3.453 \(\int \frac{(c+d \sin (e+f x))^4}{a+a \sin (e+f x)} \, dx\)
Optimal. Leaf size=189 \[ \frac{2 d \left (-16 c^2 d+3 c^3+12 c d^2-4 d^3\right ) \cos (e+f x)}{3 a f}+\frac{d^2 \left (6 c^2-20 c d+9 d^2\right ) \sin (e+f x) \cos (e+f x)}{6 a f}+\frac{d x \left (-12 c^2 d+8 c^3+12 c d^2-3 d^3\right )}{2 a}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}+\frac{d (3 c-4 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a f} \]
[Out]
(d*(8*c^3 - 12*c^2*d + 12*c*d^2 - 3*d^3)*x)/(2*a) + (2*d*(3*c^3 - 16*c^2*d + 12*c*d^2 - 4*d^3)*Cos[e + f*x])/(
3*a*f) + (d^2*(6*c^2 - 20*c*d + 9*d^2)*Cos[e + f*x]*Sin[e + f*x])/(6*a*f) + ((3*c - 4*d)*d*Cos[e + f*x]*(c + d
*Sin[e + f*x])^2)/(3*a*f) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(f*(a + a*Sin[e + f*x]))
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Rubi [A] time = 0.22736, antiderivative size = 189, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used =
{2767, 2753, 2734} \[ \frac{2 d \left (-16 c^2 d+3 c^3+12 c d^2-4 d^3\right ) \cos (e+f x)}{3 a f}+\frac{d^2 \left (6 c^2-20 c d+9 d^2\right ) \sin (e+f x) \cos (e+f x)}{6 a f}+\frac{d x \left (-12 c^2 d+8 c^3+12 c d^2-3 d^3\right )}{2 a}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}+\frac{d (3 c-4 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a f} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*Sin[e + f*x])^4/(a + a*Sin[e + f*x]),x]
[Out]
(d*(8*c^3 - 12*c^2*d + 12*c*d^2 - 3*d^3)*x)/(2*a) + (2*d*(3*c^3 - 16*c^2*d + 12*c*d^2 - 4*d^3)*Cos[e + f*x])/(
3*a*f) + (d^2*(6*c^2 - 20*c*d + 9*d^2)*Cos[e + f*x]*Sin[e + f*x])/(6*a*f) + ((3*c - 4*d)*d*Cos[e + f*x]*(c + d
*Sin[e + f*x])^2)/(3*a*f) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(f*(a + a*Sin[e + f*x]))
Rule 2767
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[((
b*c - a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n - 1))/(a*f*(a + b*Sin[e + f*x])), x] - Dist[d/(a*b), Int[(c +
d*Sin[e + f*x])^(n - 2)*Simp[b*d*(n - 1) - a*c*n + (b*c*(n - 1) - a*d*n)*Sin[e + f*x], x], x], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 1] && (IntegerQ[2
*n] || EqQ[c, 0])
Rule 2753
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[
b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
Rule 2734
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((2*a*c
+ b*d)*x)/2, x] + (-Simp[((b*c + a*d)*Cos[e + f*x])/f, x] - Simp[(b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Rubi steps
\begin{align*} \int \frac{(c+d \sin (e+f x))^4}{a+a \sin (e+f x)} \, dx &=-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a+a \sin (e+f x))}-\frac{d \int (-a (4 c-3 d)+a (3 c-4 d) \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx}{a^2}\\ &=\frac{(3 c-4 d) d \cos (e+f x) (c+d \sin (e+f x))^2}{3 a f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a+a \sin (e+f x))}-\frac{d \int (c+d \sin (e+f x)) \left (-a \left (12 c^2-15 c d+8 d^2\right )+a \left (6 c^2-20 c d+9 d^2\right ) \sin (e+f x)\right ) \, dx}{3 a^2}\\ &=\frac{d \left (8 c^3-12 c^2 d+12 c d^2-3 d^3\right ) x}{2 a}+\frac{2 d \left (3 c^3-16 c^2 d+12 c d^2-4 d^3\right ) \cos (e+f x)}{3 a f}+\frac{d^2 \left (6 c^2-20 c d+9 d^2\right ) \cos (e+f x) \sin (e+f x)}{6 a f}+\frac{(3 c-4 d) d \cos (e+f x) (c+d \sin (e+f x))^2}{3 a f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a+a \sin (e+f x))}\\ \end{align*}
Mathematica [A] time = 0.41104, size = 234, normalized size = 1.24 \[ \frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (-6 d \left (12 c^2 d-8 c^3-12 c d^2+3 d^3\right ) (e+f x) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )-3 d^2 \left (24 c^2-16 c d+7 d^2\right ) \cos (e+f x) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )-3 d^3 (4 c-d) \sin (2 (e+f x)) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )+24 (c-d)^4 \sin \left (\frac{1}{2} (e+f x)\right )+d^4 \cos (3 (e+f x)) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )}{12 a f (\sin (e+f x)+1)} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*Sin[e + f*x])^4/(a + a*Sin[e + f*x]),x]
[Out]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(24*(c - d)^4*Sin[(e + f*x)/2] - 6*d*(-8*c^3 + 12*c^2*d - 12*c*d^2 + 3*
d^3)*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 3*d^2*(24*c^2 - 16*c*d + 7*d^2)*Cos[e + f*x]*(Cos[(e +
