3.3.1 \(\int \frac {A+C \cos ^2(c+d x)}{\sqrt [3]{a+a \cos (c+d x)}} \, dx\) [201]

3.3.1.1 Optimal result

Integrand size = 27, antiderivative size = 135 \[ \int \frac {A+C \cos ^2(c+d x)}{\sqrt [3]{a+a \cos (c+d x)}} \, dx=-\frac {9 C \sin (c+d x)}{10 d \sqrt [3]{a+a \cos (c+d x)}}+\frac {3 C (a+a \cos (c+d x))^{2/3} \sin (c+d x)}{5 a d}+\frac {(10 A+7 C) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{5\ 2^{5/6} d \sqrt [6]{1+\cos (c+d x)} \sqrt [3]{a+a \cos (c+d x)}} \]

output

-9/10*C*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/3)+3/5*C*(a+a*cos(d*x+c))^(2/3)*s 
in(d*x+c)/a/d+1/10*(10*A+7*C)*hypergeom([1/2, 5/6],[3/2],1/2-1/2*cos(d*x+c 
))*sin(d*x+c)*2^(1/6)/d/(1+cos(d*x+c))^(1/6)/(a+a*cos(d*x+c))^(1/3)
 

3.3.1.2 Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.07 \[ \int \frac {A+C \cos ^2(c+d x)}{\sqrt [3]{a+a \cos (c+d x)}} \, dx=-\frac {3\ 2^{5/6} C \sqrt [6]{1-\cos \left (d x-2 \arctan \left (\cot \left (\frac {c}{2}\right )\right )\right )} (\sin (c+d x)-\sin (2 (c+d x)))+2 (10 A+7 C) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},\cos ^2\left (\frac {d x}{2}-\arctan \left (\cot \left (\frac {c}{2}\right )\right )\right )\right ) \sin \left (d x-2 \arctan \left (\cot \left (\frac {c}{2}\right )\right )\right )}{20 d \sqrt [3]{a (1+\cos (c+d x))} \sqrt [6]{\sin ^2\left (\frac {d x}{2}-\arctan \left (\cot \left (\frac {c}{2}\right )\right )\right )}} \]

input

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

output

-1/20*(3*2^(5/6)*C*(1 - Cos[d*x - 2*ArcTan[Cot[c/2]]])^(1/6)*(Sin[c + d*x] 
 - Sin[2*(c + d*x)]) + 2*(10*A + 7*C)*Hypergeometric2F1[1/2, 5/6, 3/2, Cos 
[(d*x)/2 - ArcTan[Cot[c/2]]]^2]*Sin[d*x - 2*ArcTan[Cot[c/2]]])/(d*(a*(1 + 
Cos[c + d*x]))^(1/3)*(Sin[(d*x)/2 - ArcTan[Cot[c/2]]]^2)^(1/6))
 

3.3.1.3 Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3503, 27, 3042, 3230, 3042, 3131, 3042, 3130}

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.

\(\displaystyle \int \frac {A+C \cos ^2(c+d x)}{\sqrt [3]{a \cos (c+d x)+a}} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3503

\(\displaystyle \frac {3 \int \frac {a (5 A+2 C)-3 a C \cos (c+d x)}{3 \sqrt [3]{\cos (c+d x) a+a}}dx}{5 a}+\frac {3 C \sin (c+d x) (a \cos (c+d x)+a)^{2/3}}{5 a d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a (5 A+2 C)-3 a C \cos (c+d x)}{\sqrt [3]{\cos (c+d x) a+a}}dx}{5 a}+\frac {3 C \sin (c+d x) (a \cos (c+d x)+a)^{2/3}}{5 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a (5 A+2 C)-3 a C \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt [3]{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{5 a}+\frac {3 C \sin (c+d x) (a \cos (c+d x)+a)^{2/3}}{5 a d}\)

