Integrand size = 21, antiderivative size = 131 \[ \int (a+a \sec (c+d x))^2 \sin ^7(c+d x) \, dx=\frac {2 a^2 \cos (c+d x)}{d}+\frac {3 a^2 \cos ^2(c+d x)}{d}-\frac {3 a^2 \cos ^4(c+d x)}{2 d}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {a^2 \cos ^6(c+d x)}{3 d}+\frac {a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}+\frac {a^2 \sec (c+d x)}{d} \]
2*a^2*cos(d*x+c)/d+3*a^2*cos(d*x+c)^2/d-3/2*a^2*cos(d*x+c)^4/d-2/5*a^2*cos (d*x+c)^5/d+1/3*a^2*cos(d*x+c)^6/d+1/7*a^2*cos(d*x+c)^7/d-2*a^2*ln(cos(d*x +c))/d+a^2*sec(d*x+c)/d
Time = 0.39 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82 \[ \int (a+a \sec (c+d x))^2 \sin ^7(c+d x) \, dx=\frac {a^2 (25725+11760 \cos (2 (c+d x))+5250 \cos (3 (c+d x))-588 \cos (4 (c+d x))-770 \cos (5 (c+d x))-48 \cos (6 (c+d x))+70 \cos (7 (c+d x))+15 \cos (8 (c+d x))-70 \cos (c+d x) (5+384 \log (\cos (c+d x)))) \sec (c+d x)}{13440 d} \]
(a^2*(25725 + 11760*Cos[2*(c + d*x)] + 5250*Cos[3*(c + d*x)] - 588*Cos[4*( c + d*x)] - 770*Cos[5*(c + d*x)] - 48*Cos[6*(c + d*x)] + 70*Cos[7*(c + d*x )] + 15*Cos[8*(c + d*x)] - 70*Cos[c + d*x]*(5 + 384*Log[Cos[c + d*x]]))*Se c[c + d*x])/(13440*d)
Time = 0.44 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.90, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 4360, 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.
| \(\displaystyle \int \sin ^7(c+d x) (a \sec (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos \left (c+d x-\frac {\pi }{2}\right )^7 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \sin ^5(c+d x) \tan ^2(c+d x) (a (-\cos (c+d x))-a)^2dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right )^7 \left (a \left (-\sin \left (c+d x+\frac {\pi }{2}\right )\right )-a\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )^7 \left (\sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^2}{\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^2}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle -\frac {\int (a-a \cos (c+d x))^3 (\cos (c+d x) a+a)^5 \sec ^2(c+d x)d(a \cos (c+d x))}{a^7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {(a-a \cos (c+d x))^3 (\cos (c+d x) a+a)^5 \sec ^2(c+d x)}{a^2}d(a \cos (c+d x))}{a^5 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {\int \left (-\cos ^6(c+d x) a^6-2 \cos ^5(c+d x) a^6+2 \cos ^4(c+d x) a^6+6 \cos ^3(c+d x) a^6+\sec ^2(c+d x) a^6-6 \cos (c+d x) a^6+2 \sec (c+d x) a^6-2 a^6\right )d(a \cos (c+d x))}{a^5 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {1}{7} a^7 \cos ^7(c+d x)-\frac {1}{3} a^7 \cos ^6(c+d x)+\frac {2}{5} a^7 \cos ^5(c+d x)+\frac {3}{2} a^7 \cos ^4(c+d x)-3 a^7 \cos ^2(c+d x)-2 a^7 \cos (c+d x)-a^7 \sec (c+d x)+2 a^7 \log (a \cos (c+d x))}{a^5 d}\) |
-((-2*a^7*Cos[c + d*x] - 3*a^7*Cos[c + d*x]^2 + (3*a^7*Cos[c + d*x]^4)/2 + (2*a^7*Cos[c + d*x]^5)/5 - (a^7*Cos[c + d*x]^6)/3 - (a^7*Cos[c + d*x]^7)/ 7 + 2*a^7*Log[a*Cos[c + d*x]] - a^7*Sec[c + d*x])/(a^5*d))
3.1.20.3.1 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 = 2.05 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.15
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{8}}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{2} \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{7}}{d}\) | \(151\) |
| default | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{8}}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{2} \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{7}}{d}\) | \(151\) |
| parts | \(-\frac {a^{2} \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{7 d}+\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{8}}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )\right )}{d}+\frac {2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(156\) |
| parallelrisch | \(\frac {a^{2} \left (26880 \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \cos \left (d x +c \right )-26880 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )-26880 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )+32314 \cos \left (d x +c \right )+5250 \cos \left (3 d x +3 c \right )+15 \cos \left (8 d x +8 c \right )+70 \cos \left (7 d x +7 c \right )-770 \cos \left (5 d x +5 c \right )-48 \cos \left (6 d x +6 c \right )-588 \cos \left (4 d x +4 c \right )+11760 \cos \left (2 d x +2 c \right )+25725\right )}{13440 d \cos \left (d x +c \right )}\) | \(164\) |
| risch | \(2 i a^{2} x +\frac {29 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{64 d}+\frac {117 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{128 d}+\frac {117 a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{128 d}+\frac {29 a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{64 d}+\frac {4 i a^{2} c}{d}+\frac {2 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}+\frac {a^{2} \cos \left (7 d x +7 c \right )}{448 d}+\frac {a^{2} \cos \left (6 d x +6 c \right )}{96 d}-\frac {3 a^{2} \cos \left (5 d x +5 c \right )}{320 d}-\frac {a^{2} \cos \left (4 d x +4 c \right )}{8 d}-\frac {5 a^{2} \cos \left (3 d x +3 c \right )}{64 d}\) | \(222\) |
