Integrand size = 29, antiderivative size = 154 \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {4 a^2 \csc (c+d x)}{d}-\frac {a^2 \csc ^2(c+d x)}{d}-\frac {a^2 \csc ^3(c+d x)}{3 d}-\frac {49 a^2 \log (1-\sin (c+d x))}{8 d}+\frac {6 a^2 \log (\sin (c+d x))}{d}+\frac {a^2 \log (1+\sin (c+d x))}{8 d}+\frac {a^4}{4 d (a-a \sin (c+d x))^2}+\frac {9 a^5}{4 d \left (a^3-a^3 \sin (c+d x)\right )} \] Output:
-4*a^2*csc(d*x+c)/d-a^2*csc(d*x+c)^2/d-1/3*a^2*csc(d*x+c)^3/d-49/8*a^2*ln( 1-sin(d*x+c))/d+6*a^2*ln(sin(d*x+c))/d+1/8*a^2*ln(1+sin(d*x+c))/d+1/4*a^4/ d/(a-a*sin(d*x+c))^2+9/4*a^5/d/(a^3-a^3*sin(d*x+c))
Time = 6.07 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.86 \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^5 \left (-\frac {4 \csc (c+d x)}{a^3}-\frac {\csc ^2(c+d x)}{a^3}-\frac {\csc ^3(c+d x)}{3 a^3}-\frac {49 \log (1-\sin (c+d x))}{8 a^3}+\frac {6 \log (\sin (c+d x))}{a^3}+\frac {\log (1+\sin (c+d x))}{8 a^3}+\frac {1}{4 a (a-a \sin (c+d x))^2}+\frac {9}{4 a^2 (a-a \sin (c+d x))}\right )}{d} \] Input:
Integrate[Csc[c + d*x]^4*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]
Output:
(a^5*((-4*Csc[c + d*x])/a^3 - Csc[c + d*x]^2/a^3 - Csc[c + d*x]^3/(3*a^3) - (49*Log[1 - Sin[c + d*x]])/(8*a^3) + (6*Log[Sin[c + d*x]])/a^3 + Log[1 + Sin[c + d*x]]/(8*a^3) + 1/(4*a*(a - a*Sin[c + d*x])^2) + 9/(4*a^2*(a - a* Sin[c + d*x]))))/d
Time = 0.38 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3315, 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.
| \(\displaystyle \int \csc ^4(c+d x) \sec ^5(c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^2}{\sin (c+d x)^4 \cos (c+d x)^5}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^5 \int \frac {\csc ^4(c+d x)}{(a-a \sin (c+d x))^3 (\sin (c+d x) a+a)}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^9 \int \frac {\csc ^4(c+d x)}{a^4 (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {a^9 \int \left (\frac {\csc ^4(c+d x)}{a^8}+\frac {2 \csc ^3(c+d x)}{a^8}+\frac {4 \csc ^2(c+d x)}{a^8}+\frac {6 \csc (c+d x)}{a^8}+\frac {49}{8 a^7 (a-a \sin (c+d x))}+\frac {1}{8 a^7 (\sin (c+d x) a+a)}+\frac {9}{4 a^6 (a-a \sin (c+d x))^2}+\frac {1}{2 a^5 (a-a \sin (c+d x))^3}\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^9 \left (-\frac {\csc ^3(c+d x)}{3 a^7}-\frac {\csc ^2(c+d x)}{a^7}-\frac {4 \csc (c+d x)}{a^7}+\frac {6 \log (a \sin (c+d x))}{a^7}-\frac {49 \log (a-a \sin (c+d x))}{8 a^7}+\frac {\log (a \sin (c+d x)+a)}{8 a^7}+\frac {9}{4 a^6 (a-a \sin (c+d x))}+\frac {1}{4 a^5 (a-a \sin (c+d x))^2}\right )}{d}\) |
Input:
Int[Csc[c + d*x]^4*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]
Output:
(a^9*((-4*Csc[c + d*x])/a^7 - Csc[c + d*x]^2/a^7 - Csc[c + d*x]^3/(3*a^7) + (6*Log[a*Sin[c + d*x]])/a^7 - (49*Log[a - a*Sin[c + d*x]])/(8*a^7) + Log [a + a*Sin[c + d*x]]/(8*a^7) + 1/(4*a^5*(a - a*Sin[c + d*x])^2) + 9/(4*a^6 *(a - a*Sin[c + d*x]))))/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]
Result contains complex when optimal does not.
