\(\int \frac {\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx\) [317]

Optimal result

Integrand size = 23, antiderivative size = 167 \[ \int \frac {\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b} d}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b} d}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} \sqrt [3]{b} d}-\frac {\log \left (a+b \sin ^3(c+d x)\right )}{3 b d} \] Output:

-1/3*arctan(1/3*(a^(1/3)-2*b^(1/3)*sin(d*x+c))*3^(1/2)/a^(1/3))*3^(1/2)/a^ 
(2/3)/b^(1/3)/d+1/3*ln(a^(1/3)+b^(1/3)*sin(d*x+c))/a^(2/3)/b^(1/3)/d-1/6*l 
n(a^(2/3)-a^(1/3)*b^(1/3)*sin(d*x+c)+b^(2/3)*sin(d*x+c)^2)/a^(2/3)/b^(1/3) 
/d-1/3*ln(a+b*sin(d*x+c)^3)/b/d
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.83 \[ \int \frac {\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {\left (-a^{2/3}+(-1)^{2/3} b^{2/3}\right ) \log \left (-(-1)^{2/3} \sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)\right )+\left (-a^{2/3}+b^{2/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )-\left (a^{2/3}+\sqrt [3]{-1} b^{2/3}\right ) \log \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} b d} \] Input:

Integrate[Cos[c + d*x]^3/(a + b*Sin[c + d*x]^3),x]
 

Output:

((-a^(2/3) + (-1)^(2/3)*b^(2/3))*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sin[c 
 + d*x]] + (-a^(2/3) + b^(2/3))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]] - (a^( 
2/3) + (-1)^(1/3)*b^(2/3))*Log[a^(1/3) + (-1)^(2/3)*b^(1/3)*Sin[c + d*x]]) 
/(3*a^(2/3)*b*d)
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.93, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 3702, 2410, 750, 16, 792, 1142, 25, 27, 1082, 217, 1103}

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.

\(\displaystyle \int \frac {\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (c+d x)^3}{a+b \sin (c+d x)^3}dx\)

\(\Big \downarrow \) 3702

\(\displaystyle \frac {\int \frac {1-\sin ^2(c+d x)}{b \sin ^3(c+d x)+a}d\sin (c+d x)}{d}\)

\(\Big \downarrow \) 2410

\(\displaystyle \frac {\int \frac {1}{b \sin ^3(c+d x)+a}d\sin (c+d x)-\int \frac {\sin ^2(c+d x)}{b \sin ^3(c+d x)+a}d\sin (c+d x)}{d}\)

\(\Big \downarrow \) 750

\(\displaystyle \frac {\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} \sin (c+d x)+\sqrt [3]{a}}d\sin (c+d x)}{3 a^{2/3}}-\int \frac {\sin ^2(c+d x)}{b \sin ^3(c+d x)+a}d\sin (c+d x)}{d}\)

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 792

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

\(\Big \downarrow \) 1142

\(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)\right )}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (a+b \sin ^3(c+d x)\right )}{3 b}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)\right )}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (a+b \sin ^3(c+d x)\right )}{3 b}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (a+b \sin ^3(c+d x)\right )}{3 b}}{d}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (a+b \sin ^3(c+d x)\right )}{3 b}}{d}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (a+b \sin ^3(c+d x)\right )}{3 b}}{d}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (a+b \sin ^3(c+d x)\right )}{3 b}}{d}\)

Input:

Int[Cos[c + d*x]^3/(a + b*Sin[c + d*x]^3),x]
 

Output:

(Log[a^(1/3) + b^(1/3)*Sin[c + d*x]]/(3*a^(2/3)*b^(1/3)) + (-((Sqrt[3]*Arc 
Tan[(1 - (2*b^(1/3)*Sin[c + d*x])/a^(1/3))/Sqrt[3]])/b^(1/3)) - Log[a^(2/3 
) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]/(2*b^(1/3)))/(3 
*a^(2/3)) - Log[a + b*Sin[c + d*x]^3]/(3*b))/d
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 750

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2)   Int[1/ 
(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2)   Int[(2*Rt[a, 3] - 
 Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; 
 FreeQ[{a, b}, x]
 

rule 792

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveConten 
t[a + b*x^n, x]]/(b*n), x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 2410

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B 
= Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, Int[(A + B*x)/(a + b*x^3), x] + Si 
mp[C   Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !RationalQ[ 
a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
 

rule 3042

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

rule 3702

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x 
_)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Si 
mp[ff/f   Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x], x, 
 Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 
1)/2] && (EqQ[n, 4] || GtQ[m, 0] || IGtQ[p, 0] || IntegersQ[m, p])
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.60 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.63

method	result	size
risch	\(\frac {i x}{b}+\frac {2 i c}{b d}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (27 a^{2} \textit {\_Z}^{3} d^{3} b^{3}+27 a^{2} b^{2} d^{2} \textit {\_Z}^{2}+9 a^{2} d \textit {\_Z} b +a^{2}-b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (6 i a d \textit {\_R} +\frac {2 i a}{b}\right ) {\mathrm e}^{i \left (d x +c \right )}-1\right )\right )\)	\(106\)
derivativedivides	\(\frac {\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (a +b \sin \left (d x +c \right )^{3}\right )}{3 b}}{d}\)	\(133\)
default	\(\frac {\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (a +b \sin \left (d x +c \right )^{3}\right )}{3 b}}{d}\)	\(133\)

Input:

int(cos(d*x+c)^3/(a+b*sin(d*x+c)^3),x,method=_RETURNVERBOSE)
 

Output:

I*x/b+2*I/b/d*c+sum(_R*ln(exp(2*I*(d*x+c))+(6*I*a*d*_R+2*I/b*a)*exp(I*(d*x 
+c))-1),_R=RootOf(27*_Z^3*a^2*b^3*d^3+27*_Z^2*a^2*b^2*d^2+9*_Z*a^2*b*d+a^2 
-b^2))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.76 (sec) , antiderivative size = 1049, normalized size of antiderivative = 6.28 \[ \int \frac {\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)^3/(a+b*sin(d*x+c)^3),x, algorithm="fricas")
 

Output:

-1/12*(2*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 
- b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))*b*d*log(-1/2*((1/2)^(1/3)*(I*sqrt(3 
) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2 
/(b*d))*a*b*d + b*sin(d*x + c) + a) - (((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^ 
3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))*b*d + 
 3*sqrt(1/3)*b*d*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2 
*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))^2*b^2*d^2 - 4*((1/2) 
^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3 
*d^3))^(1/3) + 2/(b*d))*b*d + 4)/(b^2*d^2)) - 6)*log(1/2*((1/2)^(1/3)*(I*s 
qrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3 
) + 2/(b*d))*a*b*d + 3/2*sqrt(1/3)*a*b*d*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 
1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b* 
d))^2*b^2*d^2 - 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3 
) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))*b*d + 4)/(b^2*d^2)) + 2*b* 
sin(d*x + c) - a) - (((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b* 
d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))*b*d - 3*sqrt(1/3)*b*d*s 
qrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b 
^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))^2*b^2*d^2 - 4*((1/2)^(1/3)*(I*sqrt(3) 
+ 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/( 
b*d))*b*d + 4)/(b^2*d^2)) - 6)*log(-1/2*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**3/(a+b*sin(d*x+c)**3),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.95 \[ \int \frac {\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {\frac {2 \, \sqrt {3} {\left (b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} - \frac {2 \, a}{b}\right )} + 2 \, a\right )} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a b} - \frac {3 \, {\left (2 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 1\right )} \log \left (\sin \left (d x + c\right )^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {6 \, {\left (\left (\frac {a}{b}\right )^{\frac {2}{3}} - 1\right )} \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right )\right )}{b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{18 \, d} \] Input:

integrate(cos(d*x+c)^3/(a+b*sin(d*x+c)^3),x, algorithm="maxima")
 

Output:

1/18*(2*sqrt(3)*(b*(3*(a/b)^(1/3) - 2*a/b) + 2*a)*arctan(-1/3*sqrt(3)*((a/ 
b)^(1/3) - 2*sin(d*x + c))/(a/b)^(1/3))/(a*b) - 3*(2*(a/b)^(2/3) + 1)*log( 
sin(d*x + c)^2 - (a/b)^(1/3)*sin(d*x + c) + (a/b)^(2/3))/(b*(a/b)^(2/3)) - 
 6*((a/b)^(2/3) - 1)*log((a/b)^(1/3) + sin(d*x + c))/(b*(a/b)^(2/3)))/d
 

Giac [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.98 \[ \int \frac {\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=-\frac {\left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | -\left (-\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right ) \right |}\right )}{3 \, a d} - \frac {\log \left ({\left | b \sin \left (d x + c\right )^{3} + a \right |}\right )}{3 \, b d} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b d} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (\sin \left (d x + c\right )^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a b d} \] Input:

integrate(cos(d*x+c)^3/(a+b*sin(d*x+c)^3),x, algorithm="giac")
 

Output:

-1/3*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x + c)))/(a*d) - 1/3*log(a 
bs(b*sin(d*x + c)^3 + a))/(b*d) + 1/3*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sq 
rt(3)*((-a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a*b*d) + 1/6*(-a*b^2) 
^(1/3)*log(sin(d*x + c)^2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a*b 
*d)
 

Mupad [B] (verification not implemented)

Time = 35.70 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {\sum _{k=1}^3\ln \left (\left (\mathrm {root}\left (27\,a^2\,b^3\,d^3+27\,a^2\,b^2\,d^2+9\,a^2\,b\,d-b^2+a^2,d,k\right )\,b\,3+1\right )\,\left (a+b\,\sin \left (c+d\,x\right )+\mathrm {root}\left (27\,a^2\,b^3\,d^3+27\,a^2\,b^2\,d^2+9\,a^2\,b\,d-b^2+a^2,d,k\right )\,a\,b\,3\right )\right )\,\mathrm {root}\left (27\,a^2\,b^3\,d^3+27\,a^2\,b^2\,d^2+9\,a^2\,b\,d-b^2+a^2,d,k\right )}{d} \] Input:

int(cos(c + d*x)^3/(a + b*sin(c + d*x)^3),x)
 

Output:

symsum(log((3*root(27*a^2*b^3*d^3 + 27*a^2*b^2*d^2 + 9*a^2*b*d - b^2 + a^2 
, d, k)*b + 1)*(a + b*sin(c + d*x) + 3*root(27*a^2*b^3*d^3 + 27*a^2*b^2*d^ 
2 + 9*a^2*b*d - b^2 + a^2, d, k)*a*b))*root(27*a^2*b^3*d^3 + 27*a^2*b^2*d^ 
2 + 9*a^2*b*d - b^2 + a^2, d, k), k, 1, 3)/d
 

Reduce [F]

\[ \int \frac {\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int \frac {\cos \left (d x +c \right )^{3}}{\sin \left (d x +c \right )^{3} b +a}d x \] Input:

int(cos(d*x+c)^3/(a+b*sin(d*x+c)^3),x)
 

Output:

int(cos(c + d*x)**3/(sin(c + d*x)**3*b + a),x)