| norman | \(\frac {-\frac {192 a^{2}}{35 d}-\frac {64 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}-\frac {4 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{d}-\frac {24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{d}-\frac {172 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{3 d}-\frac {264 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5 d}-\frac {292 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{15 d}-\frac {1012 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{35 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}+\frac {2 a^{2} \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}\) | \(237\) |
1/d*(a^2*(sin(d*x+c)^8/cos(d*x+c)+(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))+2*a^2*(-1/6*sin(d*x+c)^6-1/4*sin(d*x+c)^4-1/2*si n(d*x+c)^2-ln(cos(d*x+c)))-1/7*a^2*(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 = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98 \[ \int (a+a \sec (c+d x))^2 \sin ^7(c+d x) \, dx=\frac {120 \, a^{2} \cos \left (d x + c\right )^{8} + 280 \, a^{2} \cos \left (d x + c\right )^{7} - 336 \, a^{2} \cos \left (d x + c\right )^{6} - 1260 \, a^{2} \cos \left (d x + c\right )^{5} + 2520 \, a^{2} \cos \left (d x + c\right )^{3} + 1680 \, a^{2} \cos \left (d x + c\right )^{2} - 1680 \, a^{2} \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - 875 \, a^{2} \cos \left (d x + c\right ) + 840 \, a^{2}}{840 \, d \cos \left (d x + c\right )} \]
1/840*(120*a^2*cos(d*x + c)^8 + 280*a^2*cos(d*x + c)^7 - 336*a^2*cos(d*x + c)^6 - 1260*a^2*cos(d*x + c)^5 + 2520*a^2*cos(d*x + c)^3 + 1680*a^2*cos(d *x + c)^2 - 1680*a^2*cos(d*x + c)*log(-cos(d*x + c)) - 875*a^2*cos(d*x + c ) + 840*a^2)/(d*cos(d*x + c))
Timed out. \[ \int (a+a \sec (c+d x))^2 \sin ^7(c+d x) \, dx=\text {Timed out} \]
Time = 0.19 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82 \[ \int (a+a \sec (c+d x))^2 \sin ^7(c+d x) \, dx=\frac {30 \, a^{2} \cos \left (d x + c\right )^{7} + 70 \, a^{2} \cos \left (d x + c\right )^{6} - 84 \, a^{2} \cos \left (d x + c\right )^{5} - 315 \, a^{2} \cos \left (d x + c\right )^{4} + 630 \, a^{2} \cos \left (d x + c\right )^{2} + 420 \, a^{2} \cos \left (d x + c\right ) - 420 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) + \frac {210 \, a^{2}}{\cos \left (d x + c\right )}}{210 \, d} \]
1/210*(30*a^2*cos(d*x + c)^7 + 70*a^2*cos(d*x + c)^6 - 84*a^2*cos(d*x + c) ^5 - 315*a^2*cos(d*x + c)^4 + 630*a^2*cos(d*x + c)^2 + 420*a^2*cos(d*x + c ) - 420*a^2*log(cos(d*x + c)) + 210*a^2/cos(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (123) = 246\).
Time = 0.37 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.44 \[ \int (a+a \sec (c+d x))^2 \sin ^7(c+d x) \, dx=\frac {420 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 420 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {420 \, {\left (2 \, a^{2} + \frac {a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1} + \frac {357 \, a^{2} - \frac {3759 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {16737 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {42595 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {43855 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {25389 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {8043 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {1089 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{7}}}{210 \, d} \]
1/210*(420*a^2*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) - 420* a^2*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) + 420*(2*a^2 + a^ 2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1) + (357*a^2 - 3759*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1 6737*a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 42595*a^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 43855*a^2*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 25389*a^2*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 8043*a^ 2*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 1089*a^2*(cos(d*x + c) - 1)^ 7/(cos(d*x + c) + 1)^7)/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^7)/d
Time = 14.31 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.80 \[ \int (a+a \sec (c+d x))^2 \sin ^7(c+d x) \, dx=\frac {2\,a^2\,\cos \left (c+d\,x\right )+\frac {a^2}{\cos \left (c+d\,x\right )}+3\,a^2\,{\cos \left (c+d\,x\right )}^2-\frac {3\,a^2\,{\cos \left (c+d\,x\right )}^4}{2}-\frac {2\,a^2\,{\cos \left (c+d\,x\right )}^5}{5}+\frac {a^2\,{\cos \left (c+d\,x\right )}^6}{3}+\frac {a^2\,{\cos \left (c+d\,x\right )}^7}{7}-2\,a^2\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]