3.2.28 \(\int x^3 (a+a \sin (e+f x))^{3/2} \, dx\) [128]

Optimal. Leaf size=337 \[ -\frac {1280 a \sqrt {a+a \sin (e+f x)}}{9 f^4}+\frac {16 a x^2 \sqrt {a+a \sin (e+f x)}}{f^2}+\frac {640 a x \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{9 f^3}-\frac {8 a x^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {32 a x \cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{9 f^3}-\frac {4 a x^3 \cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f}-\frac {64 a \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{27 f^4}+\frac {8 a x^2 \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f^2} \]

[Out]

-1280/9*a*(a+a*sin(f*x+e))^(1/2)/f^4+16*a*x^2*(a+a*sin(f*x+e))^(1/2)/f^2+640/9*a*x*cot(1/2*e+1/4*Pi+1/2*f*x)*(
a+a*sin(f*x+e))^(1/2)/f^3-8/3*a*x^3*cot(1/2*e+1/4*Pi+1/2*f*x)*(a+a*sin(f*x+e))^(1/2)/f+32/9*a*x*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)/f^3-4/3*a*x^3*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)/f-64/27*a*sin(1/2*e+1/4*Pi+1/2*f*x)^2*(a+a*sin(f*x+e))^(1/2)/f^4+8/3*a
*x^2*sin(1/2*e+1/4*Pi+1/2*f*x)^2*(a+a*sin(f*x+e))^(1/2)/f^2

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Rubi [A]
time = 0.15, antiderivative size = 337, 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, 3392, 3377, 2718, 3391} \begin {gather*} -\frac {64 a \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}}{27 f^4}-\frac {1280 a \sqrt {a \sin (e+f x)+a}}{9 f^4}+\frac {32 a x \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}}{9 f^3}+\frac {640 a x \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}}{9 f^3}+\frac {8 a x^2 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}}{3 f^2}+\frac {16 a x^2 \sqrt {a \sin (e+f x)+a}}{f^2}-\frac {4 a x^3 \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}}{3 f}-\frac {8 a x^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a}}{3 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + a*Sin[e + f*x])^(3/2),x]

[Out]

(-1280*a*Sqrt[a + a*Sin[e + f*x]])/(9*f^4) + (16*a*x^2*Sqrt[a + a*Sin[e + f*x]])/f^2 + (640*a*x*Cot[e/2 + Pi/4
+ (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(9*f^3) - (8*a*x^3*Cot[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(
3*f) + (32*a*x*Cos[e/2 + Pi/4 + (f*x)/2]*Sin[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(9*f^3) - (4*a*x^
3*Cos[e/2 + Pi/4 + (f*x)/2]*Sin[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(3*f) - (64*a*Sin[e/2 + Pi/4 +
(f*x)/2]^2*Sqrt[a + a*Sin[e + f*x]])/(27*f^4) + (8*a*x^2*Sin[e/2 + Pi/4 + (f*x)/2]^2*Sqrt[a + a*Sin[e + f*x]]
)/(3*f^2)

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3391

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^
2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[b*(c + d*x)*Cos[e + f*x
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rule 3392

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*m*(c + d*x)^(m - 1)*((
b*Sin[e + f*x])^n/(f^2*n^2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[d^2*m*((m - 1)/(f^2*n^2)), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[b*(c + d*x)^m*Cos[e + f
*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[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 x^3 (a+a \sin (e+f x))^{3/2} \, 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 x^3 \sin ^3\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx\\ &=-\frac {4 a x^3 \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)}}{3 f}+\frac {8 a x^2 \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f^2}+\frac {1}{3} \left (4 a \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}\right ) \int x^3 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx-\frac {\left (16 a \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}\right ) \int x \sin ^3\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{3 f^2}\\ &=-\frac {8 a x^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {32 a x \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)}}{9 f^3}-\frac {4 a x^3 \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)}}{3 f}-\frac {64 a \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{27 f^4}+\frac {8 a x^2 \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f^2}-\frac {\left (32 a \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}\right ) \int x \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{9 f^2}+\frac {\left (8 a \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}\right ) \int x^2 \cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{f}\\ &=\frac {16 a x^2 \sqrt {a+a \sin (e+f x)}}{f^2}+\frac {64 a x \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{9 f^3}-\frac {8 a x^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {32 a x \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)}}{9 f^3}-\frac {4 a x^3 \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)}}{3 f}-\frac {64 a \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{27 f^4}+\frac {8 a x^2 \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f^2}-\frac {\left (64 a \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}\right ) \int \cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{9 f^3}-\frac {\left (32 a \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}\right ) \int x \cos \left (\frac {e}{2}-\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{f^2}\\ &=-\frac {128 a \sqrt {a+a \sin (e+f x)}}{9 f^4}+\frac {16 a x^2 \sqrt {a+a \sin (e+f x)}}{f^2}+\frac {640 a x \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{9 f^3}-\frac {8 a x^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {32 a x \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)}}{9 f^3}-\frac {4 a x^3 \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)}}{3 f}-\frac {64 a \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{27 f^4}+\frac {8 a x^2 \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f^2}-\frac {\left (64 a \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}\right ) \int \cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{f^3}\\ &=-\frac {1280 a \sqrt {a+a \sin (e+f x)}}{9 f^4}+\frac {16 a x^2 \sqrt {a+a \sin (e+f x)}}{f^2}+\frac {640 a x \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{9 f^3}-\frac {8 a x^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {32 a x \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)}}{9 f^3}-\frac {4 a x^3 \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)}}{3 f}-\frac {64 a \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{27 f^4}+\frac {8 a x^2 \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f^2}\\ \end {align*}

