Integrand size = 27, antiderivative size = 158 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {6 b \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 d}+\frac {3 \left (a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^4 d}-\frac {\cos (c+d x)}{2 a^2 d \left (1-\cos ^2(c+d x)\right )}+\frac {2 b \cot (c+d x)}{a^3 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x)}{a^3 d (a+b \sin (c+d x))} \]
3/2*(a^2-2*b^2)*arctanh(cos(d*x+c))/a^4/d-1/2*cos(d*x+c)/a^2/d/(1-cos(d*x+ c)^2)+2*b*cot(d*x+c)/a^3/d-(a^2-b^2)*cos(d*x+c)/a^3/d/(a+b*sin(d*x+c))+6*b *arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))*(a^2-b^2)^(1/2)/a^4/d
Time = 3.01 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.21 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {48 b \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+8 a b \cot \left (\frac {1}{2} (c+d x)\right )-a^2 \csc ^2\left (\frac {1}{2} (c+d x)\right )+12 \left (a^2-2 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-12 \left (a^2-2 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+a^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )+\frac {8 a \left (-a^2+b^2\right ) \cos (c+d x)}{a+b \sin (c+d x)}-8 a b \tan \left (\frac {1}{2} (c+d x)\right )}{8 a^4 d} \]
(48*b*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 8 *a*b*Cot[(c + d*x)/2] - a^2*Csc[(c + d*x)/2]^2 + 12*(a^2 - 2*b^2)*Log[Cos[ (c + d*x)/2]] - 12*(a^2 - 2*b^2)*Log[Sin[(c + d*x)/2]] + a^2*Sec[(c + d*x) /2]^2 + (8*a*(-a^2 + b^2)*Cos[c + d*x])/(a + b*Sin[c + d*x]) - 8*a*b*Tan[( c + d*x)/2])/(8*a^4*d)
Time = 1.03 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.27, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3369, 3042, 3534, 27, 3042, 3480, 3042, 3139, 1083, 217, 4257}
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.
| \(\displaystyle \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4}{\sin (c+d x)^3 (a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3369 |
| \(\displaystyle \frac {\int \frac {\csc ^2(c+d x) \left (3 b^2 \sin ^2(c+d x)-a b \sin (c+d x)+2 \left (a^2-3 b^2\right )\right )}{a+b \sin (c+d x)}dx}{2 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 b^2 \sin (c+d x)^2-a b \sin (c+d x)+2 \left (a^2-3 b^2\right )}{\sin (c+d x)^2 (a+b \sin (c+d x))}dx}{2 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int -\frac {3 \csc (c+d x) \left (b \left (a^2-2 b^2\right )-a b^2 \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{a}-\frac {2 \left (a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {3 \int \frac {\csc (c+d x) \left (b \left (a^2-2 b^2\right )-a b^2 \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{a}-\frac {2 \left (a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \int \frac {b \left (a^2-2 b^2\right )-a b^2 \sin (c+d x)}{\sin (c+d x) (a+b \sin (c+d x))}dx}{a}-\frac {2 \left (a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {-\frac {3 \left (\frac {b \left (a^2-2 b^2\right ) \int \csc (c+d x)dx}{a}-\frac {2 b^2 \left (a^2-b^2\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {2 \left (a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \left (\frac {b \left (a^2-2 b^2\right ) \int \csc (c+d x)dx}{a}-\frac {2 b^2 \left (a^2-b^2\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {2 \left (a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {-\frac {3 \left (\frac {b \left (a^2-2 b^2\right ) \int \csc (c+d x)dx}{a}-\frac {4 b^2 \left (a^2-b^2\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}-\frac {2 \left (a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-\frac {3 \left (\frac {8 b^2 \left (a^2-b^2\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}+\frac {b \left (a^2-2 b^2\right ) \int \csc (c+d x)dx}{a}\right )}{a}-\frac {2 \left (a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {3 \left (\frac {b \left (a^2-2 b^2\right ) \int \csc (c+d x)dx}{a}-\frac {4 b^2 \sqrt {a^2-b^2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}\right )}{a}-\frac {2 \left (a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {4 b^2 \sqrt {a^2-b^2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}-\frac {b \left (a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{a d}\right )}{a}-\frac {2 \left (a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}\) |
