Integrand size = 23, antiderivative size = 233 \[ \int \frac {\cos ^5(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=-\frac {2 \left (a^{4/3}+b^{4/3}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{5/3} d}-\frac {2 \left (a^{4/3}-b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{5/3} d}+\frac {\left (a^{4/3}-b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} b^{5/3} d}+\frac {2 a+b \sin (c+d x)-a \sin ^2(c+d x)}{3 a b d \left (a+b \sin ^3(c+d x)\right )} \] Output:
-2/9*(a^(4/3)+b^(4/3))*arctan(1/3*(a^(1/3)-2*b^(1/3)*sin(d*x+c))*3^(1/2)/a ^(1/3))*3^(1/2)/a^(5/3)/b^(5/3)/d-2/9*(a^(4/3)-b^(4/3))*ln(a^(1/3)+b^(1/3) *sin(d*x+c))/a^(5/3)/b^(5/3)/d+1/9*(a^(4/3)-b^(4/3))*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^(5/3)/d+1/3*(2*a+b*sin(d* x+c)-a*sin(d*x+c)^2)/a/b/d/(a+b*sin(d*x+c)^3)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.05 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.11 \[ \int \frac {\cos ^5(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {-\frac {4 \sqrt {3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{5/3} \sqrt [3]{b}}+\frac {4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{a^{5/3} \sqrt [3]{b}}-\frac {2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{a^{5/3} \sqrt [3]{b}}+\frac {9 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{a b}-\frac {9 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},2,\frac {5}{3},-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{a b}+\frac {12}{b \left (a+b \sin ^3(c+d x)\right )}+\frac {6 \sin (c+d x)}{a \left (a+b \sin ^3(c+d x)\right )}}{18 d} \] Input:
Integrate[Cos[c + d*x]^5/(a + b*Sin[c + d*x]^3)^2,x]
Output:
((-4*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)) + (4*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(a^(5/3)*b^(1 /3)) - (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^(1/3)) + (9*Hypergeometric2F1[2/3, 1, 5/3, -((b*Sin[c + d *x]^3)/a)]*Sin[c + d*x]^2)/(a*b) - (9*Hypergeometric2F1[2/3, 2, 5/3, -((b* Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(a*b) + 12/(b*(a + b*Sin[c + d*x]^3)) + (6*Sin[c + d*x])/(a*(a + b*Sin[c + d*x]^3)))/(18*d)
Time = 0.51 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 3702, 2397, 27, 2399, 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.
| \(\displaystyle \int \frac {\cos ^5(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^5}{\left (a+b \sin (c+d x)^3\right )^2}dx\) |
| \(\Big \downarrow \) 3702 |
| \(\displaystyle \frac {\int \frac {\left (1-\sin ^2(c+d x)\right )^2}{\left (b \sin ^3(c+d x)+a\right )^2}d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle \frac {\frac {\sin (c+d x) \left (-a \sin (c+d x)-2 b \sin ^2(c+d x)+b\right )}{3 a b \left (a+b \sin ^3(c+d x)\right )}-\frac {\int -\frac {2 b (b+a \sin (c+d x))}{b \sin ^3(c+d x)+a}d\sin (c+d x)}{3 a b^2}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \int \frac {b+a \sin (c+d x)}{b \sin ^3(c+d x)+a}d\sin (c+d x)}{3 a b}+\frac {\sin (c+d x) \left (-a \sin (c+d x)-2 b \sin ^2(c+d x)+b\right )}{3 a b \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 2399 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\int \frac {\sqrt [3]{a} \left (a^{4/3}+2 b^{4/3}\right )+\sqrt [3]{b} \left (a^{4/3}-b^{4/3}\right ) \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{3 a^{2/3} \sqrt [3]{b}}-\frac {1}{3} \left (\frac {a^{2/3}}{\sqrt [3]{b}}-\frac {b}{a^{2/3}}\right ) \int \frac {1}{\sqrt [3]{b} \sin (c+d x)+\sqrt [3]{a}}d\sin (c+d x)\right )}{3 a b}+\frac {\sin (c+d x) \left (-a \sin (c+d x)-2 b \sin ^2(c+d x)+b\right )}{3 a b \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\int \frac {\sqrt [3]{a} \left (a^{4/3}+2 b^{4/3}\right )+\sqrt [3]{b} \left (a^{4/3}-b^{4/3}\right ) \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left (\frac {a^{2/3}}{\sqrt [3]{b}}-\frac {b}{a^{2/3}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{b}}\right )}{3 a b}+\frac {\sin (c+d x) \left (-a \sin (c+d x)-2 b \sin ^2(c+d x)+b\right )}{3 a b \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \left (a^{4/3}+b^{4/3}\right ) \int \frac {1}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)+\frac {1}{2} \left (\frac {a^{4/3}}{\sqrt [3]{b}}-b\right ) \int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)\right )}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left (\frac {a^{2/3}}{\sqrt [3]{b}}-\frac {b}{a^{2/3}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{b}}\right )}{3 a b}+\frac {\sin (c+d x) \left (-a \sin (c+d x)-2 b \sin ^2(c+d x)+b\right )}{3 a b \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \left (a^{4/3}+b^{4/3}\right ) \int \frac {1}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)-\frac {1}{2} \left (\frac {a^{4/3}}{\sqrt [3]{b}}-b\right ) \int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)\right )}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left (\frac {a^{2/3}}{\sqrt [3]{b}}-\frac {b}{a^{2/3}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{b}}\right )}{3 a b}+\frac {\sin (c+d x) \left (-a \sin (c+d x)-2 b \sin ^2(c+d x)+b\right )}{3 a b \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \left (a^{4/3}+b^{4/3}\right ) \int \frac {1}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)-\frac {1}{2} \sqrt [3]{b} \left (\frac {a^{4/3}}{\sqrt [3]{b}}-b\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left (\frac {a^{2/3}}{\sqrt [3]{b}}-\frac {b}{a^{2/3}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{b}}\right )}{3 a b}+\frac {\sin (c+d x) \left (-a \sin (c+d x)-2 b \sin ^2(c+d x)+b\right )}{3 a b \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {3 \left (a^{4/3}+b^{4/3}\right ) \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \sqrt [3]{b} \left (\frac {a^{4/3}}{\sqrt [3]{b}}-b\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left (\frac {a^{2/3}}{\sqrt [3]{b}}-\frac {b}{a^{2/3}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{b}}\right )}{3 a b}+\frac {\sin (c+d x) \left (-a \sin (c+d x)-2 b \sin ^2(c+d x)+b\right )}{3 a b \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {2 \left (\frac {-\frac {1}{2} \sqrt [3]{b} \left (\frac {a^{4/3}}{\sqrt [3]{b}}-b\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)-\frac {\sqrt {3} \left (a^{4/3}+b^{4/3}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left (\frac {a^{2/3}}{\sqrt [3]{b}}-\frac {b}{a^{2/3}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{b}}\right )}{3 a b}+\frac {\sin (c+d x) \left (-a \sin (c+d x)-2 b \sin ^2(c+d x)+b\right )}{3 a b \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {1}{2} \left (\frac {a^{4/3}}{\sqrt [3]{b}}-b\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )-\frac {\sqrt {3} \left (a^{4/3}+b^{4/3}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left (\frac {a^{2/3}}{\sqrt [3]{b}}-\frac {b}{a^{2/3}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{b}}\right )}{3 a b}+\frac {\sin (c+d x) \left (-a \sin (c+d x)-2 b \sin ^2(c+d x)+b\right )}{3 a b \left (a+b \sin ^3(c+d x)\right )}}{d}\) |
