\(\int x (a+a \sin (e+f x))^{3/2} \, dx\) [130]

Optimal result

Integrand size = 16, antiderivative size = 165 \[ \int x (a+a \sin (e+f x))^{3/2} \, dx=\frac {16 a \sqrt {a+a \sin (e+f x)}}{3 f^2}-\frac {8 a x \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f}-\frac {4 a x \cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {8 a \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{9 f^2} \] Output:

16/3*a*(a+a*sin(f*x+e))^(1/2)/f^2-8/3*a*x*cot(1/2*e+1/4*Pi+1/2*f*x)*(a+a*s 
in(f*x+e))^(1/2)/f-4/3*a*x*cos(1/2*e+1/4*Pi+1/2*f*x)*sin(1/2*e+1/4*Pi+1/2* 
f*x)*(a+a*sin(f*x+e))^(1/2)/f+8/9*a*sin(1/2*e+1/4*Pi+1/2*f*x)^2*(a+a*sin(f 
*x+e))^(1/2)/f^2
 

Mathematica [A] (verified)

Time = 6.66 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.68 \[ \int x (a+a \sin (e+f x))^{3/2} \, dx=-\frac {\left (27 (-2+f x) \cos \left (\frac {1}{2} (e+f x)\right )+(2+3 f x) \cos \left (\frac {3}{2} (e+f x)\right )+2 (-4 (7+3 f x)+(-2+3 f x) \cos (e+f x)) \sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^{3/2}}{9 f^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \] Input:

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

Output:

-1/9*((27*(-2 + f*x)*Cos[(e + f*x)/2] + (2 + 3*f*x)*Cos[(3*(e + f*x))/2] + 
 2*(-4*(7 + 3*f*x) + (-2 + 3*f*x)*Cos[e + f*x])*Sin[(e + f*x)/2])*(a*(1 + 
Sin[e + f*x]))^(3/2))/(f^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3800, 3042, 3791, 3042, 3777, 3042, 3118}

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.

\(\displaystyle \int x (a \sin (e+f x)+a)^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int x (a \sin (e+f x)+a)^{3/2}dx\)

\(\Big \downarrow \) 3800

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \int x \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \int x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^3dx\)

\(\Big \downarrow \) 3791

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (\frac {2}{3} \int x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx+\frac {4 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}-\frac {2 x \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (\frac {2}{3} \int x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx+\frac {4 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}-\frac {2 x \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (\frac {2}{3} \left (\frac {2 \int \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx}{f}-\frac {2 x \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )+\frac {4 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}-\frac {2 x \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (\frac {2}{3} \left (\frac {2 \int \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {3 \pi }{4}\right )dx}{f}-\frac {2 x \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )+\frac {4 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}-\frac {2 x \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (\frac {4 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}+\frac {2}{3} \left (\frac {4 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 x \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )-\frac {2 x \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

Input:

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

Output:

2*a*Csc[e/2 + Pi/4 + (f*x)/2]*((-2*x*Cos[e/2 + Pi/4 + (f*x)/2]*Sin[e/2 + P 
i/4 + (f*x)/2]^2)/(3*f) + (4*Sin[e/2 + Pi/4 + (f*x)/2]^3)/(9*f^2) + (2*((- 
2*x*Cos[e/2 + Pi/4 + (f*x)/2])/f + (4*Sin[e/2 + Pi/4 + (f*x)/2])/f^2))/3)* 
Sqrt[a + a*Sin[e + f*x]]
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ 
[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3791

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> 
 Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x 
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n)   Int[(c + d* 
x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 
 1]
 

rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), 
 x_Symbol] :> Simp[(2*a)^IntPart[n]*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e 
/2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n]))   Int[(c + d*x)^m*Sin[e/2 + a 
*(Pi/(4*b)) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && 
EqQ[a^2 - b^2, 0] && IntegerQ[n + 1/2] && (GtQ[n, 0] || IGtQ[m, 0])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int x (a+a \sin (e+f x))^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.37 \[ \int x (a+a \sin (e+f x))^{3/2} \, dx=\frac {\sqrt {2} {\left (\frac {108 \, a \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{f} + \frac {4 \, a \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{f} + \frac {27 \, {\left (\pi a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - {\left (\pi - 2 \, f x - 2 \, e\right )} a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 2 \, a e \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{f} + \frac {3 \, {\left (\pi a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - {\left (\pi - 2 \, f x - 2 \, e\right )} a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 2 \, a e \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right )}{f}\right )} \sqrt {a}}{18 \, f} \] Input:

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

Output:

1/18*sqrt(2)*(108*a*cos(-1/4*pi + 1/2*f*x + 1/2*e)*sgn(cos(-1/4*pi + 1/2*f 
*x + 1/2*e))/f + 4*a*cos(-3/4*pi + 3/2*f*x + 3/2*e)*sgn(cos(-1/4*pi + 1/2* 
f*x + 1/2*e))/f + 27*(pi*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - (pi - 2*f 
*x - 2*e)*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 2*a*e*sgn(cos(-1/4*pi + 
1/2*f*x + 1/2*e)))*sin(-1/4*pi + 1/2*f*x + 1/2*e)/f + 3*(pi*a*sgn(cos(-1/4 
*pi + 1/2*f*x + 1/2*e)) - (pi - 2*f*x - 2*e)*a*sgn(cos(-1/4*pi + 1/2*f*x + 
 1/2*e)) - 2*a*e*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-3/4*pi + 3/2*f* 
x + 3/2*e)/f)*sqrt(a)/f
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

sqrt(a)*a*(int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)*x,x) + int(sqrt(sin(e + 
 f*x) + 1)*x,x))