3.176 \(\int \frac{\sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x^3} \, dx\)

Optimal. Leaf size=385 \[ -c f^2 \sin (2 e) \text{CosIntegral}(2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-\frac{1}{2} c f^2 \cos (e) \text{CosIntegral}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}+\frac{1}{2} c f^2 \sin (e) \text{Si}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-c f^2 \cos (2 e) \text{Si}(2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-\frac{c \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{2 x^2}-\frac{c \sin (2 e+2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{4 x^2}+\frac{c f \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{2 x}-\frac{c f \cos (2 e+2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{2 x} \]

[Out]

-(c*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(2*x^2) - (c*f*Cos[2*e + 2*f*x]*Sec[e + f*x]*Sqrt[a - a
*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(2*x) - (c*f^2*Cos[e]*CosIntegral[f*x]*Sec[e + f*x]*Sqrt[a - a*Sin[e
+ f*x]]*Sqrt[c + c*Sin[e + f*x]])/2 - c*f^2*CosIntegral[2*f*x]*Sec[e + f*x]*Sin[2*e]*Sqrt[a - a*Sin[e + f*x]]*
Sqrt[c + c*Sin[e + f*x]] - (c*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Sin[2*e + 2*f*x])
/(4*x^2) + (c*f^2*Sec[e + f*x]*Sin[e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[f*x])/2 -
c*f^2*Cos[2*e]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[2*f*x] + (c*f*Sqrt[a
- a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/(2*x)

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Rubi [A]  time = 0.741289, antiderivative size = 385, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {4604, 6741, 12, 6742, 3297, 3303, 3299, 3302} \[ -c f^2 \sin (2 e) \text{CosIntegral}(2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-\frac{1}{2} c f^2 \cos (e) \text{CosIntegral}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}+\frac{1}{2} c f^2 \sin (e) \text{Si}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-c f^2 \cos (2 e) \text{Si}(2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-\frac{c \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{2 x^2}-\frac{c \sin (2 e+2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{4 x^2}+\frac{c f \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{2 x}-\frac{c f \cos (2 e+2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(2*x^2) - (c*f*Cos[2*e + 2*f*x]*Sec[e + f*x]*Sqrt[a - a
*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(2*x) - (c*f^2*Cos[e]*CosIntegral[f*x]*Sec[e + f*x]*Sqrt[a - a*Sin[e
+ f*x]]*Sqrt[c + c*Sin[e + f*x]])/2 - c*f^2*CosIntegral[2*f*x]*Sec[e + f*x]*Sin[2*e]*Sqrt[a - a*Sin[e + f*x]]*
Sqrt[c + c*Sin[e + f*x]] - (c*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Sin[2*e + 2*f*x])
/(4*x^2) + (c*f^2*Sec[e + f*x]*Sin[e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[f*x])/2 -
c*f^2*Cos[2*e]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[2*f*x] + (c*f*Sqrt[a
- a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/(2*x)

Rule 4604

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]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 3297

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 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 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]

Rubi steps

\begin{align*} \int \frac{\sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x^3} \, dx &=\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) (c+c \sin (e+f x))}{x^3} \, dx\\ &=\left (\sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{c \cos (e+f x) (1+\sin (e+f x))}{x^3} \, dx\\ &=\left (c \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) (1+\sin (e+f x))}{x^3} \, dx\\ &=\left (c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \left (\frac{\cos (e+f x)}{x^3}+\frac{\sin (2 e+2 f x)}{2 x^3}\right ) \, dx\\ &=\frac{1}{2} \left (c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\sin (2 e+2 f x)}{x^3} \, dx+\left (c \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^3} \, dx\\ &=-\frac{c \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{2 x^2}-\frac{c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \sin (2 e+2 f x)}{4 x^2}+\frac{1}{2} \left (c f \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\cos (2 e+2 f x)}{x^2} \, dx-\frac{1}{2} \left (c 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^2} \, dx\\ &=-\frac{c \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{2 x^2}-\frac{c f \cos (2 e+2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{2 x}-\frac{c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \sin (2 e+2 f x)}{4 x^2}+\frac{c f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \tan (e+f x)}{2 x}-\frac{1}{2} \left (c f^2 \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} \, dx-\left (c f^2 \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\sin (2 e+2 f x)}{x} \, dx\\ &=-\frac{c \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{2 x^2}-\frac{c f \cos (2 e+2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{2 x}-\frac{c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \sin (2 e+2 f x)}{4 x^2}+\frac{c f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \tan (e+f x)}{2 x}-\frac{1}{2} \left (c f^2 \cos (e) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\cos (f x)}{x} \, dx-\left (c f^2 \cos (2 e) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\sin (2 f x)}{x} \, dx+\frac{1}{2} \left (c f^2 \sec (e+f x) \sin (e) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\sin (f x)}{x} \, dx-\left (c f^2 \sec (e+f x) \sin (2 e) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\cos (2 f x)}{x} \, dx\\ &=-\frac{c \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{2 x^2}-\frac{c f \cos (2 e+2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{2 x}-\frac{1}{2} c f^2 \cos (e) \text{Ci}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}-c f^2 \text{Ci}(2 f x) \sec (e+f x) \sin (2 e) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}-\frac{c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \sin (2 e+2 f x)}{4 x^2}+\frac{1}{2} c f^2 \sec (e+f x) \sin (e) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \text{Si}(f x)-c f^2 \cos (2 e) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \text{Si}(2 f x)+\frac{c f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \tan (e+f x)}{2 x}\\ \end{align*}

