3.8.67 \(\int \cos ^m(c+d x) (a+b \cos (c+d x)) (A+C \cos ^2(c+d x)) \, dx\) [767]

3.8.67.1 Optimal result

Integrand size = 31, antiderivative size = 217 \[ \int \cos ^m(c+d x) (a+b \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {a C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m)}+\frac {b C \cos ^{2+m}(c+d x) \sin (c+d x)}{d (3+m)}-\frac {a (C (1+m)+A (2+m)) \cos ^{1+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+m) (2+m) \sqrt {\sin ^2(c+d x)}}-\frac {b (C (2+m)+A (3+m)) \cos ^{2+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{d (2+m) (3+m) \sqrt {\sin ^2(c+d x)}} \]

output

a*C*cos(d*x+c)^(1+m)*sin(d*x+c)/d/(2+m)+b*C*cos(d*x+c)^(2+m)*sin(d*x+c)/d/ 
(3+m)-a*(C*(1+m)+A*(2+m))*cos(d*x+c)^(1+m)*hypergeom([1/2, 1/2+1/2*m],[3/2 
+1/2*m],cos(d*x+c)^2)*sin(d*x+c)/d/(1+m)/(2+m)/(sin(d*x+c)^2)^(1/2)-b*(C*( 
2+m)+A*(3+m))*cos(d*x+c)^(2+m)*hypergeom([1/2, 1+1/2*m],[2+1/2*m],cos(d*x+ 
c)^2)*sin(d*x+c)/d/(2+m)/(3+m)/(sin(d*x+c)^2)^(1/2)
 

3.8.67.2 Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.89 \[ \int \cos ^m(c+d x) (a+b \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {\cos ^{1+m}(c+d x) \csc (c+d x) \left (-\frac {a A \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(c+d x)\right )}{1+m}+\cos (c+d x) \left (-\frac {A b \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\cos ^2(c+d x)\right )}{2+m}+C \cos (c+d x) \left (-\frac {a \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},\cos ^2(c+d x)\right )}{3+m}-\frac {b \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},\cos ^2(c+d x)\right )}{4+m}\right )\right )\right ) \sqrt {\sin ^2(c+d x)}}{d} \]

input

Integrate[Cos[c + d*x]^m*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]
 

output

(Cos[c + d*x]^(1 + m)*Csc[c + d*x]*(-((a*A*Hypergeometric2F1[1/2, (1 + m)/ 
2, (3 + m)/2, Cos[c + d*x]^2])/(1 + m)) + Cos[c + d*x]*(-((A*b*Hypergeomet 
ric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2])/(2 + m)) + C*Cos[c + d* 
x]*(-((a*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Cos[c + d*x]^2])/(3 
+ m)) - (b*Cos[c + d*x]*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, Cos[c 
 + d*x]^2])/(4 + m))))*Sqrt[Sin[c + d*x]^2])/d
 

3.8.67.3 Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {3042, 3513, 3042, 3502, 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.

\(\displaystyle \int \cos ^m(c+d x) (a+b \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^m \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\)

\(\Big \downarrow \) 3513

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3502

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3227

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3122

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

input

Int[Cos[c + d*x]^m*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]
 

output

(b*C*Cos[c + d*x]^(2 + m)*Sin[c + d*x])/(d*(3 + m)) + ((a*C*(3 + m)*Cos[c 
+ d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)) + (-((a*(3 + m)*(C*(1 + m) + A*(2 
 + m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, C 
os[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*Sqrt[Sin[c + d*x]^2])) - (b*(C*(2 
+ m) + A*(3 + m))*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, ( 
4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2]))/(2 + m)) 
/(3 + m)
 

3.8.67.3.1 Defintions of rubi rules used

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]
 

rule 3513

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[ 
(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 3) 
)), x] + Simp[1/(b*(m + 3))   Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*c 
*(m + 3) + b*d*(C*(m + 2) + A*(m + 3))*Sin[e + f*x] - (2*a*C*d - b*c*C*(m + 
 3))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && 
 NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1]
 

3.8.67.4 Maple [F]

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

input

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

output

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

3.8.67.5 Fricas [F]

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

input

integrate(cos(d*x+c)^m*(a+b*cos(d*x+c))*(A+C*cos(d*x+c)^2),x, algorithm="f 
ricas")
 

output

integral((C*b*cos(d*x + c)^3 + C*a*cos(d*x + c)^2 + A*b*cos(d*x + c) + A*a 
)*cos(d*x + c)^m, x)
 

3.8.67.6 Sympy [F(-1)]

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

input

integrate(cos(d*x+c)**m*(a+b*cos(d*x+c))*(A+C*cos(d*x+c)**2),x)
 

output

Timed out
 

3.8.67.7 Maxima [F]

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

input

integrate(cos(d*x+c)^m*(a+b*cos(d*x+c))*(A+C*cos(d*x+c)^2),x, algorithm="m 
axima")
 

output

integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)*cos(d*x + c)^m, x)
 

3.8.67.8 Giac [F]

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

input

integrate(cos(d*x+c)^m*(a+b*cos(d*x+c))*(A+C*cos(d*x+c)^2),x, algorithm="g 
iac")
 

output

integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)*cos(d*x + c)^m, x)
 

3.8.67.9 Mupad [F(-1)]

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

input

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

output

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