f*x)/2] + Sin[(e + f*x)/2]) + d^4*Cos[3*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 3*(4*c - d)*d^3*(Co
s[(e + f*x)/2] + Sin[(e + f*x)/2])*Sin[2*(e + f*x)]))/(12*a*f*(1 + Sin[e + f*x]))
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Maple [B] time = 0.069, size = 673, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c+d*sin(f*x+e))^4/(a+a*sin(f*x+e)),x)
[Out]
4/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^5*c*d^3-1/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e
)^5*d^4-12/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^4*c^2*d^2+8/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/
2*f*x+1/2*e)^4*c*d^3-2/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^4*d^4-24/a/f/(1+tan(1/2*f*x+1/2*e)^2)
^3*tan(1/2*f*x+1/2*e)^2*c^2*d^2+16/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^2*c*d^3-8/a/f/(1+tan(1/2*
f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^2*d^4-4/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)*c*d^3+1/a/f/(1+ta
n(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)*d^4-12/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*c^2*d^2+8/a/f/(1+tan(1/2*f*x+1/
2*e)^2)^3*c*d^3-10/3/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*d^4+8/a/f*d*arctan(tan(1/2*f*x+1/2*e))*c^3-12/a/f*arctan(t
an(1/2*f*x+1/2*e))*c^2*d^2+12/a/f*arctan(tan(1/2*f*x+1/2*e))*c*d^3-3/a/f*arctan(tan(1/2*f*x+1/2*e))*d^4-2/a/f/
(tan(1/2*f*x+1/2*e)+1)*c^4+8/a/f/(tan(1/2*f*x+1/2*e)+1)*c^3*d-12/a/f/(tan(1/2*f*x+1/2*e)+1)*c^2*d^2+8/a/f/(tan
(1/2*f*x+1/2*e)+1)*c*d^3-2/a/f/(tan(1/2*f*x+1/2*e)+1)*d^4
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Maxima [B] time = 1.81284, size = 979, normalized size = 5.18 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*sin(f*x+e))^4/(a+a*sin(f*x+e)),x, algorithm="maxima")
[Out]
-1/3*(d^4*((7*sin(f*x + e)/(cos(f*x + e) + 1) + 39*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 24*sin(f*x + e)^3/(co
s(f*x + e) + 1)^3 + 24*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 9*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 9*sin(f*x
+ e)^6/(cos(f*x + e) + 1)^6 + 16)/(a + a*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a*sin(f*x + e)^2/(cos(f*x + e) +
1)^2 + 3*a*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*a*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 3*a*sin(f*x + e)^5
/(cos(f*x + e) + 1)^5 + a*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + a*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) + 9*arc
tan(sin(f*x + e)/(cos(f*x + e) + 1))/a) - 12*c*d^3*((sin(f*x + e)/(cos(f*x + e) + 1) + 5*sin(f*x + e)^2/(cos(f
*x + e) + 1)^2 + 3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 4)/(a + a*sin
(f*x + e)/(cos(f*x + e) + 1) + 2*a*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2*a*sin(f*x + e)^3/(cos(f*x + e) + 1)
^3 + a*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 3*arctan(sin(f*x + e)/(c
os(f*x + e) + 1))/a) + 36*c^2*d^2*((sin(f*x + e)/(cos(f*x + e) + 1) + sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2)
/(a + a*sin(f*x + e)/(cos(f*x + e) + 1) + a*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a*sin(f*x + e)^3/(cos(f*x +
e) + 1)^3) + arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a) - 24*c^3*d*(arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a
+ 1/(a + a*sin(f*x + e)/(cos(f*x + e) + 1))) + 6*c^4/(a + a*sin(f*x + e)/(cos(f*x + e) + 1)))/f
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Fricas [A] time = 1.68836, size = 799, normalized size = 4.23 \begin{align*} \frac{2 \, d^{4} \cos \left (f x + e\right )^{4} - 6 \, c^{4} + 24 \, c^{3} d - 36 \, c^{2} d^{2} + 24 \, c d^{3} - 6 \, d^{4} +{\left (12 \, c d^{3} - d^{4}\right )} \cos \left (f x + e\right )^{3} + 3 \,{\left (8 \, c^{3} d - 12 \, c^{2} d^{2} + 12 \, c d^{3} - 3 \, d^{4}\right )} f x - 12 \,{\left (3 \, c^{2} d^{2} - 2 \, c d^{3} + d^{4}\right )} \cos \left (f x + e\right )^{2} - 3 \,{\left (2 \, c^{4} - 8 \, c^{3} d + 24 \, c^{2} d^{2} - 12 \, c d^{3} + 5 \, d^{4} -{\left (8 \, c^{3} d - 12 \, c^{2} d^{2} + 12 \, c d^{3} - 3 \, d^{4}\right )} f x\right )} \cos \left (f x + e\right ) +{\left (2 \, d^{4} \cos \left (f x + e\right )^{3} + 6 \, c^{4} - 24 \, c^{3} d + 36 \, c^{2} d^{2} - 24 \, c d^{3} + 6 \, d^{4} + 3 \,{\left (8 \, c^{3} d - 12 \, c^{2} d^{2} + 12 \, c d^{3} - 3 \, d^{4}\right )} f x - 3 \,{\left (4 \, c d^{3} - d^{4}\right )} \cos \left (f x + e\right )^{2} - 3 \,{\left (12 \, c^{2} d^{2} - 4 \, c d^{3} + 3 \, d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{6 \,{\left (a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*sin(f*x+e))^4/(a+a*sin(f*x+e)),x, algorithm="fricas")
