3.461 \(\int \frac{(c+d \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx\)
Optimal. Leaf size=260 \[ \frac{2 d \left (-44 c^2 d^2+10 c^3 d+c^4+40 c d^3-12 d^4\right ) \cos (e+f x)}{3 a^2 f}+\frac{d \left (c^2+10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f}+\frac{d^2 \left (20 c^2 d+2 c^3-57 c d^2+30 d^3\right ) \sin (e+f x) \cos (e+f x)}{6 a^2 f}+\frac{5 d^2 x (2 c-d) \left (2 c^2-3 c d+2 d^2\right )}{2 a^2}-\frac{(c-d) (c+10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 a^2 f (\sin (e+f x)+1)}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a \sin (e+f x)+a)^2} \]
[Out]
(5*(2*c - d)*d^2*(2*c^2 - 3*c*d + 2*d^2)*x)/(2*a^2) + (2*d*(c^4 + 10*c^3*d - 44*c^2*d^2 + 40*c*d^3 - 12*d^4)*C
os[e + f*x])/(3*a^2*f) + (d^2*(2*c^3 + 20*c^2*d - 57*c*d^2 + 30*d^3)*Cos[e + f*x]*Sin[e + f*x])/(6*a^2*f) + (d
*(c^2 + 10*c*d - 12*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*a^2*f) - ((c - d)*(c + 10*d)*Cos[e + f*x]*(c
+ d*Sin[e + f*x])^3)/(3*a^2*f*(1 + Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(3*f*(a + a*
Sin[e + f*x])^2)
________________________________________________________________________________________
Rubi [A] time = 0.495092, antiderivative size = 260, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used =
{2765, 2977, 2753, 2734} \[ \frac{2 d \left (-44 c^2 d^2+10 c^3 d+c^4+40 c d^3-12 d^4\right ) \cos (e+f x)}{3 a^2 f}+\frac{d \left (c^2+10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f}+\frac{d^2 \left (20 c^2 d+2 c^3-57 c d^2+30 d^3\right ) \sin (e+f x) \cos (e+f x)}{6 a^2 f}+\frac{5 d^2 x (2 c-d) \left (2 c^2-3 c d+2 d^2\right )}{2 a^2}-\frac{(c-d) (c+10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 a^2 f (\sin (e+f x)+1)}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a \sin (e+f x)+a)^2} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*Sin[e + f*x])^5/(a + a*Sin[e + f*x])^2,x]
[Out]
(5*(2*c - d)*d^2*(2*c^2 - 3*c*d + 2*d^2)*x)/(2*a^2) + (2*d*(c^4 + 10*c^3*d - 44*c^2*d^2 + 40*c*d^3 - 12*d^4)*C
os[e + f*x])/(3*a^2*f) + (d^2*(2*c^3 + 20*c^2*d - 57*c*d^2 + 30*d^3)*Cos[e + f*x]*Sin[e + f*x])/(6*a^2*f) + (d
*(c^2 + 10*c*d - 12*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*a^2*f) - ((c - d)*(c + 10*d)*Cos[e + f*x]*(c
+ d*Sin[e + f*x])^3)/(3*a^2*f*(1 + Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(3*f*(a + a*
Sin[e + f*x])^2)
Rule 2765
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[((b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 1))/(a*f*(2*m + 1)), x] + Dist[1/
(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)*Simp[b*(c^2*(m + 1) + d^2*(n -
1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + 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] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ
[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
Rule 2977
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]
)^n)/(a*f*(2*m + 1)), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n -
1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ
[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (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))^5}{(a+a \sin (e+f x))^2} \, dx &=-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a+a \sin (e+f x))^2}-\frac{\int \frac{(c+d \sin (e+f x))^3 \left (-a \left (c^2+6 c d-4 d^2\right )+3 a (c-2 d) d \sin (e+f x)\right )}{a+a \sin (e+f x)} \, dx}{3 a^2}\\ &=-\frac{(c-d) (c+10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 a^2 f (1+\sin (e+f x))}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a+a \sin (e+f x))^2}-\frac{\int (c+d \sin (e+f x))^2 \left (-3 a^2 (11 c-10 d) d^2+3 a^2 d \left (c^2+10 c d-12 d^2\right ) \sin (e+f x)\right ) \, dx}{3 a^4}\\ &=\frac{d \left (c^2+10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f}-\frac{(c-d) (c+10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 a^2 f (1+\sin (e+f x))}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a+a \sin (e+f x))^2}-\frac{\int (c+d \sin (e+f x)) \left (-3 a^2 d^2 \left (31 c^2-50 c d+24 d^2\right )+3 a^2 d \left (2 c^3+20 c^2 d-57 c d^2+30 d^3\right ) \sin (e+f x)\right ) \, dx}{9 a^4}\\ &=\frac{5 (2 c-d) d^2 \left (2 c^2-3 c d+2 d^2\right ) x}{2 a^2}+\frac{2 d \left (c^4+10 c^3 d-44 c^2 d^2+40 c d^3-12 d^4\right ) \cos (e+f x)}{3 a^2 f}+\frac{d^2 \left (2 c^3+20 c^2 d-57 c d^2+30 d^3\right ) \cos (e+f x) \sin (e+f x)}{6 a^2 f}+\frac{d \left (c^2+10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f}-\frac{(c-d) (c+10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 a^2 f (1+\sin (e+f x))}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a+a \sin (e+f x))^2}\\ \end{align*}
Mathematica [B] time = 1.79741, size = 837, normalized size = 3.22 \[ \frac{\left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (48 \sin \left (\frac{1}{2} (e+f x)\right ) c^5+240 d \sin \left (\frac{1}{2} (e+f x)\right ) c^4-1440 d^2 \sin \left (\frac{1}{2} (e+f x)\right ) c^3+720 d^2 e \sin \left (\frac{1}{2} (e+f x)\right ) c^3+720 d^2 f x \sin \left (\frac{1}{2} (e+f x)\right ) c^3+240 d^2 e \sin \left (\frac{3}{2} (e+f x)\right ) c^3+240 d^2 f x \sin \left (\frac{3}{2} (e+f x)\right ) c^3+120 d^3 \cos \left (\frac{5}{2} (e+f x)\right ) c^2+2640 d^3 \sin \left (\frac{1}{2} (e+f x)\right ) c^2-1440 d^3 e \sin \left (\frac{1}{2} (e+f x)\right ) c^2-1440 d^3 f x \sin \left (\frac{1}{2} (e+f x)\right ) c^2-360 d^3 \sin \left (\frac{3}{2} (e+f x)\right ) c^2-480 d^3 e \sin \left (\frac{3}{2} (e+f x)\right ) c^2-480 d^3 f x \sin \left (\frac{3}{2} (e+f x)\right ) c^2-120 d^3 \sin \left (\frac{5}{2} (e+f x)\right ) c^2-75 d^4 \cos \left (\frac{5}{2} (e+f x)\right ) c+15 d^4 \cos \left (\frac{7}{2} (e+f x)\right ) c-1905 d^4 \sin \left (\frac{1}{2} (e+f x)\right ) c+1260 d^4 e \sin \left (\frac{1}{2} (e+f x)\right ) c+1260 d^4 f x \sin \left (\frac{1}{2} (e+f x)\right ) c+315 d^4 \sin \left (\frac{3}{2} (e+f x)\right ) c+420 d^4 e \sin \left (\frac{3}{2} (e+f x)\right ) c+420 d^4 f x \sin \left (\frac{3}{2} (e+f x)\right ) c+75 d^4 \sin \left (\frac{5}{2} (e+f x)\right ) c+15 d^4 \sin \left (\frac{7}{2} (e+f x)\right ) c+3 d \left (80 c^4+80 d (3 e+3 f x-4) c^3-80 d^2 (6 e+6 f x-5) c^2+35 d^3 (12 e+12 f x-7) c-4 d^4 (30 e+30 f x-13)\right ) \cos \left (\frac{1}{2} (e+f x)\right )-\left (16 c^5+160 d c^4+80 d^2 (3 e+3 f x-10) c^3-40 d^3 (12 e+12 f x-41) c^2+5 d^4 (84 e+84 f x-239) c-6 d^5 (20 e+20 f x-57)\right ) \cos \left (\frac{3}{2} (e+f x)\right )+30 d^5 \cos \left (\frac{5}{2} (e+f x)\right )-3 d^5 \cos \left (\frac{7}{2} (e+f x)\right )-d^5 \cos \left (\frac{9}{2} (e+f x)\right )+516 d^5 \sin \left (\frac{1}{2} (e+f x)\right )-360 d^5 e \sin \left (\frac{1}{2} (e+f x)\right )-360 d^5 f x \sin \left (\frac{1}{2} (e+f x)\right )-118 d^5 \sin \left (\frac{3}{2} (e+f x)\right )-120 d^5 e \sin \left (\frac{3}{2} (e+f x)\right )-120 d^5 f x \sin \left (\frac{3}{2} (e+f x)\right )-30 d^5 \sin \left (\frac{5}{2} (e+f x)\right )-3 d^5 \sin \left (\frac{7}{2} (e+f x)\right )+d^5 \sin \left (\frac{9}{2} (e+f x)\right )\right )}{48 a^2 f (\sin (e+f x)+1)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*Sin[e + f*x])^5/(a + a*Sin[e + f*x])^2,x]
[Out]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(3*d*(80*c^4 + 80*c^3*d*(-4 + 3*e + 3*f*x) - 80*c^2*d^2*(-5 + 6*e + 6*f
*x) + 35*c*d^3*(-7 + 12*e + 12*f*x) - 4*d^4*(-13 + 30*e + 30*f*x))*Cos[(e + f*x)/2] - (16*c^5 + 160*c^4*d + 80
*c^3*d^2*(-10 + 3*e + 3*f*x) - 40*c^2*d^3*(-41 + 12*e + 12*f*x) - 6*d^5*(-57 + 20*e + 20*f*x) + 5*c*d^4*(-239
+ 84*e + 84*f*x))*Cos[(3*(e + f*x))/2] + 120*c^2*d^3*Cos[(5*(e + f*x))/2] - 75*c*d^4*Cos[(5*(e + f*x))/2] + 30
*d^5*Cos[(5*(e + f*x))/2] + 15*c*d^4*Cos[(7*(e + f*x))/2] - 3*d^5*Cos[(7*(e + f*x))/2] - d^5*Cos[(9*(e + f*x))
/2] + 48*c^5*Sin[(e + f*x)/2] + 240*c^4*d*Sin[(e + f*x)/2] - 1440*c^3*d^2*Sin[(e + f*x)/2] + 2640*c^2*d^3*Sin[
(e + f*x)/2] - 1905*c*d^4*Sin[(e + f*x)/2] + 516*d^5*Sin[(e + f*x)/2] + 720*c^3*d^2*e*Sin[(e + f*x)/2] - 1440*
c^2*d^3*e*Sin[(e + f*x)/2] + 1260*c*d^4*e*Sin[(e + f*x)/2] - 360*d^5*e*Sin[(e + f*x)/2] + 720*c^3*d^2*f*x*Sin[
(e + f*x)/2] - 1440*c^2*d^3*f*x*Sin[(e + f*x)/2] + 1260*c*d^4*f*x*Sin[(e + f*x)/2] - 360*d^5*f*x*Sin[(e + f*x)
/2] - 360*c^2*d^3*Sin[(3*(e + f*x))/2] + 315*c*d^4*Sin[(3*(e + f*x))/2] - 118*d^5*Sin[(3*(e + f*x))/2] + 240*c
^3*d^2*e*Sin[(3*(e + f*x))/2] - 480*c^2*d^3*e*Sin[(3*(e + f*x))/2] + 420*c*d^4*e*Sin[(3*(e + f*x))/2] - 120*d^
5*e*Sin[(3*(e + f*x))/2] + 240*c^3*d^2*f*x*Sin[(3*(e + f*x))/2] - 480*c^2*d^3*f*x*Sin[(3*(e + f*x))/2] + 420*c
*d^4*f*x*Sin[(3*(e + f*x))/2] - 120*d^5*f*x*Sin[(3*(e + f*x))/2] - 120*c^2*d^3*Sin[(5*(e + f*x))/2] + 75*c*d^4
*Sin[(5*(e + f*x))/2] - 30*d^5*Sin[(5*(e + f*x))/2] + 15*c*d^4*Sin[(7*(e + f*x))/2] - 3*d^5*Sin[(7*(e + f*x))/
2] + d^5*Sin[(9*(e + f*x))/2]))/(48*a^2*f*(1 + Sin[e + f*x])^2)
________________________________________________________________________________________
Maple [B] time = 0.102, size = 982, normalized size = 3.8 \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))^5/(a+a*sin(f*x+e))^2,x)
[Out]
5/f/a^2*d^4/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^5*c-20/f/a^2*d^3/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*
f*x+1/2*e)^4*c^2+20/f/a^2*d^4/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^4*c-40/f/a^2*d^3/(1+tan(1/2*f*x+1/
