Integrand size = 27, antiderivative size = 83 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {3 a x}{2}-\frac {a \text {arctanh}(\cos (c+d x))}{d}+\frac {a \cos (c+d x)}{d}+\frac {a \cos ^3(c+d x)}{3 d}-\frac {3 a \cot (c+d x)}{2 d}+\frac {a \cos ^2(c+d x) \cot (c+d x)}{2 d} \]
-3/2*a*x-a*arctanh(cos(d*x+c))/d+a*cos(d*x+c)/d+1/3*a*cos(d*x+c)^3/d-3/2*a *cot(d*x+c)/d+1/2*a*cos(d*x+c)^2*cot(d*x+c)/d
Time = 0.45 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.93 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (15 \cos (c+d x)+\cos (3 (c+d x))-3 \left (6 c+6 d x+4 \cot (c+d x)+4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-4 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\sin (2 (c+d x))\right )\right )}{12 d} \]
(a*(15*Cos[c + d*x] + Cos[3*(c + d*x)] - 3*(6*c + 6*d*x + 4*Cot[c + d*x] + 4*Log[Cos[(c + d*x)/2]] - 4*Log[Sin[(c + d*x)/2]] + Sin[2*(c + d*x)])))/( 12*d)
Time = 0.40 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {3042, 3317, 3042, 25, 3071, 252, 262, 216, 3072, 254, 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.
| \(\displaystyle \int \cos ^2(c+d x) \cot ^2(c+d x) (a \sin (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a \sin (c+d x)+a)}{\sin (c+d x)^2}dx\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle a \int \cos ^3(c+d x) \cot (c+d x)dx+a \int \cos ^2(c+d x) \cot ^2(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int -\sin \left (c+d x+\frac {\pi }{2}\right )^3 \tan \left (c+d x+\frac {\pi }{2}\right )dx+a \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \tan \left (c+d x+\frac {\pi }{2}\right )^2dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle a \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \tan \left (c+d x+\frac {\pi }{2}\right )^2dx-a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx\) |
| \(\Big \downarrow \) 3071 |
| \(\displaystyle -\frac {a \int \frac {\cot ^4(c+d x)}{\left (\cot ^2(c+d x)+1\right )^2}d\cot (c+d x)}{d}-a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle -\frac {a \left (\frac {3}{2} \int \frac {\cot ^2(c+d x)}{\cot ^2(c+d x)+1}d\cot (c+d x)-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}-a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle -\frac {a \left (\frac {3}{2} \left (\cot (c+d x)-\int \frac {1}{\cot ^2(c+d x)+1}d\cot (c+d x)\right )-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}-a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\frac {a \left (\frac {3}{2} (\cot (c+d x)-\arctan (\cot (c+d x)))-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}\) |
| \(\Big \downarrow \) 3072 |
| \(\displaystyle -\frac {a \int \frac {\cos ^4(c+d x)}{1-\cos ^2(c+d x)}d\cos (c+d x)}{d}-\frac {a \left (\frac {3}{2} (\cot (c+d x)-\arctan (\cot (c+d x)))-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle -\frac {a \int \left (-\cos ^2(c+d x)+\frac {1}{1-\cos ^2(c+d x)}-1\right )d\cos (c+d x)}{d}-\frac {a \left (\frac {3}{2} (\cot (c+d x)-\arctan (\cot (c+d x)))-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a \left (\frac {3}{2} (\cot (c+d x)-\arctan (\cot (c+d x)))-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}-\frac {a \left (\text {arctanh}(\cos (c+d x))-\frac {1}{3} \cos ^3(c+d x)-\cos (c+d x)\right )}{d}\) |
-((a*(ArcTanh[Cos[c + d*x]] - Cos[c + d*x] - Cos[c + d*x]^3/3))/d) - (a*(( 3*(-ArcTan[Cot[c + d*x]] + Cot[c + d*x]))/2 - Cot[c + d*x]^3/(2*(1 + Cot[c + d*x]^2))))/d
3.4.70.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[b*(ff/f) Subst[I nt[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, b*(Tan[e + f*x]/ff)], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ (ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)], x ]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Time = 0.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(\frac {a \left (\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+a \left (-\frac {\cos ^{5}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )}{d}\) | \(94\) |
| default | \(\frac {a \left (\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+a \left (-\frac {\cos ^{5}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )}{d}\) | \(94\) |
