Integrand size = 23, antiderivative size = 284 \[ \int \frac {\cos ^7(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=-\frac {2 \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{7/3} d}+\frac {2 \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac {\left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac {\sin (c+d x)}{b^2 d}+\frac {3 a b-\left (a^2-b^2\right ) \sin (c+d x)-3 a b \sin ^2(c+d x)}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )} \] Output:
-2/9*(2*a^2+3*a^(4/3)*b^(2/3)+b^2)*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^(7/3)/d+2/9*(2*a^2-3*a^(4/3)*b^(2/3) +b^2)*ln(a^(1/3)+b^(1/3)*sin(d*x+c))/a^(5/3)/b^(7/3)/d-1/9*(2*a^2-3*a^(4/3 )*b^(2/3)+b^2)*ln(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^(7/3)/d-sin(d*x+c)/b^2/d+1/3*(3*a*b-(a^2-b^2)*sin(d*x+c)-3*a*b* sin(d*x+c)^2)/a/b^2/d/(a+b*sin(d*x+c)^3)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 3.46 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.55 \[ \int \frac {\cos ^7(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {\frac {4 \sqrt {3} \left (a^2-b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{5/3} b^{7/3}}+\frac {6 \sqrt [3]{-1} \left (2 \sqrt [3]{-1} a^{2/3}+3 b^{2/3}\right ) \log \left (-(-1)^{2/3} \sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)\right )}{\sqrt [3]{a} b^{7/3}}+\frac {6 \left (2 a^{2/3}-3 b^{2/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{\sqrt [3]{a} b^{7/3}}-\frac {4 \left (a^2-b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{a^{5/3} b^{7/3}}-\frac {6 \sqrt [3]{-1} \left (2 a^{2/3}+3 \sqrt [3]{-1} b^{2/3}\right ) \log \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)\right )}{\sqrt [3]{a} b^{7/3}}+\frac {2 \left (a^2-b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{a^{5/3} b^{7/3}}-\frac {18 \sin (c+d x)}{b^2}-\frac {27 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},2,\frac {5}{3},-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{a b}+\frac {18}{b \left (a+b \sin ^3(c+d x)\right )}+\frac {6 \left (1-\frac {a^2}{b^2}\right ) \sin (c+d x)}{a \left (a+b \sin ^3(c+d x)\right )}}{18 d} \] Input:
Integrate[Cos[c + d*x]^7/(a + b*Sin[c + d*x]^3)^2,x]
Output:
((4*Sqrt[3]*(a^2 - b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3] *a^(1/3))])/(a^(5/3)*b^(7/3)) + (6*(-1)^(1/3)*(2*(-1)^(1/3)*a^(2/3) + 3*b^ (2/3))*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sin[c + d*x]])/(a^(1/3)*b^(7/3) ) + (6*(2*a^(2/3) - 3*b^(2/3))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(a^(1/ 3)*b^(7/3)) - (4*(a^2 - b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(a^(5/3) *b^(7/3)) - (6*(-1)^(1/3)*(2*a^(2/3) + 3*(-1)^(1/3)*b^(2/3))*Log[a^(1/3) + (-1)^(2/3)*b^(1/3)*Sin[c + d*x]])/(a^(1/3)*b^(7/3)) + (2*(a^2 - b^2)*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) *b^(7/3)) - (18*Sin[c + d*x])/b^2 - (27*Hypergeometric2F1[2/3, 2, 5/3, -(( b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(a*b) + 18/(b*(a + b*Sin[c + d*x]^3) ) + (6*(1 - a^2/b^2)*Sin[c + d*x])/(a*(a + b*Sin[c + d*x]^3)))/(18*d)
Time = 0.57 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 3702, 2397, 25, 2426, 2009}
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.
