Integrand size = 23, antiderivative size = 200 \[ \int \frac {\cos ^3(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} \sqrt [3]{b} d}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \sqrt [3]{b} d}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} \sqrt [3]{b} d}+\frac {1}{3 b d \left (a+b \sin ^3(c+d x)\right )}+\frac {\sin (c+d x)}{3 a d \left (a+b \sin ^3(c+d x)\right )} \] Output:
-2/9*arctan(1/3*(a^(1/3)-2*b^(1/3)*sin(d*x+c))*3^(1/2)/a^(1/3))*3^(1/2)/a^ (5/3)/b^(1/3)/d+2/9*ln(a^(1/3)+b^(1/3)*sin(d*x+c))/a^(5/3)/b^(1/3)/d-1/9*l n(a^(2/3)-a^(1/3)*b^(1/3)*sin(d*x+c)+b^(2/3)*sin(d*x+c)^2)/a^(5/3)/b^(1/3) /d+1/3/b/d/(a+b*sin(d*x+c)^3)+1/3*sin(d*x+c)/a/d/(a+b*sin(d*x+c)^3)
Time = 0.84 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^3(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{5/3} \sqrt [3]{b}}+\frac {3 \sin (c+d x)}{a \left (a+b \sin ^3(c+d x)\right )}+\frac {\frac {2 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{a^{5/3}}-\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{a^{5/3}}+\frac {3}{a+b \sin ^3(c+d x)}}{b}}{9 d} \] Input:
Integrate[Cos[c + d*x]^3/(a + b*Sin[c + d*x]^3)^2,x]
Output:
((-2*Sqrt[3]*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))]) /(a^(5/3)*b^(1/3)) + (3*Sin[c + d*x])/(a*(a + b*Sin[c + d*x]^3)) + ((2*b^( 2/3)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/a^(5/3) - (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/a^(5/3) + 3/(a + b*Sin[c + d*x]^3))/b)/(9*d)
Time = 0.42 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.88, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 3702, 2393, 27, 750, 16, 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.
| \(\displaystyle \int \frac {\cos ^3(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^3}{\left (a+b \sin (c+d x)^3\right )^2}dx\) |
| \(\Big \downarrow \) 3702 |
| \(\displaystyle \frac {\int \frac {1-\sin ^2(c+d x)}{\left (b \sin ^3(c+d x)+a\right )^2}d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 2393 |
| \(\displaystyle \frac {\frac {a+b \sin (c+d x)}{3 a b \left (a+b \sin ^3(c+d x)\right )}-\frac {\int -\frac {2}{b \sin ^3(c+d x)+a}d\sin (c+d x)}{3 a}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \int \frac {1}{b \sin ^3(c+d x)+a}d\sin (c+d x)}{3 a}+\frac {a+b \sin (c+d x)}{3 a b \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} \sin (c+d x)+\sqrt [3]{a}}d\sin (c+d x)}{3 a^{2/3}}\right )}{3 a}+\frac {a+b \sin (c+d x)}{3 a b \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {a+b \sin (c+d x)}{3 a b \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)\right )}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {a+b \sin (c+d x)}{3 a b \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)\right )}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {a+b \sin (c+d x)}{3 a b \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {a+b \sin (c+d x)}{3 a b \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {a+b \sin (c+d x)}{3 a b \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {a+b \sin (c+d x)}{3 a b \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {2 \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {a+b \sin (c+d x)}{3 a b \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
Input:
Int[Cos[c + d*x]^3/(a + b*Sin[c + d*x]^3)^2,x]
Output:
((2*(Log[a^(1/3) + b^(1/3)*Sin[c + d*x]]/(3*a^(2/3)*b^(1/3)) + (-((Sqrt[3] *ArcTan[(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))))/(3*a) + (a + b*Sin[c + d*x])/(3*a*b*(a + b*Sin[c + d*x]^3) ))/d
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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]