((-3*((-4*b^2*Sqrt[a^2 - b^2]*ArcTan[(2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[ a^2 - b^2])])/(a*d) - (b*(a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(a*d)))/a - (2*(a^2 - 3*b^2)*Cot[c + d*x])/(a*d))/(2*a^2*b) + ((2*a^2 - 3*b^2)*Cot[c + d*x])/(2*a^2*b*d*(a + b*Sin[c + d*x])) - (Cot[c + d*x]*Csc[c + d*x])/(2*a *d*(a + b*Sin[c + d*x]))
3.12.32.3.1 Defintions of rubi rules used
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] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[Cos[e + f*x]*(d*Sin[ e + f*x])^(n + 1)*((a + b*Sin[e + f*x])^(m + 1)/(a*d*f*(n + 1))), x] + (-Si mp[(a^2*(n + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*(( a + b*Sin[e + f*x])^(m + 1)/(a^2*b*d^2*f*(n + 1)*(m + 1))), x] + Simp[1/(a^ 2*b*d*(n + 1)*(m + 1)) Int[(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^ (m + 1)*Simp[a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 1 )*Sin[e + f*x] - (a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] && LtQ[n, -1]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b - a*B)/(b*c - a*d) Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ (b*c - a*d) Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.55 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.32
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-4 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{3}}+\frac {\frac {4 \left (-\frac {b \left (a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{3}}{2}+\frac {a \,b^{2}}{2}\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+6 b \sqrt {a^{2}-b^{2}}\, \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{4}}-\frac {1}{8 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-6 a^{2}+12 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{4}}+\frac {b}{a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(209\) |
| default | \(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-4 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{3}}+\frac {\frac {4 \left (-\frac {b \left (a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{3}}{2}+\frac {a \,b^{2}}{2}\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+6 b \sqrt {a^{2}-b^{2}}\, \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{4}}-\frac {1}{8 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-6 a^{2}+12 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{4}}+\frac {b}{a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(209\) |
| risch | \(\frac {i \left (-3 i a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+12 i a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+6 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-9 i a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}+6 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-12 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+6 b^{3}+2 i a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-4 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+2 i a^{3} {\mathrm e}^{i \left (d x +c \right )}-2 a^{2} b \right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} b \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right ) a^{3} d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \,a^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{a^{4} d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{a^{4} d}-\frac {3 i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,a^{4}}+\frac {3 i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,a^{4}}\) | \(395\) |
1/d*(1/4/a^3*(1/2*tan(1/2*d*x+1/2*c)^2*a-4*b*tan(1/2*d*x+1/2*c))+4/a^4*((- 1/2*b*(a^2-b^2)*tan(1/2*d*x+1/2*c)-1/2*a^3+1/2*a*b^2)/(tan(1/2*d*x+1/2*c)^ 2*a+2*b*tan(1/2*d*x+1/2*c)+a)+3/2*b*(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/ 2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2)))-1/8/a^2/tan(1/2*d*x+1/2*c)^2+1/4/a^4*( -6*a^2+12*b^2)*ln(tan(1/2*d*x+1/2*c))+b/a^3/tan(1/2*d*x+1/2*c))
Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (147) = 294\).
