Integrand size = 21, antiderivative size = 165 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^6}{6 d (a-a \cos (c+d x))^3}-\frac {7 a^7}{8 d \left (a^2-a^2 \cos (c+d x)\right )^2}-\frac {31 a^7}{8 d \left (a^4-a^4 \cos (c+d x)\right )}+\frac {111 a^3 \log (1-\cos (c+d x))}{16 d}-\frac {7 a^3 \log (\cos (c+d x))}{d}+\frac {a^3 \log (1+\cos (c+d x))}{16 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \] Output:
-1/6*a^6/d/(a-a*cos(d*x+c))^3-7/8*a^7/d/(a^2-a^2*cos(d*x+c))^2-31/8*a^7/d/ (a^4-a^4*cos(d*x+c))+111/16*a^3*ln(1-cos(d*x+c))/d-7*a^3*ln(cos(d*x+c))/d+ 1/16*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 = 0.70 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.78 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^3 (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (186 \csc ^2\left (\frac {1}{2} (c+d x)\right )+21 \csc ^4\left (\frac {1}{2} (c+d x)\right )+2 \csc ^6\left (\frac {1}{2} (c+d x)\right )-12 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-56 \log (\cos (c+d x))+111 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 \sec (c+d x)+4 \sec ^2(c+d x)\right )\right )}{768 d} \] Input:
Integrate[Csc[c + d*x]^7*(a + a*Sec[c + d*x])^3,x]
Output:
-1/768*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(186*Csc[(c + d*x)/2]^ 2 + 21*Csc[(c + d*x)/2]^4 + 2*Csc[(c + d*x)/2]^6 - 12*(Log[Cos[(c + d*x)/2 ]] - 56*Log[Cos[c + d*x]] + 111*Log[Sin[(c + d*x)/2]] + 24*Sec[c + d*x] + 4*Sec[c + d*x]^2)))/d
Time = 0.48 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.88, 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.
| \(\displaystyle \int \csc ^7(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}{\cos \left (c+d x-\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \csc ^7(c+d x) \sec ^3(c+d x) \left (-(a (-\cos (c+d x))-a)^3\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -(\cos (c+d x) a+a)^3 \csc ^7(c+d x) \sec ^3(c+d x)dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \csc ^7(c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (a-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}{\sin \left (c+d x-\frac {\pi }{2}\right )^3 \cos \left (c+d x-\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )^3}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^7 \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )^3}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^7 \int -\frac {\sec ^3(c+d x)}{(a-a \cos (c+d x))^4 (\cos (c+d x) a+a)}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^7 \int \frac {\sec ^3(c+d x)}{(a-a \cos (c+d x))^4 (\cos (c+d x) a+a)}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^{10} \int \frac {\sec ^3(c+d x)}{a^3 (a-a \cos (c+d x))^4 (\cos (c+d x) a+a)}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {a^{10} \int \left (\frac {\sec ^3(c+d x)}{a^8}+\frac {3 \sec ^2(c+d x)}{a^8}+\frac {7 \sec (c+d x)}{a^8}+\frac {111}{16 a^7 (a-a \cos (c+d x))}-\frac {1}{16 a^7 (\cos (c+d x) a+a)}+\frac {31}{8 a^6 (a-a \cos (c+d x))^2}+\frac {7}{4 a^5 (a-a \cos (c+d x))^3}+\frac {1}{2 a^4 (a-a \cos (c+d x))^4}\right )d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{10} \left (-\frac {\sec ^2(c+d x)}{2 a^7}-\frac {3 \sec (c+d x)}{a^7}+\frac {7 \log (a \cos (c+d x))}{a^7}-\frac {111 \log (a-a \cos (c+d x))}{16 a^7}-\frac {\log (a \cos (c+d x)+a)}{16 a^7}+\frac {31}{8 a^6 (a-a \cos (c+d x))}+\frac {7}{8 a^5 (a-a \cos (c+d x))^2}+\frac {1}{6 a^4 (a-a \cos (c+d x))^3}\right )}{d}\) |
Input:
Int[Csc[c + d*x]^7*(a + a*Sec[c + d*x])^3,x]
Output:
-((a^10*(1/(6*a^4*(a - a*Cos[c + d*x])^3) + 7/(8*a^5*(a - a*Cos[c + d*x])^ 2) + 31/(8*a^6*(a - a*Cos[c + d*x])) + (7*Log[a*Cos[c + d*x]])/a^7 - (111* Log[a - a*Cos[c + d*x]])/(16*a^7) - Log[a + a*Cos[c + d*x]]/(16*a^7) - (3* Sec[c + d*x])/a^7 - Sec[c + d*x]^2/(2*a^7)))/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 = 0.52 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.99
| method | result | size |
| parallelrisch | \(\frac {79 a^{3} \left (\frac {112 \left (-1-\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{79}+\frac {112 \left (-1-\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{79}+\frac {222 \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{79}+\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\cos \left (d x +c \right )-\frac {161 \cos \left (2 d x +2 c \right )}{237}+\frac {71 \cos \left (3 d x +3 c \right )}{237}-\frac {449 \cos \left (4 d x +4 c \right )}{7584}-\frac {4319}{7584}\right )\right )}{16 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(164\) |
