Integrand size = 23, antiderivative size = 223 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {131072 a^7 \cos ^7(c+d x)}{969969 d (a+a \sin (c+d x))^{7/2}}-\frac {32768 a^6 \cos ^7(c+d x)}{138567 d (a+a \sin (c+d x))^{5/2}}-\frac {12288 a^5 \cos ^7(c+d x)}{46189 d (a+a \sin (c+d x))^{3/2}}-\frac {1024 a^4 \cos ^7(c+d x)}{4199 d \sqrt {a+a \sin (c+d x)}}-\frac {64 a^3 \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{323 d}-\frac {48 a^2 \cos ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{323 d}-\frac {2 a \cos ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{19 d} \]
-131072/969969*a^7*cos(d*x+c)^7/d/(a+a*sin(d*x+c))^(7/2)-32768/138567*a^6* cos(d*x+c)^7/d/(a+a*sin(d*x+c))^(5/2)-12288/46189*a^5*cos(d*x+c)^7/d/(a+a* sin(d*x+c))^(3/2)-48/323*a^2*cos(d*x+c)^7*(a+a*sin(d*x+c))^(3/2)/d-2/19*a* cos(d*x+c)^7*(a+a*sin(d*x+c))^(5/2)/d-1024/4199*a^4*cos(d*x+c)^7/d/(a+a*si n(d*x+c))^(1/2)-64/323*a^3*cos(d*x+c)^7*(a+a*sin(d*x+c))^(1/2)/d
Time = 0.35 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.46 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {2 a^3 \cos ^7(c+d x) \sqrt {a (1+\sin (c+d x))} \left (646739+1778602 \sin (c+d x)+2546901 \sin ^2(c+d x)+2244396 \sin ^3(c+d x)+1222221 \sin ^4(c+d x)+378378 \sin ^5(c+d x)+51051 \sin ^6(c+d x)\right )}{969969 d (1+\sin (c+d x))^4} \]
(-2*a^3*Cos[c + d*x]^7*Sqrt[a*(1 + Sin[c + d*x])]*(646739 + 1778602*Sin[c + d*x] + 2546901*Sin[c + d*x]^2 + 2244396*Sin[c + d*x]^3 + 1222221*Sin[c + d*x]^4 + 378378*Sin[c + d*x]^5 + 51051*Sin[c + d*x]^6))/(969969*d*(1 + Si n[c + d*x])^4)
Time = 1.15 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.09, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {3042, 3153, 3042, 3153, 3042, 3153, 3042, 3153, 3042, 3153, 3042, 3153, 3042, 3152}
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.
| \(\displaystyle \int \cos ^6(c+d x) (a \sin (c+d x)+a)^{7/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^6 (a \sin (c+d x)+a)^{7/2}dx\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {24}{19} a \int \cos ^6(c+d x) (\sin (c+d x) a+a)^{5/2}dx-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {24}{19} a \int \cos (c+d x)^6 (\sin (c+d x) a+a)^{5/2}dx-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {24}{19} a \left (\frac {20}{17} a \int \cos ^6(c+d x) (\sin (c+d x) a+a)^{3/2}dx-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{17 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {24}{19} a \left (\frac {20}{17} a \int \cos (c+d x)^6 (\sin (c+d x) a+a)^{3/2}dx-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{17 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {24}{19} a \left (\frac {20}{17} a \left (\frac {16}{15} a \int \cos ^6(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {2 a \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{17 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {24}{19} a \left (\frac {20}{17} a \left (\frac {16}{15} a \int \cos (c+d x)^6 \sqrt {\sin (c+d x) a+a}dx-\frac {2 a \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{17 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {24}{19} a \left (\frac {20}{17} a \left (\frac {16}{15} a \left (\frac {12}{13} a \int \frac {\cos ^6(c+d x)}{\sqrt {\sin (c+d x) a+a}}dx-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {2 a \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{17 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {24}{19} a \left (\frac {20}{17} a \left (\frac {16}{15} a \left (\frac {12}{13} a \int \frac {\cos (c+d x)^6}{\sqrt {\sin (c+d x) a+a}}dx-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {2 a \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{17 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {24}{19} a \left (\frac {20}{17} a \left (\frac {16}{15} a \left (\frac {12}{13} a \left (\frac {8}{11} a \int \frac {\cos ^6(c+d x)}{(\sin (c+d x) a+a)^{3/2}}dx-\frac {2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}}\right )-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {2 a \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{17 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {24}{19} a \left (\frac {20}{17} a \left (\frac {16}{15} a \left (\frac {12}{13} a \left (\frac {8}{11} a \int \frac {\cos (c+d x)^6}{(\sin (c+d x) a+a)^{3/2}}dx-\frac {2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}}\right )-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {2 a \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{17 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {24}{19} a \left (\frac {20}{17} a \left (\frac {16}{15} a \left (\frac {12}{13} a \left (\frac {8}{11} a \left (\frac {4}{9} a \int \frac {\cos ^6(c+d x)}{(\sin (c+d x) a+a)^{5/2}}dx-\frac {2 a \cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^{5/2}}\right )-\frac {2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}}\right )-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {2 a \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{17 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {24}{19} a \left (\frac {20}{17} a \left (\frac {16}{15} a \left (\frac {12}{13} a \left (\frac {8}{11} a \left (\frac {4}{9} a \int \frac {\cos (c+d x)^6}{(\sin (c+d x) a+a)^{5/2}}dx-\frac {2 a \cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^{5/2}}\right )-\frac {2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}}\right )-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {2 a \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{17 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d}\) |
