∫ (a+a sin(e+fx))3∕2 ____________________________

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

Optimal. Leaf size=332 \[ -\frac {3}{16} a f^2 \sin \left (\frac {1}{4} (2 e+\pi )\right ) \text {Ci}\left (\frac {f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}-\frac {9}{16} a f^2 \cos \left (\frac {3}{4} (2 e-\pi )\right ) \text {Ci}\left (\frac {3 f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}+\frac {9}{16} a f^2 \sin \left (\frac {3}{4} (2 e-\pi )\right ) \text {Si}\left (\frac {3 f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}-\frac {3}{16} a f^2 \cos \left (\frac {1}{4} (2 e+\pi )\right ) \text {Si}\left (\frac {f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}-\frac {a \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}}{x^2}-\frac {3 a f \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}}{2 x} \]

[Out]

9/16*a*f^2*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)-3/16*a*f^2*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)-9/16*a*f^2*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)-3/16*a*f^2*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)-3/2*a*f*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)/x-a*sin(1/2*e+1/4*Pi+1/2*f*x)^2*(a+a*sin(f*x+e))^(1/2)/x^2

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Rubi [A] time = 0.38, antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3319, 3314, 3303, 3299, 3302, 3312} \[ -\frac {3}{16} a f^2 \sin \left (\frac {1}{4} (2 e+\pi )\right ) \text {CosIntegral}\left (\frac {f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}-\frac {9}{16} a f^2 \cos \left (\frac {3}{4} (2 e-\pi )\right ) \text {CosIntegral}\left (\frac {3 f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}+\frac {9}{16} a f^2 \sin \left (\frac {3}{4} (2 e-\pi )\right ) \text {Si}\left (\frac {3 f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}-\frac {3}{16} a f^2 \cos \left (\frac {1}{4} (2 e+\pi )\right ) \text {Si}\left (\frac {f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}-\frac {a \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}}{x^2}-\frac {3 a f \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}}{2 x} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-9*a*f^2*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]])/16

- (3*a*f^2*CosIntegral[(f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]*Sin[(2*e + Pi)/4]*Sqrt[a + a*Sin[e + f*x]])/16 - (3*

a*f*Cos[e/2 + Pi/4 + (f*x)/2]*Sin[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(2*x) - (a*Sin[e/2 + Pi/4 +

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

[e + f*x]]*SinIntegral[(f*x)/2])/16 + (9*a*f^2*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])/16

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 3314

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(b*Si

n[e + f*x])^n)/(d*(m + 1)), x] + (Dist[(b^2*f^2*n*(n - 1))/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin

[e + f*x])^(n - 2), x], x] - Dist[(f^2*n^2)/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x

], x] - Simp[(b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1))/(d^2*(m + 1)*(m + 2)), x]) /; Fre

eQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]

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^3} \, 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^3} \, dx\\ &=-\frac {3 a f \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)}}{2 x}-\frac {a \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{x^2}+\frac {1}{2} \left (3 a f^2 \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}{4} \left (9 a f^2 \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\\ &=-\frac {3 a f \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)}}{2 x}-\frac {a \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{x^2}-\frac {1}{4} \left (9 a f^2 \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 (3 a f^2 \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 f^2 \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 {3}{2} a f^2 \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 a f \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)}}{2 x}-\frac {a \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{x^2}+\frac {3}{2} a f^2 \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}{16} \left (9 a f^2 \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}{16} \left (27 a f^2 \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 {3}{2} a f^2 \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 a f \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)}}{2 x}-\frac {a \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{x^2}+\frac {3}{2} a f^2 \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}{16} \left (9 a f^2 \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}{16} \left (9 a f^2 \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}{16} \left (27 a f^2 \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}{16} \left (27 a f^2 \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 {9}{16} a f^2 \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}{16} a f^2 \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 a f \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)}}{2 x}-\frac {a \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{x^2}-\frac {3}{16} a f^2 \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 {9}{16} a f^2 \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*}

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Mathematica [C] time = 0.91, size = 295, normalized size = 0.89 \[ -\frac {i \left (-i a e^{-i (e+f x)} \left (e^{i (e+f x)}+i\right )^2\right )^{3/2} \left (3 i f^2 x^2 e^{i e+\frac {3 i f x}{2}} \text {Ei}\left (-\frac {1}{2} i f x\right )+3 f^2 x^2 e^{2 i e+\frac {3 i f x}{2}} \text {Ei}\left (\frac {i f x}{2}\right )-9 i f^2 x^2 e^{\frac {3}{2} i (2 e+f x)} \text {Ei}\left (\frac {3 i f x}{2}\right )+6 f x e^{i (e+f x)}+6 i f x e^{2 i (e+f x)}+6 f x e^{3 i (e+f x)}+12 i e^{i (e+f x)}+12 e^{2 i (e+f x)}-4 i e^{3 i (e+f x)}-9 f^2 x^2 e^{\frac {3 i f x}{2}} \text {Ei}\left (-\frac {3}{2} i f x\right )+6 i f x-4\right )}{16 \sqrt {2} x^2 \left (e^{i (e+f x)}+i\right )^3} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-1/16*I)*(((-I)*a*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x)))^(3/2)*(-4 + (12*I)*E^(I*(e + f*x)) + 12*E^((2*I)

*(e + f*x)) - (4*I)*E^((3*I)*(e + f*x)) + (6*I)*f*x + 6*E^(I*(e + f*x))*f*x + (6*I)*E^((2*I)*(e + f*x))*f*x +

6*E^((3*I)*(e + f*x))*f*x + (3*I)*E^(I*e + ((3*I)/2)*f*x)*f^2*x^2*ExpIntegralEi[(-1/2*I)*f*x] + 3*E^((2*I)*e +

((3*I)/2)*f*x)*f^2*x^2*ExpIntegralEi[(I/2)*f*x] - 9*E^(((3*I)/2)*f*x)*f^2*x^2*ExpIntegralEi[((-3*I)/2)*f*x] -

(9*I)*E^(((3*I)/2)*(2*e + f*x))*f^2*x^2*ExpIntegralEi[((3*I)/2)*f*x]))/(Sqrt[2]*(I + E^(I*(e + f*x)))^3*x^2)

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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^3,x, algorithm="fricas")

[Out]

Exception raised: TypeError >> Error detected within library code: integrate: implementation incomplete (ha

s polynomial part)

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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^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/