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]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[(a*Coeff[Pq, x, q] - b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q , x])*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) In t[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^( p + 1), x], x] /; q == n - 1] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n , 0] && LtQ[p, -1]
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])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.23 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.76
| method | result | size |
| risch | \(-\frac {4 \left (2 i a \,{\mathrm e}^{3 i \left (d x +c \right )}+b \,{\mathrm e}^{4 i \left (d x +c \right )}-b \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 a b d \left ({\mathrm e}^{6 i \left (d x +c \right )} b -3 b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{2 i \left (d x +c \right )}-8 i a \,{\mathrm e}^{3 i \left (d x +c \right )}-b \right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (729 a^{5} b \,d^{3} \textit {\_Z}^{3}-8\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+9 i a^{2} d \textit {\_R} \,{\mathrm e}^{i \left (d x +c \right )}-1\right )\right )\) | \(153\) |
| derivativedivides | \(\frac {\frac {\sin \left (d x +c \right )}{3 a \left (a +b \sin \left (d x +c \right )^{3}\right )}+\frac {\frac {2 \ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{a}+\frac {1}{3 b \left (a +b \sin \left (d x +c \right )^{3}\right )}}{d}\) | \(165\) |
| default | \(\frac {\frac {\sin \left (d x +c \right )}{3 a \left (a +b \sin \left (d x +c \right )^{3}\right )}+\frac {\frac {2 \ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{a}+\frac {1}{3 b \left (a +b \sin \left (d x +c \right )^{3}\right )}}{d}\) | \(165\) |
Input:
int(cos(d*x+c)^3/(a+b*sin(d*x+c)^3)^2,x,method=_RETURNVERBOSE)
Output:
-4/3/a/b/d/(exp(6*I*(d*x+c))*b-3*b*exp(4*I*(d*x+c))+3*b*exp(2*I*(d*x+c))-8 *I*a*exp(3*I*(d*x+c))-b)*(2*I*a*exp(3*I*(d*x+c))+b*exp(4*I*(d*x+c))-b*exp( 2*I*(d*x+c)))+sum(_R*ln(exp(2*I*(d*x+c))+9*I*a^2*d*_R*exp(I*(d*x+c))-1),_R =RootOf(729*_Z^3*a^5*b*d^3-8))
Time = 0.15 (sec) , antiderivative size = 665, normalized size of antiderivative = 3.32 \[ \int \frac {\cos ^3(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^3/(a+b*sin(d*x+c)^3)^2,x, algorithm="fricas")
Output:
[1/9*(3*a^2*b*sin(d*x + c) + 3*a^3 + 3*sqrt(1/3)*(a^2*b - (a*b^2*cos(d*x + c)^2 - a*b^2)*sin(d*x + c))*sqrt(-(a^2*b)^(1/3)/b)*log(-(3*(a^2*b)^(1/3)* a*sin(d*x + c) + a^2 + 3*sqrt(1/3)*(2*a*b*cos(d*x + c)^2 - 2*a*b - (a^2*b) ^(2/3)*sin(d*x + c) + (a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^(1/3)/b) + 2*(a*b*cos (d*x + c)^2 - a*b)*sin(d*x + c))/((b*cos(d*x + c)^2 - b)*sin(d*x + c) - a) ) + (a^2*b)^(2/3)*((b*cos(d*x + c)^2 - b)*sin(d*x + c) - a)*log(-a*b*cos(d *x + c)^2 + a*b - (a^2*b)^(2/3)*sin(d*x + c) + (a^2*b)^(1/3)*a) - 2*(a^2*b )^(2/3)*((b*cos(d*x + c)^2 - b)*sin(d*x + c) - a)*log(a*b*sin(d*x + c) + ( a^2*b)^(2/3)))/(a^4*b*d - (a^3*b^2*d*cos(d*x + c)^2 - a^3*b^2*d)*sin(d*x + c)), 1/9*(3*a^2*b*sin(d*x + c) + 3*a^3 + 6*sqrt(1/3)*(a^2*b - (a*b^2*cos( d*x + c)^2 - a*b^2)*sin(d*x + c))*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*( 2*(a^2*b)^(2/3)*sin(d*x + c) - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) + (a^2*b)^(2/3)*((b*cos(d*x + c)^2 - b)*sin(d*x + c) - a)*log(-a*b*cos(d* x + c)^2 + a*b - (a^2*b)^(2/3)*sin(d*x + c) + (a^2*b)^(1/3)*a) - 2*(a^2*b) ^(2/3)*((b*cos(d*x + c)^2 - b)*sin(d*x + c) - a)*log(a*b*sin(d*x + c) + (a ^2*b)^(2/3)))/(a^4*b*d - (a^3*b^2*d*cos(d*x + c)^2 - a^3*b^2*d)*sin(d*x + c))]