Input:
Int[Cos[c + d*x]^5/(a + b*Sin[c + d*x]^3)^2,x]
Output:
((2*(-1/3*((a^(2/3)/b^(1/3) - b/a^(2/3))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x ]])/b^(1/3) + (-((Sqrt[3]*(a^(4/3) + b^(4/3))*ArcTan[(1 - (2*b^(1/3)*Sin[c + d*x])/a^(1/3))/Sqrt[3]])/b^(1/3)) + ((a^(4/3)/b^(1/3) - b)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/2)/(3*a^(2/3)*b^ (1/3))))/(3*a*b) + (Sin[c + d*x]*(b - a*Sin[c + d*x] - 2*b*Sin[c + d*x]^2) )/(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_) + (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] :> 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[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numer ator[Rt[a/b, 3]], s = Denominator[Rt[a/b, 3]]}, Simp[(-r)*((B*r - A*s)/(3*a *s)) Int[1/(r + s*x), x], x] + Simp[r/(3*a*s) Int[(r*(B*r + 2*A*s) + s* (B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] & & NeQ[a*B^3 - b*A^3, 0] && PosQ[a/b]
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 = 1.42 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.22
| method | result | size |
| derivativedivides | \(\frac {\frac {-\frac {\sin \left (d x +c \right )^{2}}{3 b}+\frac {\sin \left (d x +c \right )}{3 a}+\frac {2}{3 b}}{a +b \sin \left (d x +c \right )^{3}}+\frac {\frac {2 b \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {2 a \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}}{a b}}{d}\) | \(284\) |
| default | \(\frac {\frac {-\frac {\sin \left (d x +c \right )^{2}}{3 b}+\frac {\sin \left (d x +c \right )}{3 a}+\frac {2}{3 b}}{a +b \sin \left (d x +c \right )^{3}}+\frac {\frac {2 b \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {2 a \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}}{a b}}{d}\) | \(284\) |
| risch | \(-\frac {2 i \left (a \,{\mathrm e}^{5 i \left (d x +c \right )}+6 \,{\mathrm e}^{3 i \left (d x +c \right )} a -2 i b \,{\mathrm e}^{4 i \left (d x +c \right )}+a \,{\mathrm e}^{i \left (d x +c \right )}+2 i b \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 a b d \left ({\mathrm e}^{6 i \left (d x +c \right )} b -3 b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{2 i \left (d x +c \right )}-8 i a \,{\mathrm e}^{3 i \left (d x +c \right )}-b \right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (729 a^{5} b^{5} d^{3} \textit {\_Z}^{3}+108 a^{3} b^{3} d \textit {\_Z} +8 a^{4}-8 b^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {81 i a^{5} b^{3} d^{2} \textit {\_R}^{2}}{2 a^{4}+2 b^{4}}+\frac {18 i a^{2} b^{4} d \textit {\_R}}{2 a^{4}+2 b^{4}}+\frac {8 i a^{3} b}{2 a^{4}+2 b^{4}}\right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {2 a^{4}}{2 a^{4}+2 b^{4}}-\frac {2 b^{4}}{2 a^{4}+2 b^{4}}\right )\right )\) | \(300\) |
Input:
int(cos(d*x+c)^5/(a+b*sin(d*x+c)^3)^2,x,method=_RETURNVERBOSE)
Output:
1/d*((-1/3*sin(d*x+c)^2/b+1/3/a*sin(d*x+c)+2/3/b)/(a+b*sin(d*x+c)^3)+2/3/a /b*(b*(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)))+a*(-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)*a rctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*sin(d*x+c)-1)))))
Result contains complex when optimal does not.
