3.131 \(\int \frac {(a+a \sin (e+f x))^{3/2}}{x} \, dx\)
Optimal. Leaf size=221 \[ \frac {3}{2} a \sin \left (\frac {1}{4} (2 e+\pi )\right ) \text {Ci}\left (\frac {f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}+\frac {1}{2} a \cos \left (\frac {3}{4} (2 e-\pi )\right ) \text {Ci}\left (\frac {3 f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}-\frac {1}{2} a \sin \left (\frac {3}{4} (2 e-\pi )\right ) \text {Si}\left (\frac {3 f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}+\frac {3}{2} a \cos \left (\frac {1}{4} (2 e+\pi )\right ) \text {Si}\left (\frac {f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \]
[Out]
-1/2*a*Ci(3/2*f*x)*cos(3/2*e+1/4*Pi)*csc(1/2*e+1/4*Pi+1/2*f*x)*(a+a*sin(f*x+e))^(1/2)+3/2*a*cos(1/2*e+1/4*Pi)*
csc(1/2*e+1/4*Pi+1/2*f*x)*Si(1/2*f*x)*(a+a*sin(f*x+e))^(1/2)+1/2*a*csc(1/2*e+1/4*Pi+1/2*f*x)*Si(3/2*f*x)*sin(3
/2*e+1/4*Pi)*(a+a*sin(f*x+e))^(1/2)+3/2*a*Ci(1/2*f*x)*csc(1/2*e+1/4*Pi+1/2*f*x)*sin(1/2*e+1/4*Pi)*(a+a*sin(f*x
+e))^(1/2)
________________________________________________________________________________________
Rubi [A] time = 0.28, antiderivative size = 221, normalized size of antiderivative = 1.00,
number of steps used = 9, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used =
{3319, 3312, 3303, 3299, 3302} \[ \frac {3}{2} a \sin \left (\frac {1}{4} (2 e+\pi )\right ) \text {CosIntegral}\left (\frac {f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}+\frac {1}{2} a \cos \left (\frac {3}{4} (2 e-\pi )\right ) \text {CosIntegral}\left (\frac {3 f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}-\frac {1}{2} a \sin \left (\frac {3}{4} (2 e-\pi )\right ) \text {Si}\left (\frac {3 f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}+\frac {3}{2} a \cos \left (\frac {1}{4} (2 e+\pi )\right ) \text {Si}\left (\frac {f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Sin[e + f*x])^(3/2)/x,x]
[Out]
(a*Cos[(3*(2*e - Pi))/4]*CosIntegral[(3*f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/2 + (3*a*C
osIntegral[(f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]*Sin[(2*e + Pi)/4]*Sqrt[a + a*Sin[e + f*x]])/2 + (3*a*Cos[(2*e +
Pi)/4]*Csc[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]]*SinIntegral[(f*x)/2])/2 - (a*Csc[e/2 + Pi/4 + (f*x)/
2]*Sin[(3*(2*e - Pi))/4]*Sqrt[a + a*Sin[e + f*x]]*SinIntegral[(3*f*x)/2])/2
Rule 3299
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]
Rule 3302
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Rule 3303
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]
Rule 3312
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)
Rule 3319
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[((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])
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^{3/2}}{x} \, dx &=\left (2 a \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}\right ) \int \frac {\sin ^3\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{x} \, dx\\ &=\left (2 a \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}\right ) \int \left (\frac {3 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{4 x}+\frac {\sin \left (\frac {3 e}{2}-\frac {\pi }{4}+\frac {3 f x}{2}\right )}{4 x}\right ) \, dx\\ &=\frac {1}{2} \left (a \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}\right ) \int \frac {\sin \left (\frac {3 e}{2}-\frac {\pi }{4}+\frac {3 f x}{2}\right )}{x} \, dx+\frac {1}{2} \left (3 a \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}\right ) \int \frac {\sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{x} \, dx\\ &=\frac {1}{2} \left (a \cos \left (\frac {3}{4} (2 e-\pi )\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}\right ) \int \frac {\cos \left (\frac {3 f x}{2}\right )}{x} \, dx+\frac {1}{2} \left (a \cos \left (\frac {3 e}{2}-\frac {\pi }{4}\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}\right ) \int \frac {\sin \left (\frac {3 f x}{2}\right )}{x} \, dx+\frac {1}{2} \left (3 a \cos \left (\frac {1}{4} (2 e+\pi )\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}\right ) \int \frac {\sin \left (\frac {f x}{2}\right )}{x} \, dx+\frac {1}{2} \left (3 a \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {1}{4} (2 e+\pi )\right ) \sqrt {a+a \sin (e+f x)}\right ) \int \frac {\cos \left (\frac {f x}{2}\right )}{x} \, dx\\ &=\frac {1}{2} a \cos \left (\frac {3}{4} (2 e-\pi )\right ) \text {Ci}\left (\frac {3 f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}+\frac {3}{2} a \text {Ci}\left (\frac {f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {1}{4} (2 e+\pi )\right ) \sqrt {a+a \sin (e+f x)}+\frac {3}{2} a \cos \left (\frac {1}{4} (2 e+\pi )\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)} \text {Si}\left (\frac {f x}{2}\right )-\frac {1}{2} a \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {3}{4} (2 e-\pi )\right ) \sqrt {a+a \sin (e+f x)} \text {Si}\left (\frac {3 f x}{2}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.73, size = 127, normalized size = 0.57 \[ \frac {(a (\sin (e+f x)+1))^{3/2} \left (3 \text {Ci}\left (\frac {f x}{2}\right ) \left (\sin \left (\frac {e}{2}\right )+\cos \left (\frac {e}{2}\right )\right )+\text {Ci}\left (\frac {3 f x}{2}\right ) \left (\sin \left (\frac {3 e}{2}\right )-\cos \left (\frac {3 e}{2}\right )\right )+\left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left ((2 \sin (e)+1) \text {Si}\left (\frac {3 f x}{2}\right )+3 \text {Si}\left (\frac {f x}{2}\right )\right )\right )}{2 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Sin[e + f*x])^(3/2)/x,x]
[Out]
((a*(1 + Sin[e + f*x]))^(3/2)*(3*CosIntegral[(f*x)/2]*(Cos[e/2] + Sin[e/2]) + CosIntegral[(3*f*x)/2]*(-Cos[(3*
e)/2] + Sin[(3*e)/2]) + (Cos[e/2] - Sin[e/2])*(3*SinIntegral[(f*x)/2] + (1 + 2*Sin[e])*SinIntegral[(3*f*x)/2])
))/(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^(3/2)/x,x, algorithm="fricas")
[Out]
Exception raised: TypeError >> Error detected within library code: integrate: implementation incomplete (ha
s polynomial part)
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^(3/2)/x,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)sqrt(2*a)*(6*a*Si(1/2*f*x)*sign(cos(1/2*
(f*x+exp(1))-1/4*pi))+2*a*Si(3/2*f*x)*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+3*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi)
)*im(Ci(1/2*f*x))+a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*im(Ci(3/2*f*x))-3*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*
im(Ci(-1/2*f*x))-a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*im(Ci(-3/2*f*x))+3*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*
re(Ci(1/2*f*x))-a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*re(Ci(3/2*f*x))+3*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*re
(Ci(-1/2*f*x))-a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*re(Ci(-3/2*f*x))-6*a*Si(1/2*f*x)*sign(cos(1/2*(f*x+exp(1))
-1/4*pi))*tan(1/4*exp(1))^2+6*a*Si(1/2*f*x)*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(3/4*exp(1))^2-12*a*Si(1/2*f
*x)*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))+2*a*Si(3/2*f*x)*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(
1/4*exp(1))^2-2*a*Si(3/2*f*x)*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(3/4*exp(1))^2+4*a*Si(3/2*f*x)*sign(cos(1/
2*(f*x+exp(1))-1/4*pi))*tan(3/4*exp(1))-3*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*im(Ci(1/2*f*x))*tan(1/4*exp(1))
^2+3*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*im(Ci(1/2*f*x))*tan(3/4*exp(1))^2-6*a*sign(cos(1/2*(f*x+exp(1))-1/4*
pi))*im(Ci(1/2*f*x))*tan(1/4*exp(1))+a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*im(Ci(3/2*f*x))*tan(1/4*exp(1))^2-a*
sign(cos(1/2*(f*x+exp(1))-1/4*pi))*im(Ci(3/2*f*x))*tan(3/4*exp(1))^2+2*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*im
(Ci(3/2*f*x))*tan(3/4*exp(1))+3*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*im(Ci(-1/2*f*x))*tan(1/4*exp(1))^2-3*a*si
gn(cos(1/2*(f*x+exp(1))-1/4*pi))*im(Ci(-1/2*f*x))*tan(3/4*exp(1))^2+6*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*im(
