Integrand size = 19, antiderivative size = 87 \[ \int (a+a \sec (c+d x)) \sin ^5(c+d x) \, dx=-\frac {a \cos (c+d x)}{d}+\frac {a \cos ^2(c+d x)}{d}+\frac {2 a \cos ^3(c+d x)}{3 d}-\frac {a \cos ^4(c+d x)}{4 d}-\frac {a \cos ^5(c+d x)}{5 d}-\frac {a \log (\cos (c+d x))}{d} \]
-a*cos(d*x+c)/d+a*cos(d*x+c)^2/d+2/3*a*cos(d*x+c)^3/d-1/4*a*cos(d*x+c)^4/d -1/5*a*cos(d*x+c)^5/d-a*ln(cos(d*x+c))/d
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.95 \[ \int (a+a \sec (c+d x)) \sin ^5(c+d x) \, dx=-\frac {5 a \cos (c+d x)}{8 d}+\frac {5 a \cos (3 (c+d x))}{48 d}-\frac {a \cos (5 (c+d x))}{80 d}-\frac {a \left (-\cos ^2(c+d x)+\frac {1}{4} \cos ^4(c+d x)+\log (\cos (c+d x))\right )}{d} \]
(-5*a*Cos[c + d*x])/(8*d) + (5*a*Cos[3*(c + d*x)])/(48*d) - (a*Cos[5*(c + d*x)])/(80*d) - (a*(-Cos[c + d*x]^2 + Cos[c + d*x]^4/4 + Log[Cos[c + d*x]] ))/d
Time = 0.35 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {3042, 4360, 25, 25, 3042, 25, 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.
\(\displaystyle \int \sin ^5(c+d x) (a \sec (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cos \left (c+d x-\frac {\pi }{2}\right )^5 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )dx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \int -\left (\sin ^4(c+d x) \tan (c+d x) (a (-\cos (c+d x))-a)\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -\left ((\cos (c+d x) a+a) \sin ^4(c+d x) \tan (c+d x)\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \sin ^4(c+d x) \tan (c+d x) (a \cos (c+d x)+a)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right )^5 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )^5 \left (\sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )}{\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle -\frac {\int (a-a \cos (c+d x))^2 (\cos (c+d x) a+a)^3 \sec (c+d x)d(a \cos (c+d x))}{a^5 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {(a-a \cos (c+d x))^2 (\cos (c+d x) a+a)^3 \sec (c+d x)}{a}d(a \cos (c+d x))}{a^4 d}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle -\frac {\int \left (\cos ^4(c+d x) a^4+\cos ^3(c+d x) a^4-2 \cos ^2(c+d x) a^4-2 \cos (c+d x) a^4+\sec (c+d x) a^4+a^4\right )d(a \cos (c+d x))}{a^4 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {1}{5} a^5 \cos ^5(c+d x)+\frac {1}{4} a^5 \cos ^4(c+d x)-\frac {2}{3} a^5 \cos ^3(c+d x)-a^5 \cos ^2(c+d x)+a^5 \cos (c+d x)+a^5 \log (a \cos (c+d x))}{a^4 d}\) |
-((a^5*Cos[c + d*x] - a^5*Cos[c + d*x]^2 - (2*a^5*Cos[c + d*x]^3)/3 + (a^5 *Cos[c + d*x]^4)/4 + (a^5*Cos[c + d*x]^5)/5 + a^5*Log[a*Cos[c + d*x]])/(a^ 4*d))
3.1.3.3.1 Defintions of rubi rules used
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 = 1.36 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a \left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{5}}{d}\) | \(67\) |
default | \(\frac {a \left (-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a \left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{5}}{d}\) | \(67\) |
parts | \(-\frac {a \left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{5 d}+\frac {a \left (-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(69\) |
parallelrisch | \(-\frac {a \left (-480 \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+480 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+480 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+421+6 \cos \left (5 d x +5 c \right )+15 \cos \left (4 d x +4 c \right )-50 \cos \left (3 d x +3 c \right )-180 \cos \left (2 d x +2 c \right )+300 \cos \left (d x +c \right )\right )}{480 d}\) | \(103\) |
