\(\int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x)) \, dx\) [531]

Optimal result

Integrand size = 25, antiderivative size = 97 \[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x)) \, dx=-\frac {24 (5 c+3 d) \cos (e+f x)}{5 f \sqrt {3+3 \sin (e+f x)}}-\frac {2 (5 c+3 d) \cos (e+f x) \sqrt {3+3 \sin (e+f x)}}{5 f}-\frac {2 d \cos (e+f x) (3+3 \sin (e+f x))^{3/2}}{5 f} \]

[Out]

-2/5*d*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/f-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)*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/f

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2830, 2726, 2725} \[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x)) \, dx=-\frac {8 a^2 (5 c+3 d) \cos (e+f x)}{15 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a (5 c+3 d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{15 f}-\frac {2 d \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f} \]

[In]

Int[(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x]),x]

[Out]

(-8*a^2*(5*c + 3*d)*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*(5*c + 3*d)*Cos[e + f*x]*Sqrt[a + a*S
in[e + f*x]])/(15*f) - (2*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*f)

Rule 2725

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] && EqQ[a^2 - b^2, 0]

Rule 2726

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] + Dist[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]

Rule 2830

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] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*S
in[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)]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 d \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 f}+\frac {1}{5} (5 c+3 d) \int (a+a \sin (e+f x))^{3/2} \, dx \\ & = -\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}+\frac {1}{15} (4 a (5 c+3 d)) \int \sqrt {a+a \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.24 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.06 \[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x)) \, dx=-\frac {\sqrt {3} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^{3/2} (50 c+39 d-3 d \cos (2 (e+f x))+2 (5 c+9 d) \sin (e+f x))}{5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \]

[In]

Integrate[(3 + 3*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x]),x]

[Out]

-1/5*(Sqrt[3]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^(3/2)*(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])^3)

Maple [A] (verified)

Time = 1.82 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79

method	result	size
default	\(\frac {2 \left (\sin \left (f x +e \right )+1\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (-3 \left (\cos ^{2}\left (f x +e \right )\right ) d +\sin \left (f x +e \right ) \left (5 c +9 d \right )+25 c +21 d \right )}{15 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(77\)
parts	\(\frac {2 c \left (\sin \left (f x +e \right )+1\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (\sin \left (f x +e \right )+5\right )}{3 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 d \left (\sin \left (f x +e \right )+1\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (\sin ^{2}\left (f x +e \right )+3 \sin \left (f x +e \right )+6\right )}{5 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(118\)

[In]

int((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

2/15*(sin(f*x+e)+1)*a^2*(sin(f*x+e)-1)*(-3*cos(f*x+e)^2*d+sin(f*x+e)*(5*c+9*d)+25*c+21*d)/cos(f*x+e)/(a+a*sin(
f*x+e))^(1/2)/f

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.40 \[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x)) \, dx=\frac {2 \, {\left (3 \, a d \cos \left (f x + e\right )^{3} - {\left (5 \, a c + 6 \, a d\right )} \cos \left (f x + e\right )^{2} - 20 \, a c - 12 \, a d - {\left (25 \, a c + 21 \, a d\right )} \cos \left (f x + e\right ) - {\left (3 \, a d \cos \left (f x + e\right )^{2} - 20 \, a c - 12 \, a d + {\left (5 \, a c + 9 \, a d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{15 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \]

[In]

integrate((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e)),x, algorithm="fricas")

[Out]

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)

Sympy [F]

\[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x)) \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (c + d \sin {\left (e + f x \right )}\right )\, dx \]

[In]

integrate((a+a*sin(f*x+e))**(3/2)*(c+d*sin(f*x+e)),x)

[Out]

Integral((a*(sin(e + f*x) + 1))**(3/2)*(c + d*sin(e + f*x)), x)

Maxima [F]

\[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x)) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \sin \left (f x + e\right ) + c\right )} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(3/2)*(d*sin(f*x + e) + c), x)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.43 \[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x)) \, dx=\frac {\sqrt {2} {\left (3 \, a d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) + 30 \, {\left (3 \, a c \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2 \, a d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 5 \, {\left (2 \, a c \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, a d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right )\right )} \sqrt {a}}{30 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e)),x, algorithm="giac")

[Out]

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(cos(-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

Mupad [F(-1)]

Timed out. \[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x)) \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,\left (c+d\,\sin \left (e+f\,x\right )\right ) \,d x \]

[In]

int((a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x)),x)

[Out]

int((a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x)), x)