Time = 0.92 (sec) , antiderivative size = 2255, normalized size of antiderivative = 9.68 \[ \int \frac {\cos ^5(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^5/(a+b*sin(d*x+c)^3)^2,x, algorithm="fricas")
Output:
1/36*(12*a*cos(d*x + c)^2 - 2*(a^2*b*d - (a*b^2*d*cos(d*x + c)^2 - a*b^2*d )*sin(d*x + c))*(4^(1/3)*(I*sqrt(3) + 1)*((a^4 + b^4)/(a^5*b^5*d^3) - (a^4 - b^4)/(a^5*b^5*d^3))^(1/3) - 4^(2/3)*(-I*sqrt(3) + 1)/(a^2*b^2*d^2*((a^4 + b^4)/(a^5*b^5*d^3) - (a^4 - b^4)/(a^5*b^5*d^3))^(1/3)))*log(1/4*(4^(1/3 )*(I*sqrt(3) + 1)*((a^4 + b^4)/(a^5*b^5*d^3) - (a^4 - b^4)/(a^5*b^5*d^3))^ (1/3) - 4^(2/3)*(-I*sqrt(3) + 1)/(a^2*b^2*d^2*((a^4 + b^4)/(a^5*b^5*d^3) - (a^4 - b^4)/(a^5*b^5*d^3))^(1/3)))^2*a^5*b^3*d^2 - (4^(1/3)*(I*sqrt(3) + 1)*((a^4 + b^4)/(a^5*b^5*d^3) - (a^4 - b^4)/(a^5*b^5*d^3))^(1/3) - 4^(2/3) *(-I*sqrt(3) + 1)/(a^2*b^2*d^2*((a^4 + b^4)/(a^5*b^5*d^3) - (a^4 - b^4)/(a ^5*b^5*d^3))^(1/3)))*a^2*b^4*d + 8*a^3*b + 4*(a^4 + b^4)*sin(d*x + c)) + ( (a^2*b*d - (a*b^2*d*cos(d*x + c)^2 - a*b^2*d)*sin(d*x + c))*(4^(1/3)*(I*sq rt(3) + 1)*((a^4 + b^4)/(a^5*b^5*d^3) - (a^4 - b^4)/(a^5*b^5*d^3))^(1/3) - 4^(2/3)*(-I*sqrt(3) + 1)/(a^2*b^2*d^2*((a^4 + b^4)/(a^5*b^5*d^3) - (a^4 - b^4)/(a^5*b^5*d^3))^(1/3))) + 3*sqrt(1/3)*(a^2*b*d - (a*b^2*d*cos(d*x + c )^2 - a*b^2*d)*sin(d*x + c))*sqrt(-((4^(1/3)*(I*sqrt(3) + 1)*((a^4 + b^4)/ (a^5*b^5*d^3) - (a^4 - b^4)/(a^5*b^5*d^3))^(1/3) - 4^(2/3)*(-I*sqrt(3) + 1 )/(a^2*b^2*d^2*((a^4 + b^4)/(a^5*b^5*d^3) - (a^4 - b^4)/(a^5*b^5*d^3))^(1/ 3)))^2*a^2*b^2*d^2 + 64)/(a^2*b^2*d^2)))*log(1/4*(4^(1/3)*(I*sqrt(3) + 1)* ((a^4 + b^4)/(a^5*b^5*d^3) - (a^4 - b^4)/(a^5*b^5*d^3))^(1/3) - 4^(2/3)*(- I*sqrt(3) + 1)/(a^2*b^2*d^2*((a^4 + b^4)/(a^5*b^5*d^3) - (a^4 - b^4)/(a...