| \(\displaystyle \int \frac {\cos ^7(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^7}{\left (a+b \sin (c+d x)^3\right )^2}dx\) |
| \(\Big \downarrow \) 3702 |
| \(\displaystyle \frac {\int \frac {\left (1-\sin ^2(c+d x)\right )^3}{\left (b \sin ^3(c+d x)+a\right )^2}d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle \frac {-\frac {\int -\frac {-3 a b \sin ^3(c+d x)+6 a b \sin (c+d x)+a^2+2 b^2}{b \sin ^3(c+d x)+a}d\sin (c+d x)}{3 a b^2}-\frac {\sin (c+d x) \left (a^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)-b^2\right )}{3 a b^2 \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {-3 a b \sin ^3(c+d x)+6 a b \sin (c+d x)+a^2+2 b^2}{b \sin ^3(c+d x)+a}d\sin (c+d x)}{3 a b^2}-\frac {\sin (c+d x) \left (a^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)-b^2\right )}{3 a b^2 \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 2426 |
| \(\displaystyle \frac {\frac {\int \left (\frac {2 \left (2 a^2+3 b \sin (c+d x) a+b^2\right )}{b \sin ^3(c+d x)+a}-3 a\right )d\sin (c+d x)}{3 a b^2}-\frac {\sin (c+d x) \left (a^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)-b^2\right )}{3 a b^2 \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {-\frac {2 \left (3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b}}-\frac {\left (-3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {2 \left (-3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}-3 a \sin (c+d x)}{3 a b^2}-\frac {\sin (c+d x) \left (a^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)-b^2\right )}{3 a b^2 \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
Input:
Int[Cos[c + d*x]^7/(a + b*Sin[c + d*x]^3)^2,x]
Output:
(((-2*(2*a^2 + 3*a^(4/3)*b^(2/3) + b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*b^(1/3)) + (2*(2*a^2 - 3*a^(4 /3)*b^(2/3) + b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(3*a^(2/3)*b^(1/3) ) - ((2*a^2 - 3*a^(4/3)*b^(2/3) + b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(3*a^(2/3)*b^(1/3)) - 3*a*Sin[c + d*x]) /(3*a*b^2) - (Sin[c + d*x]*(a^2 - b^2 + 3*a*b*Sin[c + d*x] + 3*b^2*Sin[c + d*x]^2))/(3*a*b^2*(a + b*Sin[c + d*x]^3)))/d
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x]}, S imp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[(a + b*x^n)^(p + 1)* ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]
Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IntegerQ[n]
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])
Time = 3.51 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.10
| method | result | size |
| derivativedivides | \(-\frac {\frac {\sin \left (d x +c \right )}{b^{2}}-\frac {\frac {-b \sin \left (d x +c \right )^{2}-\frac {\left (a^{2}-b^{2}\right ) \sin \left (d x +c \right )}{3 a}+b}{a +b \sin \left (d x +c \right )^{3}}+\frac {\frac {2 \left (2 a^{2}+b^{2}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+2 a b \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a}}{b^{2}}}{d}\) | \(311\) |
| default | \(-\frac {\frac {\sin \left (d x +c \right )}{b^{2}}-\frac {\frac {-b \sin \left (d x +c \right )^{2}-\frac {\left (a^{2}-b^{2}\right ) \sin \left (d x +c \right )}{3 a}+b}{a +b \sin \left (d x +c \right )^{3}}+\frac {\frac {2 \left (2 a^{2}+b^{2}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+2 a b \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a}}{b^{2}}}{d}\) | \(311\) |
