Integrand size = 16, antiderivative size = 127 \[ \int (-a+a \sin (c+d x))^{7/2} \, dx=\frac {256 a^4 \cos (c+d x)}{35 d \sqrt {-a+a \sin (c+d x)}}-\frac {64 a^3 \cos (c+d x) \sqrt {-a+a \sin (c+d x)}}{35 d}+\frac {24 a^2 \cos (c+d x) (-a+a \sin (c+d x))^{3/2}}{35 d}-\frac {2 a \cos (c+d x) (-a+a \sin (c+d x))^{5/2}}{7 d} \] Output:
256/35*a^4*cos(d*x+c)/d/(-a+a*sin(d*x+c))^(1/2)-64/35*a^3*cos(d*x+c)*(-a+a *sin(d*x+c))^(1/2)/d+24/35*a^2*cos(d*x+c)*(-a+a*sin(d*x+c))^(3/2)/d-2/7*a* cos(d*x+c)*(-a+a*sin(d*x+c))^(5/2)/d
Time = 2.86 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.23 \[ \int (-a+a \sin (c+d x))^{7/2} \, dx=\frac {a^3 (-1+\sin (c+d x))^3 \sqrt {a (-1+\sin (c+d x))} \left (1225 \cos \left (\frac {1}{2} (c+d x)\right )+245 \cos \left (\frac {3}{2} (c+d x)\right )-49 \cos \left (\frac {5}{2} (c+d x)\right )-5 \cos \left (\frac {7}{2} (c+d x)\right )+1225 \sin \left (\frac {1}{2} (c+d x)\right )-245 \sin \left (\frac {3}{2} (c+d x)\right )-49 \sin \left (\frac {5}{2} (c+d x)\right )+5 \sin \left (\frac {7}{2} (c+d x)\right )\right )}{140 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^7} \] Input:
Integrate[(-a + a*Sin[c + d*x])^(7/2),x]
Output:
(a^3*(-1 + Sin[c + d*x])^3*Sqrt[a*(-1 + Sin[c + d*x])]*(1225*Cos[(c + d*x) /2] + 245*Cos[(3*(c + d*x))/2] - 49*Cos[(5*(c + d*x))/2] - 5*Cos[(7*(c + d *x))/2] + 1225*Sin[(c + d*x)/2] - 245*Sin[(3*(c + d*x))/2] - 49*Sin[(5*(c + d*x))/2] + 5*Sin[(7*(c + d*x))/2]))/(140*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^7)
Time = 0.46 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3126, 3042, 3126, 3042, 3126, 3042, 3125}
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.
| \(\displaystyle \int (a \sin (c+d x)-a)^{7/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (c+d x)-a)^{7/2}dx\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle -\frac {12}{7} a \int (a \sin (c+d x)-a)^{5/2}dx-\frac {2 a \cos (c+d x) (a \sin (c+d x)-a)^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {12}{7} a \int (a \sin (c+d x)-a)^{5/2}dx-\frac {2 a \cos (c+d x) (a \sin (c+d x)-a)^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle -\frac {12}{7} a \left (-\frac {8}{5} a \int (a \sin (c+d x)-a)^{3/2}dx-\frac {2 a \cos (c+d x) (a \sin (c+d x)-a)^{3/2}}{5 d}\right )-\frac {2 a \cos (c+d x) (a \sin (c+d x)-a)^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {12}{7} a \left (-\frac {8}{5} a \int (a \sin (c+d x)-a)^{3/2}dx-\frac {2 a \cos (c+d x) (a \sin (c+d x)-a)^{3/2}}{5 d}\right )-\frac {2 a \cos (c+d x) (a \sin (c+d x)-a)^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle -\frac {12}{7} a \left (-\frac {8}{5} a \left (-\frac {4}{3} a \int \sqrt {a \sin (c+d x)-a}dx-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)-a}}{3 d}\right )-\frac {2 a \cos (c+d x) (a \sin (c+d x)-a)^{3/2}}{5 d}\right )-\frac {2 a \cos (c+d x) (a \sin (c+d x)-a)^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {12}{7} a \left (-\frac {8}{5} a \left (-\frac {4}{3} a \int \sqrt {a \sin (c+d x)-a}dx-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)-a}}{3 d}\right )-\frac {2 a \cos (c+d x) (a \sin (c+d x)-a)^{3/2}}{5 d}\right )-\frac {2 a \cos (c+d x) (a \sin (c+d x)-a)^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle -\frac {12}{7} a \left (-\frac {8}{5} a \left (\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)-a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)-a}}{3 d}\right )-\frac {2 a \cos (c+d x) (a \sin (c+d x)-a)^{3/2}}{5 d}\right )-\frac {2 a \cos (c+d x) (a \sin (c+d x)-a)^{5/2}}{7 d}\) |
Input:
Int[(-a + a*Sin[c + d*x])^(7/2),x]
Output:
(-2*a*Cos[c + d*x]*(-a + a*Sin[c + d*x])^(5/2))/(7*d) - (12*a*((-2*a*Cos[c + d*x]*(-a + a*Sin[c + d*x])^(3/2))/(5*d) - (8*a*((8*a^2*Cos[c + d*x])/(3 *d*Sqrt[-a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]*Sqrt[-a + a*Sin[c + d*x] ])/(3*d)))/5))/7