Time = 0.98 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.32
| method | result | size |
| risch | \(-\frac {i a^{2} \left (-228 i {\mathrm e}^{8 i \left (d x +c \right )}+75 \,{\mathrm e}^{9 i \left (d x +c \right )}+652 i {\mathrm e}^{6 i \left (d x +c \right )}-412 \,{\mathrm e}^{7 i \left (d x +c \right )}-652 i {\mathrm e}^{4 i \left (d x +c \right )}+738 \,{\mathrm e}^{5 i \left (d x +c \right )}+228 i {\mathrm e}^{2 i \left (d x +c \right )}-412 \,{\mathrm e}^{3 i \left (d x +c \right )}+75 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{6 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{4} d}-\frac {49 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{4 d}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{4 d}+\frac {6 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(204\) |
| derivativedivides | \(\frac {a^{2} \left (\frac {1}{4 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {5}{8 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {15}{8 \sin \left (d x +c \right )}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+2 a^{2} \left (\frac {1}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}+\frac {3}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )^{2}}+3 \ln \left (\tan \left (d x +c \right )\right )\right )+a^{2} \left (\frac {1}{4 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{4}}-\frac {7}{12 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {35}{24 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {35}{8 \sin \left (d x +c \right )}+\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(219\) |
| default | \(\frac {a^{2} \left (\frac {1}{4 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {5}{8 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {15}{8 \sin \left (d x +c \right )}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+2 a^{2} \left (\frac {1}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}+\frac {3}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )^{2}}+3 \ln \left (\tan \left (d x +c \right )\right )\right )+a^{2} \left (\frac {1}{4 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{4}}-\frac {7}{12 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {35}{24 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {35}{8 \sin \left (d x +c \right )}+\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(219\) |
| parallelrisch | \(-\frac {53 \left (\frac {588 \left (-3+\cos \left (2 d x +2 c \right )+4 \sin \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{53}+\frac {12 \left (3-\cos \left (2 d x +2 c \right )-4 \sin \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{53}+\frac {288 \left (3-\cos \left (2 d x +2 c \right )-4 \sin \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{53}+\frac {4 \left (-\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-28 \sec \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{53}+\left (\left (\cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-\frac {55 \cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )}{212}-\frac {55 \cos \left (\frac {9 d x}{2}+\frac {9 c}{2}\right )}{212}-\frac {173 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{106}+\cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\frac {2432 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{53}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\frac {1160 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (\cos \left (d x +c \right )-3\right )}{53}\right ) a^{2}}{48 d \left (-3+\cos \left (2 d x +2 c \right )+4 \sin \left (d x +c \right )\right )}\) | \(279\) |
Input:
int(csc(d*x+c)^4*sec(d*x+c)^5*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-1/6*I*a^2/(exp(2*I*(d*x+c))-1)^3/(exp(I*(d*x+c))-I)^4/d*(-228*I*exp(8*I*( d*x+c))+75*exp(9*I*(d*x+c))+652*I*exp(6*I*(d*x+c))-412*exp(7*I*(d*x+c))-65 2*I*exp(4*I*(d*x+c))+738*exp(5*I*(d*x+c))+228*I*exp(2*I*(d*x+c))-412*exp(3 *I*(d*x+c))+75*exp(I*(d*x+c)))-49/4*a^2/d*ln(exp(I*(d*x+c))-I)+1/4*a^2/d*l n(exp(I*(d*x+c))+I)+6*a^2/d*ln(exp(2*I*(d*x+c))-1)
Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (146) = 292\).