Ci(-1/2*f*x))*tan(1/4*exp(1))-a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*im(Ci(-3/2*f*x))*tan(1/4*exp(1))^2+a*sign(c
os(1/2*(f*x+exp(1))-1/4*pi))*im(Ci(-3/2*f*x))*tan(3/4*exp(1))^2-2*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*im(Ci(-
3/2*f*x))*tan(3/4*exp(1))-3*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*re(Ci(1/2*f*x))*tan(1/4*exp(1))^2+3*a*sign(co
s(1/2*(f*x+exp(1))-1/4*pi))*re(Ci(1/2*f*x))*tan(3/4*exp(1))^2+6*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*re(Ci(1/2
*f*x))*tan(1/4*exp(1))-a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*re(Ci(3/2*f*x))*tan(1/4*exp(1))^2+a*sign(cos(1/2*(
f*x+exp(1))-1/4*pi))*re(Ci(3/2*f*x))*tan(3/4*exp(1))^2+2*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*re(Ci(3/2*f*x))*
tan(3/4*exp(1))-3*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*re(Ci(-1/2*f*x))*tan(1/4*exp(1))^2+3*a*sign(cos(1/2*(f*
x+exp(1))-1/4*pi))*re(Ci(-1/2*f*x))*tan(3/4*exp(1))^2+6*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*re(Ci(-1/2*f*x))*
tan(1/4*exp(1))-a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*re(Ci(-3/2*f*x))*tan(1/4*exp(1))^2+a*sign(cos(1/2*(f*x+ex
p(1))-1/4*pi))*re(Ci(-3/2*f*x))*tan(3/4*exp(1))^2+2*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*re(Ci(-3/2*f*x))*tan(
3/4*exp(1))-6*a*Si(1/2*f*x)*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^2*tan(3/4*exp(1))^2-12*a*Si(1/2
*f*x)*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))*tan(3/4*exp(1))^2-2*a*Si(3/2*f*x)*sign(cos(1/2*(f*x+e
xp(1))-1/4*pi))*tan(1/4*exp(1))^2*tan(3/4*exp(1))^2+4*a*Si(3/2*f*x)*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4
*exp(1))^2*tan(3/4*exp(1))-3*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*im(Ci(1/2*f*x))*tan(1/4*exp(1))^2*tan(3/4*ex
p(1))^2-6*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*im(Ci(1/2*f*x))*tan(1/4*exp(1))*tan(3/4*exp(1))^2-a*sign(cos(1/
2*(f*x+exp(1))-1/4*pi))*im(Ci(3/2*f*x))*tan(1/4*exp(1))^2*tan(3/4*exp(1))^2+2*a*sign(cos(1/2*(f*x+exp(1))-1/4*
pi))*im(Ci(3/2*f*x))*tan(1/4*exp(1))^2*tan(3/4*exp(1))+3*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*im(Ci(-1/2*f*x))
*tan(1/4*exp(1))^2*tan(3/4*exp(1))^2+6*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*im(Ci(-1/2*f*x))*tan(1/4*exp(1))*t
an(3/4*exp(1))^2+a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*im(Ci(-3/2*f*x))*tan(1/4*exp(1))^2*tan(3/4*exp(1))^2-2*a
*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*im(Ci(-3/2*f*x))*tan(1/4*exp(1))^2*tan(3/4*exp(1))-3*a*sign(cos(1/2*(f*x+e
xp(1))-1/4*pi))*re(Ci(1/2*f*x))*tan(1/4*exp(1))^2*tan(3/4*exp(1))^2+6*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*re(
Ci(1/2*f*x))*tan(1/4*exp(1))*tan(3/4*exp(1))^2+a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*re(Ci(3/2*f*x))*tan(1/4*ex
p(1))^2*tan(3/4*exp(1))^2+2*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*re(Ci(3/2*f*x))*tan(1/4*exp(1))^2*tan(3/4*exp
(1))-3*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*re(Ci(-1/2*f*x))*tan(1/4*exp(1))^2*tan(3/4*exp(1))^2+6*a*sign(cos(
1/2*(f*x+exp(1))-1/4*pi))*re(Ci(-1/2*f*x))*tan(1/4*exp(1))*tan(3/4*exp(1))^2+a*sign(cos(1/2*(f*x+exp(1))-1/4*p
i))*re(Ci(-3/2*f*x))*tan(1/4*exp(1))^2*tan(3/4*exp(1))^2+2*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*re(Ci(-3/2*f*x
))*tan(1/4*exp(1))^2*tan(3/4*exp(1)))/(4*sqrt(2)*tan(1/4*exp(1))^2*tan(3/4*exp(1))^2+4*sqrt(2)*tan(1/4*exp(1))
^2+4*sqrt(2)*tan(3/4*exp(1))^2+4*sqrt(2))
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sin(f*x+e))^(3/2)/x,x)
[Out]
int((a+a*sin(f*x+e))^(3/2)/x,x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^(3/2)/x,x, algorithm="maxima")
[Out]
integrate((a*sin(f*x + e) + a)^(3/2)/x, x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + a*sin(e + f*x))^(3/2)/x,x)
[Out]
int((a + a*sin(e + f*x))^(3/2)/x, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))**(3/2)/x,x)
[Out]
Integral((a*(sin(e + f*x) + 1))**(3/2)/x, x)
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