Integrand size = 25, antiderivative size = 101 \[ \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x)) \, dx=-\frac {8 a^2 (5 c+3 d) \cos (e+f x)}{15 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a (5 c+3 d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 f}-\frac {2 d \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 f} \] Output:
-8/15*a^2*(5*c+3*d)*cos(f*x+e)/f/(a+a*sin(f*x+e))^(1/2)-2/15*a*(5*c+3*d)*c os(f*x+e)*(a+a*sin(f*x+e))^(1/2)/f-2/5*d*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2) /f
Time = 1.30 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00 \[ \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x)) \, dx=-\frac {a \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} (50 c+39 d-3 d \cos (2 (e+f x))+2 (5 c+9 d) \sin (e+f x))}{15 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \] Input:
Integrate[(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x]),x]
Output:
-1/15*(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]* (50*c + 39*d - 3*d*Cos[2*(e + f*x)] + 2*(5*c + 9*d)*Sin[e + f*x]))/(f*(Cos [(e + f*x)/2] + Sin[(e + f*x)/2]))
Time = 0.40 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 3230, 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.
| \(\displaystyle \int (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))dx\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {1}{5} (5 c+3 d) \int (\sin (e+f x) a+a)^{3/2}dx-\frac {2 d \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} (5 c+3 d) \int (\sin (e+f x) a+a)^{3/2}dx-\frac {2 d \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \frac {1}{5} (5 c+3 d) \left (\frac {4}{3} a \int \sqrt {\sin (e+f x) a+a}dx-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 d \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} (5 c+3 d) \left (\frac {4}{3} a \int \sqrt {\sin (e+f x) a+a}dx-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 d \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle \frac {1}{5} (5 c+3 d) \left (-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 d \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\) |
Input:
Int[(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x]),x]
Output:
(-2*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*f) + ((5*c + 3*d)*((-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]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Time = 0.99 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.76
| method | result | size |
| default | \(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (-1+\sin \left (f x +e \right )\right ) \left (-3 \cos \left (f x +e \right )^{2} d +\sin \left (f x +e \right ) \left (5 c +9 d \right )+25 c +21 d \right )}{15 \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}\) | \(77\) |
| parts | \(\frac {2 c \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (-1+\sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )+5\right )}{3 \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}+\frac {2 d \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (-1+\sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+6\right )}{5 \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}\) | \(118\) |
Input:
int((a+sin(f*x+e)*a)^(3/2)*(c+d*sin(f*x+e)),x,method=_RETURNVERBOSE)
Output:
2/15*(1+sin(f*x+e))*a^2*(-1+sin(f*x+e))*(-3*cos(f*x+e)^2*d+sin(f*x+e)*(5*c +9*d)+25*c+21*d)/cos(f*x+e)/(a+sin(f*x+e)*a)^(1/2)/f
Time = 0.08 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.35 \[ \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x)) \, dx=\frac {2 \, {\left (3 \, a d \cos \left (f x + e\right )^{3} - {\left (5 \, a c + 6 \, a d\right )} \cos \left (f x + e\right )^{2} - 20 \, a c - 12 \, a d - {\left (25 \, a c + 21 \, a d\right )} \cos \left (f x + e\right ) - {\left (3 \, a d \cos \left (f x + e\right )^{2} - 20 \, a c - 12 \, a d + {\left (5 \, a c + 9 \, a d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{15 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \] Input:
integrate((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e)),x, algorithm="fricas")
Output:
2/15*(3*a*d*cos(f*x + e)^3 - (5*a*c + 6*a*d)*cos(f*x + e)^2 - 20*a*c - 12* a*d - (25*a*c + 21*a*d)*cos(f*x + e) - (3*a*d*cos(f*x + e)^2 - 20*a*c - 12 *a*d + (5*a*c + 9*a*d)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a )/(f*cos(f*x + e) + f*sin(f*x + e) + f)
\[ \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x)) \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (c + d \sin {\left (e + f x \right )}\right )\, dx \] Input:
integrate((a+a*sin(f*x+e))**(3/2)*(c+d*sin(f*x+e)),x)
Output:
Integral((a*(sin(e + f*x) + 1))**(3/2)*(c + d*sin(e + f*x)), x)
\[ \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x)) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \sin \left (f x + e\right ) + c\right )} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e)),x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)^(3/2)*(d*sin(f*x + e) + c), x)
Time = 0.15 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.38 \[ \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x)) \, dx=\frac {\sqrt {2} {\left (3 \, a d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) + 30 \, {\left (3 \, a c \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2 \, a d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 5 \, {\left (2 \, a c \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, a d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right )\right )} \sqrt {a}}{30 \, f} \] Input:
integrate((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e)),x, algorithm="giac")
Output:
1/30*sqrt(2)*(3*a*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-5/4*pi + 5/2* f*x + 5/2*e) + 30*(3*a*c*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 2*a*d*sgn(c os(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 5*(2*a*c* sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*a*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/ 2*e)))*sin(-3/4*pi + 3/2*f*x + 3/2*e))*sqrt(a)/f
Timed out. \[ \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x)) \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,\left (c+d\,\sin \left (e+f\,x\right )\right ) \,d x \] Input:
int((a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x)),x)
Output:
int((a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x)), x)
\[ \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x)) \, dx=\sqrt {a}\, a \left (\left (\int \sqrt {\sin \left (f x +e \right )+1}d x \right ) c +\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) d +\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) c +\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) d \right ) \] Input:
int((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e)),x)
Output:
sqrt(a)*a*(int(sqrt(sin(e + f*x) + 1),x)*c + int(sqrt(sin(e + f*x) + 1)*si n(e + f*x)**2,x)*d + int(sqrt(sin(e + f*x) + 1)*sin(e + f*x),x)*c + int(sq rt(sin(e + f*x) + 1)*sin(e + f*x),x)*d)