| norman | \(\frac {-\frac {a^{3}}{48 d}-\frac {23 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{96 d}-\frac {91 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{48 d}-\frac {103 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16 d}+\frac {339 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{32 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}+\frac {111 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {7 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {7 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(172\) |
| risch | \(\frac {a^{3} \left (165 \,{\mathrm e}^{9 i \left (d x +c \right )}-822 \,{\mathrm e}^{8 i \left (d x +c \right )}+1852 \,{\mathrm e}^{7 i \left (d x +c \right )}-2754 \,{\mathrm e}^{6 i \left (d x +c \right )}+3182 \,{\mathrm e}^{5 i \left (d x +c \right )}-2754 \,{\mathrm e}^{4 i \left (d x +c \right )}+1852 \,{\mathrm e}^{3 i \left (d x +c \right )}-822 \,{\mathrm e}^{2 i \left (d x +c \right )}+165 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{12 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{6} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}+\frac {111 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}-\frac {7 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(196\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {1}{6 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )^{2}}-\frac {1}{3 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{2}}+\frac {1}{\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {2}{\sin \left (d x +c \right )^{2}}+4 \ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {1}{6 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )}-\frac {7}{24 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {35}{48 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {35}{16 \cos \left (d x +c \right )}+\frac {35 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+3 a^{3} \left (-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{3} \left (\left (-\frac {\csc \left (d x +c \right )^{5}}{6}-\frac {5 \csc \left (d x +c \right )^{3}}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\) | \(273\) |
| default | \(\frac {a^{3} \left (-\frac {1}{6 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )^{2}}-\frac {1}{3 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{2}}+\frac {1}{\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {2}{\sin \left (d x +c \right )^{2}}+4 \ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {1}{6 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )}-\frac {7}{24 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {35}{48 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {35}{16 \cos \left (d x +c \right )}+\frac {35 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+3 a^{3} \left (-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{3} \left (\left (-\frac {\csc \left (d x +c \right )^{5}}{6}-\frac {5 \csc \left (d x +c \right )^{3}}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\) | \(273\) |
Input:
int(csc(d*x+c)^7*(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
79/16*a^3*(112/79*(-1-cos(2*d*x+2*c))*ln(tan(1/2*d*x+1/2*c)-1)+112/79*(-1- cos(2*d*x+2*c))*ln(tan(1/2*d*x+1/2*c)+1)+222/79*(1+cos(2*d*x+2*c))*ln(tan( 1/2*d*x+1/2*c))+cot(1/2*d*x+1/2*c)^2*csc(1/2*d*x+1/2*c)^4*(cos(d*x+c)-161/ 237*cos(2*d*x+2*c)+71/237*cos(3*d*x+3*c)-449/7584*cos(4*d*x+4*c)-4319/7584 ))/d/(1+cos(2*d*x+2*c))
Time = 0.10 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.80 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {330 \, a^{3} \cos \left (d x + c\right )^{4} - 822 \, a^{3} \cos \left (d x + c\right )^{3} + 596 \, a^{3} \cos \left (d x + c\right )^{2} - 72 \, a^{3} \cos \left (d x + c\right ) - 24 \, a^{3} - 336 \, {\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{3} - a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\cos \left (d x + c\right )\right ) + 3 \, {\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{3} - a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 333 \, {\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{3} - a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{48 \, {\left (d \cos \left (d x + c\right )^{5} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2}\right )}} \] Input:
integrate(csc(d*x+c)^7*(a+a*sec(d*x+c))^3,x, algorithm="fricas")
Output:
1/48*(330*a^3*cos(d*x + c)^4 - 822*a^3*cos(d*x + c)^3 + 596*a^3*cos(d*x + c)^2 - 72*a^3*cos(d*x + c) - 24*a^3 - 336*(a^3*cos(d*x + c)^5 - 3*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)) + 3*(a^3*cos(d*x + c)^5 - 3*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) + 333*(a^3*cos(d*x + c)^5 - 3*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)^5 - 3*d*cos(d*x + c)^4 + 3*d*cos(d* x + c)^3 - d*cos(d*x + c)^2)
