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

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

Optimal. Leaf size=263 \[ -\frac {3}{4} a f \sin \left (\frac {1}{4} (2 e-\pi )\right ) \text {Ci}\left (\frac {f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}+\frac {3}{4} a f \sin \left (\frac {1}{4} (6 e+\pi )\right ) \text {Ci}\left (\frac {3 f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}-\frac {3}{4} a f \sin \left (\frac {1}{4} (2 e+\pi )\right ) \text {Si}\left (\frac {f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}+\frac {3}{4} a f \cos \left (\frac {1}{4} (6 e+\pi )\right ) \text {Si}\left (\frac {3 f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}-\frac {2 a \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}}{x} \]

[Out]

3/4*a*f*cos(3/2*e+1/4*Pi)*csc(1/2*e+1/4*Pi+1/2*f*x)*Si(3/2*f*x)*(a+a*sin(f*x+e))^(1/2)+3/4*a*f*Ci(1/2*f*x)*csc

(1/2*e+1/4*Pi+1/2*f*x)*cos(1/2*e+1/4*Pi)*(a+a*sin(f*x+e))^(1/2)-3/4*a*f*csc(1/2*e+1/4*Pi+1/2*f*x)*Si(1/2*f*x)*

sin(1/2*e+1/4*Pi)*(a+a*sin(f*x+e))^(1/2)+3/4*a*f*Ci(3/2*f*x)*csc(1/2*e+1/4*Pi+1/2*f*x)*sin(3/2*e+1/4*Pi)*(a+a*

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

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Rubi [A] time = 0.30, antiderivative size = 263, 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, 3313, 3303, 3299, 3302} \[ -\frac {3}{4} a f \sin \left (\frac {1}{4} (2 e-\pi )\right ) \text {CosIntegral}\left (\frac {f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}+\frac {3}{4} a f \sin \left (\frac {1}{4} (6 e+\pi )\right ) \text {CosIntegral}\left (\frac {3 f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}-\frac {3}{4} a f \sin \left (\frac {1}{4} (2 e+\pi )\right ) \text {Si}\left (\frac {f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}+\frac {3}{4} a f \cos \left (\frac {1}{4} (6 e+\pi )\right ) \text {Si}\left (\frac {3 f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}-\frac {2 a \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}}{x} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

CosIntegral[(3*f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]*Sin[(6*e + Pi)/4]*Sqrt[a + a*Sin[e + f*x]])/4 - (2*a*Sin[e/2

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

a*Sin[e + f*x]]*SinIntegral[(f*x)/2])/4 + (3*a*f*Cos[(6*e + Pi)/4]*Csc[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e

+ f*x]]*SinIntegral[(3*f*x)/2])/4

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 3313

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

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

^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && 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^2} \, 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^2} \, dx\\ &=-\frac {2 a \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{x}+\left (3 a f \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}\right ) \int \left (\frac {\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{4 x}+\frac {\cos \left (\frac {3 e}{2}-\frac {\pi }{4}+\frac {3 f x}{2}\right )}{4 x}\right ) \, dx\\ &=-\frac {2 a \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{x}+\frac {1}{4} \left (3 a f \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 {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{x} \, dx+\frac {1}{4} \left (3 a f \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 e}{2}-\frac {\pi }{4}+\frac {3 f x}{2}\right )}{x} \, dx\\ &=-\frac {2 a \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{x}+\frac {1}{4} \left (3 a f \cos \left (\frac {1}{4} (6 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 {3 f x}{2}\right )}{x} \, dx-\frac {1}{4} \left (3 a f \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}{4} \left (3 a f \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 {\sin \left (\frac {f x}{2}\right )}{x} \, dx+\frac {1}{4} \left (3 a f \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {1}{4} (6 e+\pi )\right ) \sqrt {a+a \sin (e+f x)}\right ) \int \frac {\cos \left (\frac {3 f x}{2}\right )}{x} \, dx\\ &=-\frac {3}{4} a f \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}{4} a f \text {Ci}\left (\frac {3 f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {1}{4} (6 e+\pi )\right ) \sqrt {a+a \sin (e+f x)}-\frac {2 a \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{x}-\frac {3}{4} a f \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)} \text {Si}\left (\frac {f x}{2}\right )+\frac {3}{4} a f \cos \left (\frac {1}{4} (6 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 {3 f x}{2}\right )\\ \end {align*}

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Mathematica [C] time = 0.99, size = 226, normalized size = 0.86 \[ \frac {i \left (-i a e^{-i (e+f x)} \left (e^{i (e+f x)}+i\right )^2\right )^{3/2} \left (3 f x e^{i e+\frac {3 i f x}{2}} \text {Ei}\left (-\frac {1}{2} i f x\right )+3 i f x e^{2 i e+\frac {3 i f x}{2}} \text {Ei}\left (\frac {i f x}{2}\right )+3 f x e^{\frac {3}{2} i (2 e+f x)} \text {Ei}\left (\frac {3 i f x}{2}\right )-6 i e^{i (e+f x)}-6 e^{2 i (e+f x)}+2 i e^{3 i (e+f x)}+3 i f x e^{\frac {3 i f x}{2}} \text {Ei}\left (-\frac {3}{2} i f x\right )+2\right )}{4 \sqrt {2} x \left (e^{i (e+f x)}+i\right )^3} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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

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

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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^2,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^2,x, algorithm="giac")

[Out]

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