Integrand size = 30, antiderivative size = 89 \[ \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2} \, dx=-\frac {a^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{3 f \sqrt {a+a \sin (e+f x)}}-\frac {a \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}{3 f} \] Output:
-1/3*a^2*cos(f*x+e)*(c-c*sin(f*x+e))^(3/2)/f/(a+a*sin(f*x+e))^(1/2)-1/3*a* cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(3/2)/f
Time = 1.38 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.78 \[ \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2} \, dx=\frac {c \sec ^2(e+f x) (-1+\sin (e+f x)) (a (1+\sin (e+f x)))^{3/2} \sqrt {c-c \sin (e+f x)} \left (-3+\sin ^2(e+f x)\right ) \tan (e+f x)}{3 f} \] Input:
Integrate[(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2),x]
Output:
(c*Sec[e + f*x]^2*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]]*(-3 + Sin[e + f*x]^2)*Tan[e + f*x])/(3*f)
Time = 0.45 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3042, 3219, 3042, 3217}
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.
\(\displaystyle \int (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}dx\) |
\(\Big \downarrow \) 3219 |
\(\displaystyle \frac {2}{3} a \int \sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}dx-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}{3 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{3} a \int \sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}dx-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}{3 f}\) |
\(\Big \downarrow \) 3217 |
\(\displaystyle -\frac {a^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}{3 f}\) |
Input:
Int[(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2),x]
Output:
-1/3*(a^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(f*Sqrt[a + a*Sin[e + f *x]]) - (a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2 ))/(3*f)
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f _.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] /; FreeQ[{a, b, c, d, e, f, n }, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -2^(-1)]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*(a + b*Sin[e + f*x])^ (m - 1)*((c + d*Sin[e + f*x])^n/(f*(m + n))), x] + Simp[a*((2*m - 1)/(m + n )) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; Fre eQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I GtQ[m - 1/2, 0] && !LtQ[n, -1] && !(IGtQ[n - 1/2, 0] && LtQ[n, m]) && !( ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0])
Time = 1.77 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {2 \sin \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2} \tan \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right ) \sqrt {a \sin \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, \sqrt {c \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, \left (2 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}+1\right ) a c \sqrt {4}}{3 f}\) | \(91\) |
Input:
int((a+sin(f*x+e)*a)^(3/2)*(c-c*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
Output:
2/3/f*sin(1/4*Pi+1/2*f*x+1/2*e)^2*tan(1/4*Pi+1/2*f*x+1/2*e)*(a*sin(1/4*Pi+ 1/2*f*x+1/2*e)^2)^(1/2)*(c*cos(1/4*Pi+1/2*f*x+1/2*e)^2)^(1/2)*(2*cos(1/4*P i+1/2*f*x+1/2*e)^2+1)*a*c*4^(1/2)
Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.67 \[ \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2} \, dx=\frac {{\left (a c \cos \left (f x + e\right )^{2} + 2 \, a c\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{3 \, f \cos \left (f x + e\right )} \] Input:
integrate((a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(3/2),x, algorithm="fric as")
Output:
1/3*(a*c*cos(f*x + e)^2 + 2*a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e)/(f*cos(f*x + e))
\[ \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2} \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}\, dx \] Input:
integrate((a+a*sin(f*x+e))**(3/2)*(c-c*sin(f*x+e))**(3/2),x)
Output:
Integral((a*(sin(e + f*x) + 1))**(3/2)*(-c*(sin(e + f*x) - 1))**(3/2), x)
\[ \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(3/2),x, algorithm="maxi ma")
Output:
integrate((a*sin(f*x + e) + a)^(3/2)*(-c*sin(f*x + e) + c)^(3/2), x)
Time = 0.15 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.12 \[ \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2} \, dx=-\frac {4 \, {\left (2 \, a c \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 3 \, a c \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}\right )} \sqrt {a} \sqrt {c}}{3 \, f} \] Input:
integrate((a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(3/2),x, algorithm="giac ")
Output:
-4/3*(2*a*c*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^6 - 3*a*c*sgn(cos(-1/4*pi + 1/2*f *x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1 /2*e)^4)*sqrt(a)*sqrt(c)/f
Time = 0.55 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.74 \[ \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2} \, dx=\frac {a\,c\,\left (10\,\sin \left (2\,e+2\,f\,x\right )+\sin \left (4\,e+4\,f\,x\right )\right )\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}}{12\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \] Input:
int((a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(3/2),x)
Output:
(a*c*(10*sin(2*e + 2*f*x) + sin(4*e + 4*f*x))*(a*(sin(e + f*x) + 1))^(1/2) *(-c*(sin(e + f*x) - 1))^(1/2))/(12*f*(cos(2*e + 2*f*x) + 1))
\[ \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2} \, dx=\sqrt {c}\, \sqrt {a}\, a c \left (-\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right )+\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}d x \right ) \] Input:
int((a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(3/2),x)
Output:
sqrt(c)*sqrt(a)*a*c*( - int(sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**2,x) + int(sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1),x))