Integrand size = 29, antiderivative size = 115 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {3 a^3 \csc (c+d x)}{d}-\frac {a^3 \csc ^2(c+d x)}{2 d}-\frac {6 a^3 \log (1-\sin (c+d x))}{d}+\frac {6 a^3 \log (\sin (c+d x))}{d}+\frac {a^5}{2 d (a-a \sin (c+d x))^2}+\frac {3 a^5}{d \left (a^2-a^2 \sin (c+d x)\right )} \] Output:
-3*a^3*csc(d*x+c)/d-1/2*a^3*csc(d*x+c)^2/d-6*a^3*ln(1-sin(d*x+c))/d+6*a^3* ln(sin(d*x+c))/d+1/2*a^5/d/(a-a*sin(d*x+c))^2+3*a^5/d/(a^2-a^2*sin(d*x+c))
Time = 1.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.63 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \left (6 \csc (c+d x)+\csc ^2(c+d x)+12 \log (1-\sin (c+d x))-12 \log (\sin (c+d x))-\frac {1}{(-1+\sin (c+d x))^2}+\frac {6}{-1+\sin (c+d x)}\right )}{2 d} \] Input:
Integrate[Csc[c + d*x]^3*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]
Output:
-1/2*(a^3*(6*Csc[c + d*x] + Csc[c + d*x]^2 + 12*Log[1 - Sin[c + d*x]] - 12 *Log[Sin[c + d*x]] - (-1 + Sin[c + d*x])^(-2) + 6/(-1 + Sin[c + d*x])))/d
Time = 0.35 (sec) , antiderivative size = 103, 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, 54, 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.
| \(\displaystyle \int \csc ^3(c+d x) \sec ^5(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^3}{\sin (c+d x)^3 \cos (c+d x)^5}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^5 \int \frac {\csc ^3(c+d x)}{(a-a \sin (c+d x))^3}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^8 \int \frac {\csc ^3(c+d x)}{a^3 (a-a \sin (c+d x))^3}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {a^8 \int \left (\frac {\csc ^3(c+d x)}{a^6}+\frac {3 \csc ^2(c+d x)}{a^6}+\frac {6 \csc (c+d x)}{a^6}+\frac {6}{a^5 (a-a \sin (c+d x))}+\frac {3}{a^4 (a-a \sin (c+d x))^2}+\frac {1}{a^3 (a-a \sin (c+d x))^3}\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^8 \left (-\frac {\csc ^2(c+d x)}{2 a^5}-\frac {3 \csc (c+d x)}{a^5}+\frac {6 \log (a \sin (c+d x))}{a^5}-\frac {6 \log (a-a \sin (c+d x))}{a^5}+\frac {3}{a^4 (a-a \sin (c+d x))}+\frac {1}{2 a^3 (a-a \sin (c+d x))^2}\right )}{d}\) |
Input:
Int[Csc[c + d*x]^3*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]
Output:
(a^8*((-3*Csc[c + d*x])/a^5 - Csc[c + d*x]^2/(2*a^5) + (6*Log[a*Sin[c + d* x]])/a^5 - (6*Log[a - a*Sin[c + d*x]])/a^5 + 1/(2*a^3*(a - a*Sin[c + d*x]) ^2) + 3/(a^4*(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_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
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 = 0.90 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.55
| method | result | size |
| risch | \(-\frac {4 i \left (-9 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+3 a^{3} {\mathrm e}^{7 i \left (d x +c \right )}+16 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-13 a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-9 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+13 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-3 a^{3} {\mathrm e}^{i \left (d x +c \right )}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{4} d}-\frac {12 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {6 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(178\) |
| parallelrisch | \(\frac {\left (\left (144-48 \cos \left (2 d x +2 c \right )-192 \sin \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-72+24 \cos \left (2 d x +2 c \right )+96 \sin \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-136 \cos \left (d x +c \right )+34 \cos \left (2 d x +2 c \right )+106\right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (88 \cos \left (d x +c \right )-88\right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (\left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-4\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+8 \sec \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{4 d \left (-3+\cos \left (2 d x +2 c \right )+4 \sin \left (d x +c \right )\right )}\) | \(195\) |
| derivativedivides | \(\frac {a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+3 a^{3} \left (\frac {1}{4 \cos \left (d x +c \right )^{4}}+\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {1}{4 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {5}{8 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {15}{8 \sin \left (d x +c \right )}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{3} \left (\frac {1}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}+\frac {3}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )^{2}}+3 \ln \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(215\) |
| default | \(\frac {a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+3 a^{3} \left (\frac {1}{4 \cos \left (d x +c \right )^{4}}+\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {1}{4 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {5}{8 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {15}{8 \sin \left (d x +c \right )}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{3} \left (\frac {1}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}+\frac {3}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )^{2}}+3 \ln \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(215\) |
Input:
int(csc(d*x+c)^3*sec(d*x+c)^5*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-4*I*(-9*I*a^3*exp(6*I*(d*x+c))+3*a^3*exp(7*I*(d*x+c))+16*I*a^3*exp(4*I*(d *x+c))-13*a^3*exp(5*I*(d*x+c))-9*I*a^3*exp(2*I*(d*x+c))+13*a^3*exp(3*I*(d* x+c))-3*a^3*exp(I*(d*x+c)))/(exp(2*I*(d*x+c))-1)^2/(exp(I*(d*x+c))-I)^4/d- 12*a^3/d*ln(exp(I*(d*x+c))-I)+6*a^3/d*ln(exp(2*I*(d*x+c))-1)
Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (113) = 226\).
