3.2.78 \(\int \cos ^m(c+d x) \sqrt [3]{b \cos (c+d x)} (A+C \cos ^2(c+d x)) \, dx\) [178]

3.2.78.1 Optimal result

Integrand size = 33, antiderivative size = 146 \[ \int \cos ^m(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {3 C \cos ^{1+m}(c+d x) \sqrt [3]{b \cos (c+d x)} \sin (c+d x)}{d (7+3 m)}-\frac {3 (C (4+3 m)+A (7+3 m)) \cos ^{1+m}(c+d x) \sqrt [3]{b \cos (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (4+3 m),\frac {1}{6} (10+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (4+3 m) (7+3 m) \sqrt {\sin ^2(c+d x)}} \]

output

3*C*cos(d*x+c)^(1+m)*(b*cos(d*x+c))^(1/3)*sin(d*x+c)/d/(7+3*m)-3*(C*(4+3*m 
)+A*(7+3*m))*cos(d*x+c)^(1+m)*(b*cos(d*x+c))^(1/3)*hypergeom([1/2, 2/3+1/2 
*m],[5/3+1/2*m],cos(d*x+c)^2)*sin(d*x+c)/d/(9*m^2+33*m+28)/(sin(d*x+c)^2)^ 
(1/2)
 

3.2.78.2 Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.97 \[ \int \cos ^m(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {3 \cos ^{1+m}(c+d x) \sqrt [3]{b \cos (c+d x)} \csc (c+d x) \left (C (4+3 m) \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{3}+\frac {m}{2},\frac {8}{3}+\frac {m}{2},\cos ^2(c+d x)\right )+A (10+3 m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (4+3 m),\frac {5}{3}+\frac {m}{2},\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (4+3 m) (10+3 m)} \]

input

Integrate[Cos[c + d*x]^m*(b*Cos[c + d*x])^(1/3)*(A + C*Cos[c + d*x]^2),x]
 

output

(-3*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(1/3)*Csc[c + d*x]*(C*(4 + 3*m)* 
Cos[c + d*x]^2*Hypergeometric2F1[1/2, 5/3 + m/2, 8/3 + m/2, Cos[c + d*x]^2 
] + A*(10 + 3*m)*Hypergeometric2F1[1/2, (4 + 3*m)/6, 5/3 + m/2, Cos[c + d* 
x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(4 + 3*m)*(10 + 3*m))
 

3.2.78.3 Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2034, 3042, 3493, 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.

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

\(\Big \downarrow \) 2034

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3493

\(\displaystyle \frac {\sqrt [3]{b \cos (c+d x)} \left (\frac {(A (3 m+7)+C (3 m+4)) \int \cos ^{m+\frac {1}{3}}(c+d x)dx}{3 m+7}+\frac {3 C \sin (c+d x) \cos ^{m+\frac {4}{3}}(c+d x)}{d (3 m+7)}\right )}{\sqrt [3]{\cos (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt [3]{b \cos (c+d x)} \left (\frac {(A (3 m+7)+C (3 m+4)) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{m+\frac {1}{3}}dx}{3 m+7}+\frac {3 C \sin (c+d x) \cos ^{m+\frac {4}{3}}(c+d x)}{d (3 m+7)}\right )}{\sqrt [3]{\cos (c+d x)}}\)

\(\Big \downarrow \) 3122

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

input

Int[Cos[c + d*x]^m*(b*Cos[c + d*x])^(1/3)*(A + C*Cos[c + d*x]^2),x]
 

output

((b*Cos[c + d*x])^(1/3)*((3*C*Cos[c + d*x]^(4/3 + m)*Sin[c + d*x])/(d*(7 + 
 3*m)) - (3*(C*(4 + 3*m) + A*(7 + 3*m))*Cos[c + d*x]^(4/3 + m)*Hypergeomet 
ric2F1[1/2, (4 + 3*m)/6, (10 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(4 
 + 3*m)*(7 + 3*m)*Sqrt[Sin[c + d*x]^2])))/Cos[c + d*x]^(1/3)
 

3.2.78.3.1 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 3493

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

3.2.78.4 Maple [F]

\[\int \left (\cos ^{m}\left (d x +c \right )\right ) \left (\cos \left (d x +c \right ) b \right )^{\frac {1}{3}} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )d x\]

input

int(cos(d*x+c)^m*(cos(d*x+c)*b)^(1/3)*(A+C*cos(d*x+c)^2),x)
 

output

int(cos(d*x+c)^m*(cos(d*x+c)*b)^(1/3)*(A+C*cos(d*x+c)^2),x)
 

3.2.78.5 Fricas [F]

\[ \int \cos ^m(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}} \cos \left (d x + c\right )^{m} \,d x } \]

input

integrate(cos(d*x+c)^m*(b*cos(d*x+c))^(1/3)*(A+C*cos(d*x+c)^2),x, algorith 
m="fricas")
 

output

integral((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c))^(1/3)*cos(d*x + c)^m, x)
 

3.2.78.6 Sympy [F]

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

input

integrate(cos(d*x+c)**m*(b*cos(d*x+c))**(1/3)*(A+C*cos(d*x+c)**2),x)
 

output

Integral((b*cos(c + d*x))**(1/3)*(A + C*cos(c + d*x)**2)*cos(c + d*x)**m, 
x)
 

3.2.78.7 Maxima [F]

\[ \int \cos ^m(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}} \cos \left (d x + c\right )^{m} \,d x } \]

input

integrate(cos(d*x+c)^m*(b*cos(d*x+c))^(1/3)*(A+C*cos(d*x+c)^2),x, algorith 
m="maxima")
 

output

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

3.2.78.8 Giac [F]

\[ \int \cos ^m(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}} \cos \left (d x + c\right )^{m} \,d x } \]

input

integrate(cos(d*x+c)^m*(b*cos(d*x+c))^(1/3)*(A+C*cos(d*x+c)^2),x, algorith 
m="giac")
 

output

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

3.2.78.9 Mupad [F(-1)]

Timed out. \[ \int \cos ^m(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^m\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{1/3} \,d x \]

input

int(cos(c + d*x)^m*(A + C*cos(c + d*x)^2)*(b*cos(c + d*x))^(1/3),x)
 

output

int(cos(c + d*x)^m*(A + C*cos(c + d*x)^2)*(b*cos(c + d*x))^(1/3), x)