Integrand size = 21, antiderivative size = 183 \[ \int (a+a \sec (c+d x))^2 \sin ^9(c+d x) \, dx=\frac {3 a^2 \cos (c+d x)}{d}+\frac {4 a^2 \cos ^2(c+d x)}{d}-\frac {2 a^2 \cos ^3(c+d x)}{3 d}-\frac {3 a^2 \cos ^4(c+d x)}{d}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {4 a^2 \cos ^6(c+d x)}{3 d}+\frac {3 a^2 \cos ^7(c+d x)}{7 d}-\frac {a^2 \cos ^8(c+d x)}{4 d}-\frac {a^2 \cos ^9(c+d x)}{9 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}+\frac {a^2 \sec (c+d x)}{d} \]
3*a^2*cos(d*x+c)/d+4*a^2*cos(d*x+c)^2/d-2/3*a^2*cos(d*x+c)^3/d-3*a^2*cos(d *x+c)^4/d-2/5*a^2*cos(d*x+c)^5/d+4/3*a^2*cos(d*x+c)^6/d+3/7*a^2*cos(d*x+c) ^7/d-1/4*a^2*cos(d*x+c)^8/d-1/9*a^2*cos(d*x+c)^9/d-2*a^2*ln(cos(d*x+c))/d+ a^2*sec(d*x+c)/d
Time = 1.21 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.69 \[ \int (a+a \sec (c+d x))^2 \sin ^9(c+d x) \, dx=-\frac {a^2 (-714420-361620 \cos (2 (c+d x))-134820 \cos (3 (c+d x))+29232 \cos (4 (c+d x))+24780 \cos (5 (c+d x))-1458 \cos (6 (c+d x))-3885 \cos (7 (c+d x))-380 \cos (8 (c+d x))+315 \cos (9 (c+d x))+70 \cos (10 (c+d x))+210 \cos (c+d x) (205+3072 \log (\cos (c+d x)))) \sec (c+d x)}{322560 d} \]
-1/322560*(a^2*(-714420 - 361620*Cos[2*(c + d*x)] - 134820*Cos[3*(c + d*x) ] + 29232*Cos[4*(c + d*x)] + 24780*Cos[5*(c + d*x)] - 1458*Cos[6*(c + d*x) ] - 3885*Cos[7*(c + d*x)] - 380*Cos[8*(c + d*x)] + 315*Cos[9*(c + d*x)] + 70*Cos[10*(c + d*x)] + 210*Cos[c + d*x]*(205 + 3072*Log[Cos[c + d*x]]))*Se c[c + d*x])/d
Time = 0.47 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.88, 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.
| \(\displaystyle \int \sin ^9(c+d x) (a \sec (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos \left (c+d x-\frac {\pi }{2}\right )^9 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \sin ^7(c+d x) \tan ^2(c+d x) (a (-\cos (c+d x))-a)^2dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right )^9 \left (a \left (-\sin \left (c+d x+\frac {\pi }{2}\right )\right )-a\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )^9 \left (\sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^2}{\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^2}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle -\frac {\int (a-a \cos (c+d x))^4 (\cos (c+d x) a+a)^6 \sec ^2(c+d x)d(a \cos (c+d x))}{a^9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {(a-a \cos (c+d x))^4 (\cos (c+d x) a+a)^6 \sec ^2(c+d x)}{a^2}d(a \cos (c+d x))}{a^7 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {\int \left (\cos ^8(c+d x) a^8+2 \cos ^7(c+d x) a^8-3 \cos ^6(c+d x) a^8-8 \cos ^5(c+d x) a^8+2 \cos ^4(c+d x) a^8+12 \cos ^3(c+d x) a^8+2 \cos ^2(c+d x) a^8+\sec ^2(c+d x) a^8-8 \cos (c+d x) a^8+2 \sec (c+d x) a^8-3 a^8\right )d(a \cos (c+d x))}{a^7 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {1}{9} a^9 \cos ^9(c+d x)+\frac {1}{4} a^9 \cos ^8(c+d x)-\frac {3}{7} a^9 \cos ^7(c+d x)-\frac {4}{3} a^9 \cos ^6(c+d x)+\frac {2}{5} a^9 \cos ^5(c+d x)+3 a^9 \cos ^4(c+d x)+\frac {2}{3} a^9 \cos ^3(c+d x)-4 a^9 \cos ^2(c+d x)-3 a^9 \cos (c+d x)-a^9 \sec (c+d x)+2 a^9 \log (a \cos (c+d x))}{a^7 d}\) |
-((-3*a^9*Cos[c + d*x] - 4*a^9*Cos[c + d*x]^2 + (2*a^9*Cos[c + d*x]^3)/3 + 3*a^9*Cos[c + d*x]^4 + (2*a^9*Cos[c + d*x]^5)/5 - (4*a^9*Cos[c + d*x]^6)/ 3 - (3*a^9*Cos[c + d*x]^7)/7 + (a^9*Cos[c + d*x]^8)/4 + (a^9*Cos[c + d*x]^ 9)/9 + 2*a^9*Log[a*Cos[c + d*x]] - a^9*Sec[c + d*x])/(a^7*d))
3.1.19.3.1 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 = 2.36 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.99