| \(\Big \downarrow \) 3152 |
| \(\displaystyle \frac {24}{19} a \left (\frac {20}{17} a \left (\frac {16}{15} a \left (\frac {12}{13} a \left (\frac {8}{11} a \left (-\frac {8 a^2 \cos ^7(c+d x)}{63 d (a \sin (c+d x)+a)^{7/2}}-\frac {2 a \cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^{5/2}}\right )-\frac {2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}}\right )-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {2 a \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{17 d}\right )-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d}\) |
(-2*a*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^(5/2))/(19*d) + (24*a*((-2*a*Cos [c + d*x]^7*(a + a*Sin[c + d*x])^(3/2))/(17*d) + (20*a*((-2*a*Cos[c + d*x] ^7*Sqrt[a + a*Sin[c + d*x]])/(15*d) + (16*a*((-2*a*Cos[c + d*x]^7)/(13*d*S qrt[a + a*Sin[c + d*x]]) + (12*a*((-2*a*Cos[c + d*x]^7)/(11*d*(a + a*Sin[c + d*x])^(3/2)) + (8*a*((-8*a^2*Cos[c + d*x]^7)/(63*d*(a + a*Sin[c + d*x]) ^(7/2)) - (2*a*Cos[c + d*x]^7)/(9*d*(a + a*Sin[c + d*x])^(5/2))))/11))/13) )/15))/17))/19
3.2.40.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^(m - 1)/(f*g*(m - 1))), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[a*((2*m + p - 1)/(m + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]
Time = 153.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.48
| method | result | size |
| default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{4} \left (\sin \left (d x +c \right )-1\right )^{4} \left (51051 \left (\sin ^{6}\left (d x +c \right )\right )+378378 \left (\sin ^{5}\left (d x +c \right )\right )+1222221 \left (\sin ^{4}\left (d x +c \right )\right )+2244396 \left (\sin ^{3}\left (d x +c \right )\right )+2546901 \left (\sin ^{2}\left (d x +c \right )\right )+1778602 \sin \left (d x +c \right )+646739\right )}{969969 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(107\) |
-2/969969*(1+sin(d*x+c))*a^4*(sin(d*x+c)-1)^4*(51051*sin(d*x+c)^6+378378*s in(d*x+c)^5+1222221*sin(d*x+c)^4+2244396*sin(d*x+c)^3+2546901*sin(d*x+c)^2 +1778602*sin(d*x+c)+646739)/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d
Time = 0.28 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.33 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {2 \, {\left (51051 \, a^{3} \cos \left (d x + c\right )^{10} + 225225 \, a^{3} \cos \left (d x + c\right )^{9} - 270270 \, a^{3} \cos \left (d x + c\right )^{8} - 562716 \, a^{3} \cos \left (d x + c\right )^{7} + 10752 \, a^{3} \cos \left (d x + c\right )^{6} - 14336 \, a^{3} \cos \left (d x + c\right )^{5} + 20480 \, a^{3} \cos \left (d x + c\right )^{4} - 32768 \, a^{3} \cos \left (d x + c\right )^{3} + 65536 \, a^{3} \cos \left (d x + c\right )^{2} - 262144 \, a^{3} \cos \left (d x + c\right ) - 524288 \, a^{3} + {\left (51051 \, a^{3} \cos \left (d x + c\right )^{9} - 174174 \, a^{3} \cos \left (d x + c\right )^{8} - 444444 \, a^{3} \cos \left (d x + c\right )^{7} + 118272 \, a^{3} \cos \left (d x + c\right )^{6} + 129024 \, a^{3} \cos \left (d x + c\right )^{5} + 143360 \, a^{3} \cos \left (d x + c\right )^{4} + 163840 \, a^{3} \cos \left (d x + c\right )^{3} + 196608 \, a^{3} \cos \left (d x + c\right )^{2} + 262144 \, a^{3} \cos \left (d x + c\right ) + 524288 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{969969 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
2/969969*(51051*a^3*cos(d*x + c)^10 + 225225*a^3*cos(d*x + c)^9 - 270270*a ^3*cos(d*x + c)^8 - 562716*a^3*cos(d*x + c)^7 + 10752*a^3*cos(d*x + c)^6 - 14336*a^3*cos(d*x + c)^5 + 20480*a^3*cos(d*x + c)^4 - 32768*a^3*cos(d*x + c)^3 + 65536*a^3*cos(d*x + c)^2 - 262144*a^3*cos(d*x + c) - 524288*a^3 + (51051*a^3*cos(d*x + c)^9 - 174174*a^3*cos(d*x + c)^8 - 444444*a^3*cos(d*x + c)^7 + 118272*a^3*cos(d*x + c)^6 + 129024*a^3*cos(d*x + c)^5 + 143360*a ^3*cos(d*x + c)^4 + 163840*a^3*cos(d*x + c)^3 + 196608*a^3*cos(d*x + c)^2 + 262144*a^3*cos(d*x + c) + 524288*a^3)*sin(d*x + c))*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c) + d*sin(d*x + c) + d)
Timed out. \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\text {Timed out} \]
\[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \cos \left (d x + c\right )^{6} \,d x } \]
Time = 0.33 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.06 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {1024 \, \sqrt {2} {\left (51051 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{19} - 342342 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 969969 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} - 1492260 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 1322685 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 646646 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 138567 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}\right )} \sqrt {a}}{969969 \, d} \]
1024/969969*sqrt(2)*(51051*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/ 4*pi + 1/2*d*x + 1/2*c)^19 - 342342*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c) )*sin(-1/4*pi + 1/2*d*x + 1/2*c)^17 + 969969*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^15 - 1492260*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^13 + 1322685*a^3*sgn(c os(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^11 - 646646* a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^9 + 138567*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/ 2*c)^7)*sqrt(a)/d
Timed out. \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\int {\cos \left (c+d\,x\right )}^6\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2} \,d x \]