Integrand size = 19, antiderivative size = 119 \[ \int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx=-\frac {a \cos (c+d x)}{d}+\frac {3 a \cos ^2(c+d x)}{2 d}+\frac {a \cos ^3(c+d x)}{d}-\frac {3 a \cos ^4(c+d x)}{4 d}-\frac {3 a \cos ^5(c+d x)}{5 d}+\frac {a \cos ^6(c+d x)}{6 d}+\frac {a \cos ^7(c+d x)}{7 d}-\frac {a \log (\cos (c+d x))}{d} \]
-a*cos(d*x+c)/d+3/2*a*cos(d*x+c)^2/d+a*cos(d*x+c)^3/d-3/4*a*cos(d*x+c)^4/d -3/5*a*cos(d*x+c)^5/d+1/6*a*cos(d*x+c)^6/d+1/7*a*cos(d*x+c)^7/d-a*ln(cos(d *x+c))/d
Time = 0.18 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.95 \[ \int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx=-\frac {35 a \cos (c+d x)}{64 d}+\frac {7 a \cos (3 (c+d x))}{64 d}-\frac {7 a \cos (5 (c+d x))}{320 d}+\frac {a \cos (7 (c+d x))}{448 d}-\frac {a \left (-\frac {3}{2} \cos ^2(c+d x)+\frac {3}{4} \cos ^4(c+d x)-\frac {1}{6} \cos ^6(c+d x)+\log (\cos (c+d x))\right )}{d} \]
(-35*a*Cos[c + d*x])/(64*d) + (7*a*Cos[3*(c + d*x)])/(64*d) - (7*a*Cos[5*( c + d*x)])/(320*d) + (a*Cos[7*(c + d*x)])/(448*d) - (a*((-3*Cos[c + d*x]^2 )/2 + (3*Cos[c + d*x]^4)/4 - Cos[c + d*x]^6/6 + Log[Cos[c + d*x]]))/d
Time = 0.38 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.01, 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.
| \(\displaystyle \int \sin ^7(c+d x) (a \sec (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos \left (c+d x-\frac {\pi }{2}\right )^7 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\left (\sin ^6(c+d x) \tan (c+d x) (a (-\cos (c+d x))-a)\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\left ((\cos (c+d x) a+a) \sin ^6(c+d x) \tan (c+d x)\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \sin ^6(c+d x) \tan (c+d x) (a \cos (c+d x)+a)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right )^7 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )^7 \left (\sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )}{\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle -\frac {\int (a-a \cos (c+d x))^3 (\cos (c+d x) a+a)^4 \sec (c+d x)d(a \cos (c+d x))}{a^7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {(a-a \cos (c+d x))^3 (\cos (c+d x) a+a)^4 \sec (c+d x)}{a}d(a \cos (c+d x))}{a^6 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {\int \left (-\cos ^6(c+d x) a^6-\cos ^5(c+d x) a^6+3 \cos ^4(c+d x) a^6+3 \cos ^3(c+d x) a^6-3 \cos ^2(c+d x) a^6-3 \cos (c+d x) a^6+\sec (c+d x) a^6+a^6\right )d(a \cos (c+d x))}{a^6 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {1}{7} a^7 \cos ^7(c+d x)-\frac {1}{6} a^7 \cos ^6(c+d x)+\frac {3}{5} a^7 \cos ^5(c+d x)+\frac {3}{4} a^7 \cos ^4(c+d x)-a^7 \cos ^3(c+d x)-\frac {3}{2} a^7 \cos ^2(c+d x)+a^7 \cos (c+d x)+a^7 \log (a \cos (c+d x))}{a^6 d}\) |
-((a^7*Cos[c + d*x] - (3*a^7*Cos[c + d*x]^2)/2 - a^7*Cos[c + d*x]^3 + (3*a ^7*Cos[c + d*x]^4)/4 + (3*a^7*Cos[c + d*x]^5)/5 - (a^7*Cos[c + d*x]^6)/6 - (a^7*Cos[c + d*x]^7)/7 + a^7*Log[a*Cos[c + d*x]])/(a^6*d))
3.1.2.3.1 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 = 1.46 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.73
| method | result | size |
| derivativedivides | \(\frac {a \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{7}}{d}\) | \(87\) |
| default | \(\frac {a \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{7}}{d}\) | \(87\) |
| parts | \(-\frac {a \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{7 d}+\frac {a \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(89\) |
| parallelrisch | \(\frac {\left (192 \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-192 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-192 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {5732}{35}-12 \cos \left (4 d x +4 c \right )+21 \cos \left (3 d x +3 c \right )-105 \cos \left (d x +c \right )+\frac {3 \cos \left (7 d x +7 c \right )}{7}+\cos \left (6 d x +6 c \right )-\frac {21 \cos \left (5 d x +5 c \right )}{5}+87 \cos \left (2 d x +2 c \right )\right ) a}{192 d}\) | \(123\) |
| risch | \(i a x +\frac {2 i a c}{d}+\frac {29 a \,{\mathrm e}^{2 i \left (d x +c \right )}}{128 d}+\frac {29 a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {35 a \cos \left (d x +c \right )}{64 d}+\frac {a \cos \left (7 d x +7 c \right )}{448 d}+\frac {a \cos \left (6 d x +6 c \right )}{192 d}-\frac {7 a \cos \left (5 d x +5 c \right )}{320 d}-\frac {a \cos \left (4 d x +4 c \right )}{16 d}+\frac {7 a \cos \left (3 d x +3 c \right )}{64 d}\) | \(150\) |
