Integrand size = 21, antiderivative size = 149 \[ \int \cot ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^3}{16 d (1-\cos (c+d x))^4}+\frac {5 a^3}{12 d (1-\cos (c+d x))^3}-\frac {39 a^3}{32 d (1-\cos (c+d x))^2}+\frac {9 a^3}{4 d (1-\cos (c+d x))}+\frac {a^3}{32 d (1+\cos (c+d x))}+\frac {57 a^3 \log (1-\cos (c+d x))}{64 d}+\frac {7 a^3 \log (1+\cos (c+d x))}{64 d} \] Output:
-1/16*a^3/d/(1-cos(d*x+c))^4+5/12*a^3/d/(1-cos(d*x+c))^3-39/32*a^3/d/(1-co s(d*x+c))^2+9/4*a^3/d/(1-cos(d*x+c))+1/32*a^3/d/(1+cos(d*x+c))+57/64*a^3*l n(1-cos(d*x+c))/d+7/64*a^3*ln(1+cos(d*x+c))/d
Time = 0.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.87 \[ \int \cot ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {a^3 (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (864 \csc ^2\left (\frac {1}{2} (c+d x)\right )-234 \csc ^4\left (\frac {1}{2} (c+d x)\right )+40 \csc ^6\left (\frac {1}{2} (c+d x)\right )-3 \csc ^8\left (\frac {1}{2} (c+d x)\right )+12 \left (14 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+114 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\sec ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )}{6144 d} \] Input:
Integrate[Cot[c + d*x]^9*(a + a*Sec[c + d*x])^3,x]
Output:
(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(864*Csc[(c + d*x)/2]^2 - 234 *Csc[(c + d*x)/2]^4 + 40*Csc[(c + d*x)/2]^6 - 3*Csc[(c + d*x)/2]^8 + 12*(1 4*Log[Cos[(c + d*x)/2]] + 114*Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^2)) )/(6144*d)
Time = 0.31 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.77, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 25, 4367, 27, 99, 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.
| \(\displaystyle \int \cot ^9(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}{\cot \left (c+d x+\frac {\pi }{2}\right )^9}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^3}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^9}dx\) |
| \(\Big \downarrow \) 4367 |
| \(\displaystyle -\frac {a^{10} \int \frac {\cos ^6(c+d x)}{a^7 (1-\cos (c+d x))^5 (\cos (c+d x)+1)^2}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \int \frac {\cos ^6(c+d x)}{(1-\cos (c+d x))^5 (\cos (c+d x)+1)^2}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {a^3 \int \left (-\frac {7}{64 (\cos (c+d x)+1)}+\frac {1}{32 (\cos (c+d x)+1)^2}-\frac {57}{64 (\cos (c+d x)-1)}-\frac {9}{4 (\cos (c+d x)-1)^2}-\frac {39}{16 (\cos (c+d x)-1)^3}-\frac {5}{4 (\cos (c+d x)-1)^4}-\frac {1}{4 (\cos (c+d x)-1)^5}\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 \left (-\frac {9}{4 (1-\cos (c+d x))}-\frac {1}{32 (\cos (c+d x)+1)}+\frac {39}{32 (1-\cos (c+d x))^2}-\frac {5}{12 (1-\cos (c+d x))^3}+\frac {1}{16 (1-\cos (c+d x))^4}-\frac {57}{64} \log (1-\cos (c+d x))-\frac {7}{64} \log (\cos (c+d x)+1)\right )}{d}\) |
Input:
Int[Cot[c + d*x]^9*(a + a*Sec[c + d*x])^3,x]
Output:
-((a^3*(1/(16*(1 - Cos[c + d*x])^4) - 5/(12*(1 - Cos[c + d*x])^3) + 39/(32 *(1 - Cos[c + d*x])^2) - 9/(4*(1 - Cos[c + d*x])) - 1/(32*(1 + Cos[c + d*x ])) - (57*Log[1 - Cos[c + d*x]])/64 - (7*Log[1 + Cos[c + d*x]])/64))/d)
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_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _.), x_Symbol] :> Simp[1/(a^(m - n - 1)*b^n*d) Subst[Int[(a - b*x)^((m - 1)/2)*((a + b*x)^((m - 1)/2 + n)/x^(m + n)), x], x, Sin[c + d*x]], x] /; Fr eeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && Integer Q[n]
Result contains complex when optimal does not.