Time = 0.08 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.04 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {18 \, a^{3} \cos \left (d x + c\right )^{2} - 17 \, a^{3} - 12 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3} + 2 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 12 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3} + 2 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 4 \, {\left (3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )}{2 \, {\left (d \cos \left (d x + c\right )^{4} - 3 \, d \cos \left (d x + c\right )^{2} + 2 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right ) + 2 \, d\right )}} \] Input:
integrate(csc(d*x+c)^3*sec(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
-1/2*(18*a^3*cos(d*x + c)^2 - 17*a^3 - 12*(a^3*cos(d*x + c)^4 - 3*a^3*cos( d*x + c)^2 + 2*a^3 + 2*(a^3*cos(d*x + c)^2 - a^3)*sin(d*x + c))*log(1/2*si n(d*x + c)) + 12*(a^3*cos(d*x + c)^4 - 3*a^3*cos(d*x + c)^2 + 2*a^3 + 2*(a ^3*cos(d*x + c)^2 - a^3)*sin(d*x + c))*log(-sin(d*x + c) + 1) - 4*(3*a^3*c os(d*x + c)^2 - 4*a^3)*sin(d*x + c))/(d*cos(d*x + c)^4 - 3*d*cos(d*x + c)^ 2 + 2*(d*cos(d*x + c)^2 - d)*sin(d*x + c) + 2*d)
Timed out. \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(csc(d*x+c)**3*sec(d*x+c)**5*(a+a*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {12 \, a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) - 12 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + \frac {12 \, a^{3} \sin \left (d x + c\right )^{3} - 18 \, a^{3} \sin \left (d x + c\right )^{2} + 4 \, a^{3} \sin \left (d x + c\right ) + a^{3}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )^{2}}}{2 \, d} \] Input:
integrate(csc(d*x+c)^3*sec(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
-1/2*(12*a^3*log(sin(d*x + c) - 1) - 12*a^3*log(sin(d*x + c)) + (12*a^3*si n(d*x + c)^3 - 18*a^3*sin(d*x + c)^2 + 4*a^3*sin(d*x + c) + a^3)/(sin(d*x + c)^4 - 2*sin(d*x + c)^3 + sin(d*x + c)^2))/d
Time = 0.15 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.76 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {1}{2} \, a^{3} {\left (\frac {12 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{d} - \frac {12 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{d} + \frac {12 \, \sin \left (d x + c\right )^{3} - 18 \, \sin \left (d x + c\right )^{2} + 4 \, \sin \left (d x + c\right ) + 1}{{\left (\sin \left (d x + c\right )^{2} - \sin \left (d x + c\right )\right )}^{2} d}\right )} \] Input:
integrate(csc(d*x+c)^3*sec(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
-1/2*a^3*(12*log(abs(sin(d*x + c) - 1))/d - 12*log(abs(sin(d*x + c)))/d + (12*sin(d*x + c)^3 - 18*sin(d*x + c)^2 + 4*sin(d*x + c) + 1)/((sin(d*x + c )^2 - sin(d*x + c))^2*d))
Time = 0.10 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.84 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {12\,a^3\,\mathrm {atanh}\left (2\,\sin \left (c+d\,x\right )-1\right )}{d}-\frac {6\,a^3\,{\sin \left (c+d\,x\right )}^3-9\,a^3\,{\sin \left (c+d\,x\right )}^2+2\,a^3\,\sin \left (c+d\,x\right )+\frac {a^3}{2}}{d\,\left ({\sin \left (c+d\,x\right )}^4-2\,{\sin \left (c+d\,x\right )}^3+{\sin \left (c+d\,x\right )}^2\right )} \] Input:
int((a + a*sin(c + d*x))^3/(cos(c + d*x)^5*sin(c + d*x)^3),x)
Output:
(12*a^3*atanh(2*sin(c + d*x) - 1))/d - (2*a^3*sin(c + d*x) + a^3/2 - 9*a^3 *sin(c + d*x)^2 + 6*a^3*sin(c + d*x)^3)/(d*(sin(c + d*x)^2 - 2*sin(c + d*x )^3 + sin(c + d*x)^4))
Time = 0.18 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.76 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (-48 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{4}+96 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{3}-48 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2}+24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4}-48 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{3}+24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2}-15 \sin \left (d x +c \right )^{4}+6 \sin \left (d x +c \right )^{3}+21 \sin \left (d x +c \right )^{2}-8 \sin \left (d x +c \right )-2\right )}{4 \sin \left (d x +c \right )^{2} d \left (\sin \left (d x +c \right )^{2}-2 \sin \left (d x +c \right )+1\right )} \] Input:
int(csc(d*x+c)^3*sec(d*x+c)^5*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*( - 48*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4 + 96*log(tan((c + d *x)/2) - 1)*sin(c + d*x)**3 - 48*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2 + 24*log(tan((c + d*x)/2))*sin(c + d*x)**4 - 48*log(tan((c + d*x)/2))*sin (c + d*x)**3 + 24*log(tan((c + d*x)/2))*sin(c + d*x)**2 - 15*sin(c + d*x)* *4 + 6*sin(c + d*x)**3 + 21*sin(c + d*x)**2 - 8*sin(c + d*x) - 2))/(4*sin( c + d*x)**2*d*(sin(c + d*x)**2 - 2*sin(c + d*x) + 1))