Integrand size = 21, antiderivative size = 218 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^7}{16 d (a-a \cos (c+d x))^4}-\frac {a^9}{3 d \left (a^2-a^2 \cos (c+d x)\right )^3}-\frac {39 a^9}{32 d \left (a^3-a^3 \cos (c+d x)\right )^2}-\frac {75 a^9}{16 d \left (a^6-a^6 \cos (c+d x)\right )}-\frac {a^9}{32 d \left (a^6+a^6 \cos (c+d x)\right )}+\frac {501 a^3 \log (1-\cos (c+d x))}{64 d}-\frac {8 a^3 \log (\cos (c+d x))}{d}+\frac {11 a^3 \log (1+\cos (c+d x))}{64 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \] Output:
-1/16*a^7/d/(a-a*cos(d*x+c))^4-1/3*a^9/d/(a^2-a^2*cos(d*x+c))^3-39/32*a^9/ d/(a^3-a^3*cos(d*x+c))^2-75/16*a^9/d/(a^6-a^6*cos(d*x+c))-1/32*a^9/d/(a^6+ a^6*cos(d*x+c))+501/64*a^3*ln(1-cos(d*x+c))/d-8*a^3*ln(cos(d*x+c))/d+11/64 *a^3*ln(1+cos(d*x+c))/d+3*a^3*sec(d*x+c)/d+1/2*a^3*sec(d*x+c)^2/d
Time = 0.81 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.73 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^3 (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (1800 \csc ^2\left (\frac {1}{2} (c+d x)\right )+234 \csc ^4\left (\frac {1}{2} (c+d x)\right )+32 \csc ^6\left (\frac {1}{2} (c+d x)\right )+3 \csc ^8\left (\frac {1}{2} (c+d x)\right )-12 \left (22 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-512 \log (\cos (c+d x))+1002 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\sec ^2\left (\frac {1}{2} (c+d x)\right )+192 \sec (c+d x)+32 \sec ^2(c+d x)\right )\right )}{6144 d} \] Input:
Integrate[Csc[c + d*x]^9*(a + a*Sec[c + d*x])^3,x]
Output:
-1/6144*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(1800*Csc[(c + d*x)/2 ]^2 + 234*Csc[(c + d*x)/2]^4 + 32*Csc[(c + d*x)/2]^6 + 3*Csc[(c + d*x)/2]^ 8 - 12*(22*Log[Cos[(c + d*x)/2]] - 512*Log[Cos[c + d*x]] + 1002*Log[Sin[(c + d*x)/2]] - Sec[(c + d*x)/2]^2 + 192*Sec[c + d*x] + 32*Sec[c + d*x]^2))) /d
Time = 0.52 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.85, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3042, 4360, 25, 25, 3042, 25, 3315, 25, 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.
| \(\displaystyle \int \csc ^9(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}{\cos \left (c+d x-\frac {\pi }{2}\right )^9}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \csc ^9(c+d x) \sec ^3(c+d x) \left (-(a (-\cos (c+d x))-a)^3\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -(\cos (c+d x) a+a)^3 \csc ^9(c+d x) \sec ^3(c+d x)dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \csc ^9(c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (a-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}{\sin \left (c+d x-\frac {\pi }{2}\right )^3 \cos \left (c+d x-\frac {\pi }{2}\right )^9}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )^3}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^9 \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )^3}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^9 \int -\frac {\sec ^3(c+d x)}{(a-a \cos (c+d x))^5 (\cos (c+d x) a+a)^2}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^9 \int \frac {\sec ^3(c+d x)}{(a-a \cos (c+d x))^5 (\cos (c+d x) a+a)^2}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^{12} \int \frac {\sec ^3(c+d x)}{a^3 (a-a \cos (c+d x))^5 (\cos (c+d x) a+a)^2}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {a^{12} \int \left (\frac {\sec ^3(c+d x)}{a^{10}}+\frac {3 \sec ^2(c+d x)}{a^{10}}+\frac {8 \sec (c+d x)}{a^{10}}+\frac {501}{64 a^9 (a-a \cos (c+d x))}-\frac {11}{64 a^9 (\cos (c+d x) a+a)}+\frac {75}{16 a^8 (a-a \cos (c+d x))^2}-\frac {1}{32 a^8 (\cos (c+d x) a+a)^2}+\frac {39}{16 a^7 (a-a \cos (c+d x))^3}+\frac {1}{a^6 (a-a \cos (c+d x))^4}+\frac {1}{4 a^5 (a-a \cos (c+d x))^5}\right )d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{12} \left (-\frac {\sec ^2(c+d x)}{2 a^9}-\frac {3 \sec (c+d x)}{a^9}+\frac {8 \log (a \cos (c+d x))}{a^9}-\frac {501 \log (a-a \cos (c+d x))}{64 a^9}-\frac {11 \log (a \cos (c+d x)+a)}{64 a^9}+\frac {75}{16 a^8 (a-a \cos (c+d x))}+\frac {1}{32 a^8 (a \cos (c+d x)+a)}+\frac {39}{32 a^7 (a-a \cos (c+d x))^2}+\frac {1}{3 a^6 (a-a \cos (c+d x))^3}+\frac {1}{16 a^5 (a-a \cos (c+d x))^4}\right )}{d}\) |
Input:
Int[Csc[c + d*x]^9*(a + a*Sec[c + d*x])^3,x]
Output:
-((a^12*(1/(16*a^5*(a - a*Cos[c + d*x])^4) + 1/(3*a^6*(a - a*Cos[c + d*x]) ^3) + 39/(32*a^7*(a - a*Cos[c + d*x])^2) + 75/(16*a^8*(a - a*Cos[c + d*x]) ) + 1/(32*a^8*(a + a*Cos[c + d*x])) + (8*Log[a*Cos[c + d*x]])/a^9 - (501*L og[a - a*Cos[c + d*x]])/(64*a^9) - (11*Log[a + a*Cos[c + d*x]])/(64*a^9) - (3*Sec[c + d*x])/a^9 - Sec[c + d*x]^2/(2*a^9)))/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[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 0.69 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.96
| method | result | size |
| norman | \(\frac {-\frac {a^{3}}{256 d}-\frac {19 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{384 d}-\frac {263 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{768 d}-\frac {431 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{192 d}-\frac {a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{64 d}-\frac {451 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{64 d}+\frac {749 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}+\frac {501 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {8 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {8 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(210\) |
| parallelrisch | \(-\frac {a^{3} \left (2048 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+2048 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-4008 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {38 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3}+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {263 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3}+\frac {1724 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3}+1804 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2996\right )}{256 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}\) | \(231\) |
| risch | \(\frac {a^{3} \left (735 \,{\mathrm e}^{13 i \left (d x +c \right )}-3642 \,{\mathrm e}^{12 i \left (d x +c \right )}+6662 \,{\mathrm e}^{11 i \left (d x +c \right )}-4650 \,{\mathrm e}^{10 i \left (d x +c \right )}-1983 \,{\mathrm e}^{9 i \left (d x +c \right )}+8868 \,{\mathrm e}^{8 i \left (d x +c \right )}-12748 \,{\mathrm e}^{7 i \left (d x +c \right )}+8868 \,{\mathrm e}^{6 i \left (d x +c \right )}-1983 \,{\mathrm e}^{5 i \left (d x +c \right )}-4650 \,{\mathrm e}^{4 i \left (d x +c \right )}+6662 \,{\mathrm e}^{3 i \left (d x +c \right )}-3642 \,{\mathrm e}^{2 i \left (d x +c \right )}+735 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{48 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {501 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{32 d}+\frac {11 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{32 d}-\frac {8 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(253\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8} \cos \left (d x +c \right )^{2}}-\frac {5}{24 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )^{2}}-\frac {5}{12 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{2}}+\frac {5}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {5}{2 \sin \left (d x +c \right )^{2}}+5 \ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8} \cos \left (d x +c \right )}-\frac {3}{16 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )}-\frac {21}{64 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {105}{128 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {315}{128 \cos \left (d x +c \right )}+\frac {315 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+3 a^{3} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8}}-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{3} \left (\left (-\frac {\csc \left (d x +c \right )^{7}}{8}-\frac {7 \csc \left (d x +c \right )^{5}}{48}-\frac {35 \csc \left (d x +c \right )^{3}}{192}-\frac {35 \csc \left (d x +c \right )}{128}\right ) \cot \left (d x +c \right )+\frac {35 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\) | \(330\) |
| default | \(\frac {a^{3} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8} \cos \left (d x +c \right )^{2}}-\frac {5}{24 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )^{2}}-\frac {5}{12 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{2}}+\frac {5}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {5}{2 \sin \left (d x +c \right )^{2}}+5 \ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8} \cos \left (d x +c \right )}-\frac {3}{16 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )}-\frac {21}{64 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {105}{128 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {315}{128 \cos \left (d x +c \right )}+\frac {315 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+3 a^{3} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8}}-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{3} \left (\left (-\frac {\csc \left (d x +c \right )^{7}}{8}-\frac {7 \csc \left (d x +c \right )^{5}}{48}-\frac {35 \csc \left (d x +c \right )^{3}}{192}-\frac {35 \csc \left (d x +c \right )}{128}\right ) \cot \left (d x +c \right )+\frac {35 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\) | \(330\) |
Input:
int(csc(d*x+c)^9*(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
(-1/256*a^3/d-19/384*a^3/d*tan(1/2*d*x+1/2*c)^2-263/768*a^3/d*tan(1/2*d*x+ 1/2*c)^4-431/192*a^3/d*tan(1/2*d*x+1/2*c)^6-1/64*a^3/d*tan(1/2*d*x+1/2*c)^ 14-451/64*a^3/d*tan(1/2*d*x+1/2*c)^10+749/64*a^3/d*tan(1/2*d*x+1/2*c)^8)/t an(1/2*d*x+1/2*c)^8/(tan(1/2*d*x+1/2*c)^2-1)^2+501/32*a^3/d*ln(tan(1/2*d*x +1/2*c))-8*a^3/d*ln(tan(1/2*d*x+1/2*c)-1)-8*a^3/d*ln(tan(1/2*d*x+1/2*c)+1)
Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (206) = 412\).
Time = 0.10 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.92 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {1470 \, a^{3} \cos \left (d x + c\right )^{6} - 3642 \, a^{3} \cos \left (d x + c\right )^{5} + 1126 \, a^{3} \cos \left (d x + c\right )^{4} + 3390 \, a^{3} \cos \left (d x + c\right )^{3} - 2752 \, a^{3} \cos \left (d x + c\right )^{2} + 288 \, a^{3} \cos \left (d x + c\right ) + 96 \, a^{3} - 1536 \, {\left (a^{3} \cos \left (d x + c\right )^{7} - 3 \, a^{3} \cos \left (d x + c\right )^{6} + 2 \, a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\cos \left (d x + c\right )\right ) + 33 \, {\left (a^{3} \cos \left (d x + c\right )^{7} - 3 \, a^{3} \cos \left (d x + c\right )^{6} + 2 \, a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 1503 \, {\left (a^{3} \cos \left (d x + c\right )^{7} - 3 \, a^{3} \cos \left (d x + c\right )^{6} + 2 \, a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{192 \, {\left (d \cos \left (d x + c\right )^{7} - 3 \, d \cos \left (d x + c\right )^{6} + 2 \, d \cos \left (d x + c\right )^{5} + 2 \, d \cos \left (d x + c\right )^{4} - 3 \, d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )}} \] Input:
integrate(csc(d*x+c)^9*(a+a*sec(d*x+c))^3,x, algorithm="fricas")
Output:
1/192*(1470*a^3*cos(d*x + c)^6 - 3642*a^3*cos(d*x + c)^5 + 1126*a^3*cos(d* x + c)^4 + 3390*a^3*cos(d*x + c)^3 - 2752*a^3*cos(d*x + c)^2 + 288*a^3*cos (d*x + c) + 96*a^3 - 1536*(a^3*cos(d*x + c)^7 - 3*a^3*cos(d*x + c)^6 + 2*a ^3*cos(d*x + c)^5 + 2*a^3*cos(d*x + c)^4 - 3*a^3*cos(d*x + c)^3 + a^3*cos( d*x + c)^2)*log(-cos(d*x + c)) + 33*(a^3*cos(d*x + c)^7 - 3*a^3*cos(d*x + c)^6 + 2*a^3*cos(d*x + c)^5 + 2*a^3*cos(d*x + c)^4 - 3*a^3*cos(d*x + c)^3 + a^3*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) + 1503*(a^3*cos(d*x + c) ^7 - 3*a^3*cos(d*x + c)^6 + 2*a^3*cos(d*x + c)^5 + 2*a^3*cos(d*x + c)^4 - 3*a^3*cos(d*x + c)^3 + a^3*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2))/( d*cos(d*x + c)^7 - 3*d*cos(d*x + c)^6 + 2*d*cos(d*x + c)^5 + 2*d*cos(d*x + c)^4 - 3*d*cos(d*x + c)^3 + d*cos(d*x + c)^2)
Timed out. \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(csc(d*x+c)**9*(a+a*sec(d*x+c))**3,x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.87 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {33 \, a^{3} \log \left (\cos \left (d x + c\right ) + 1\right ) + 1503 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 1536 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac {2 \, {\left (735 \, a^{3} \cos \left (d x + c\right )^{6} - 1821 \, a^{3} \cos \left (d x + c\right )^{5} + 563 \, a^{3} \cos \left (d x + c\right )^{4} + 1695 \, a^{3} \cos \left (d x + c\right )^{3} - 1376 \, a^{3} \cos \left (d x + c\right )^{2} + 144 \, a^{3} \cos \left (d x + c\right ) + 48 \, a^{3}\right )}}{\cos \left (d x + c\right )^{7} - 3 \, \cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} + 2 \, \cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}}{192 \, d} \] Input:
integrate(csc(d*x+c)^9*(a+a*sec(d*x+c))^3,x, algorithm="maxima")
Output:
1/192*(33*a^3*log(cos(d*x + c) + 1) + 1503*a^3*log(cos(d*x + c) - 1) - 153 6*a^3*log(cos(d*x + c)) + 2*(735*a^3*cos(d*x + c)^6 - 1821*a^3*cos(d*x + c )^5 + 563*a^3*cos(d*x + c)^4 + 1695*a^3*cos(d*x + c)^3 - 1376*a^3*cos(d*x + c)^2 + 144*a^3*cos(d*x + c) + 48*a^3)/(cos(d*x + c)^7 - 3*cos(d*x + c)^6 + 2*cos(d*x + c)^5 + 2*cos(d*x + c)^4 - 3*cos(d*x + c)^3 + cos(d*x + c)^2 ))/d
Time = 0.14 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.65 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {1}{192} \, a^{3} {\left (\frac {33 \, \log \left ({\left | \cos \left (d x + c\right ) + 1 \right |}\right )}{d} + \frac {1503 \, \log \left ({\left | \cos \left (d x + c\right ) - 1 \right |}\right )}{d} - \frac {1536 \, \log \left ({\left | \cos \left (d x + c\right ) \right |}\right )}{d} + \frac {2 \, {\left (735 \, \cos \left (d x + c\right )^{6} - 1821 \, \cos \left (d x + c\right )^{5} + 563 \, \cos \left (d x + c\right )^{4} + 1695 \, \cos \left (d x + c\right )^{3} - 1376 \, \cos \left (d x + c\right )^{2} + 144 \, \cos \left (d x + c\right ) + 48\right )}}{d {\left (\cos \left (d x + c\right ) + 1\right )} {\left (\cos \left (d x + c\right ) - 1\right )}^{4} \cos \left (d x + c\right )^{2}}\right )} \] Input:
integrate(csc(d*x+c)^9*(a+a*sec(d*x+c))^3,x, algorithm="giac")