Time = 5.52 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.30
| method | result | size |
| risch | \(-i a^{3} x -\frac {2 i a^{3} c}{d}-\frac {a^{3} \left (213 \,{\mathrm e}^{9 i \left (d x +c \right )}-606 \,{\mathrm e}^{8 i \left (d x +c \right )}+472 \,{\mathrm e}^{7 i \left (d x +c \right )}+846 \,{\mathrm e}^{6 i \left (d x +c \right )}-1658 \,{\mathrm e}^{5 i \left (d x +c \right )}+846 \,{\mathrm e}^{4 i \left (d x +c \right )}+472 \,{\mathrm e}^{3 i \left (d x +c \right )}-606 \,{\mathrm e}^{2 i \left (d x +c \right )}+213 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{48 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{2}}+\frac {57 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{32 d}+\frac {7 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{32 d}\) | \(193\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{128 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{128}-\frac {5 \cos \left (d x +c \right )^{3}}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )-\frac {3 a^{3} \cos \left (d x +c \right )^{8}}{8 \sin \left (d x +c \right )^{8}}+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{9}}{8 \sin \left (d x +c \right )^{8}}+\frac {\cos \left (d x +c \right )^{9}}{48 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{9}}{64 \sin \left (d x +c \right )^{4}}+\frac {5 \cos \left (d x +c \right )^{9}}{128 \sin \left (d x +c \right )^{2}}+\frac {5 \cos \left (d x +c \right )^{7}}{128}+\frac {7 \cos \left (d x +c \right )^{5}}{128}+\frac {35 \cos \left (d x +c \right )^{3}}{384}+\frac {35 \cos \left (d x +c \right )}{128}+\frac {35 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+a^{3} \left (-\frac {\cot \left (d x +c \right )^{8}}{8}+\frac {\cot \left (d x +c \right )^{6}}{6}-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(336\) |
| default | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{128 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{128}-\frac {5 \cos \left (d x +c \right )^{3}}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )-\frac {3 a^{3} \cos \left (d x +c \right )^{8}}{8 \sin \left (d x +c \right )^{8}}+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{9}}{8 \sin \left (d x +c \right )^{8}}+\frac {\cos \left (d x +c \right )^{9}}{48 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{9}}{64 \sin \left (d x +c \right )^{4}}+\frac {5 \cos \left (d x +c \right )^{9}}{128 \sin \left (d x +c \right )^{2}}+\frac {5 \cos \left (d x +c \right )^{7}}{128}+\frac {7 \cos \left (d x +c \right )^{5}}{128}+\frac {35 \cos \left (d x +c \right )^{3}}{384}+\frac {35 \cos \left (d x +c \right )}{128}+\frac {35 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+a^{3} \left (-\frac {\cot \left (d x +c \right )^{8}}{8}+\frac {\cot \left (d x +c \right )^{6}}{6}-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(336\) |
Input:
int(cot(d*x+c)^9*(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-I*a^3*x-2*I/d*a^3*c-1/48*a^3/d/(exp(I*(d*x+c))-1)^8/(exp(I*(d*x+c))+1)^2* (213*exp(9*I*(d*x+c))-606*exp(8*I*(d*x+c))+472*exp(7*I*(d*x+c))+846*exp(6* I*(d*x+c))-1658*exp(5*I*(d*x+c))+846*exp(4*I*(d*x+c))+472*exp(3*I*(d*x+c)) -606*exp(2*I*(d*x+c))+213*exp(I*(d*x+c)))+57/32/d*a^3*ln(exp(I*(d*x+c))-1) +7/32/d*a^3*ln(exp(I*(d*x+c))+1)
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (127) = 254\).