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{10}}{\cos \left (d x +c \right )}+\left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{8}}{8}-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{2} \left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )}{9}}{d}\) | \(181\) |
| default | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{10}}{\cos \left (d x +c \right )}+\left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{8}}{8}-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{2} \left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )}{9}}{d}\) | \(181\) |
| parallelrisch | \(\frac {a^{2} \left (645120 \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \cos \left (d x +c \right )-645120 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )-645120 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )+934966 \cos \left (d x +c \right )+3885 \cos \left (7 d x +7 c \right )-315 \cos \left (9 d x +9 c \right )-24780 \cos \left (5 d x +5 c \right )+134820 \cos \left (3 d x +3 c \right )-70 \cos \left (10 d x +10 c \right )+380 \cos \left (8 d x +8 c \right )+1458 \cos \left (6 d x +6 c \right )-29232 \cos \left (4 d x +4 c \right )+361620 \cos \left (2 d x +2 c \right )+714420\right )}{322560 d \cos \left (d x +c \right )}\) | \(186\) |
| parts | \(-\frac {a^{2} \left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )}{9 d}+\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{10}}{\cos \left (d x +c \right )}+\left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )\right )}{d}+\frac {2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{8}}{8}-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(186\) |
| risch | \(\frac {2 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {65 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{128 d}+2 i a^{2} x +\frac {4 i a^{2} c}{d}+\frac {65 a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}+\frac {311 a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{256 d}+\frac {311 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{256 d}-\frac {a^{2} \cos \left (9 d x +9 c \right )}{2304 d}-\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {a^{2} \cos \left (8 d x +8 c \right )}{512 d}+\frac {5 a^{2} \cos \left (7 d x +7 c \right )}{1792 d}+\frac {5 a^{2} \cos \left (6 d x +6 c \right )}{192 d}+\frac {a^{2} \cos \left (5 d x +5 c \right )}{160 d}-\frac {23 a^{2} \cos \left (4 d x +4 c \right )}{128 d}-\frac {3 a^{2} \cos \left (3 d x +3 c \right )}{16 d}\) | \(256\) |
1/d*(a^2*(sin(d*x+c)^10/cos(d*x+c)+(128/35+sin(d*x+c)^8+8/7*sin(d*x+c)^6+4 8/35*sin(d*x+c)^4+64/35*sin(d*x+c)^2)*cos(d*x+c))+2*a^2*(-1/8*sin(d*x+c)^8 -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)))-1/9*a^ 2*(128/35+sin(d*x+c)^8+8/7*sin(d*x+c)^6+48/35*sin(d*x+c)^4+64/35*sin(d*x+c )^2)*cos(d*x+c))
Time = 0.30 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.91 \[ \int (a+a \sec (c+d x))^2 \sin ^9(c+d x) \, dx=-\frac {17920 \, a^{2} \cos \left (d x + c\right )^{10} + 40320 \, a^{2} \cos \left (d x + c\right )^{9} - 69120 \, a^{2} \cos \left (d x + c\right )^{8} - 215040 \, a^{2} \cos \left (d x + c\right )^{7} + 64512 \, a^{2} \cos \left (d x + c\right )^{6} + 483840 \, a^{2} \cos \left (d x + c\right )^{5} + 107520 \, a^{2} \cos \left (d x + c\right )^{4} - 645120 \, a^{2} \cos \left (d x + c\right )^{3} - 483840 \, a^{2} \cos \left (d x + c\right )^{2} + 322560 \, a^{2} \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) + 197295 \, a^{2} \cos \left (d x + c\right ) - 161280 \, a^{2}}{161280 \, d \cos \left (d x + c\right )} \]
-1/161280*(17920*a^2*cos(d*x + c)^10 + 40320*a^2*cos(d*x + c)^9 - 69120*a^ 2*cos(d*x + c)^8 - 215040*a^2*cos(d*x + c)^7 + 64512*a^2*cos(d*x + c)^6 + 483840*a^2*cos(d*x + c)^5 + 107520*a^2*cos(d*x + c)^4 - 645120*a^2*cos(d*x + c)^3 - 483840*a^2*cos(d*x + c)^2 + 322560*a^2*cos(d*x + c)*log(-cos(d*x + c)) + 197295*a^2*cos(d*x + c) - 161280*a^2)/(d*cos(d*x + c))