2*e)^2)^3*tan(1/2*f*x+1/2*e)^2*c^2+40/f/a^2*d^4/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^2*c-5/f/a^2*d^4/
(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)*c+10/f/a^2/(tan(1/2*f*x+1/2*e)+1)^2*c*d^4-2/f/a^2*d^5/(1+tan(1/2
*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^5-6/f/a^2*d^5/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^4-16/f/a^2*d^5
/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^2+2/f/a^2*d^5/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)-20/
f/a^2*d^3/(1+tan(1/2*f*x+1/2*e)^2)^3*c^2+20/f/a^2*d^4/(1+tan(1/2*f*x+1/2*e)^2)^3*c+20/f/a^2/(tan(1/2*f*x+1/2*e
)+1)*c^3*d^2+20/f/a^2*d^2*arctan(tan(1/2*f*x+1/2*e))*c^3-10/f/a^2/(tan(1/2*f*x+1/2*e)+1)^2*c^4*d+20/f/a^2/(tan
(1/2*f*x+1/2*e)+1)^2*c^3*d^2+2/f/a^2/(tan(1/2*f*x+1/2*e)+1)^2*c^5-2/f/a^2/(tan(1/2*f*x+1/2*e)+1)^2*d^5-4/3/f/a
^2/(tan(1/2*f*x+1/2*e)+1)^3*c^5+4/3/f/a^2/(tan(1/2*f*x+1/2*e)+1)^3*d^5-22/3/f/a^2*d^5/(1+tan(1/2*f*x+1/2*e)^2)
^3-10/f/a^2*d^5*arctan(tan(1/2*f*x+1/2*e))-2/f/a^2/(tan(1/2*f*x+1/2*e)+1)*c^5-8/f/a^2/(tan(1/2*f*x+1/2*e)+1)*d
^5-40/f/a^2*d^3*arctan(tan(1/2*f*x+1/2*e))*c^2+35/f/a^2*d^4*arctan(tan(1/2*f*x+1/2*e))*c-40/f/a^2/(tan(1/2*f*x
+1/2*e)+1)*c^2*d^3+30/f/a^2/(tan(1/2*f*x+1/2*e)+1)*c*d^4+20/3/f/a^2/(tan(1/2*f*x+1/2*e)+1)^3*c^4*d-40/3/f/a^2/
(tan(1/2*f*x+1/2*e)+1)^3*c^3*d^2+40/3/f/a^2/(tan(1/2*f*x+1/2*e)+1)^3*c^2*d^3-20/3/f/a^2/(tan(1/2*f*x+1/2*e)+1)
^3*c*d^4-20/f/a^2/(tan(1/2*f*x+1/2*e)+1)^2*c^2*d^3
________________________________________________________________________________________
Maxima [B] time = 1.97013, size = 1771, normalized size = 6.81 \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))^5/(a+a*sin(f*x+e))^2,x, algorithm="maxima")
[Out]
1/3*(5*c*d^4*((75*sin(f*x + e)/(cos(f*x + e) + 1) + 97*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 126*sin(f*x + e)^
3/(cos(f*x + e) + 1)^3 + 98*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 63*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 21*
sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 32)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 5*a^2*sin(f*x + e)^2/
(cos(f*x + e) + 1)^2 + 7*a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 7*a^2*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 +
5*a^2*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 3*a^2*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + a^2*sin(f*x + e)^7/(c
os(f*x + e) + 1)^7) + 21*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) - 2*d^5*((57*sin(f*x + e)/(cos(f*x + e)
+ 1) + 99*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 155*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 153*sin(f*x + e)^4/(
cos(f*x + e) + 1)^4 + 135*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 85*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 45*si
n(f*x + e)^7/(cos(f*x + e) + 1)^7 + 15*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 24)/(a^2 + 3*a^2*sin(f*x + e)/(co
s(f*x + e) + 1) + 6*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 12*
a^2*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 12*a^2*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 10*a^2*sin(f*x + e)^6/(