| risch | \(\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 b^{2} d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 b^{2} d}-\frac {2 i \left (3 a b \,{\mathrm e}^{5 i \left (d x +c \right )}+6 a b \,{\mathrm e}^{3 i \left (d x +c \right )}+2 i a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-2 i b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+3 a b \,{\mathrm e}^{i \left (d x +c \right )}-2 i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+2 i b^{2} {\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 b^{2} d a \left ({\mathrm e}^{6 i \left (d x +c \right )} b -3 b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{2 i \left (d x +c \right )}-8 i a \,{\mathrm e}^{3 i \left (d x +c \right )}-b \right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (729 a^{5} b^{7} d^{3} \textit {\_Z}^{3}+\left (648 a^{5} b^{3} d +324 a^{3} b^{5} d \right ) \textit {\_Z} -64 a^{6}+120 a^{4} b^{2}-48 b^{4} a^{2}-8 b^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {243 i a^{5} b^{5} d^{2} \textit {\_R}^{2}}{16 a^{6}+78 a^{4} b^{2}+12 b^{4} a^{2}+2 b^{6}}+\left (\frac {72 i a^{6} b^{2} d}{16 a^{6}+78 a^{4} b^{2}+12 b^{4} a^{2}+2 b^{6}}+\frac {72 i a^{4} b^{4} d}{16 a^{6}+78 a^{4} b^{2}+12 b^{4} a^{2}+2 b^{6}}+\frac {18 i a^{2} b^{6} d}{16 a^{6}+78 a^{4} b^{2}+12 b^{4} a^{2}+2 b^{6}}\right ) \textit {\_R} +\frac {144 i a^{5} b}{16 a^{6}+78 a^{4} b^{2}+12 b^{4} a^{2}+2 b^{6}}+\frac {72 i a^{3} b^{3}}{16 a^{6}+78 a^{4} b^{2}+12 b^{4} a^{2}+2 b^{6}}\right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {16 a^{6}}{16 a^{6}+78 a^{4} b^{2}+12 b^{4} a^{2}+2 b^{6}}-\frac {78 a^{4} b^{2}}{16 a^{6}+78 a^{4} b^{2}+12 b^{4} a^{2}+2 b^{6}}-\frac {12 b^{4} a^{2}}{16 a^{6}+78 a^{4} b^{2}+12 b^{4} a^{2}+2 b^{6}}-\frac {2 b^{6}}{16 a^{6}+78 a^{4} b^{2}+12 b^{4} a^{2}+2 b^{6}}\right )\right )\) | \(674\) |
Input:
int(cos(d*x+c)^7/(a+b*sin(d*x+c)^3)^2,x,method=_RETURNVERBOSE)
Output:
-1/d*(1/b^2*sin(d*x+c)-1/b^2*((-b*sin(d*x+c)^2-1/3*(a^2-b^2)/a*sin(d*x+c)+ b)/(a+b*sin(d*x+c)^3)+2/3/a*((2*a^2+b^2)*(1/3/b/(1/b*a)^(2/3)*ln(sin(d*x+c )+(1/b*a)^(1/3))-1/6/b/(1/b*a)^(2/3)*ln(sin(d*x+c)^2-(1/b*a)^(1/3)*sin(d*x +c)+(1/b*a)^(2/3))+1/3/b/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b* a)^(1/3)*sin(d*x+c)-1)))+3*a*b*(-1/3/b/(1/b*a)^(1/3)*ln(sin(d*x+c)+(1/b*a) ^(1/3))+1/6/b/(1/b*a)^(1/3)*ln(sin(d*x+c)^2-(1/b*a)^(1/3)*sin(d*x+c)+(1/b* a)^(2/3))+1/3*3^(1/2)/b/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)* sin(d*x+c)-1))))))
Result contains complex when optimal does not.
Time = 1.22 (sec) , antiderivative size = 3624, normalized size of antiderivative = 12.76 \[ \int \frac {\cos ^7(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^7/(a+b*sin(d*x+c)^3)^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\cos ^7(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**7/(a+b*sin(d*x+c)**3)**2,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.93 \[ \int \frac {\cos ^7(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=-\frac {\frac {3 \, {\left (3 \, a b \sin \left (d x + c\right )^{2} - 3 \, a b + {\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )\right )}}{a b^{3} \sin \left (d x + c\right )^{3} + a^{2} b^{2}} + \frac {9 \, \sin \left (d x + c\right )}{b^{2}} - \frac {2 \, \sqrt {3} {\left (3 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a^{2} + b^{2}\right )} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (3 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{2} - b^{2}\right )} \log \left (\sin \left (d x + c\right )^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, {\left (3 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{2} - b^{2}\right )} \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right )\right )}{a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{9 \, d} \] Input:
integrate(cos(d*x+c)^7/(a+b*sin(d*x+c)^3)^2,x, algorithm="maxima")
Output:
-1/9*(3*(3*a*b*sin(d*x + c)^2 - 3*a*b + (a^2 - b^2)*sin(d*x + c))/(a*b^3*s in(d*x + c)^3 + a^2*b^2) + 9*sin(d*x + c)/b^2 - 2*sqrt(3)*(3*a*b*(a/b)^(1/ 3) + 2*a^2 + b^2)*arctan(-1/3*sqrt(3)*((a/b)^(1/3) - 2*sin(d*x + c))/(a/b) ^(1/3))/(a*b^3*(a/b)^(2/3)) - (3*a*b*(a/b)^(1/3) - 2*a^2 - b^2)*log(sin(d* x + c)^2 - (a/b)^(1/3)*sin(d*x + c) + (a/b)^(2/3))/(a*b^3*(a/b)^(2/3)) + 2 *(3*a*b*(a/b)^(1/3) - 2*a^2 - b^2)*log((a/b)^(1/3) + sin(d*x + c))/(a*b^3* (a/b)^(2/3)))/d
Time = 0.43 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.01 \[ \int \frac {\cos ^7(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=-\frac {\sin \left (d x + c\right )}{b^{2} d} - \frac {2 \, {\left (3 \, a b \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a^{2} + b^{2}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | -\left (-\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right ) \right |}\right )}{9 \, a^{2} b^{2} d} - \frac {2 \, \sqrt {3} {\left (3 \, \left (-a b^{2}\right )^{\frac {2}{3}} a - \left (-a b^{2}\right )^{\frac {1}{3}} {\left (2 \, a^{2} + b^{2}\right )}\right )} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} b^{3} d} - \frac {3 \, a b \sin \left (d x + c\right )^{2} + a^{2} \sin \left (d x + c\right ) - b^{2} \sin \left (d x + c\right ) - 3 \, a b}{3 \, {\left (b \sin \left (d x + c\right )^{3} + a\right )} a b^{2} d} + \frac {{\left (3 \, \left (-a b^{2}\right )^{\frac {2}{3}} a + \left (-a b^{2}\right )^{\frac {1}{3}} {\left (2 \, a^{2} + b^{2}\right )}\right )} \log \left (\sin \left (d x + c\right )^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a^{2} b^{3} d} \] Input:
integrate(cos(d*x+c)^7/(a+b*sin(d*x+c)^3)^2,x, algorithm="giac")
Output:
-sin(d*x + c)/(b^2*d) - 2/9*(3*a*b*(-a/b)^(1/3) + 2*a^2 + b^2)*(-a/b)^(1/3 )*log(abs(-(-a/b)^(1/3) + sin(d*x + c)))/(a^2*b^2*d) - 2/9*sqrt(3)*(3*(-a* b^2)^(2/3)*a - (-a*b^2)^(1/3)*(2*a^2 + b^2))*arctan(1/3*sqrt(3)*((-a/b)^(1 /3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a^2*b^3*d) - 1/3*(3*a*b*sin(d*x + c)^ 2 + a^2*sin(d*x + c) - b^2*sin(d*x + c) - 3*a*b)/((b*sin(d*x + c)^3 + a)*a *b^2*d) + 1/9*(3*(-a*b^2)^(2/3)*a + (-a*b^2)^(1/3)*(2*a^2 + b^2))*log(sin( d*x + c)^2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a^2*b^3*d)