[Out]
1/6*(2*d^4*cos(f*x + e)^4 - 6*c^4 + 24*c^3*d - 36*c^2*d^2 + 24*c*d^3 - 6*d^4 + (12*c*d^3 - d^4)*cos(f*x + e)^3
+ 3*(8*c^3*d - 12*c^2*d^2 + 12*c*d^3 - 3*d^4)*f*x - 12*(3*c^2*d^2 - 2*c*d^3 + d^4)*cos(f*x + e)^2 - 3*(2*c^4
- 8*c^3*d + 24*c^2*d^2 - 12*c*d^3 + 5*d^4 - (8*c^3*d - 12*c^2*d^2 + 12*c*d^3 - 3*d^4)*f*x)*cos(f*x + e) + (2*d
^4*cos(f*x + e)^3 + 6*c^4 - 24*c^3*d + 36*c^2*d^2 - 24*c*d^3 + 6*d^4 + 3*(8*c^3*d - 12*c^2*d^2 + 12*c*d^3 - 3*
d^4)*f*x - 3*(4*c*d^3 - d^4)*cos(f*x + e)^2 - 3*(12*c^2*d^2 - 4*c*d^3 + 3*d^4)*cos(f*x + e))*sin(f*x + e))/(a*
f*cos(f*x + e) + a*f*sin(f*x + e) + a*f)
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Sympy [A] time = 25.654, size = 8344, normalized size = 44.15 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*sin(f*x+e))**4/(a+a*sin(f*x+e)),x)
[Out]
Piecewise((12*c**4*tan(e/2 + f*x/2)**7/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2
+ f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*ta
n(e/2 + f*x/2) + 6*a*f) + 36*c**4*tan(e/2 + f*x/2)**5/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 +
18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2
)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 36*c**4*tan(e/2 + f*x/2)**3/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/
2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*
tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 12*c**4*tan(e/2 + f*x/2)/(6*a*f*tan(e/2 + f*x/2)**7 +
6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)*
*3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 24*c**3*d*f*x*tan(e/2 + f*x/2)**7/(6*a*f*t
an(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*
a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 24*c**3*d*f*x*tan(e/2
+ f*x/2)**6/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(
e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) +
72*c**3*d*f*x*tan(e/2 + f*x/2)**5/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*
x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2
+ f*x/2) + 6*a*f) + 72*c**3*d*f*x*tan(e/2 + f*x/2)**4/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6
+ 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/
2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 72*c**3*d*f*x*tan(e/2 + f*x/2)**3/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f
*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 +
18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 72*c**3*d*f*x*tan(e/2 + f*x/2)**2/(6*a*f*tan(e/
2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*t
an(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 24*c**3*d*f*x*tan(e/2 + f*
x/2)/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*
x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 24*c**3*
d*f*x/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f
*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 48*c**3
*d*tan(e/2 + f*x/2)**7/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 1
8*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) +
6*a*f) - 144*c**3*d*tan(e/2 + f*x/2)**5/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e
/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*
tan(e/2 + f*x/2) + 6*a*f) - 144*c**3*d*tan(e/2 + f*x/2)**3/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)
**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 +
f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 48*c**3*d*tan(e/2 + f*x/2)/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*ta
n(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*
a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 36*c**2*d**2*f*x*tan(e/2 + f*x/2)**7/(6*a*f*tan(e/
2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*t
an(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 36*c**2*d**2*f*x*tan(e/2 +
f*x/2)**6/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/
2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 10
8*c**2*d**2*f*x*tan(e/2 + f*x/2)**5/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 +