Time = 0.38 (sec) , antiderivative size = 804, normalized size of antiderivative = 5.09 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\left [-\frac {6 \, a^{2} b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 4 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 6 \, {\left (a b \cos \left (d x + c\right )^{2} - a b + {\left (b^{2} \cos \left (d x + c\right )^{2} - b^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 6 \, {\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right ) + 3 \, {\left (a^{3} - 2 \, a b^{2} - {\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{2} b - 2 \, b^{3} - {\left (a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left (a^{3} - 2 \, a b^{2} - {\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{2} b - 2 \, b^{3} - {\left (a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left (a^{5} d \cos \left (d x + c\right )^{2} - a^{5} d + {\left (a^{4} b d \cos \left (d x + c\right )^{2} - a^{4} b d\right )} \sin \left (d x + c\right )\right )}}, -\frac {6 \, a^{2} b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 4 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 12 \, {\left (a b \cos \left (d x + c\right )^{2} - a b + {\left (b^{2} \cos \left (d x + c\right )^{2} - b^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - 6 \, {\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right ) + 3 \, {\left (a^{3} - 2 \, a b^{2} - {\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{2} b - 2 \, b^{3} - {\left (a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left (a^{3} - 2 \, a b^{2} - {\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{2} b - 2 \, b^{3} - {\left (a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left (a^{5} d \cos \left (d x + c\right )^{2} - a^{5} d + {\left (a^{4} b d \cos \left (d x + c\right )^{2} - a^{4} b d\right )} \sin \left (d x + c\right )\right )}}\right ] \]
[-1/4*(6*a^2*b*cos(d*x + c)*sin(d*x + c) + 4*(a^3 - 3*a*b^2)*cos(d*x + c)^ 3 - 6*(a*b*cos(d*x + c)^2 - a*b + (b^2*cos(d*x + c)^2 - b^2)*sin(d*x + c)) *sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) - 6*(a^3 - 2 *a*b^2)*cos(d*x + c) + 3*(a^3 - 2*a*b^2 - (a^3 - 2*a*b^2)*cos(d*x + c)^2 + (a^2*b - 2*b^3 - (a^2*b - 2*b^3)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*co s(d*x + c) + 1/2) - 3*(a^3 - 2*a*b^2 - (a^3 - 2*a*b^2)*cos(d*x + c)^2 + (a ^2*b - 2*b^3 - (a^2*b - 2*b^3)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos( d*x + c) + 1/2))/(a^5*d*cos(d*x + c)^2 - a^5*d + (a^4*b*d*cos(d*x + c)^2 - a^4*b*d)*sin(d*x + c)), -1/4*(6*a^2*b*cos(d*x + c)*sin(d*x + c) + 4*(a^3 - 3*a*b^2)*cos(d*x + c)^3 + 12*(a*b*cos(d*x + c)^2 - a*b + (b^2*cos(d*x + c)^2 - b^2)*sin(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sq rt(a^2 - b^2)*cos(d*x + c))) - 6*(a^3 - 2*a*b^2)*cos(d*x + c) + 3*(a^3 - 2 *a*b^2 - (a^3 - 2*a*b^2)*cos(d*x + c)^2 + (a^2*b - 2*b^3 - (a^2*b - 2*b^3) *cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 3*(a^3 - 2*a* b^2 - (a^3 - 2*a*b^2)*cos(d*x + c)^2 + (a^2*b - 2*b^3 - (a^2*b - 2*b^3)*co s(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2))/(a^5*d*cos(d*x + c)^2 - a^5*d + (a^4*b*d*cos(d*x + c)^2 - a^4*b*d)*sin(d*x + c))]
\[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\int \frac {\cos ^{4}{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.44 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.74 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {\frac {12 \, {\left (a^{2} - 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{4}} - \frac {48 \, {\left (a^{2} b - b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{4}} - \frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{4}} + \frac {16 \, {\left (a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3} - a b^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )} a^{4}} - \frac {18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}}{a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