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Mathematica [A]
time = 0.84, size = 231, normalized size = 0.69 \begin {gather*} \frac {2 a \left (-\frac {2 \left (\left (968-480 f x-117 f^2 x^2+18 f^3 x^3\right ) \cos \left (\frac {e}{2}\right )+\left (968+480 f x-117 f^2 x^2-18 f^3 x^3\right ) \sin \left (\frac {e}{2}\right )\right )}{\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )}-\cos (f x) \left (3 f x \left (-8+3 f^2 x^2\right ) \cos (e)+2 \left (8-9 f^2 x^2\right ) \sin (e)\right )+\left (2 \left (-8+9 f^2 x^2\right ) \cos (e)+3 f x \left (-8+3 f^2 x^2\right ) \sin (e)\right ) \sin (f x)+\frac {24 f x \left (-80+3 f^2 x^2\right ) \sin \left (\frac {f x}{2}\right )}{\left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right ) \sqrt {a (1+\sin (e+f x))}}{27 f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*((-2*((968 - 480*f*x - 117*f^2*x^2 + 18*f^3*x^3)*Cos[e/2] + (968 + 480*f*x - 117*f^2*x^2 - 18*f^3*x^3)*Si
n[e/2]))/(Cos[e/2] + Sin[e/2]) - Cos[f*x]*(3*f*x*(-8 + 3*f^2*x^2)*Cos[e] + 2*(8 - 9*f^2*x^2)*Sin[e]) + (2*(-8
+ 9*f^2*x^2)*Cos[e] + 3*f*x*(-8 + 3*f^2*x^2)*Sin[e])*Sin[f*x] + (24*f*x*(-80 + 3*f^2*x^2)*Sin[(f*x)/2])/((Cos[
e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))*Sqrt[a*(1 + Sin[e + f*x])])/(27*f^4)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3*(a+a*sin(f*x+e))^(3/2),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(x^3*(a+a*sin(f*x+e))^(3/2),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(x^3*(a+a*sin(f*x+e))^(3/2),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 x^{3} \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1210 vs. \(2 (275) = 550\).
time = 3.07, size = 1210, normalized size = 3.59 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/216*sqrt(2)*(972*(4*a*cos(-1/4*pi + 1/2*f*x + 1/2*e)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))/f^3 + (pi*a*sgn(cos
(-1/4*pi + 1/2*f*x + 1/2*e)) - (pi - 2*f*x - 2*e)*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-1/4*pi + 1/2*f*x
+ 1/2*e)/f^3)*e^2 + 36*(4*a*cos(-3/4*pi + 3/2*f*x + 3/2*e)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))/f^3 + 3*(pi*a*
sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - (pi - 2*f*x - 2*e)*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-3/4*pi +
3/2*f*x + 3/2*e)/f^3)*e^2 - 486*(8*(pi*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - (pi - 2*f*x - 2*e)*a*sgn(cos(-1
/4*pi + 1/2*f*x + 1/2*e)))*cos(-1/4*pi + 1/2*f*x + 1/2*e)/f^3 + (pi^2*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) -
2*pi*(pi - 2*f*x - 2*e)*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + (pi - 2*f*x - 2*e)^2*a*sgn(cos(-1/4*pi + 1/2*f
*x + 1/2*e)) - 32*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-1/4*pi + 1/2*f*x + 1/2*e)/f^3)*e - 6*(24*(pi*a*s
gn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - (pi - 2*f*x - 2*e)*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-3/4*pi + 3
/2*f*x + 3/2*e)/f^3 + (9*pi^2*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 18*pi*(pi - 2*f*x - 2*e)*a*sgn(cos(-1/4*
pi + 1/2*f*x + 1/2*e)) + 9*(pi - 2*f*x - 2*e)^2*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 32*a*sgn(cos(-1/4*pi +
1/2*f*x + 1/2*e)))*sin(-3/4*pi + 3/2*f*x + 3/2*e)/f^3)*e - 648*a*e^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(
-1/4*pi + 1/2*f*x + 1/2*e)/f^3 - 72*a*e^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-3/4*pi + 3/2*f*x + 3/2*e)/f
^3 + 972*(pi^2*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 2*pi*(pi - 2*f*x - 2*e)*a*sgn(cos(-1/4*pi + 1/2*f*x + 1
/2*e)) + (pi - 2*f*x - 2*e)^2*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 32*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))
)*cos(-1/4*pi + 1/2*f*x + 1/2*e)/f^3 + 4*(9*pi^2*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 18*pi*(pi - 2*f*x - 2
*e)*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 9*(pi - 2*f*x - 2*e)^2*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 32*
a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-3/4*pi + 3/2*f*x + 3/2*e)/f^3 + 81*(pi^3*a*sgn(cos(-1/4*pi + 1/2*f
*x + 1/2*e)) - 3*pi^2*(pi - 2*f*x - 2*e)*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*pi*(pi - 2*f*x - 2*e)^2*a*s
gn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - (pi - 2*f*x - 2*e)^3*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 96*pi*a*sgn(
cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 96*(pi - 2*f*x - 2*e)*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-1/4*pi + 1
/2*f*x + 1/2*e)/f^3 + 3*(3*pi^3*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 9*pi^2*(pi - 2*f*x - 2*e)*a*sgn(cos(-1
/4*pi + 1/2*f*x + 1/2*e)) + 9*pi*(pi - 2*f*x - 2*e)^2*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 3*(pi - 2*f*x -
2*e)^3*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 32*pi*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 32*(pi - 2*f*x -
2*e)*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-3/4*pi + 3/2*f*x + 3/2*e)/f^3)*sqrt(a)/f

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + a*sin(e + f*x))^(3/2),x)

[Out]

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

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