risch | \(i a x +\frac {3 a \,{\mathrm e}^{2 i \left (d x +c \right )}}{16 d}+\frac {3 a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{16 d}+\frac {2 i a c}{d}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {5 a \cos \left (d x +c \right )}{8 d}-\frac {a \cos \left (5 d x +5 c \right )}{80 d}-\frac {a \cos \left (4 d x +4 c \right )}{32 d}+\frac {5 a \cos \left (3 d x +3 c \right )}{48 d}\) | \(120\) |
norman | \(\frac {-\frac {16 a}{15 d}-\frac {22 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3 d}-\frac {62 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3 d}-\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}-\frac {10 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}+\frac {a \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(148\) |
1/d*(a*(-1/4*sin(d*x+c)^4-1/2*sin(d*x+c)^2-ln(cos(d*x+c)))-1/5*a*(8/3+sin( d*x+c)^4+4/3*sin(d*x+c)^2)*cos(d*x+c))
Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.82 \[ \int (a+a \sec (c+d x)) \sin ^5(c+d x) \, dx=-\frac {12 \, a \cos \left (d x + c\right )^{5} + 15 \, a \cos \left (d x + c\right )^{4} - 40 \, a \cos \left (d x + c\right )^{3} - 60 \, a \cos \left (d x + c\right )^{2} + 60 \, a \cos \left (d x + c\right ) + 60 \, a \log \left (-\cos \left (d x + c\right )\right )}{60 \, d} \]
-1/60*(12*a*cos(d*x + c)^5 + 15*a*cos(d*x + c)^4 - 40*a*cos(d*x + c)^3 - 6 0*a*cos(d*x + c)^2 + 60*a*cos(d*x + c) + 60*a*log(-cos(d*x + c)))/d
\[ \int (a+a \sec (c+d x)) \sin ^5(c+d x) \, dx=a \left (\int \sin ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sin ^{5}{\left (c + d x \right )}\, dx\right ) \]
Time = 0.19 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.79 \[ \int (a+a \sec (c+d x)) \sin ^5(c+d x) \, dx=-\frac {12 \, a \cos \left (d x + c\right )^{5} + 15 \, a \cos \left (d x + c\right )^{4} - 40 \, a \cos \left (d x + c\right )^{3} - 60 \, a \cos \left (d x + c\right )^{2} + 60 \, a \cos \left (d x + c\right ) + 60 \, a \log \left (\cos \left (d x + c\right )\right )}{60 \, d} \]
-1/60*(12*a*cos(d*x + c)^5 + 15*a*cos(d*x + c)^4 - 40*a*cos(d*x + c)^3 - 6 0*a*cos(d*x + c)^2 + 60*a*cos(d*x + c) + 60*a*log(cos(d*x + c)))/d
Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (81) = 162\).
Time = 0.31 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.31 \[ \int (a+a \sec (c+d x)) \sin ^5(c+d x) \, dx=\frac {60 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 60 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {201 \, a - \frac {1125 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {2610 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1970 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {805 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {137 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{5}}}{60 \, d} \]
1/60*(60*a*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) - 60*a*log (abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) + (201*a - 1125*a*(cos(d *x + c) - 1)/(cos(d*x + c) + 1) + 2610*a*(cos(d*x + c) - 1)^2/(cos(d*x + c ) + 1)^2 - 1970*a*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 805*a*(cos(d *x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 137*a*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5)/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^5)/d
Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.77 \[ \int (a+a \sec (c+d x)) \sin ^5(c+d x) \, dx=-\frac {a\,\cos \left (c+d\,x\right )-a\,{\cos \left (c+d\,x\right )}^2-\frac {2\,a\,{\cos \left (c+d\,x\right )}^3}{3}+\frac {a\,{\cos \left (c+d\,x\right )}^4}{4}+\frac {a\,{\cos \left (c+d\,x\right )}^5}{5}+a\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]