| parallelrisch | \(\frac {a \left (3 \cot \left (\frac {d x}{2}+\frac {c}{2}\right ) \cos \left (2 d x +2 c \right )-6 \cot \left (\frac {d x}{2}+\frac {c}{2}\right ) \cos \left (d x +c \right )+6 \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )-18 d x +15 \cos \left (d x +c \right )+\cos \left (3 d x +3 c \right )-9 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+16\right )}{12 d}\) | \(110\) |
| risch | \(-\frac {3 a x}{2}+\frac {i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {5 a \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {5 a \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {2 i a}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}+\frac {a \cos \left (3 d x +3 c \right )}{12 d}\) | \(138\) |
| norman | \(\frac {\frac {4 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a}{2 d}-\frac {2 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {3 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {9 a x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {9 a x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {3 a x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(209\) |
1/d*(a*(1/3*cos(d*x+c)^3+cos(d*x+c)+ln(csc(d*x+c)-cot(d*x+c)))+a*(-1/sin(d *x+c)*cos(d*x+c)^5-(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)-3/2*d*x-3/2*c) )
Time = 0.29 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.29 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 \, a \cos \left (d x + c\right )^{3} - 3 \, a \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 3 \, a \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 9 \, a \cos \left (d x + c\right ) + {\left (2 \, a \cos \left (d x + c\right )^{3} - 9 \, a d x + 6 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, d \sin \left (d x + c\right )} \]
1/6*(3*a*cos(d*x + c)^3 - 3*a*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 3 *a*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 9*a*cos(d*x + c) + (2*a*cos (d*x + c)^3 - 9*a*d*x + 6*a*cos(d*x + c))*sin(d*x + c))/(d*sin(d*x + c))
\[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=a \left (\int \cos ^{4}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx\right ) \]
a*(Integral(cos(c + d*x)**4*csc(c + d*x)**2, x) + Integral(sin(c + d*x)*co s(c + d*x)**4*csc(c + d*x)**2, x))
Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.08 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {{\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a - 3 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a}{6 \, d} \]
1/6*((2*cos(d*x + c)^3 + 6*cos(d*x + c) - 3*log(cos(d*x + c) + 1) + 3*log( cos(d*x + c) - 1))*a - 3*(3*d*x + 3*c + (3*tan(d*x + c)^2 + 2)/(tan(d*x + c)^3 + tan(d*x + c)))*a)/d
Time = 0.33 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.71 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {9 \, {\left (d x + c\right )} a - 6 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {3 \, {\left (2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - \frac {2 \, {\left (3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 12 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
-1/6*(9*(d*x + c)*a - 6*a*log(abs(tan(1/2*d*x + 1/2*c))) - 3*a*tan(1/2*d*x + 1/2*c) + 3*(2*a*tan(1/2*d*x + 1/2*c) + a)/tan(1/2*d*x + 1/2*c) - 2*(3*a *tan(1/2*d*x + 1/2*c)^5 + 12*a*tan(1/2*d*x + 1/2*c)^4 + 12*a*tan(1/2*d*x + 1/2*c)^2 - 3*a*tan(1/2*d*x + 1/2*c) + 8*a)/(tan(1/2*d*x + 1/2*c)^2 + 1)^3 )/d
Time = 9.90 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.94 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {16\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}-a}{d\,\left (2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}+\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}+\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {3\,a\,\mathrm {atan}\left (\frac {9\,a^2}{6\,a^2+9\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {6\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{6\,a^2+9\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
((16*a*tan(c/2 + (d*x)/2))/3 - a - 5*a*tan(c/2 + (d*x)/2)^2 + 8*a*tan(c/2 + (d*x)/2)^3 - 3*a*tan(c/2 + (d*x)/2)^4 + 8*a*tan(c/2 + (d*x)/2)^5 + a*tan (c/2 + (d*x)/2)^6)/(d*(2*tan(c/2 + (d*x)/2) + 6*tan(c/2 + (d*x)/2)^3 + 6*t an(c/2 + (d*x)/2)^5 + 2*tan(c/2 + (d*x)/2)^7)) + (a*tan(c/2 + (d*x)/2))/(2 *d) + (a*log(tan(c/2 + (d*x)/2)))/d + (3*a*atan((9*a^2)/(6*a^2 + 9*a^2*tan (c/2 + (d*x)/2)) - (6*a^2*tan(c/2 + (d*x)/2))/(6*a^2 + 9*a^2*tan(c/2 + (d* x)/2))))/d