Time = 0.40 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.35 \[ \int \frac {\cos ^7(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {\sum _{k=1}^3\ln \left (\frac {8\,a^2+4\,b^2+{\mathrm {root}\left (729\,a^5\,b^7\,d^3+648\,a^5\,b^3\,d+324\,a^3\,b^5\,d+120\,a^4\,b^2-48\,a^2\,b^4-8\,b^6-64\,a^6,d,k\right )}^2\,a^2\,b^4\,27+12\,a\,b\,\sin \left (c+d\,x\right )+\mathrm {root}\left (729\,a^5\,b^7\,d^3+648\,a^5\,b^3\,d+324\,a^3\,b^5\,d+120\,a^4\,b^2-48\,a^2\,b^4-8\,b^6-64\,a^6,d,k\right )\,b^4\,\sin \left (c+d\,x\right )\,6+\mathrm {root}\left (729\,a^5\,b^7\,d^3+648\,a^5\,b^3\,d+324\,a^3\,b^5\,d+120\,a^4\,b^2-48\,a^2\,b^4-8\,b^6-64\,a^6,d,k\right )\,a^2\,b^2\,\sin \left (c+d\,x\right )\,12}{a\,b^2\,3}\right )\,\mathrm {root}\left (729\,a^5\,b^7\,d^3+648\,a^5\,b^3\,d+324\,a^3\,b^5\,d+120\,a^4\,b^2-48\,a^2\,b^4-8\,b^6-64\,a^6,d,k\right )}{d}-\frac {\sin \left (c+d\,x\right )}{b^2\,d}-\frac {b\,{\sin \left (c+d\,x\right )}^2-b+\frac {\sin \left (c+d\,x\right )\,\left (a^2-b^2\right )}{3\,a}}{d\,\left (b^3\,{\sin \left (c+d\,x\right )}^3+a\,b^2\right )} \] Input:
int(cos(c + d*x)^7/(a + b*sin(c + d*x)^3)^2,x)
Output:
symsum(log((8*a^2 + 4*b^2 + 27*root(729*a^5*b^7*d^3 + 648*a^5*b^3*d + 324* a^3*b^5*d + 120*a^4*b^2 - 48*a^2*b^4 - 8*b^6 - 64*a^6, d, k)^2*a^2*b^4 + 1 2*a*b*sin(c + d*x) + 6*root(729*a^5*b^7*d^3 + 648*a^5*b^3*d + 324*a^3*b^5* d + 120*a^4*b^2 - 48*a^2*b^4 - 8*b^6 - 64*a^6, d, k)*b^4*sin(c + d*x) + 12 *root(729*a^5*b^7*d^3 + 648*a^5*b^3*d + 324*a^3*b^5*d + 120*a^4*b^2 - 48*a ^2*b^4 - 8*b^6 - 64*a^6, d, k)*a^2*b^2*sin(c + d*x))/(3*a*b^2))*root(729*a ^5*b^7*d^3 + 648*a^5*b^3*d + 324*a^3*b^5*d + 120*a^4*b^2 - 48*a^2*b^4 - 8* b^6 - 64*a^6, d, k), k, 1, 3)/d - sin(c + d*x)/(b^2*d) - (b*sin(c + d*x)^2 - b + (sin(c + d*x)*(a^2 - b^2))/(3*a))/(d*(a*b^2 + b^3*sin(c + d*x)^3))
\[ \int \frac {\cos ^7(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {too large to display} \] Input:
int(cos(d*x+c)^7/(a+b*sin(d*x+c)^3)^2,x)
Output:
(1050*cos(c + d*x)*sin(c + d*x)**3*a**5*b + 2316*cos(c + d*x)*sin(c + d*x) **3*a**3*b**3 + 288*cos(c + d*x)*sin(c + d*x)**3*a*b**5 - 22480*cos(c + d* x)*sin(c + d*x)**2*a**4*b**2 - 58336*cos(c + d*x)*sin(c + d*x)**2*a**2*b** 4 - 7168*cos(c + d*x)*sin(c + d*x)**2*b**6 - 10185*cos(c + d*x)*sin(c + d* x)*a**5*b - 27822*cos(c + d*x)*sin(c + d*x)*a**3*b**3 - 3456*cos(c + d*x)* sin(c + d*x)*a*b**5 + 14512*cos(c + d*x)*a**4*b**2 + 31648*cos(c + d*x)*a* *2*b**4 + 4096*cos(c + d*x)*b**6 + 80640*int(tan((c + d*x)/2)**4/(tan((c + d*x)/2)**14*a**4 + tan((c + d*x)/2)**14*a**2*b**2 + 7*tan((c + d*x)/2)**1 2*a**4 + 7*tan((c + d*x)/2)**12*a**2*b**2 + 16*tan((c + d*x)/2)**11*a**3*b + 16*tan((c + d*x)/2)**11*a*b**3 + 21*tan((c + d*x)/2)**10*a**4 + 21*tan( (c + d*x)/2)**10*a**2*b**2 + 64*tan((c + d*x)/2)**9*a**3*b + 64*tan((c + d *x)/2)**9*a*b**3 + 35*tan((c + d*x)/2)**8*a**4 + 99*tan((c + d*x)/2)**8*a* *2*b**2 + 64*tan((c + d*x)/2)**8*b**4 + 96*tan((c + d*x)/2)**7*a**3*b + 96 *tan((c + d*x)/2)**7*a*b**3 + 35*tan((c + d*x)/2)**6*a**4 + 99*tan((c + d* x)/2)**6*a**2*b**2 + 64*tan((c + d*x)/2)**6*b**4 + 64*tan((c + d*x)/2)**5* a**3*b + 64*tan((c + d*x)/2)**5*a*b**3 + 21*tan((c + d*x)/2)**4*a**4 + 21* tan((c + d*x)/2)**4*a**2*b**2 + 16*tan((c + d*x)/2)**3*a**3*b + 16*tan((c + d*x)/2)**3*a*b**3 + 7*tan((c + d*x)/2)**2*a**4 + 7*tan((c + d*x)/2)**2*a **2*b**2 + a**4 + a**2*b**2),x)*sin(c + d*x)**3*a**8*b**2*d + 474624*int(t an((c + d*x)/2)**4/(tan((c + d*x)/2)**14*a**4 + tan((c + d*x)/2)**14*a*...