Timed out. \[ \int \frac {\cos ^5(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**5/(a+b*sin(d*x+c)**3)**2,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.91 \[ \int \frac {\cos ^5(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=-\frac {\frac {3 \, {\left (a \sin \left (d x + c\right )^{2} - b \sin \left (d x + c\right ) - 2 \, a\right )}}{a b^{2} \sin \left (d x + c\right )^{3} + a^{2} b} - \frac {2 \, \sqrt {3} {\left (a \left (\frac {a}{b}\right )^{\frac {1}{3}} + b\right )} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (a \left (\frac {a}{b}\right )^{\frac {1}{3}} - b\right )} \log \left (\sin \left (d x + c\right )^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, {\left (a \left (\frac {a}{b}\right )^{\frac {1}{3}} - b\right )} \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right )\right )}{a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{9 \, d} \] Input:
integrate(cos(d*x+c)^5/(a+b*sin(d*x+c)^3)^2,x, algorithm="maxima")
Output:
-1/9*(3*(a*sin(d*x + c)^2 - b*sin(d*x + c) - 2*a)/(a*b^2*sin(d*x + c)^3 + a^2*b) - 2*sqrt(3)*(a*(a/b)^(1/3) + b)*arctan(-1/3*sqrt(3)*((a/b)^(1/3) - 2*sin(d*x + c))/(a/b)^(1/3))/(a*b^2*(a/b)^(2/3)) - (a*(a/b)^(1/3) - b)*log (sin(d*x + c)^2 - (a/b)^(1/3)*sin(d*x + c) + (a/b)^(2/3))/(a*b^2*(a/b)^(2/ 3)) + 2*(a*(a/b)^(1/3) - b)*log((a/b)^(1/3) + sin(d*x + c))/(a*b^2*(a/b)^( 2/3)))/d
Time = 0.39 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.01 \[ \int \frac {\cos ^5(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=-\frac {2 \, {\left (a \left (-\frac {a}{b}\right )^{\frac {1}{3}} + b\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | -\left (-\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right ) \right |}\right )}{9 \, a^{2} b d} - \frac {a \sin \left (d x + c\right )^{2} - b \sin \left (d x + c\right ) - 2 \, a}{3 \, {\left (b \sin \left (d x + c\right )^{3} + a\right )} a b d} + \frac {2 \, \sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} - \left (-a b^{2}\right )^{\frac {2}{3}} a\right )} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} b^{3} d} + \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} + \left (-a b^{2}\right )^{\frac {2}{3}} a\right )} \log \left (\sin \left (d x + c\right )^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a^{2} b^{3} d} \] Input:
integrate(cos(d*x+c)^5/(a+b*sin(d*x+c)^3)^2,x, algorithm="giac")
Output:
-2/9*(a*(-a/b)^(1/3) + b)*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x + c )))/(a^2*b*d) - 1/3*(a*sin(d*x + c)^2 - b*sin(d*x + c) - 2*a)/((b*sin(d*x + c)^3 + a)*a*b*d) + 2/9*sqrt(3)*((-a*b^2)^(1/3)*b^2 - (-a*b^2)^(2/3)*a)*a rctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a^2*b^3*d ) + 1/9*((-a*b^2)^(1/3)*b^2 + (-a*b^2)^(2/3)*a)*log(sin(d*x + c)^2 + (-a/b )^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a^2*b^3*d)