Timed out. \[ \int \frac {\cos ^3(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**3/(a+b*sin(d*x+c)**3)**2,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.82 \[ \int \frac {\cos ^3(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {\frac {3 \, {\left (b \sin \left (d x + c\right ) + a\right )}}{a b^{2} \sin \left (d x + c\right )^{3} + a^{2} b} + \frac {2 \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (\sin \left (d x + c\right )^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right )\right )}{a b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{9 \, d} \] Input:
integrate(cos(d*x+c)^3/(a+b*sin(d*x+c)^3)^2,x, algorithm="maxima")
Output:
1/9*(3*(b*sin(d*x + c) + a)/(a*b^2*sin(d*x + c)^3 + a^2*b) + 2*sqrt(3)*arc tan(-1/3*sqrt(3)*((a/b)^(1/3) - 2*sin(d*x + c))/(a/b)^(1/3))/(a*b*(a/b)^(2 /3)) - log(sin(d*x + c)^2 - (a/b)^(1/3)*sin(d*x + c) + (a/b)^(2/3))/(a*b*( a/b)^(2/3)) + 2*log((a/b)^(1/3) + sin(d*x + c))/(a*b*(a/b)^(2/3)))/d
Time = 0.39 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.88 \[ \int \frac {\cos ^3(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=-\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | -\left (-\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right ) \right |}\right )}{9 \, a^{2} d} + \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} b d} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (\sin \left (d x + c\right )^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a^{2} b d} + \frac {b \sin \left (d x + c\right ) + a}{3 \, {\left (b \sin \left (d x + c\right )^{3} + a\right )} a b d} \] Input:
integrate(cos(d*x+c)^3/(a+b*sin(d*x+c)^3)^2,x, algorithm="giac")
Output:
-2/9*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x + c)))/(a^2*d) + 2/9*sqr t(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(d*x + c))/(-a /b)^(1/3))/(a^2*b*d) + 1/9*(-a*b^2)^(1/3)*log(sin(d*x + c)^2 + (-a/b)^(1/3 )*sin(d*x + c) + (-a/b)^(2/3))/(a^2*b*d) + 1/3*(b*sin(d*x + c) + a)/((b*si n(d*x + c)^3 + a)*a*b*d)
Time = 0.36 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.86 \[ \int \frac {\cos ^3(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {\frac {\sin \left (c+d\,x\right )}{3\,a}+\frac {1}{3\,b}}{d\,\left (b\,{\sin \left (c+d\,x\right )}^3+a\right )}+\frac {2\,\ln \left (\frac {2\,b^{5/3}}{a^{2/3}}+\frac {2\,b^2\,\sin \left (c+d\,x\right )}{a}\right )}{9\,a^{5/3}\,b^{1/3}\,d}+\frac {\ln \left (\frac {2\,b^2\,\sin \left (c+d\,x\right )}{a}+\frac {b^{5/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{a^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,a^{5/3}\,b^{1/3}\,d}-\frac {\ln \left (\frac {2\,b^2\,\sin \left (c+d\,x\right )}{a}-\frac {b^{5/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{a^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,a^{5/3}\,b^{1/3}\,d} \] Input:
int(cos(c + d*x)^3/(a + b*sin(c + d*x)^3)^2,x)
Output:
(sin(c + d*x)/(3*a) + 1/(3*b))/(d*(a + b*sin(c + d*x)^3)) + (2*log((2*b^(5 /3))/a^(2/3) + (2*b^2*sin(c + d*x))/a))/(9*a^(5/3)*b^(1/3)*d) + (log((2*b^ 2*sin(c + d*x))/a + (b^(5/3)*(3^(1/2)*1i - 1))/a^(2/3))*(3^(1/2)*1i - 1))/ (9*a^(5/3)*b^(1/3)*d) - (log((2*b^2*sin(c + d*x))/a - (b^(5/3)*(3^(1/2)*1i + 1))/a^(2/3))*(3^(1/2)*1i + 1))/(9*a^(5/3)*b^(1/3)*d)
\[ \int \frac {\cos ^3(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int \frac {\cos \left (d x +c \right )^{3}}{\sin \left (d x +c \right )^{6} b^{2}+2 \sin \left (d x +c \right )^{3} a b +a^{2}}d x \] Input:
int(cos(d*x+c)^3/(a+b*sin(d*x+c)^3)^2,x)
Output:
int(cos(c + d*x)**3/(sin(c + d*x)**6*b**2 + 2*sin(c + d*x)**3*a*b + a**2), x)