-1/8*(12*(a^2 - 2*b^2)*log(abs(tan(1/2*d*x + 1/2*c)))/a^4 - 48*(a^2*b - b^ 3)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2* c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*a^4) - (a^2*tan(1/2*d*x + 1/2*c )^2 - 8*a*b*tan(1/2*d*x + 1/2*c))/a^4 + 16*(a^2*b*tan(1/2*d*x + 1/2*c) - b ^3*tan(1/2*d*x + 1/2*c) + a^3 - a*b^2)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*ta n(1/2*d*x + 1/2*c) + a)*a^4) - (18*a^2*tan(1/2*d*x + 1/2*c)^2 - 36*b^2*tan (1/2*d*x + 1/2*c)^2 + 8*a*b*tan(1/2*d*x + 1/2*c) - a^2)/(a^4*tan(1/2*d*x + 1/2*c)^2))/d
Time = 10.32 (sec) , antiderivative size = 675, normalized size of antiderivative = 4.27 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {17\,a^2}{2}-16\,b^2\right )+\frac {a^2}{2}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (a^2\,b-2\,b^3\right )}{a}-3\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (4\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+8\,b\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^3\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (3\,a^2-6\,b^2\right )}{2\,a^4\,d}-\frac {6\,b\,\mathrm {atanh}\left (\frac {72\,b^4\,\sqrt {b^2-a^2}}{18\,a^4\,b+72\,b^5-90\,a^2\,b^3-216\,a\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+72\,a^3\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {144\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a}}-\frac {54\,b^2\,\sqrt {b^2-a^2}}{18\,a^2\,b-90\,b^3+\frac {72\,b^5}{a^2}+72\,a\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {216\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a}+\frac {144\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^3}}+\frac {18\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b^2-a^2}}{18\,a\,b-\frac {90\,b^3}{a}+\frac {72\,b^5}{a^3}+72\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {216\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^2}+\frac {144\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^4}}-\frac {144\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b^2-a^2}}{18\,a^3\,b-90\,a\,b^3+\frac {72\,b^5}{a}-216\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+72\,a^2\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {144\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^2}}+\frac {144\,b^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b^2-a^2}}{18\,a^5\,b+72\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^4\,b^2-90\,a^3\,b^3-216\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^4+72\,a\,b^5+144\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^6}\right )\,\sqrt {b^2-a^2}}{a^4\,d} \]
tan(c/2 + (d*x)/2)^2/(8*a^2*d) - (tan(c/2 + (d*x)/2)^2*((17*a^2)/2 - 16*b^ 2) + a^2/2 + (4*tan(c/2 + (d*x)/2)^3*(a^2*b - 2*b^3))/a - 3*a*b*tan(c/2 + (d*x)/2))/(d*(4*a^4*tan(c/2 + (d*x)/2)^2 + 4*a^4*tan(c/2 + (d*x)/2)^4 + 8* a^3*b*tan(c/2 + (d*x)/2)^3)) - (b*tan(c/2 + (d*x)/2))/(a^3*d) - (log(tan(c /2 + (d*x)/2))*(3*a^2 - 6*b^2))/(2*a^4*d) - (6*b*atanh((72*b^4*(b^2 - a^2) ^(1/2))/(18*a^4*b + 72*b^5 - 90*a^2*b^3 - 216*a*b^4*tan(c/2 + (d*x)/2) + 7 2*a^3*b^2*tan(c/2 + (d*x)/2) + (144*b^6*tan(c/2 + (d*x)/2))/a) - (54*b^2*( b^2 - a^2)^(1/2))/(18*a^2*b - 90*b^3 + (72*b^5)/a^2 + 72*a*b^2*tan(c/2 + ( d*x)/2) - (216*b^4*tan(c/2 + (d*x)/2))/a + (144*b^6*tan(c/2 + (d*x)/2))/a^ 3) + (18*b*tan(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2))/(18*a*b - (90*b^3)/a + (7 2*b^5)/a^3 + 72*b^2*tan(c/2 + (d*x)/2) - (216*b^4*tan(c/2 + (d*x)/2))/a^2 + (144*b^6*tan(c/2 + (d*x)/2))/a^4) - (144*b^3*tan(c/2 + (d*x)/2)*(b^2 - a ^2)^(1/2))/(18*a^3*b - 90*a*b^3 + (72*b^5)/a - 216*b^4*tan(c/2 + (d*x)/2) + 72*a^2*b^2*tan(c/2 + (d*x)/2) + (144*b^6*tan(c/2 + (d*x)/2))/a^2) + (144 *b^5*tan(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2))/(72*a*b^5 + 18*a^5*b - 90*a^3*b ^3 + 144*b^6*tan(c/2 + (d*x)/2) - 216*a^2*b^4*tan(c/2 + (d*x)/2) + 72*a^4* b^2*tan(c/2 + (d*x)/2)))*(b^2 - a^2)^(1/2))/(a^4*d)