Mathematica [C]  time = 1.81722, size = 317, normalized size = 0.82 \[ \frac{c^2 e^{-2 i (e+f x)} \left (e^{i (e+f x)}+i\right ) \left (2 i f^2 x^2 e^{i (e+2 f x)} \text{ExpIntegralEi}(-i f x)+2 i f^2 x^2 e^{3 i e+2 i f x} \text{ExpIntegralEi}(i f x)+4 f^2 x^2 e^{2 i (2 e+f x)} \text{ExpIntegralEi}(2 i f x)+2 f x e^{i (e+f x)}-2 f x e^{3 i (e+f x)}+2 i f x e^{4 i (e+f x)}+2 i e^{i (e+f x)}+2 i e^{3 i (e+f x)}+e^{4 i (e+f x)}-4 f^2 x^2 e^{2 i f x} \text{ExpIntegralEi}(-2 i f x)+2 i f x-1\right ) \sqrt{a-a \sin (e+f x)}}{4 \sqrt{2} x^2 \left (e^{i (e+f x)}-i\right ) \sqrt{-i c e^{-i (e+f x)} \left (e^{i (e+f x)}+i\right )^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(I + E^(I*(e + f*x)))*(-1 + (2*I)*E^(I*(e + f*x)) + (2*I)*E^((3*I)*(e + f*x)) + E^((4*I)*(e + f*x)) + (2*
I)*f*x + 2*E^(I*(e + f*x))*f*x - 2*E^((3*I)*(e + f*x))*f*x + (2*I)*E^((4*I)*(e + f*x))*f*x + (2*I)*E^(I*(e + 2
*f*x))*f^2*x^2*ExpIntegralEi[(-I)*f*x] + (2*I)*E^((3*I)*e + (2*I)*f*x)*f^2*x^2*ExpIntegralEi[I*f*x] - 4*E^((2*
I)*f*x)*f^2*x^2*ExpIntegralEi[(-2*I)*f*x] + 4*E^((2*I)*(2*e + f*x))*f^2*x^2*ExpIntegralEi[(2*I)*f*x])*Sqrt[a -
a*Sin[e + f*x]])/(4*Sqrt[2]*E^((2*I)*(e + f*x))*(-I + E^(I*(e + f*x)))*Sqrt[((-I)*c*(I + E^(I*(e + f*x)))^2)/
E^(I*(e + f*x))]*x^2)

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Maple [F]  time = 0.084, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( c+c\sin \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}\sqrt{a-a\sin \left ( fx+e \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+c*sin(f*x+e))^(3/2)*(a-a*sin(f*x+e))^(1/2)/x^3,x)

[Out]

int((c+c*sin(f*x+e))^(3/2)*(a-a*sin(f*x+e))^(1/2)/x^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a \sin \left (f x + e\right ) + a}{\left (c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+c*sin(f*x+e))^(3/2)*(a-a*sin(f*x+e))^(1/2)/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+c*sin(f*x+e))^(3/2)*(a-a*sin(f*x+e))^(1/2)/x^3,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+c*sin(f*x+e))**(3/2)*(a-a*sin(f*x+e))**(1/2)/x**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a \sin \left (f x + e\right ) + a}{\left (c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+c*sin(f*x+e))^(3/2)*(a-a*sin(f*x+e))^(1/2)/x^3,x, algorithm="giac")

[Out]

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