cos(f*x + e) + 1)^6 + 6*a^2*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 3*a^2*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 +
a^2*sin(f*x + e)^9/(cos(f*x + e) + 1)^9) + 15*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) - 40*c^2*d^3*((12*s
in(f*x + e)/(cos(f*x + e) + 1) + 11*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 9*sin(f*x + e)^3/(cos(f*x + e) + 1)^
3 + 3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 5)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 4*a^2*sin(f*x +
e)^2/(cos(f*x + e) + 1)^2 + 4*a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*a^2*sin(f*x + e)^4/(cos(f*x + e) + 1
)^4 + a^2*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 3*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) + 20*c^3*d^2*(
(9*sin(f*x + e)/(cos(f*x + e) + 1) + 3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 4)/(a^2 + 3*a^2*sin(f*x + e)/(cos
(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3) + 3*arct
an(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) - 2*c^5*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 3*sin(f*x + e)^2/(cos(f*
x + e) + 1)^2 + 2)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 +
a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3) - 10*c^4*d*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 1)/(a^2 + 3*a^2*sin(f
*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)
^3))/f
________________________________________________________________________________________
Fricas [B] time = 1.81477, size = 1314, normalized size = 5.05 \begin{align*} \frac{2 \, d^{5} \cos \left (f x + e\right )^{5} + 2 \, c^{5} - 10 \, c^{4} d + 20 \, c^{3} d^{2} - 20 \, c^{2} d^{3} + 10 \, c d^{4} - 2 \, d^{5} -{\left (15 \, c d^{4} - 4 \, d^{5}\right )} \cos \left (f x + e\right )^{4} - 2 \,{\left (30 \, c^{2} d^{3} - 15 \, c d^{4} + 8 \, d^{5}\right )} \cos \left (f x + e\right )^{3} - 30 \,{\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} f x +{\left (2 \, c^{5} + 20 \, c^{4} d - 100 \, c^{3} d^{2} + 220 \, c^{2} d^{3} - 155 \, c d^{4} + 46 \, d^{5} + 15 \,{\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} f x\right )} \cos \left (f x + e\right )^{2} +{\left (4 \, c^{5} + 10 \, c^{4} d - 80 \, c^{3} d^{2} + 260 \, c^{2} d^{3} - 190 \, c d^{4} + 62 \, d^{5} - 15 \,{\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} f x\right )} \cos \left (f x + e\right ) -{\left (2 \, d^{5} \cos \left (f x + e\right )^{4} + 2 \, c^{5} - 10 \, c^{4} d + 20 \, c^{3} d^{2} - 20 \, c^{2} d^{3} + 10 \, c d^{4} - 2 \, d^{5} +{\left (15 \, c d^{4} - 2 \, d^{5}\right )} \cos \left (f x + e\right )^{3} + 30 \,{\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} f x - 3 \,{\left (20 \, c^{2} d^{3} - 15 \, c d^{4} + 6 \, d^{5}\right )} \cos \left (f x + e\right )^{2} -{\left (2 \, c^{5} + 20 \, c^{4} d - 100 \, c^{3} d^{2} + 280 \, c^{2} d^{3} - 200 \, c d^{4} + 64 \, d^{5} - 15 \,{\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} f x\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{6 \,{\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f -{\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x, algorithm="fricas")
