m)*(b*Cos[c + d*x])^(2/3))
 

3.3.14.3 Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {2034, 3042, 3489, 3042, 3227, 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.

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

\(\Big \downarrow \) 2034

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3489

\(\displaystyle \frac {\cos ^{\frac {2}{3}}(c+d x) \int \cos ^{m+\frac {1}{3}}(c+d x) (B+C \cos (c+d x))dx}{(b \cos (c+d x))^{2/3}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cos ^{\frac {2}{3}}(c+d x) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{m+\frac {1}{3}} \left (B+C \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{(b \cos (c+d x))^{2/3}}\)

\(\Big \downarrow \) 3227

\(\displaystyle \frac {\cos ^{\frac {2}{3}}(c+d x) \left (B \int \cos ^{m+\frac {1}{3}}(c+d x)dx+C \int \cos ^{m+\frac {4}{3}}(c+d x)dx\right )}{(b \cos (c+d x))^{2/3}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cos ^{\frac {2}{3}}(c+d x) \left (B \int \sin \left (c+d x+\frac {\pi }{2}\right )^{m+\frac {1}{3}}dx+C \int \sin \left (c+d x+\frac {\pi }{2}\right )^{m+\frac {4}{3}}dx\right )}{(b \cos (c+d x))^{2/3}}\)

\(\Big \downarrow \) 3122

\(\displaystyle \frac {\cos ^{\frac {2}{3}}(c+d x) \left (-\frac {3 B \sin (c+d x) \cos ^{m+\frac {4}{3}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (3 m+4),\frac {1}{6} (3 m+10),\cos ^2(c+d x)\right )}{d (3 m+4) \sqrt {\sin ^2(c+d x)}}-\frac {3 C \sin (c+d x) \cos ^{m+\frac {7}{3}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (3 m+7),\frac {1}{6} (3 m+13),\cos ^2(c+d x)\right )}{d (3 m+7) \sqrt {\sin ^2(c+d x)}}\right )}{(b \cos (c+d x))^{2/3}}\)

input

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

output

(Cos[c + d*x]^(2/3)*((-3*B*Cos[c + d*x]^(4/3 + m)*Hypergeometric2F1[1/2, ( 
4 + 3*m)/6, (10 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(4 + 3*m)*Sqrt[ 
Sin[c + d*x]^2]) - (3*C*Cos[c + d*x]^(7/3 + m)*Hypergeometric2F1[1/2, (7 + 
 3*m)/6, (13 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 3*m)*Sqrt[Sin 
[c + d*x]^2])))/(b*Cos[c + d*x])^(2/3)
 

3.3.14.3.1 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]
 

rule 3227

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x 
_)]), x_Symbol] :> Simp[c   Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b   Int 
[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
 

rule 3489

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((B_.)*sin[(e_.) + (f_.)*(x_)] + 
(C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[1/b   Int[(b*Sin[e + f* 
x])^(m + 1)*(B + C*Sin[e + f*x]), x], x] /; FreeQ[{b, e, f, B, C, m}, x]
 

3.3.14.4 Maple [F]

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

input

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

output

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

3.3.14.5 Fricas [F]

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

input

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

output

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

3.3.14.6 Sympy [F]

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

input

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

output

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

3.3.14.7 Maxima [F]

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

input

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

output

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

3.3.14.8 Giac [F]

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

input

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

output

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

3.3.14.9 Mupad [F(-1)]

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

input

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

output

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