f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e
/2 + f*x/2) + 6*a*f) - 108*c**2*d**2*f*x*tan(e/2 + f*x/2)**4/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/
2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2
+ f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 108*c**2*d**2*f*x*tan(e/2 + f*x/2)**3/(6*a*f*tan(e/2 + f*x/2)*
*7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*
x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 108*c**2*d**2*f*x*tan(e/2 + f*x/2)**2
/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)
**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 36*c**2*d**2
*f*x*tan(e/2 + f*x/2)/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18
*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) +
6*a*f) - 36*c**2*d**2*f*x/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5
+ 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2
) + 6*a*f) + 72*c**2*d**2*tan(e/2 + f*x/2)**7/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*
tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6
*a*f*tan(e/2 + f*x/2) + 6*a*f) + 144*c**2*d**2*tan(e/2 + f*x/2)**5/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2
+ f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*ta
n(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 72*c**2*d**2*tan(e/2 + f*x/2)**4/(6*a*f*tan(e/2 + f*x/2)
**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f
*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 72*c**2*d**2*tan(e/2 + f*x/2)**3/(6*
a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4
+ 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 144*c**2*d**2*ta
n(e/2 + f*x/2)**2/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f
*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*
f) - 72*c**2*d**2/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f
*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*
f) + 36*c*d**3*f*x*tan(e/2 + f*x/2)**7/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2
+ f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*ta
n(e/2 + f*x/2) + 6*a*f) + 36*c*d**3*f*x*tan(e/2 + f*x/2)**6/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2
)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 +
f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 108*c*d**3*f*x*tan(e/2 + f*x/2)**5/(6*a*f*tan(e/2 + f*x/2)**7 +
6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)
**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 108*c*d**3*f*x*tan(e/2 + f*x/2)**4/(6*a*f
*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 1
8*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 108*c*d**3*f*x*tan(
e/2 + f*x/2)**3/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*t
an(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f)
+ 108*c*d**3*f*x*tan(e/2 + f*x/2)**2/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2
+ f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan
(e/2 + f*x/2) + 6*a*f) + 36*c*d**3*f*x*tan(e/2 + f*x/2)/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6
+ 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x
/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 36*c*d**3*f*x/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**
6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*
x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 72*c*d**3*tan(e/2 + f*x/2)**7/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*t
an(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18
*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 144*c*d**3*tan(e/2 + f*x/2)**5/(6*a*f*tan(e/2 + f
*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/
2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 24*c*d**3*tan(e/2 + f*x/2)**4/(
6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**
4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 120*c*d**3*tan
(e/2 + f*x/2)**3/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*
tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f
) - 48*c*d**3*tan(e/2 + f*x/2)/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2
)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 +
f*x/2) + 6*a*f) + 24*c*d**3/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**
5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x
/2) + 6*a*f) - 9*d**4*f*x*tan(e/2 + f*x/2)**7/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*
tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6
*a*f*tan(e/2 + f*x/2) + 6*a*f) - 9*d**4*f*x*tan(e/2 + f*x/2)**6/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f
*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e
/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 27*d**4*f*x*tan(e/2 + f*x/2)**5/(6*a*f*tan(e/2 + f*x/2)**7
+ 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2
)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 27*d**4*f*x*tan(e/2 + f*x/2)**4/(6*a*f*t
an(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*
a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 27*d**4*f*x*tan(e/2 +
f*x/2)**3/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/
2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 27
*d**4*f*x*tan(e/2 + f*x/2)**2/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)
**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f
*x/2) + 6*a*f) - 9*d**4*f*x*tan(e/2 + f*x/2)/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*t
an(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*
a*f*tan(e/2 + f*x/2) + 6*a*f) - 9*d**4*f*x/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan
(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*
f*tan(e/2 + f*x/2) + 6*a*f) + 18*d**4*tan(e/2 + f*x/2)**7/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)*
*6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f
*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 36*d**4*tan(e/2 + f*x/2)**5/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*ta
n(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*
a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 6*d**4*tan(e/2 + f*x/2)**4/(6*a*f*tan(e/2 + f*x/2)
**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f
*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 6*d**4*tan(e/2 + f*x/2)**3/(6*a*f*ta
n(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a
*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 24*d**4*tan(e/2 + f*x/
2)**2/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f
*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 4*d**4*
tan(e/2 + f*x/2)/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*
tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f
) - 14*d**4/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e
/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f), Ne
(f, 0)), (x*(c + d*sin(e))**4/(a*sin(e) + a), True))
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Giac [A] time = 1.36612, size = 413, normalized size = 2.19 \begin{align*} \frac{\frac{3 \,{\left (8 \, c^{3} d - 12 \, c^{2} d^{2} + 12 \, c d^{3} - 3 \, d^{4}\right )}{\left (f x + e\right )}}{a} - \frac{12 \,{\left (c^{4} - 4 \, c^{3} d + 6 \, c^{2} d^{2} - 4 \, c d^{3} + d^{4}\right )}}{a{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )}} + \frac{2 \,{\left (12 \, c d^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 3 \, d^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 36 \, c^{2} d^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 24 \, c d^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 6 \, d^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 72 \, c^{2} d^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 48 \, c d^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 24 \, d^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 12 \, c d^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 3 \, d^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 36 \, c^{2} d^{2} + 24 \, c d^{3} - 10 \, d^{4}\right )}}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1\right )}^{3} a}}{6 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*sin(f*x+e))^4/(a+a*sin(f*x+e)),x, algorithm="giac")
[Out]
1/6*(3*(8*c^3*d - 12*c^2*d^2 + 12*c*d^3 - 3*d^4)*(f*x + e)/a - 12*(c^4 - 4*c^3*d + 6*c^2*d^2 - 4*c*d^3 + d^4)/
(a*(tan(1/2*f*x + 1/2*e) + 1)) + 2*(12*c*d^3*tan(1/2*f*x + 1/2*e)^5 - 3*d^4*tan(1/2*f*x + 1/2*e)^5 - 36*c^2*d^
2*tan(1/2*f*x + 1/2*e)^4 + 24*c*d^3*tan(1/2*f*x + 1/2*e)^4 - 6*d^4*tan(1/2*f*x + 1/2*e)^4 - 72*c^2*d^2*tan(1/2
*f*x + 1/2*e)^2 + 48*c*d^3*tan(1/2*f*x + 1/2*e)^2 - 24*d^4*tan(1/2*f*x + 1/2*e)^2 - 12*c*d^3*tan(1/2*f*x + 1/2
*e) + 3*d^4*tan(1/2*f*x + 1/2*e) - 36*c^2*d^2 + 24*c*d^3 - 10*d^4)/((tan(1/2*f*x + 1/2*e)^2 + 1)^3*a))/f