Time = 35.82 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.87 \[ \int \frac {\cos ^5(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {\sum _{k=1}^3\ln \left (\frac {4\,b+4\,a\,\sin \left (c+d\,x\right )+{\mathrm {root}\left (729\,a^5\,b^5\,d^3+108\,a^3\,b^3\,d-8\,b^4+8\,a^4,d,k\right )}^2\,a^2\,b^3\,81+\mathrm {root}\left (729\,a^5\,b^5\,d^3+108\,a^3\,b^3\,d-8\,b^4+8\,a^4,d,k\right )\,b^3\,\sin \left (c+d\,x\right )\,18}{a\,b\,9}\right )\,\mathrm {root}\left (729\,a^5\,b^5\,d^3+108\,a^3\,b^3\,d-8\,b^4+8\,a^4,d,k\right )}{d}+\frac {\frac {\sin \left (c+d\,x\right )}{3\,a}+\frac {2}{3\,b}-\frac {{\sin \left (c+d\,x\right )}^2}{3\,b}}{d\,\left (b\,{\sin \left (c+d\,x\right )}^3+a\right )} \] Input:
int(cos(c + d*x)^5/(a + b*sin(c + d*x)^3)^2,x)
Output:
symsum(log((4*b + 4*a*sin(c + d*x) + 81*root(729*a^5*b^5*d^3 + 108*a^3*b^3 *d - 8*b^4 + 8*a^4, d, k)^2*a^2*b^3 + 18*root(729*a^5*b^5*d^3 + 108*a^3*b^ 3*d - 8*b^4 + 8*a^4, d, k)*b^3*sin(c + d*x))/(9*a*b))*root(729*a^5*b^5*d^3 + 108*a^3*b^3*d - 8*b^4 + 8*a^4, d, k), k, 1, 3)/d + (sin(c + d*x)/(3*a) + 2/(3*b) - sin(c + d*x)^2/(3*b))/(d*(a + b*sin(c + d*x)^3))
\[ \int \frac {\cos ^5(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {too large to display} \] Input:
int(cos(d*x+c)^5/(a+b*sin(d*x+c)^3)^2,x)
Output:
(452*cos(c + d*x)*sin(c + d*x)**2*a**2*b**2 + 304*cos(c + d*x)*sin(c + d*x )**2*b**4 + 120*cos(c + d*x)*sin(c + d*x)*a**3*b + 72*cos(c + d*x)*sin(c + d*x)*a*b**3 - 416*cos(c + d*x)*a**2*b**2 - 256*cos(c + d*x)*b**4 - 960*in t(tan((c + d*x)/2)**4/(tan((c + d*x)/2)**12*a**2 + 6*tan((c + d*x)/2)**10* a**2 + 16*tan((c + d*x)/2)**9*a*b + 15*tan((c + d*x)/2)**8*a**2 + 48*tan(( c + d*x)/2)**7*a*b + 20*tan((c + d*x)/2)**6*a**2 + 64*tan((c + d*x)/2)**6* b**2 + 48*tan((c + d*x)/2)**5*a*b + 15*tan((c + d*x)/2)**4*a**2 + 16*tan(( c + d*x)/2)**3*a*b + 6*tan((c + d*x)/2)**2*a**2 + a**2),x)*sin(c + d*x)**3 *a**4*b**2*d + 1536*int(tan((c + d*x)/2)**4/(tan((c + d*x)/2)**12*a**2 + 6 *tan((c + d*x)/2)**10*a**2 + 16*tan((c + d*x)/2)**9*a*b + 15*tan((c + d*x) /2)**8*a**2 + 48*tan((c + d*x)/2)**7*a*b + 20*tan((c + d*x)/2)**6*a**2 + 6 4*tan((c + d*x)/2)**6*b**2 + 48*tan((c + d*x)/2)**5*a*b + 15*tan((c + d*x) /2)**4*a**2 + 16*tan((c + d*x)/2)**3*a*b + 6*tan((c + d*x)/2)**2*a**2 + a* *2),x)*sin(c + d*x)**3*b**6*d - 960*int(tan((c + d*x)/2)**4/(tan((c + d*x) /2)**12*a**2 + 6*tan((c + d*x)/2)**10*a**2 + 16*tan((c + d*x)/2)**9*a*b + 15*tan((c + d*x)/2)**8*a**2 + 48*tan((c + d*x)/2)**7*a*b + 20*tan((c + d*x )/2)**6*a**2 + 64*tan((c + d*x)/2)**6*b**2 + 48*tan((c + d*x)/2)**5*a*b + 15*tan((c + d*x)/2)**4*a**2 + 16*tan((c + d*x)/2)**3*a*b + 6*tan((c + d*x) /2)**2*a**2 + a**2),x)*a**5*b*d + 1536*int(tan((c + d*x)/2)**4/(tan((c + d *x)/2)**12*a**2 + 6*tan((c + d*x)/2)**10*a**2 + 16*tan((c + d*x)/2)**9*...