3.2.32 \(\int \frac {(a+a \sin (e+f x))^{3/2}}{x^2} \, dx\) [132]

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

[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 = 0.18, 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 = {3400, 3394, 3384, 3380, 3383} \begin {gather*} -\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} \end {gather*}

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 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 3394

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 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^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] Result contains complex when optimal does not.
time = 0.54, size = 226, normalized size = 0.86 \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 (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+\frac {3 i f x}{2}} f x \text {Ei}\left (-\frac {1}{2} i f x\right )+3 i e^{2 i e+\frac {3 i f x}{2}} f x \text {Ei}\left (\frac {i f x}{2}\right )+3 i e^{\frac {3 i f x}{2}} f x \text {Ei}\left (-\frac {3}{2} i f x\right )+3 e^{\frac {3}{2} i (2 e+f x)} f x \text {Ei}\left (\frac {3 i f x}{2}\right )\right )}{4 \sqrt {2} \left (i+e^{i (e+f x)}\right )^3 x} \end {gather*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((a*sin(f*x + e) + a)^(3/2)/x^2, 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^2,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^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (211) = 422\).
time = 4.00, size = 541, normalized size = 2.06 \begin {gather*} \frac {\sqrt {2} {\left (3 \, \pi a f^{2} \operatorname {Ci}\left (\frac {3}{2} \, f x\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (\frac {3}{4} \, \pi - \frac {3}{2} \, e\right ) - 3 \, {\left (\pi - 2 \, f x - 2 \, e\right )} a f^{2} \operatorname {Ci}\left (\frac {3}{2} \, f x\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (\frac {3}{4} \, \pi - \frac {3}{2} \, e\right ) - 6 \, a f^{2} \operatorname {Ci}\left (\frac {3}{2} \, f x\right ) e \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (\frac {3}{4} \, \pi - \frac {3}{2} \, e\right ) + 3 \, \pi a f^{2} \operatorname {Ci}\left (\frac {1}{2} \, f x\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (\frac {1}{4} \, \pi - \frac {1}{2} \, e\right ) - 3 \, {\left (\pi - 2 \, f x - 2 \, e\right )} a f^{2} \operatorname {Ci}\left (\frac {1}{2} \, f x\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (\frac {1}{4} \, \pi - \frac {1}{2} \, e\right ) - 6 \, a f^{2} \operatorname {Ci}\left (\frac {1}{2} \, f x\right ) e \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (\frac {1}{4} \, \pi - \frac {1}{2} \, e\right ) - 3 \, \pi a f^{2} \cos \left (\frac {3}{4} \, \pi - \frac {3}{2} \, e\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \operatorname {Si}\left (\frac {3}{2} \, f x\right ) + 3 \, {\left (\pi - 2 \, f x - 2 \, e\right )} a f^{2} \cos \left (\frac {3}{4} \, \pi - \frac {3}{2} \, e\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \operatorname {Si}\left (\frac {3}{2} \, f x\right ) + 6 \, a f^{2} \cos \left (\frac {3}{4} \, \pi - \frac {3}{2} \, e\right ) e \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \operatorname {Si}\left (\frac {3}{2} \, f x\right ) - 3 \, \pi a f^{2} \cos \left (\frac {1}{4} \, \pi - \frac {1}{2} \, e\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \operatorname {Si}\left (\frac {1}{2} \, f x\right ) + 3 \, {\left (\pi - 2 \, f x - 2 \, e\right )} a f^{2} \cos \left (\frac {1}{4} \, \pi - \frac {1}{2} \, e\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \operatorname {Si}\left (\frac {1}{2} \, f x\right ) + 6 \, a f^{2} \cos \left (\frac {1}{4} \, \pi - \frac {1}{2} \, e\right ) e \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \operatorname {Si}\left (\frac {1}{2} \, f x\right ) - 12 \, a f^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 4 \, a f^{2} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a}}{8 \, f^{2} x} \end {gather*}

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]

1/8*sqrt(2)*(3*pi*a*f^2*cos_integral(3/2*f*x)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(3/4*pi - 3/2*e) - 3*(pi
- 2*f*x - 2*e)*a*f^2*cos_integral(3/2*f*x)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(3/4*pi - 3/2*e) - 6*a*f^2*c
os_integral(3/2*f*x)*e*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(3/4*pi - 3/2*e) + 3*pi*a*f^2*cos_integral(1/2*f
*x)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(1/4*pi - 1/2*e) - 3*(pi - 2*f*x - 2*e)*a*f^2*cos_integral(1/2*f*x)
*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(1/4*pi - 1/2*e) - 6*a*f^2*cos_integral(1/2*f*x)*e*sgn(cos(-1/4*pi + 1
/2*f*x + 1/2*e))*sin(1/4*pi - 1/2*e) - 3*pi*a*f^2*cos(3/4*pi - 3/2*e)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin_
integral(3/2*f*x) + 3*(pi - 2*f*x - 2*e)*a*f^2*cos(3/4*pi - 3/2*e)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin_int
egral(3/2*f*x) + 6*a*f^2*cos(3/4*pi - 3/2*e)*e*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin_integral(3/2*f*x) - 3*p
i*a*f^2*cos(1/4*pi - 1/2*e)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin_integral(1/2*f*x) + 3*(pi - 2*f*x - 2*e)*a
*f^2*cos(1/4*pi - 1/2*e)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin_integral(1/2*f*x) + 6*a*f^2*cos(1/4*pi - 1/2*
e)*e*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin_integral(1/2*f*x) - 12*a*f^2*cos(-1/4*pi + 1/2*f*x + 1/2*e)*sgn(c
os(-1/4*pi + 1/2*f*x + 1/2*e)) - 4*a*f^2*cos(-3/4*pi + 3/2*f*x + 3/2*e)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*s
qrt(a)/(f^2*x)

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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^2} \,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^2,x)

[Out]

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

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