Integrand size = 14, antiderivative size = 89 \[ \int (a+a \sin (e+f x))^{5/2} \, dx=-\frac {64 a^3 \cos (e+f x)}{15 f \sqrt {a+a \sin (e+f x)}}-\frac {16 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 f}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 f} \] Output:
-64/15*a^3*cos(f*x+e)/f/(a+a*sin(f*x+e))^(1/2)-16/15*a^2*cos(f*x+e)*(a+a*s in(f*x+e))^(1/2)/f-2/5*a*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/f
Time = 1.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.31 \[ \int (a+a \sin (e+f x))^{5/2} \, dx=-\frac {(a (1+\sin (e+f x)))^{5/2} \left (150 \cos \left (\frac {1}{2} (e+f x)\right )+25 \cos \left (\frac {3}{2} (e+f x)\right )-3 \cos \left (\frac {5}{2} (e+f x)\right )-150 \sin \left (\frac {1}{2} (e+f x)\right )+25 \sin \left (\frac {3}{2} (e+f x)\right )+3 \sin \left (\frac {5}{2} (e+f x)\right )\right )}{30 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \] Input:
Integrate[(a + a*Sin[e + f*x])^(5/2),x]
Output:
-1/30*((a*(1 + Sin[e + f*x]))^(5/2)*(150*Cos[(e + f*x)/2] + 25*Cos[(3*(e + f*x))/2] - 3*Cos[(5*(e + f*x))/2] - 150*Sin[(e + f*x)/2] + 25*Sin[(3*(e + f*x))/2] + 3*Sin[(5*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/ 2])^5)
Time = 0.36 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {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.
\(\displaystyle \int (a \sin (e+f x)+a)^{5/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a \sin (e+f x)+a)^{5/2}dx\) |
\(\Big \downarrow \) 3126 |
\(\displaystyle \frac {8}{5} a \int (\sin (e+f x) a+a)^{3/2}dx-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {8}{5} a \int (\sin (e+f x) a+a)^{3/2}dx-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\) |
\(\Big \downarrow \) 3126 |
\(\displaystyle \frac {8}{5} a \left (\frac {4}{3} a \int \sqrt {\sin (e+f x) a+a}dx-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {8}{5} a \left (\frac {4}{3} a \int \sqrt {\sin (e+f x) a+a}dx-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\) |
\(\Big \downarrow \) 3125 |
\(\displaystyle \frac {8}{5} a \left (-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\) |
Input:
Int[(a + a*Sin[e + f*x])^(5/2),x]
Output:
(-2*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*f) + (8*a*((-8*a^2*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*Sqrt[a + a*Sin [e + f*x]])/(3*f)))/5
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 = 0.37 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (-1+\sin \left (f x +e \right )\right ) \left (3 \sin \left (f x +e \right )^{2}+14 \sin \left (f x +e \right )+43\right )}{15 \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}\) | \(65\) |
Input:
int((a+sin(f*x+e)*a)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/15*(1+sin(f*x+e))*a^3*(-1+sin(f*x+e))*(3*sin(f*x+e)^2+14*sin(f*x+e)+43)/ cos(f*x+e)/(a+sin(f*x+e)*a)^(1/2)/f
Time = 0.07 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.29 \[ \int (a+a \sin (e+f x))^{5/2} \, dx=\frac {2 \, {\left (3 \, a^{2} \cos \left (f x + e\right )^{3} - 11 \, a^{2} \cos \left (f x + e\right )^{2} - 46 \, a^{2} \cos \left (f x + e\right ) - 32 \, a^{2} - {\left (3 \, a^{2} \cos \left (f x + e\right )^{2} + 14 \, a^{2} \cos \left (f x + e\right ) - 32 \, a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{15 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \] Input:
integrate((a+a*sin(f*x+e))^(5/2),x, algorithm="fricas")
Output:
2/15*(3*a^2*cos(f*x + e)^3 - 11*a^2*cos(f*x + e)^2 - 46*a^2*cos(f*x + e) - 32*a^2 - (3*a^2*cos(f*x + e)^2 + 14*a^2*cos(f*x + e) - 32*a^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e) + f*sin(f*x + e) + f)
\[ \int (a+a \sin (e+f x))^{5/2} \, dx=\int \left (a \sin {\left (e + f x \right )} + a\right )^{\frac {5}{2}}\, dx \] Input:
integrate((a+a*sin(f*x+e))**(5/2),x)
Output:
Integral((a*sin(e + f*x) + a)**(5/2), x)
\[ \int (a+a \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(5/2),x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)^(5/2), x)
Time = 0.14 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.15 \[ \int (a+a \sin (e+f x))^{5/2} \, dx=\frac {\sqrt {2} {\left (150 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 25 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) + 3 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right )\right )} \sqrt {a}}{30 \, f} \] Input:
integrate((a+a*sin(f*x+e))^(5/2),x, algorithm="giac")
Output:
1/30*sqrt(2)*(150*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/ 2*f*x + 1/2*e) + 25*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-3/4*pi + 3/2*f*x + 3/2*e) + 3*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-5/4*pi + 5/2*f*x + 5/2*e))*sqrt(a)/f
Timed out. \[ \int (a+a \sin (e+f x))^{5/2} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \] Input:
int((a + a*sin(e + f*x))^(5/2),x)
Output:
int((a + a*sin(e + f*x))^(5/2), x)
\[ \int (a+a \sin (e+f x))^{5/2} \, dx=\sqrt {a}\, a^{2} \left (\int \sqrt {\sin \left (f x +e \right )+1}d x +\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x +2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right )\right ) \] Input:
int((a+a*sin(f*x+e))^(5/2),x)
Output:
sqrt(a)*a**2*(int(sqrt(sin(e + f*x) + 1),x) + int(sqrt(sin(e + f*x) + 1)*s in(e + f*x)**2,x) + 2*int(sqrt(sin(e + f*x) + 1)*sin(e + f*x),x))