3.4.67 \(\int \frac {\cos ^m(c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{(b \cos (c+d x))^{2/3}} \, dx\) [367]

3.4.67.1 Optimal result

Integrand size = 41, antiderivative size = 227 \[ \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{2/3}} \, dx=\frac {3 C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (4+3 m) (b \cos (c+d x))^{2/3}}-\frac {3 (C+3 C m+A (4+3 m)) \cos ^{1+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (1+3 m),\frac {1}{6} (7+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+3 m) (4+3 m) (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}}-\frac {3 B \cos ^{2+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (4+3 m),\frac {1}{6} (10+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (4+3 m) (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}} \]

output

3*C*cos(d*x+c)^(1+m)*sin(d*x+c)/d/(4+3*m)/(b*cos(d*x+c))^(2/3)-3*(C+3*C*m+ 
A*(4+3*m))*cos(d*x+c)^(1+m)*hypergeom([1/2, 1/6+1/2*m],[7/6+1/2*m],cos(d*x 
+c)^2)*sin(d*x+c)/d/(9*m^2+15*m+4)/(b*cos(d*x+c))^(2/3)/(sin(d*x+c)^2)^(1/ 
2)-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)
 

3.4.67.2 Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{2/3}} \, dx=\frac {3 \cos ^{1+m}(c+d x) \csc (c+d x) \left (-\left ((C+3 C m+A (4+3 m)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (1+3 m),\frac {1}{6} (7+3 m),\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )+(1+3 m) \left (C \sin ^2(c+d x)-B \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (4+3 m),\frac {5}{3}+\frac {m}{2},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )\right )}{d (1+3 m) (4+3 m) (b \cos (c+d x))^{2/3}} \]

input

Integrate[(Cos[c + d*x]^m*(A + 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]^(1 + m)*Csc[c + d*x]*(-((C + 3*C*m + A*(4 + 3*m))*Hypergeo 
metric2F1[1/2, (1 + 3*m)/6, (7 + 3*m)/6, Cos[c + d*x]^2]*Sqrt[Sin[c + d*x] 
^2]) + (1 + 3*m)*(C*Sin[c + d*x]^2 - B*Cos[c + d*x]*Hypergeometric2F1[1/2, 
 (4 + 3*m)/6, 5/3 + m/2, Cos[c + d*x]^2]*Sqrt[Sin[c + d*x]^2])))/(d*(1 + 3 
*m)*(4 + 3*m)*(b*Cos[c + d*x])^(2/3))
 

3.4.67.3 Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {2034, 3042, 3502, 27, 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.

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

\(\Big \downarrow \) 2034

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3502

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3227

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3122

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

input

Int[(Cos[c + d*x]^m*(A + 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*C*Cos[c + d*x]^(1/3 + m)*Sin[c + d*x])/(d*(4 + 3*m 
)) + ((-3*(C + 3*C*m + A*(4 + 3*m))*Cos[c + d*x]^(1/3 + m)*Hypergeometric2 
F1[1/2, (1 + 3*m)/6, (7 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + 3* 
m)*Sqrt[Sin[c + d*x]^2]) - (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*Sqrt[Sin[c 
 + d*x]^2]))/(4 + 3*m)))/(b*Cos[c + d*x])^(2/3)
 

3.4.67.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 3502

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co 
s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m 
+ 2))   Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m 
 + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] 
 &&  !LtQ[m, -1]
 

3.4.67.4 Maple [F]

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

input

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

3.4.67.5 Fricas [F]

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

input

integrate(cos(d*x+c)^m*(A+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)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c))^(1/3)*co 
s(d*x + c)^m/(b*cos(d*x + c)), x)
 

3.4.67.6 Sympy [F]

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

input

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

output

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

3.4.67.7 Maxima [F]

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

input

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

3.4.67.8 Giac [F]

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

input

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

3.4.67.9 Mupad [F(-1)]

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

input

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