Time = 0.09 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.83 \[ \int \cot ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {426 \, a^{3} \cos \left (d x + c\right )^{4} - 606 \, a^{3} \cos \left (d x + c\right )^{3} - 190 \, a^{3} \cos \left (d x + c\right )^{2} + 666 \, a^{3} \cos \left (d x + c\right ) - 272 \, a^{3} - 21 \, {\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 2 \, a^{3} \cos \left (d x + c\right )^{3} + 2 \, a^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 171 \, {\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 2 \, a^{3} \cos \left (d x + c\right )^{3} + 2 \, a^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{192 \, {\left (d \cos \left (d x + c\right )^{5} - 3 \, d \cos \left (d x + c\right )^{4} + 2 \, d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right )^{2} - 3 \, d \cos \left (d x + c\right ) + d\right )}} \] Input:
integrate(cot(d*x+c)^9*(a+a*sec(d*x+c))^3,x, algorithm="fricas")
Output:
-1/192*(426*a^3*cos(d*x + c)^4 - 606*a^3*cos(d*x + c)^3 - 190*a^3*cos(d*x + c)^2 + 666*a^3*cos(d*x + c) - 272*a^3 - 21*(a^3*cos(d*x + c)^5 - 3*a^3*c os(d*x + c)^4 + 2*a^3*cos(d*x + c)^3 + 2*a^3*cos(d*x + c)^2 - 3*a^3*cos(d* x + c) + a^3)*log(1/2*cos(d*x + c) + 1/2) - 171*(a^3*cos(d*x + c)^5 - 3*a^ 3*cos(d*x + c)^4 + 2*a^3*cos(d*x + c)^3 + 2*a^3*cos(d*x + c)^2 - 3*a^3*cos (d*x + c) + a^3)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^5 - 3*d*cos (d*x + c)^4 + 2*d*cos(d*x + c)^3 + 2*d*cos(d*x + c)^2 - 3*d*cos(d*x + c) + d)
Timed out. \[ \int \cot ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**9*(a+a*sec(d*x+c))**3,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.95 \[ \int \cot ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {21 \, a^{3} \log \left (\cos \left (d x + c\right ) + 1\right ) + 171 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (213 \, a^{3} \cos \left (d x + c\right )^{4} - 303 \, a^{3} \cos \left (d x + c\right )^{3} - 95 \, a^{3} \cos \left (d x + c\right )^{2} + 333 \, a^{3} \cos \left (d x + c\right ) - 136 \, a^{3}\right )}}{\cos \left (d x + c\right )^{5} - 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) + 1}}{192 \, d} \] Input:
integrate(cot(d*x+c)^9*(a+a*sec(d*x+c))^3,x, algorithm="maxima")
Output:
1/192*(21*a^3*log(cos(d*x + c) + 1) + 171*a^3*log(cos(d*x + c) - 1) - 2*(2 13*a^3*cos(d*x + c)^4 - 303*a^3*cos(d*x + c)^3 - 95*a^3*cos(d*x + c)^2 + 3 33*a^3*cos(d*x + c) - 136*a^3)/(cos(d*x + c)^5 - 3*cos(d*x + c)^4 + 2*cos( d*x + c)^3 + 2*cos(d*x + c)^2 - 3*cos(d*x + c) + 1))/d
Time = 0.16 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.68 \[ \int \cot ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {1}{192} \, a^{3} {\left (\frac {21 \, \log \left ({\left | \cos \left (d x + c\right ) + 1 \right |}\right )}{d} + \frac {171 \, \log \left ({\left | \cos \left (d x + c\right ) - 1 \right |}\right )}{d} - \frac {2 \, {\left (213 \, \cos \left (d x + c\right )^{4} - 303 \, \cos \left (d x + c\right )^{3} - 95 \, \cos \left (d x + c\right )^{2} + 333 \, \cos \left (d x + c\right ) - 136\right )}}{d {\left (\cos \left (d x + c\right ) + 1\right )} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}\right )} \] Input:
integrate(cot(d*x+c)^9*(a+a*sec(d*x+c))^3,x, algorithm="giac")
Output:
1/192*a^3*(21*log(abs(cos(d*x + c) + 1))/d + 171*log(abs(cos(d*x + c) - 1) )/d - 2*(213*cos(d*x + c)^4 - 303*cos(d*x + c)^3 - 95*cos(d*x + c)^2 + 333 *cos(d*x + c) - 136)/(d*(cos(d*x + c) + 1)*(cos(d*x + c) - 1)^4))
Time = 12.45 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.87 \[ \int \cot ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {57\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{32\,d}+\frac {21\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{6}-\frac {a^3}{8}}{32\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}-\frac {a^3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \] Input:
int(cot(c + d*x)^9*(a + a/cos(c + d*x))^3,x)
Output:
(a^3*tan(c/2 + (d*x)/2)^2)/(64*d) + (57*a^3*log(tan(c/2 + (d*x)/2)))/(32*d ) + ((7*a^3*tan(c/2 + (d*x)/2)^2)/6 - (11*a^3*tan(c/2 + (d*x)/2)^4)/2 + 21 *a^3*tan(c/2 + (d*x)/2)^6 - a^3/8)/(32*d*tan(c/2 + (d*x)/2)^8) - (a^3*log( tan(c/2 + (d*x)/2)^2 + 1))/d
Time = 0.18 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.83 \[ \int \cot ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {a^{3} \left (-768 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+1368 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+504 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-132 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+28 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-3\right )}{768 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} d} \] Input:
int(cot(d*x+c)^9*(a+a*sec(d*x+c))^3,x)
Output:
(a**3*( - 768*log(tan((c + d*x)/2)**2 + 1)*tan((c + d*x)/2)**8 + 1368*log( tan((c + d*x)/2))*tan((c + d*x)/2)**8 + 12*tan((c + d*x)/2)**10 + 504*tan( (c + d*x)/2)**6 - 132*tan((c + d*x)/2)**4 + 28*tan((c + d*x)/2)**2 - 3))/( 768*tan((c + d*x)/2)**8*d)