[Out]
1/6*(2*d^5*cos(f*x + e)^5 + 2*c^5 - 10*c^4*d + 20*c^3*d^2 - 20*c^2*d^3 + 10*c*d^4 - 2*d^5 - (15*c*d^4 - 4*d^5)
*cos(f*x + e)^4 - 2*(30*c^2*d^3 - 15*c*d^4 + 8*d^5)*cos(f*x + e)^3 - 30*(4*c^3*d^2 - 8*c^2*d^3 + 7*c*d^4 - 2*d
^5)*f*x + (2*c^5 + 20*c^4*d - 100*c^3*d^2 + 220*c^2*d^3 - 155*c*d^4 + 46*d^5 + 15*(4*c^3*d^2 - 8*c^2*d^3 + 7*c
*d^4 - 2*d^5)*f*x)*cos(f*x + e)^2 + (4*c^5 + 10*c^4*d - 80*c^3*d^2 + 260*c^2*d^3 - 190*c*d^4 + 62*d^5 - 15*(4*
c^3*d^2 - 8*c^2*d^3 + 7*c*d^4 - 2*d^5)*f*x)*cos(f*x + e) - (2*d^5*cos(f*x + e)^4 + 2*c^5 - 10*c^4*d + 20*c^3*d
^2 - 20*c^2*d^3 + 10*c*d^4 - 2*d^5 + (15*c*d^4 - 2*d^5)*cos(f*x + e)^3 + 30*(4*c^3*d^2 - 8*c^2*d^3 + 7*c*d^4 -
2*d^5)*f*x - 3*(20*c^2*d^3 - 15*c*d^4 + 6*d^5)*cos(f*x + e)^2 - (2*c^5 + 20*c^4*d - 100*c^3*d^2 + 280*c^2*d^3
- 200*c*d^4 + 64*d^5 - 15*(4*c^3*d^2 - 8*c^2*d^3 + 7*c*d^4 - 2*d^5)*f*x)*cos(f*x + e))*sin(f*x + e))/(a^2*f*c
os(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*sin(f*x+e))**5/(a+a*sin(f*x+e))**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.53243, size = 1319, normalized size = 5.07 \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))^5/(a+a*sin(f*x+e))^2,x, algorithm="giac")
[Out]
1/6*(15*(4*c^3*d^2 - 8*c^2*d^3 + 7*c*d^4 - 2*d^5)*(f*x + e)/a^2 - 2*(6*c^5*tan(1/2*f*x + 1/2*e)^8 - 60*c^3*d^2
*tan(1/2*f*x + 1/2*e)^8 + 120*c^2*d^3*tan(1/2*f*x + 1/2*e)^8 - 105*c*d^4*tan(1/2*f*x + 1/2*e)^8 + 30*d^5*tan(1
/2*f*x + 1/2*e)^8 + 6*c^5*tan(1/2*f*x + 1/2*e)^7 + 30*c^4*d*tan(1/2*f*x + 1/2*e)^7 - 180*c^3*d^2*tan(1/2*f*x +
1/2*e)^7 + 360*c^2*d^3*tan(1/2*f*x + 1/2*e)^7 - 315*c*d^4*tan(1/2*f*x + 1/2*e)^7 + 90*d^5*tan(1/2*f*x + 1/2*e
)^7 + 22*c^5*tan(1/2*f*x + 1/2*e)^6 + 10*c^4*d*tan(1/2*f*x + 1/2*e)^6 - 260*c^3*d^2*tan(1/2*f*x + 1/2*e)^6 + 6
80*c^2*d^3*tan(1/2*f*x + 1/2*e)^6 - 595*c*d^4*tan(1/2*f*x + 1/2*e)^6 + 170*d^5*tan(1/2*f*x + 1/2*e)^6 + 18*c^5
*tan(1/2*f*x + 1/2*e)^5 + 90*c^4*d*tan(1/2*f*x + 1/2*e)^5 - 540*c^3*d^2*tan(1/2*f*x + 1/2*e)^5 + 1200*c^2*d^3*
tan(1/2*f*x + 1/2*e)^5 - 945*c*d^4*tan(1/2*f*x + 1/2*e)^5 + 270*d^5*tan(1/2*f*x + 1/2*e)^5 + 30*c^5*tan(1/2*f*
x + 1/2*e)^4 + 30*c^4*d*tan(1/2*f*x + 1/2*e)^4 - 420*c^3*d^2*tan(1/2*f*x + 1/2*e)^4 + 1200*c^2*d^3*tan(1/2*f*x
+ 1/2*e)^4 - 975*c*d^4*tan(1/2*f*x + 1/2*e)^4 + 306*d^5*tan(1/2*f*x + 1/2*e)^4 + 18*c^5*tan(1/2*f*x + 1/2*e)^
3 + 90*c^4*d*tan(1/2*f*x + 1/2*e)^3 - 540*c^3*d^2*tan(1/2*f*x + 1/2*e)^3 + 1320*c^2*d^3*tan(1/2*f*x + 1/2*e)^3
- 1005*c*d^4*tan(1/2*f*x + 1/2*e)^3 + 310*d^5*tan(1/2*f*x + 1/2*e)^3 + 18*c^5*tan(1/2*f*x + 1/2*e)^2 + 30*c^4
*d*tan(1/2*f*x + 1/2*e)^2 - 300*c^3*d^2*tan(1/2*f*x + 1/2*e)^2 + 840*c^2*d^3*tan(1/2*f*x + 1/2*e)^2 - 645*c*d^
4*tan(1/2*f*x + 1/2*e)^2 + 198*d^5*tan(1/2*f*x + 1/2*e)^2 + 6*c^5*tan(1/2*f*x + 1/2*e) + 30*c^4*d*tan(1/2*f*x
+ 1/2*e) - 180*c^3*d^2*tan(1/2*f*x + 1/2*e) + 480*c^2*d^3*tan(1/2*f*x + 1/2*e) - 375*c*d^4*tan(1/2*f*x + 1/2*e
) + 114*d^5*tan(1/2*f*x + 1/2*e) + 4*c^5 + 10*c^4*d - 80*c^3*d^2 + 200*c^2*d^3 - 160*c*d^4 + 48*d^5)/((tan(1/2
*f*x + 1/2*e)^3 + tan(1/2*f*x + 1/2*e)^2 + tan(1/2*f*x + 1/2*e) + 1)^3*a^2))/f