Timed out. \[ \int (a+a \sec (c+d x))^2 \sin ^9(c+d x) \, dx=\text {Timed out} \]
Time = 0.23 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.80 \[ \int (a+a \sec (c+d x))^2 \sin ^9(c+d x) \, dx=-\frac {140 \, a^{2} \cos \left (d x + c\right )^{9} + 315 \, a^{2} \cos \left (d x + c\right )^{8} - 540 \, a^{2} \cos \left (d x + c\right )^{7} - 1680 \, a^{2} \cos \left (d x + c\right )^{6} + 504 \, a^{2} \cos \left (d x + c\right )^{5} + 3780 \, a^{2} \cos \left (d x + c\right )^{4} + 840 \, a^{2} \cos \left (d x + c\right )^{3} - 5040 \, a^{2} \cos \left (d x + c\right )^{2} - 3780 \, a^{2} \cos \left (d x + c\right ) + 2520 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac {1260 \, a^{2}}{\cos \left (d x + c\right )}}{1260 \, d} \]
-1/1260*(140*a^2*cos(d*x + c)^9 + 315*a^2*cos(d*x + c)^8 - 540*a^2*cos(d*x + c)^7 - 1680*a^2*cos(d*x + c)^6 + 504*a^2*cos(d*x + c)^5 + 3780*a^2*cos( d*x + c)^4 + 840*a^2*cos(d*x + c)^3 - 5040*a^2*cos(d*x + c)^2 - 3780*a^2*c os(d*x + c) + 2520*a^2*log(cos(d*x + c)) - 1260*a^2/cos(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (171) = 342\).
Time = 0.42 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.02 \[ \int (a+a \sec (c+d x))^2 \sin ^9(c+d x) \, dx=\frac {2520 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 2520 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {2520 \, {\left (2 \, a^{2} + \frac {a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1} + \frac {1457 \, a^{2} - \frac {20673 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {123012 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {421428 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {949662 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {1009134 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {666036 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {276804 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {66681 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {7129 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{9}}}{1260 \, d} \]
1/1260*(2520*a^2*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) - 25 20*a^2*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) + 2520*(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) + (1457*a^2 - 20673*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 123012*a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 421428*a^2*(co s(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 949662*a^2*(cos(d*x + c) - 1)^4/( cos(d*x + c) + 1)^4 - 1009134*a^2*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^ 5 + 666036*a^2*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 276804*a^2*(cos (d*x + c) - 1)^7/(cos(d*x + c) + 1)^7 + 66681*a^2*(cos(d*x + c) - 1)^8/(co s(d*x + c) + 1)^8 - 7129*a^2*(cos(d*x + c) - 1)^9/(cos(d*x + c) + 1)^9)/(( cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^9)/d
Time = 14.38 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.80 \[ \int (a+a \sec (c+d x))^2 \sin ^9(c+d x) \, dx=-\frac {\frac {2\,a^2\,{\cos \left (c+d\,x\right )}^3}{3}-\frac {a^2}{\cos \left (c+d\,x\right )}-4\,a^2\,{\cos \left (c+d\,x\right )}^2-3\,a^2\,\cos \left (c+d\,x\right )+3\,a^2\,{\cos \left (c+d\,x\right )}^4+\frac {2\,a^2\,{\cos \left (c+d\,x\right )}^5}{5}-\frac {4\,a^2\,{\cos \left (c+d\,x\right )}^6}{3}-\frac {3\,a^2\,{\cos \left (c+d\,x\right )}^7}{7}+\frac {a^2\,{\cos \left (c+d\,x\right )}^8}{4}+\frac {a^2\,{\cos \left (c+d\,x\right )}^9}{9}+2\,a^2\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]