Output:
1/192*a^3*(33*log(abs(cos(d*x + c) + 1))/d + 1503*log(abs(cos(d*x + c) - 1 ))/d - 1536*log(abs(cos(d*x + c)))/d + 2*(735*cos(d*x + c)^6 - 1821*cos(d* x + c)^5 + 563*cos(d*x + c)^4 + 1695*cos(d*x + c)^3 - 1376*cos(d*x + c)^2 + 144*cos(d*x + c) + 48)/(d*(cos(d*x + c) + 1)*(cos(d*x + c) - 1)^4*cos(d* x + c)^2))
Time = 10.75 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.89 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {501\,a^3\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{64\,d}+\frac {11\,a^3\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{64\,d}+\frac {\frac {245\,a^3\,{\cos \left (c+d\,x\right )}^6}{32}-\frac {607\,a^3\,{\cos \left (c+d\,x\right )}^5}{32}+\frac {563\,a^3\,{\cos \left (c+d\,x\right )}^4}{96}+\frac {565\,a^3\,{\cos \left (c+d\,x\right )}^3}{32}-\frac {43\,a^3\,{\cos \left (c+d\,x\right )}^2}{3}+\frac {3\,a^3\,\cos \left (c+d\,x\right )}{2}+\frac {a^3}{2}}{d\,\left ({\cos \left (c+d\,x\right )}^7-3\,{\cos \left (c+d\,x\right )}^6+2\,{\cos \left (c+d\,x\right )}^5+2\,{\cos \left (c+d\,x\right )}^4-3\,{\cos \left (c+d\,x\right )}^3+{\cos \left (c+d\,x\right )}^2\right )}-\frac {8\,a^3\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \] Input:
int((a + a/cos(c + d*x))^3/sin(c + d*x)^9,x)
Output:
(501*a^3*log(cos(c + d*x) - 1))/(64*d) + (11*a^3*log(cos(c + d*x) + 1))/(6 4*d) + ((3*a^3*cos(c + d*x))/2 + a^3/2 - (43*a^3*cos(c + d*x)^2)/3 + (565* a^3*cos(c + d*x)^3)/32 + (563*a^3*cos(c + d*x)^4)/96 - (607*a^3*cos(c + d* x)^5)/32 + (245*a^3*cos(c + d*x)^6)/32)/(d*(cos(c + d*x)^2 - 3*cos(c + d*x )^3 + 2*cos(c + d*x)^4 + 2*cos(c + d*x)^5 - 3*cos(c + d*x)^6 + cos(c + d*x )^7)) - (8*a^3*log(cos(c + d*x)))/d
Time = 0.18 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.59 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {a^{3} \left (-6144 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+12288 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-6144 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-6144 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+12288 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-6144 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+12024 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}-24048 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+12024 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}-8988 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+12564 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-1724 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-263 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-38 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-3\right )}{768 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )} \] Input:
int(csc(d*x+c)^9*(a+a*sec(d*x+c))^3,x)
Output:
(a**3*( - 6144*log(tan((c + d*x)/2) - 1)*tan((c + d*x)/2)**12 + 12288*log( tan((c + d*x)/2) - 1)*tan((c + d*x)/2)**10 - 6144*log(tan((c + d*x)/2) - 1 )*tan((c + d*x)/2)**8 - 6144*log(tan((c + d*x)/2) + 1)*tan((c + d*x)/2)**1 2 + 12288*log(tan((c + d*x)/2) + 1)*tan((c + d*x)/2)**10 - 6144*log(tan((c + d*x)/2) + 1)*tan((c + d*x)/2)**8 + 12024*log(tan((c + d*x)/2))*tan((c + d*x)/2)**12 - 24048*log(tan((c + d*x)/2))*tan((c + d*x)/2)**10 + 12024*lo g(tan((c + d*x)/2))*tan((c + d*x)/2)**8 - 12*tan((c + d*x)/2)**14 - 8988*t an((c + d*x)/2)**12 + 12564*tan((c + d*x)/2)**10 - 1724*tan((c + d*x)/2)** 6 - 263*tan((c + d*x)/2)**4 - 38*tan((c + d*x)/2)**2 - 3))/(768*tan((c + d *x)/2)**8*d*(tan((c + d*x)/2)**4 - 2*tan((c + d*x)/2)**2 + 1))