\(\int (b \cos (c+d x))^{4/3} (A+C \cos ^2(c+d x)) \sec ^3(c+d x) \, dx\) [157]

Optimal result

Integrand size = 33, antiderivative size = 90 \[ \int (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {3 A b^2 \sin (c+d x)}{2 d (b \cos (c+d x))^{2/3}}+\frac {3 (A-2 C) (b \cos (c+d x))^{4/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{8 d \sqrt {\sin ^2(c+d x)}} \] Output:

3/2*A*b^2*sin(d*x+c)/d/(b*cos(d*x+c))^(2/3)+3/8*(A-2*C)*(b*cos(d*x+c))^(4/ 
3)*hypergeom([1/2, 2/3],[5/3],cos(d*x+c)^2)*sin(d*x+c)/d/(sin(d*x+c)^2)^(1 
/2)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=-\frac {3 b^2 \csc (c+d x) \left (-2 A \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{2},\frac {2}{3},\cos ^2(c+d x)\right )+C \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{4 d (b \cos (c+d x))^{2/3}} \] Input:

Integrate[(b*Cos[c + d*x])^(4/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]
 

Output:

(-3*b^2*Csc[c + d*x]*(-2*A*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^ 
2] + C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2])*Sq 
rt[Sin[c + d*x]^2])/(4*d*(b*Cos[c + d*x])^(2/3))
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {3042, 2030, 3491, 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.

\(\displaystyle \int \sec ^3(c+d x) (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 2030

\(\displaystyle b^3 \int \frac {C \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^2+A}{\left (b \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )^{5/3}}dx\)

\(\Big \downarrow \) 3491

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3122

\(\displaystyle b^3 \left (\frac {3 (A-2 C) \sin (c+d x) (b \cos (c+d x))^{4/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(c+d x)\right )}{8 b^3 d \sqrt {\sin ^2(c+d x)}}+\frac {3 A \sin (c+d x)}{2 b d (b \cos (c+d x))^{2/3}}\right )\)

Input:

Int[(b*Cos[c + d*x])^(4/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]
 

Output:

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

Defintions of rubi rules used

rule 2030

Int[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m   Int[(b*v) 
^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
 

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 3491

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

Maple [F]

\[\int \left (b \cos \left (d x +c \right )\right )^{\frac {4}{3}} \left (A +C \cos \left (d x +c \right )^{2}\right ) \sec \left (d x +c \right )^{3}d x\]

Input:

int((b*cos(d*x+c))^(4/3)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^3,x)
 

Output:

int((b*cos(d*x+c))^(4/3)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^3,x)
 

Fricas [F]

\[ \int (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}} \sec \left (d x + c\right )^{3} \,d x } \] Input:

integrate((b*cos(d*x+c))^(4/3)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^3,x, algorith 
m="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\text {Timed out} \] Input:

integrate((b*cos(d*x+c))**(4/3)*(A+C*cos(d*x+c)**2)*sec(d*x+c)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}} \sec \left (d x + c\right )^{3} \,d x } \] Input:

integrate((b*cos(d*x+c))^(4/3)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^3,x, algorith 
m="maxima")
 

Output:

integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c))^(4/3)*sec(d*x + c)^3, x)
 

Giac [F]

\[ \int (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}} \sec \left (d x + c\right )^{3} \,d x } \] Input:

integrate((b*cos(d*x+c))^(4/3)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^3,x, algorith 
m="giac")
 

Output:

integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c))^(4/3)*sec(d*x + c)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{4/3}}{{\cos \left (c+d\,x\right )}^3} \,d x \] Input:

int(((A + C*cos(c + d*x)^2)*(b*cos(c + d*x))^(4/3))/cos(c + d*x)^3,x)
 

Output:

int(((A + C*cos(c + d*x)^2)*(b*cos(c + d*x))^(4/3))/cos(c + d*x)^3, x)
 

Reduce [F]

\[ \int (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=b^{\frac {4}{3}} \left (\left (\int \cos \left (d x +c \right )^{\frac {4}{3}} \sec \left (d x +c \right )^{3}d x \right ) a +\left (\int \cos \left (d x +c \right )^{\frac {10}{3}} \sec \left (d x +c \right )^{3}d x \right ) c \right ) \] Input:

int((b*cos(d*x+c))^(4/3)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^3,x)
 

Output:

b**(1/3)*b*(int(cos(c + d*x)**(1/3)*cos(c + d*x)*sec(c + d*x)**3,x)*a + in 
t(cos(c + d*x)**(1/3)*cos(c + d*x)**3*sec(c + d*x)**3,x)*c)