Timed out. \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(csc(d*x+c)**7*(a+a*sec(d*x+c))**3,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.88 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {3 \, a^{3} \log \left (\cos \left (d x + c\right ) + 1\right ) + 333 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 336 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac {2 \, {\left (165 \, a^{3} \cos \left (d x + c\right )^{4} - 411 \, a^{3} \cos \left (d x + c\right )^{3} + 298 \, a^{3} \cos \left (d x + c\right )^{2} - 36 \, a^{3} \cos \left (d x + c\right ) - 12 \, a^{3}\right )}}{\cos \left (d x + c\right )^{5} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2}}}{48 \, d} \] Input:
integrate(csc(d*x+c)^7*(a+a*sec(d*x+c))^3,x, algorithm="maxima")
Output:
1/48*(3*a^3*log(cos(d*x + c) + 1) + 333*a^3*log(cos(d*x + c) - 1) - 336*a^ 3*log(cos(d*x + c)) + 2*(165*a^3*cos(d*x + c)^4 - 411*a^3*cos(d*x + c)^3 + 298*a^3*cos(d*x + c)^2 - 36*a^3*cos(d*x + c) - 12*a^3)/(cos(d*x + c)^5 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^3 - cos(d*x + c)^2))/d
Time = 0.14 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.68 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {1}{48} \, a^{3} {\left (\frac {3 \, \log \left ({\left | \cos \left (d x + c\right ) + 1 \right |}\right )}{d} + \frac {333 \, \log \left ({\left | \cos \left (d x + c\right ) - 1 \right |}\right )}{d} - \frac {336 \, \log \left ({\left | \cos \left (d x + c\right ) \right |}\right )}{d} + \frac {2 \, {\left (165 \, \cos \left (d x + c\right )^{4} - 411 \, \cos \left (d x + c\right )^{3} + 298 \, \cos \left (d x + c\right )^{2} - 36 \, \cos \left (d x + c\right ) - 12\right )}}{d {\left (\cos \left (d x + c\right ) - 1\right )}^{3} \cos \left (d x + c\right )^{2}}\right )} \] Input:
integrate(csc(d*x+c)^7*(a+a*sec(d*x+c))^3,x, algorithm="giac")
Output:
1/48*a^3*(3*log(abs(cos(d*x + c) + 1))/d + 333*log(abs(cos(d*x + c) - 1))/ d - 336*log(abs(cos(d*x + c)))/d + 2*(165*cos(d*x + c)^4 - 411*cos(d*x + c )^3 + 298*cos(d*x + c)^2 - 36*cos(d*x + c) - 12)/(d*(cos(d*x + c) - 1)^3*c os(d*x + c)^2))
Time = 10.20 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.92 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {111\,a^3\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{16\,d}+\frac {a^3\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{16\,d}+\frac {-\frac {55\,a^3\,{\cos \left (c+d\,x\right )}^4}{8}+\frac {137\,a^3\,{\cos \left (c+d\,x\right )}^3}{8}-\frac {149\,a^3\,{\cos \left (c+d\,x\right )}^2}{12}+\frac {3\,a^3\,\cos \left (c+d\,x\right )}{2}+\frac {a^3}{2}}{d\,\left (-{\cos \left (c+d\,x\right )}^5+3\,{\cos \left (c+d\,x\right )}^4-3\,{\cos \left (c+d\,x\right )}^3+{\cos \left (c+d\,x\right )}^2\right )}-\frac {7\,a^3\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \] Input:
int((a + a/cos(c + d*x))^3/sin(c + d*x)^7,x)
Output:
(111*a^3*log(cos(c + d*x) - 1))/(16*d) + (a^3*log(cos(c + d*x) + 1))/(16*d ) + ((3*a^3*cos(c + d*x))/2 + a^3/2 - (149*a^3*cos(c + d*x)^2)/12 + (137*a ^3*cos(c + d*x)^3)/8 - (55*a^3*cos(c + d*x)^4)/8)/(d*(cos(c + d*x)^2 - 3*c os(c + d*x)^3 + 3*cos(c + d*x)^4 - cos(c + d*x)^5)) - (7*a^3*log(cos(c + d *x)))/d
Time = 0.17 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.94 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {a^{3} \left (-672 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+1344 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-672 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-672 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+1344 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-672 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+1332 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-2664 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+1332 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-1017 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+1416 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-182 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2\right )}{96 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )} \] Input:
int(csc(d*x+c)^7*(a+a*sec(d*x+c))^3,x)
Output:
(a**3*( - 672*log(tan((c + d*x)/2) - 1)*tan((c + d*x)/2)**10 + 1344*log(ta n((c + d*x)/2) - 1)*tan((c + d*x)/2)**8 - 672*log(tan((c + d*x)/2) - 1)*ta n((c + d*x)/2)**6 - 672*log(tan((c + d*x)/2) + 1)*tan((c + d*x)/2)**10 + 1 344*log(tan((c + d*x)/2) + 1)*tan((c + d*x)/2)**8 - 672*log(tan((c + d*x)/ 2) + 1)*tan((c + d*x)/2)**6 + 1332*log(tan((c + d*x)/2))*tan((c + d*x)/2)* *10 - 2664*log(tan((c + d*x)/2))*tan((c + d*x)/2)**8 + 1332*log(tan((c + d *x)/2))*tan((c + d*x)/2)**6 - 1017*tan((c + d*x)/2)**10 + 1416*tan((c + d* x)/2)**8 - 182*tan((c + d*x)/2)**4 - 23*tan((c + d*x)/2)**2 - 2))/(96*tan( (c + d*x)/2)**6*d*(tan((c + d*x)/2)**4 - 2*tan((c + d*x)/2)**2 + 1))