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos [c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[a*((2*n - 1)/n) Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ a^2 - b^2, 0] && IGtQ[n - 1/2, 0]
Time = 0.17 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.61
| method | result | size |
| default | \(\frac {2 \left (\sin \left (d x +c \right )-1\right ) a^{4} \left (1+\sin \left (d x +c \right )\right ) \left (5 \sin \left (d x +c \right )^{3}-27 \sin \left (d x +c \right )^{2}+71 \sin \left (d x +c \right )-177\right )}{35 \cos \left (d x +c \right ) \sqrt {-a +a \sin \left (d x +c \right )}\, d}\) | \(77\) |
Input:
int((-a+a*sin(d*x+c))^(7/2),x,method=_RETURNVERBOSE)
Output:
2/35*(sin(d*x+c)-1)*a^4*(1+sin(d*x+c))*(5*sin(d*x+c)^3-27*sin(d*x+c)^2+71* sin(d*x+c)-177)/cos(d*x+c)/(-a+a*sin(d*x+c))^(1/2)/d
Time = 0.08 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.13 \[ \int (-a+a \sin (c+d x))^{7/2} \, dx=\frac {2 \, {\left (5 \, a^{3} \cos \left (d x + c\right )^{4} + 27 \, a^{3} \cos \left (d x + c\right )^{3} - 54 \, a^{3} \cos \left (d x + c\right )^{2} - 204 \, a^{3} \cos \left (d x + c\right ) - 128 \, a^{3} - {\left (5 \, a^{3} \cos \left (d x + c\right )^{3} - 22 \, a^{3} \cos \left (d x + c\right )^{2} - 76 \, a^{3} \cos \left (d x + c\right ) + 128 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) - a}}{35 \, {\left (d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d\right )}} \] Input:
integrate((-a+a*sin(d*x+c))^(7/2),x, algorithm="fricas")
Output:
2/35*(5*a^3*cos(d*x + c)^4 + 27*a^3*cos(d*x + c)^3 - 54*a^3*cos(d*x + c)^2 - 204*a^3*cos(d*x + c) - 128*a^3 - (5*a^3*cos(d*x + c)^3 - 22*a^3*cos(d*x + c)^2 - 76*a^3*cos(d*x + c) + 128*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. \[ \int (-a+a \sin (c+d x))^{7/2} \, dx=\text {Timed out} \] Input:
integrate((-a+a*sin(d*x+c))**(7/2),x)
Output:
Timed out
\[ \int (-a+a \sin (c+d x))^{7/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) - a\right )}^{\frac {7}{2}} \,d x } \] Input:
integrate((-a+a*sin(d*x+c))^(7/2),x, algorithm="maxima")
Output:
integrate((a*sin(d*x + c) - a)^(7/2), x)
Time = 0.20 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06 \[ \int (-a+a \sin (c+d x))^{7/2} \, dx=\frac {\sqrt {2} {\left (1225 \, a^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 245 \, a^{3} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 49 \, a^{3} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 5 \, a^{3} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sqrt {-a}}{140 \, d} \] Input:
integrate((-a+a*sin(d*x+c))^(7/2),x, algorithm="giac")
Output:
1/140*sqrt(2)*(1225*a^3*cos(-1/4*pi + 1/2*d*x + 1/2*c)*sgn(sin(-1/4*pi + 1 /2*d*x + 1/2*c)) - 245*a^3*cos(-3/4*pi + 3/2*d*x + 3/2*c)*sgn(sin(-1/4*pi + 1/2*d*x + 1/2*c)) + 49*a^3*cos(-5/4*pi + 5/2*d*x + 5/2*c)*sgn(sin(-1/4*p i + 1/2*d*x + 1/2*c)) - 5*a^3*cos(-7/4*pi + 7/2*d*x + 7/2*c)*sgn(sin(-1/4* pi + 1/2*d*x + 1/2*c)))*sqrt(-a)/d
Timed out. \[ \int (-a+a \sin (c+d x))^{7/2} \, dx=\int {\left (a\,\sin \left (c+d\,x\right )-a\right )}^{7/2} \,d x \] Input:
int((a*sin(c + d*x) - a)^(7/2),x)
Output:
int((a*sin(c + d*x) - a)^(7/2), x)
\[ \int (-a+a \sin (c+d x))^{7/2} \, dx=\sqrt {a}\, a^{3} \left (-\left (\int \sqrt {\sin \left (d x +c \right )-1}d x \right )+\int \sqrt {\sin \left (d x +c \right )-1}\, \sin \left (d x +c \right )^{3}d x -3 \left (\int \sqrt {\sin \left (d x +c \right )-1}\, \sin \left (d x +c \right )^{2}d x \right )+3 \left (\int \sqrt {\sin \left (d x +c \right )-1}\, \sin \left (d x +c \right )d x \right )\right ) \] Input:
int((-a+a*sin(d*x+c))^(7/2),x)
Output:
sqrt(a)*a**3*( - int(sqrt(sin(c + d*x) - 1),x) + int(sqrt(sin(c + d*x) - 1 )*sin(c + d*x)**3,x) - 3*int(sqrt(sin(c + d*x) - 1)*sin(c + d*x)**2,x) + 3 *int(sqrt(sin(c + d*x) - 1)*sin(c + d*x),x))