Integrand size = 27, antiderivative size = 80 \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {4 a^4 \csc (c+d x)}{d}-\frac {a^4 \csc ^2(c+d x)}{2 d}+\frac {6 a^4 \log (\sin (c+d x))}{d}+\frac {4 a^4 \sin (c+d x)}{d}+\frac {a^4 \sin ^2(c+d x)}{2 d} \]
-4*a^4*csc(d*x+c)/d-1/2*a^4*csc(d*x+c)^2/d+6*a^4*ln(sin(d*x+c))/d+4*a^4*si n(d*x+c)/d+1/2*a^4*sin(d*x+c)^2/d
Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.84 \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {1}{2} a^4 \left (-\frac {8 \csc (c+d x)}{d}-\frac {\csc ^2(c+d x)}{d}+\frac {12 \log (\sin (c+d x))}{d}+\frac {8 \sin (c+d x)}{d}+\frac {\sin ^2(c+d x)}{d}\right ) \]
(a^4*((-8*Csc[c + d*x])/d - Csc[c + d*x]^2/d + (12*Log[Sin[c + d*x]])/d + (8*Sin[c + d*x])/d + Sin[c + d*x]^2/d))/2
Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3312, 27, 49, 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.
\(\displaystyle \int \cot (c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^4 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x) (a \sin (c+d x)+a)^4}{\sin (c+d x)^3}dx\) |
\(\Big \downarrow \) 3312 |
\(\displaystyle \frac {\int \csc ^3(c+d x) (\sin (c+d x) a+a)^4d(a \sin (c+d x))}{a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a^2 \int \frac {\csc ^3(c+d x) (\sin (c+d x) a+a)^4}{a^3}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {a^2 \int \left (a \csc ^3(c+d x)+4 a \csc ^2(c+d x)+6 a \csc (c+d x)+4 a+a \sin (c+d x)\right )d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 \left (\frac {1}{2} a^2 \sin ^2(c+d x)+4 a^2 \sin (c+d x)-\frac {1}{2} a^2 \csc ^2(c+d x)-4 a^2 \csc (c+d x)+6 a^2 \log (a \sin (c+d x))\right )}{d}\) |
(a^2*(-4*a^2*Csc[c + d*x] - (a^2*Csc[c + d*x]^2)/2 + 6*a^2*Log[a*Sin[c + d *x]] + 4*a^2*Sin[c + d*x] + (a^2*Sin[c + d*x]^2)/2))/d
3.3.23.3.1 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_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*(( c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b*f) Su bst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(-\frac {a^{4} \left (\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}+4 \csc \left (d x +c \right )+6 \ln \left (\csc \left (d x +c \right )\right )-\frac {4}{\csc \left (d x +c \right )}-\frac {1}{2 \csc \left (d x +c \right )^{2}}\right )}{d}\) | \(57\) |
default | \(-\frac {a^{4} \left (\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}+4 \csc \left (d x +c \right )+6 \ln \left (\csc \left (d x +c \right )\right )-\frac {4}{\csc \left (d x +c \right )}-\frac {1}{2 \csc \left (d x +c \right )^{2}}\right )}{d}\) | \(57\) |
risch | \(-6 i a^{4} x -\frac {a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {2 i a^{4} {\mathrm e}^{i \left (d x +c \right )}}{d}+\frac {2 i a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{d}-\frac {a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {12 i a^{4} c}{d}-\frac {2 i a^{4} \left (i {\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{3 i \left (d x +c \right )}-4 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+\frac {6 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(166\) |
parallelrisch | \(\frac {a^{4} \left (48 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-48 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (2 d x +2 c \right )-8 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (d x +c \right )+2 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-32 \cot \left (\frac {d x}{2}+\frac {c}{2}\right ) \cos \left (d x +c \right )-16 \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+4 \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+32 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}\) | \(175\) |
norman | \(\frac {-\frac {a^{4}}{8 d}-\frac {2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a^{4} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a^{4} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a^{4} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a^{4} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{4} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {6 a^{4} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {25 a^{4} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {25 a^{4} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {6 a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {6 a^{4} \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(266\) |
Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.22 \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {2 \, a^{4} \cos \left (d x + c\right )^{4} - 16 \, a^{4} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 3 \, a^{4} \cos \left (d x + c\right )^{2} - a^{4} - 24 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right )}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
-1/4*(2*a^4*cos(d*x + c)^4 - 16*a^4*cos(d*x + c)^2*sin(d*x + c) - 3*a^4*co s(d*x + c)^2 - a^4 - 24*(a^4*cos(d*x + c)^2 - a^4)*log(1/2*sin(d*x + c)))/ (d*cos(d*x + c)^2 - d)
Timed out. \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.82 \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^{4} \sin \left (d x + c\right )^{2} + 12 \, a^{4} \log \left (\sin \left (d x + c\right )\right ) + 8 \, a^{4} \sin \left (d x + c\right ) - \frac {8 \, a^{4} \sin \left (d x + c\right ) + a^{4}}{\sin \left (d x + c\right )^{2}}}{2 \, d} \]
1/2*(a^4*sin(d*x + c)^2 + 12*a^4*log(sin(d*x + c)) + 8*a^4*sin(d*x + c) - (8*a^4*sin(d*x + c) + a^4)/sin(d*x + c)^2)/d
Time = 0.37 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.84 \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^{4} \sin \left (d x + c\right )^{2} + 12 \, a^{4} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 8 \, a^{4} \sin \left (d x + c\right ) - \frac {8 \, a^{4} \sin \left (d x + c\right ) + a^{4}}{\sin \left (d x + c\right )^{2}}}{2 \, d} \]
1/2*(a^4*sin(d*x + c)^2 + 12*a^4*log(abs(sin(d*x + c))) + 8*a^4*sin(d*x + c) - (8*a^4*sin(d*x + c) + a^4)/sin(d*x + c)^2)/d
Time = 9.53 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.59 \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {6\,a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {-24\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {15\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}-16\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+8\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^4}{2}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {2\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d}-\frac {6\,a^4\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \]
(6*a^4*log(tan(c/2 + (d*x)/2)))/d - (a^4*tan(c/2 + (d*x)/2)^2)/(8*d) - (a^ 4*tan(c/2 + (d*x)/2)^2 - 16*a^4*tan(c/2 + (d*x)/2)^3 - (15*a^4*tan(c/2 + ( d*x)/2)^4)/2 - 24*a^4*tan(c/2 + (d*x)/2)^5 + a^4/2 + 8*a^4*tan(c/2 + (d*x) /2))/(d*(4*tan(c/2 + (d*x)/2)^2 + 8*tan(c/2 + (d*x)/2)^4 + 4*tan(c/2 + (d* x)/2)^6)) - (2*a^4*tan(c/2 + (d*x)/2))/d - (6*a^4*log(tan(c/2 + (d*x)/2)^2 + 1))/d