\(\int \frac {\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{x^2} \, dx\) [169]

Optimal result

Integrand size = 33, antiderivative size = 123 \[ \int \frac {\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{x^2} \, dx=-\frac {\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{x}-f \operatorname {CosIntegral}(f x) \sec (e+f x) \sin (e) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}-f \cos (e) \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \text {Si}(f x) \]

[Out]

-(a-a*sin(f*x+e))^(1/2)*(c+c*sin(f*x+e))^(1/2)/x-f*cos(e)*sec(f*x+e)*Si(f*x)*(a-a*sin(f*x+e))^(1/2)*(c+c*sin(f
*x+e))^(1/2)-f*Ci(f*x)*sec(f*x+e)*sin(e)*(a-a*sin(f*x+e))^(1/2)*(c+c*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {4700, 3378, 3384, 3380, 3383} \[ \int \frac {\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{x^2} \, dx=-f \sin (e) \operatorname {CosIntegral}(f x) \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}-f \cos (e) \text {Si}(f x) \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}-\frac {\sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}{x} \]

[In]

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

[Out]

-((Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/x) - f*CosIntegral[f*x]*Sec[e + f*x]*Sin[e]*Sqrt[a - a*S
in[e + f*x]]*Sqrt[c + c*Sin[e + f*x]] - f*Cos[e]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]
]*SinIntegral[f*x]

Rule 3378

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

Rule 3380

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 3383

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 3384

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 4700

Int[((g_.) + (h_.)*(x_))^(p_.)*((a_) + (b_.)*Sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*Sin[(e_.) + (f_.)*(x_
)])^(n_), x_Symbol] :> Dist[a^IntPart[m]*c^IntPart[m]*(a + b*Sin[e + f*x])^FracPart[m]*((c + d*Sin[e + f*x])^F
racPart[m]/Cos[e + f*x]^(2*FracPart[m])), Int[(g + h*x)^p*Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x],
x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p] && IntegerQ
[2*m] && IGeQ[n - m, 0]

Rubi steps \begin{align*} \text {integral}& = \left (\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}\right ) \int \frac {\cos (e+f x)}{x^2} \, dx \\ & = -\frac {\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{x}-\left (f \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}\right ) \int \frac {\sin (e+f x)}{x} \, dx \\ & = -\frac {\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{x}-\left (f \cos (e) \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}\right ) \int \frac {\sin (f x)}{x} \, dx-\left (f \sec (e+f x) \sin (e) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}\right ) \int \frac {\cos (f x)}{x} \, dx \\ & = -\frac {\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{x}-f \operatorname {CosIntegral}(f x) \sec (e+f x) \sin (e) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}-f \cos (e) \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \text {Si}(f x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.92 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{x^2} \, dx=-\frac {\sec (e+f x) \sqrt {c (1+\sin (e+f x))} \sqrt {a-a \sin (e+f x)} (\cos (e+f x)+f x \operatorname {CosIntegral}(f x) \sin (e)+f x \cos (e) \text {Si}(f x))}{x} \]

[In]

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

[Out]

-((Sec[e + f*x]*Sqrt[c*(1 + Sin[e + f*x])]*Sqrt[a - a*Sin[e + f*x]]*(Cos[e + f*x] + f*x*CosIntegral[f*x]*Sin[e
] + f*x*Cos[e]*SinIntegral[f*x]))/x)

Maple [F]

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

[In]

int((a-sin(f*x+e)*a)^(1/2)*(c+c*sin(f*x+e))^(1/2)/x^2,x)

[Out]

int((a-sin(f*x+e)*a)^(1/2)*(c+c*sin(f*x+e))^(1/2)/x^2,x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a-a*sin(f*x+e))^(1/2)*(c+c*sin(f*x+e))^(1/2)/x^2,x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

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

[In]

integrate((a-a*sin(f*x+e))**(1/2)*(c+c*sin(f*x+e))**(1/2)/x**2,x)

[Out]

Integral(sqrt(c*(sin(e + f*x) + 1))*sqrt(-a*(sin(e + f*x) - 1))/x**2, x)

Maxima [F]

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

[In]

integrate((a-a*sin(f*x+e))^(1/2)*(c+c*sin(f*x+e))^(1/2)/x^2,x, algorithm="maxima")

[Out]

integrate(sqrt(-a*sin(f*x + e) + a)*sqrt(c*sin(f*x + e) + c)/x^2, x)

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.41 (sec) , antiderivative size = 886, normalized size of antiderivative = 7.20 \[ \int \frac {\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{x^2} \, dx=\text {Too large to display} \]

[In]

integrate((a-a*sin(f*x+e))^(1/2)*(c+c*sin(f*x+e))^(1/2)/x^2,x, algorithm="giac")

[Out]

-1/2*(f*x*imag_part(cos_integral(f*x))*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))
*tan(1/2*f*x)^2*tan(1/2*e)^2 - f*x*imag_part(cos_integral(-f*x))*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-
1/4*pi + 1/2*f*x + 1/2*e))*tan(1/2*f*x)^2*tan(1/2*e)^2 + 2*f*x*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_integral(f*x)*tan(1/2*f*x)^2*tan(1/2*e)^2 - 2*f*x*real_part(cos_integral(f*x))*sg
n(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*tan(1/2*f*x)^2*tan(1/2*e) - 2*f*x*real_p
art(cos_integral(-f*x))*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*tan(1/2*f*x)^2
*tan(1/2*e) - f*x*imag_part(cos_integral(f*x))*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x +
1/2*e))*tan(1/2*f*x)^2 + f*x*imag_part(cos_integral(-f*x))*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*p
i + 1/2*f*x + 1/2*e))*tan(1/2*f*x)^2 - 2*f*x*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_integral(f*x)*tan(1/2*f*x)^2 + f*x*imag_part(cos_integral(f*x))*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))
*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*tan(1/2*e)^2 - f*x*imag_part(cos_integral(-f*x))*sgn(cos(-1/4*pi + 1/2*f*
x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*tan(1/2*e)^2 + 2*f*x*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(s
in(-1/4*pi + 1/2*f*x + 1/2*e))*sin_integral(f*x)*tan(1/2*e)^2 - 2*f*x*real_part(cos_integral(f*x))*sgn(cos(-1/
4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*tan(1/2*e) - 2*f*x*real_part(cos_integral(-f*x))*
sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*tan(1/2*e) - 2*sgn(cos(-1/4*pi + 1/2*f
*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*tan(1/2*f*x)^2*tan(1/2*e)^2 - f*x*imag_part(cos_integral(f*x)
)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + f*x*imag_part(cos_integral(-f*x))*
sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 2*f*x*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_integral(f*x) + 2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1
/4*pi + 1/2*f*x + 1/2*e))*tan(1/2*f*x)^2 + 8*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1
/2*e))*tan(1/2*f*x)*tan(1/2*e) + 2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*tan
(1/2*e)^2 - 2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*sqrt(a)*sqrt(c)/(x*tan(
1/2*f*x)^2*tan(1/2*e)^2 + x*tan(1/2*f*x)^2 + x*tan(1/2*e)^2 + x)

Mupad [F(-1)]

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

[In]

int(((a - a*sin(e + f*x))^(1/2)*(c + c*sin(e + f*x))^(1/2))/x^2,x)

[Out]

int(((a - a*sin(e + f*x))^(1/2)*(c + c*sin(e + f*x))^(1/2))/x^2, x)