3.2.33 \(\int \frac {(a+a \sin (e+f x))^{3/2}}{x^3} \, dx\) [133]

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

[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 = 0.23, 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 = {3400, 3395, 3384, 3380, 3383, 3393} \begin {gather*} -\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} \end {gather*}

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 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 3393

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 3395

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 3400

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] Result contains complex when optimal does not.
time = 0.46, size = 295, normalized size = 0.89 \begin {gather*} -\frac {i \left (-i a e^{-i (e+f x)} \left (i+e^{i (e+f x)}\right )^2\right )^{3/2} \left (-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+\frac {3 i f x}{2}} f^2 x^2 \text {Ei}\left (-\frac {1}{2} i f x\right )+3 e^{2 i e+\frac {3 i f x}{2}} f^2 x^2 \text {Ei}\left (\frac {i f x}{2}\right )-9 e^{\frac {3 i f x}{2}} f^2 x^2 \text {Ei}\left (-\frac {3}{2} i f x\right )-9 i e^{\frac {3}{2} i (2 e+f x)} f^2 x^2 \text {Ei}\left (\frac {3 i f x}{2}\right )\right )}{16 \sqrt {2} \left (i+e^{i (e+f x)}\right )^3 x^2} \end {gather*}

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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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*(sin(e + f*x) + 1))**(3/2)/x**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1344 vs. \(2 (265) = 530\).
time = 2.81, size = 1344, normalized size = 4.05 \begin {gather*} \text {Too large to display} \end {gather*}

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]

-1/16*sqrt(2)*(9*pi^2*a*f^3*cos(3/4*pi - 3/2*e)*cos_integral(3/2*f*x)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 18
*pi*(pi - 2*f*x - 2*e)*a*f^3*cos(3/4*pi - 3/2*e)*cos_integral(3/2*f*x)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 9
*(pi - 2*f*x - 2*e)^2*a*f^3*cos(3/4*pi - 3/2*e)*cos_integral(3/2*f*x)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*
pi^2*a*f^3*cos(1/4*pi - 1/2*e)*cos_integral(1/2*f*x)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 6*pi*(pi - 2*f*x -
2*e)*a*f^3*cos(1/4*pi - 1/2*e)*cos_integral(1/2*f*x)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*(pi - 2*f*x - 2*e
)^2*a*f^3*cos(1/4*pi - 1/2*e)*cos_integral(1/2*f*x)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 36*pi*a*f^3*cos(3/4*
pi - 3/2*e)*cos_integral(3/2*f*x)*e*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 36*(pi - 2*f*x - 2*e)*a*f^3*cos(3/4*
pi - 3/2*e)*cos_integral(3/2*f*x)*e*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 12*pi*a*f^3*cos(1/4*pi - 1/2*e)*cos_
integral(1/2*f*x)*e*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 12*(pi - 2*f*x - 2*e)*a*f^3*cos(1/4*pi - 1/2*e)*cos_
integral(1/2*f*x)*e*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 9*pi^2*a*f^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin
(3/4*pi - 3/2*e)*sin_integral(3/2*f*x) - 18*pi*(pi - 2*f*x - 2*e)*a*f^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*si
n(3/4*pi - 3/2*e)*sin_integral(3/2*f*x) + 9*(pi - 2*f*x - 2*e)^2*a*f^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin
(3/4*pi - 3/2*e)*sin_integral(3/2*f*x) - 36*pi*a*f^3*e*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(3/4*pi - 3/2*e)
*sin_integral(3/2*f*x) + 36*(pi - 2*f*x - 2*e)*a*f^3*e*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(3/4*pi - 3/2*e)
*sin_integral(3/2*f*x) + 3*pi^2*a*f^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(1/4*pi - 1/2*e)*sin_integral(1/2
*f*x) - 6*pi*(pi - 2*f*x - 2*e)*a*f^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(1/4*pi - 1/2*e)*sin_integral(1/2
*f*x) + 3*(pi - 2*f*x - 2*e)^2*a*f^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(1/4*pi - 1/2*e)*sin_integral(1/2*
f*x) - 12*pi*a*f^3*e*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(1/4*pi - 1/2*e)*sin_integral(1/2*f*x) + 12*(pi -
2*f*x - 2*e)*a*f^3*e*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(1/4*pi - 1/2*e)*sin_integral(1/2*f*x) + 36*a*f^3*
cos(3/4*pi - 3/2*e)*cos_integral(3/2*f*x)*e^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 12*a*f^3*cos(1/4*pi - 1/2*
e)*cos_integral(1/2*f*x)*e^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 36*a*f^3*e^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/
2*e))*sin(3/4*pi - 3/2*e)*sin_integral(3/2*f*x) + 12*a*f^3*e^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(1/4*pi
- 1/2*e)*sin_integral(1/2*f*x) - 12*pi*a*f^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e
) + 12*(pi - 2*f*x - 2*e)*a*f^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 24*a*f^3*
e*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e) - 12*pi*a*f^3*sgn(cos(-1/4*pi + 1/2*f*x +
1/2*e))*sin(-3/4*pi + 3/2*f*x + 3/2*e) + 12*(pi - 2*f*x - 2*e)*a*f^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(
-3/4*pi + 3/2*f*x + 3/2*e) + 24*a*f^3*e*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-3/4*pi + 3/2*f*x + 3/2*e) + 4
8*a*f^3*cos(-1/4*pi + 1/2*f*x + 1/2*e)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 16*a*f^3*cos(-3/4*pi + 3/2*f*x +
3/2*e)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sqrt(a)/((pi^2 - 2*pi*(pi - 2*f*x - 2*e) + (pi - 2*f*x - 2*e)^2 -
4*pi*e + 4*(pi - 2*f*x - 2*e)*e + 4*e^2)*f)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(e + f*x))^(3/2)/x^3,x)

[Out]

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

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