| norman | \(\frac {-\frac {32 a}{35 d}-\frac {128 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d}-\frac {166 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5 d}-\frac {224 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}-\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{d}-\frac {42 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{5 d}-\frac {14 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}+\frac {a \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(182\) |
1/d*(a*(-1/6*sin(d*x+c)^6-1/4*sin(d*x+c)^4-1/2*sin(d*x+c)^2-ln(cos(d*x+c)) )-1/7*a*(16/5+sin(d*x+c)^6+6/5*sin(d*x+c)^4+8/5*sin(d*x+c)^2)*cos(d*x+c))
Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.78 \[ \int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx=\frac {60 \, a \cos \left (d x + c\right )^{7} + 70 \, a \cos \left (d x + c\right )^{6} - 252 \, a \cos \left (d x + c\right )^{5} - 315 \, a \cos \left (d x + c\right )^{4} + 420 \, a \cos \left (d x + c\right )^{3} + 630 \, a \cos \left (d x + c\right )^{2} - 420 \, a \cos \left (d x + c\right ) - 420 \, a \log \left (-\cos \left (d x + c\right )\right )}{420 \, d} \]
1/420*(60*a*cos(d*x + c)^7 + 70*a*cos(d*x + c)^6 - 252*a*cos(d*x + c)^5 - 315*a*cos(d*x + c)^4 + 420*a*cos(d*x + c)^3 + 630*a*cos(d*x + c)^2 - 420*a *cos(d*x + c) - 420*a*log(-cos(d*x + c)))/d
Timed out. \[ \int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx=\text {Timed out} \]
Time = 0.19 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.76 \[ \int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx=\frac {60 \, a \cos \left (d x + c\right )^{7} + 70 \, a \cos \left (d x + c\right )^{6} - 252 \, a \cos \left (d x + c\right )^{5} - 315 \, a \cos \left (d x + c\right )^{4} + 420 \, a \cos \left (d x + c\right )^{3} + 630 \, a \cos \left (d x + c\right )^{2} - 420 \, a \cos \left (d x + c\right ) - 420 \, a \log \left (\cos \left (d x + c\right )\right )}{420 \, d} \]
1/420*(60*a*cos(d*x + c)^7 + 70*a*cos(d*x + c)^6 - 252*a*cos(d*x + c)^5 - 315*a*cos(d*x + c)^4 + 420*a*cos(d*x + c)^3 + 630*a*cos(d*x + c)^2 - 420*a *cos(d*x + c) - 420*a*log(cos(d*x + c)))/d
Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (109) = 218\).
Time = 0.31 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.08 \[ \int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx=\frac {420 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 420 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {1473 \, a - \frac {11151 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {36813 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {69475 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {56035 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {28749 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {8463 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {1089 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{7}}}{420 \, d} \]
1/420*(420*a*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) - 420*a* log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) + (1473*a - 11151*a*( cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 36813*a*(cos(d*x + c) - 1)^2/(cos(d *x + c) + 1)^2 - 69475*a*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 56035 *a*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 28749*a*(cos(d*x + c) - 1)^ 5/(cos(d*x + c) + 1)^5 + 8463*a*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 1089*a*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7)/((cos(d*x + c) - 1)/(c os(d*x + c) + 1) - 1)^7)/d
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.75 \[ \int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx=-\frac {a\,\cos \left (c+d\,x\right )-\frac {3\,a\,{\cos \left (c+d\,x\right )}^2}{2}-a\,{\cos \left (c+d\,x\right )}^3+\frac {3\,a\,{\cos \left (c+d\,x\right )}^4}{4}+\frac {3\,a\,{\cos \left (c+d\,x\right )}^5}{5}-\frac {a\,{\cos \left (c+d\,x\right )}^6}{6}-\frac {a\,{\cos \left (c+d\,x\right )}^7}{7}+a\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]