Time = 0.09 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.40 \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {150 \, a^{2} \cos \left (d x + c\right )^{4} - 356 \, a^{2} \cos \left (d x + c\right )^{2} + 214 \, a^{2} + 144 \, {\left (2 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} - {\left (a^{2} \cos \left (d x + c\right )^{4} - 3 \, a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 3 \, {\left (2 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} - {\left (a^{2} \cos \left (d x + c\right )^{4} - 3 \, a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 147 \, {\left (2 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} - {\left (a^{2} \cos \left (d x + c\right )^{4} - 3 \, a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 4 \, {\left (57 \, a^{2} \cos \left (d x + c\right )^{2} - 55 \, a^{2}\right )} \sin \left (d x + c\right )}{24 \, {\left (2 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right )^{4} - 3 \, d \cos \left (d x + c\right )^{2} + 2 \, d\right )} \sin \left (d x + c\right ) + 2 \, d\right )}} \] Input:
integrate(csc(d*x+c)^4*sec(d*x+c)^5*(a+a*sin(d*x+c))^2,x, algorithm="frica s")
Output:
1/24*(150*a^2*cos(d*x + c)^4 - 356*a^2*cos(d*x + c)^2 + 214*a^2 + 144*(2*a ^2*cos(d*x + c)^4 - 4*a^2*cos(d*x + c)^2 + 2*a^2 - (a^2*cos(d*x + c)^4 - 3 *a^2*cos(d*x + c)^2 + 2*a^2)*sin(d*x + c))*log(1/2*sin(d*x + c)) + 3*(2*a^ 2*cos(d*x + c)^4 - 4*a^2*cos(d*x + c)^2 + 2*a^2 - (a^2*cos(d*x + c)^4 - 3* a^2*cos(d*x + c)^2 + 2*a^2)*sin(d*x + c))*log(sin(d*x + c) + 1) - 147*(2*a ^2*cos(d*x + c)^4 - 4*a^2*cos(d*x + c)^2 + 2*a^2 - (a^2*cos(d*x + c)^4 - 3 *a^2*cos(d*x + c)^2 + 2*a^2)*sin(d*x + c))*log(-sin(d*x + c) + 1) + 4*(57* a^2*cos(d*x + c)^2 - 55*a^2)*sin(d*x + c))/(2*d*cos(d*x + c)^4 - 4*d*cos(d *x + c)^2 - (d*cos(d*x + c)^4 - 3*d*cos(d*x + c)^2 + 2*d)*sin(d*x + c) + 2 *d)
Timed out. \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate(csc(d*x+c)**4*sec(d*x+c)**5*(a+a*sin(d*x+c))**2,x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.86 \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3 \, a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 147 \, a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) + 144 \, a^{2} \log \left (\sin \left (d x + c\right )\right ) - \frac {2 \, {\left (75 \, a^{2} \sin \left (d x + c\right )^{4} - 114 \, a^{2} \sin \left (d x + c\right )^{3} + 28 \, a^{2} \sin \left (d x + c\right )^{2} + 4 \, a^{2} \sin \left (d x + c\right ) + 4 \, a^{2}\right )}}{\sin \left (d x + c\right )^{5} - 2 \, \sin \left (d x + c\right )^{4} + \sin \left (d x + c\right )^{3}}}{24 \, d} \] Input:
integrate(csc(d*x+c)^4*sec(d*x+c)^5*(a+a*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
1/24*(3*a^2*log(sin(d*x + c) + 1) - 147*a^2*log(sin(d*x + c) - 1) + 144*a^ 2*log(sin(d*x + c)) - 2*(75*a^2*sin(d*x + c)^4 - 114*a^2*sin(d*x + c)^3 + 28*a^2*sin(d*x + c)^2 + 4*a^2*sin(d*x + c) + 4*a^2)/(sin(d*x + c)^5 - 2*si n(d*x + c)^4 + sin(d*x + c)^3))/d
Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.73 \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {1}{24} \, a^{2} {\left (\frac {3 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{d} - \frac {147 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{d} + \frac {144 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{d} - \frac {2 \, {\left (75 \, \sin \left (d x + c\right )^{4} - 114 \, \sin \left (d x + c\right )^{3} + 28 \, \sin \left (d x + c\right )^{2} + 4 \, \sin \left (d x + c\right ) + 4\right )}}{d {\left (\sin \left (d x + c\right ) - 1\right )}^{2} \sin \left (d x + c\right )^{3}}\right )} \] Input:
integrate(csc(d*x+c)^4*sec(d*x+c)^5*(a+a*sin(d*x+c))^2,x, algorithm="giac" )
Output:
1/24*a^2*(3*log(abs(sin(d*x + c) + 1))/d - 147*log(abs(sin(d*x + c) - 1))/ d + 144*log(abs(sin(d*x + c)))/d - 2*(75*sin(d*x + c)^4 - 114*sin(d*x + c) ^3 + 28*sin(d*x + c)^2 + 4*sin(d*x + c) + 4)/(d*(sin(d*x + c) - 1)^2*sin(d *x + c)^3))
Time = 32.34 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.91 \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{8\,d}-\frac {49\,a^2\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{8\,d}-\frac {\frac {25\,a^2\,{\sin \left (c+d\,x\right )}^4}{4}-\frac {19\,a^2\,{\sin \left (c+d\,x\right )}^3}{2}+\frac {7\,a^2\,{\sin \left (c+d\,x\right )}^2}{3}+\frac {a^2\,\sin \left (c+d\,x\right )}{3}+\frac {a^2}{3}}{d\,\left ({\sin \left (c+d\,x\right )}^5-2\,{\sin \left (c+d\,x\right )}^4+{\sin \left (c+d\,x\right )}^3\right )}+\frac {6\,a^2\,\ln \left (\sin \left (c+d\,x\right )\right )}{d} \] Input:
int((a + a*sin(c + d*x))^2/(cos(c + d*x)^5*sin(c + d*x)^4),x)
Output:
(a^2*log(sin(c + d*x) + 1))/(8*d) - (49*a^2*log(sin(c + d*x) - 1))/(8*d) - ((a^2*sin(c + d*x))/3 + a^2/3 + (7*a^2*sin(c + d*x)^2)/3 - (19*a^2*sin(c + d*x)^3)/2 + (25*a^2*sin(c + d*x)^4)/4)/(d*(sin(c + d*x)^3 - 2*sin(c + d* x)^4 + sin(c + d*x)^5)) + (6*a^2*log(sin(c + d*x)))/d
Time = 0.22 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.81 \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \left (-588 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{5}+1176 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{4}-588 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{3}+12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{5}-24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{4}+12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{3}+288 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{5}-576 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4}+288 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{3}-197 \sin \left (d x +c \right )^{5}+94 \sin \left (d x +c \right )^{4}+259 \sin \left (d x +c \right )^{3}-112 \sin \left (d x +c \right )^{2}-16 \sin \left (d x +c \right )-16\right )}{48 \sin \left (d x +c \right )^{3} d \left (\sin \left (d x +c \right )^{2}-2 \sin \left (d x +c \right )+1\right )} \] Input:
int(csc(d*x+c)^4*sec(d*x+c)^5*(a+a*sin(d*x+c))^2,x)
Output:
(a**2*( - 588*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**5 + 1176*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4 - 588*log(tan((c + d*x)/2) - 1)*sin(c + d*x )**3 + 12*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**5 - 24*log(tan((c + d*x) /2) + 1)*sin(c + d*x)**4 + 12*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**3 + 288*log(tan((c + d*x)/2))*sin(c + d*x)**5 - 576*log(tan((c + d*x)/2))*sin( c + d*x)**4 + 288*log(tan((c + d*x)/2))*sin(c + d*x)**3 - 197*sin(c + d*x) **5 + 94*sin(c + d*x)**4 + 259*sin(c + d*x)**3 - 112*sin(c + d*x)**2 - 16* sin(c + d*x) - 16))/(48*sin(c + d*x)**3*d*(sin(c + d*x)**2 - 2*sin(c + d*x ) + 1))