\(\Big \downarrow \) 3230

\(\displaystyle \frac {\frac {1}{2} a (10 A+7 C) \int \frac {1}{\sqrt [3]{\cos (c+d x) a+a}}dx-\frac {9 a C \sin (c+d x)}{2 d \sqrt [3]{a \cos (c+d x)+a}}}{5 a}+\frac {3 C \sin (c+d x) (a \cos (c+d x)+a)^{2/3}}{5 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{2} a (10 A+7 C) \int \frac {1}{\sqrt [3]{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx-\frac {9 a C \sin (c+d x)}{2 d \sqrt [3]{a \cos (c+d x)+a}}}{5 a}+\frac {3 C \sin (c+d x) (a \cos (c+d x)+a)^{2/3}}{5 a d}\)

\(\Big \downarrow \) 3131

\(\displaystyle \frac {\frac {a (10 A+7 C) \sqrt [3]{\cos (c+d x)+1} \int \frac {1}{\sqrt [3]{\cos (c+d x)+1}}dx}{2 \sqrt [3]{a \cos (c+d x)+a}}-\frac {9 a C \sin (c+d x)}{2 d \sqrt [3]{a \cos (c+d x)+a}}}{5 a}+\frac {3 C \sin (c+d x) (a \cos (c+d x)+a)^{2/3}}{5 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {a (10 A+7 C) \sqrt [3]{\cos (c+d x)+1} \int \frac {1}{\sqrt [3]{\sin \left (c+d x+\frac {\pi }{2}\right )+1}}dx}{2 \sqrt [3]{a \cos (c+d x)+a}}-\frac {9 a C \sin (c+d x)}{2 d \sqrt [3]{a \cos (c+d x)+a}}}{5 a}+\frac {3 C \sin (c+d x) (a \cos (c+d x)+a)^{2/3}}{5 a d}\)

\(\Big \downarrow \) 3130

\(\displaystyle \frac {\frac {a (10 A+7 C) \sin (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x))\right )}{2^{5/6} d \sqrt [6]{\cos (c+d x)+1} \sqrt [3]{a \cos (c+d x)+a}}-\frac {9 a C \sin (c+d x)}{2 d \sqrt [3]{a \cos (c+d x)+a}}}{5 a}+\frac {3 C \sin (c+d x) (a \cos (c+d x)+a)^{2/3}}{5 a d}\)

input

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

output

(3*C*(a + a*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*a*d) + ((-9*a*C*Sin[c + d 
*x])/(2*d*(a + a*Cos[c + d*x])^(1/3)) + (a*(10*A + 7*C)*Hypergeometric2F1[ 
1/2, 5/6, 3/2, (1 - Cos[c + d*x])/2]*Sin[c + d*x])/(2^(5/6)*d*(1 + Cos[c + 
 d*x])^(1/6)*(a + a*Cos[c + d*x])^(1/3)))/(5*a)
 

3.3.1.3.1 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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3130

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2^(n + 
 1/2))*a^(n - 1/2)*b*(Cos[c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]]))*Hypergeome 
tric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - b*(Sin[c + d*x]/a))], x] /; FreeQ[{a, 
 b, c, d, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] && GtQ[a, 0]
 

rule 3131

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a^IntPar 
t[n]*((a + b*Sin[c + d*x])^FracPart[n]/(1 + (b/a)*Sin[c + d*x])^FracPart[n] 
)   Int[(1 + (b/a)*Sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && 
EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] &&  !GtQ[a, 0]
 

rule 3230

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

rule 3503

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

3.3.1.4 Maple [F]

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

input

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

output

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

3.3.1.5 Fricas [F]

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

input

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

output

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

3.3.1.6 Sympy [F]

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

input

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

output

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

3.3.1.7 Maxima [F]

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

input

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

output

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

3.3.1.8 Giac [F]

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

input

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

output

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

3.3.1.9 Mupad [F(-1)]

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

input

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

output

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