3.470 \(\int \frac{(c+d \sin (e+f x))^6}{(a+a \sin (e+f x))^3} \, dx\)
Optimal. Leaf size=354 \[ \frac{2 d \left (107 c^3 d^2-472 c^2 d^3+18 c^4 d+2 c^5+456 c d^4-136 d^5\right ) \cos (e+f x)}{15 a^3 f}-\frac{(c-d) \left (2 c^2+18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{15 f \left (a^3 \sin (e+f x)+a^3\right )}+\frac{d \left (18 c^2 d+2 c^3+111 c d^2-136 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a^3 f}+\frac{d^2 \left (216 c^2 d^2+36 c^3 d+4 c^4-626 c d^3+345 d^4\right ) \sin (e+f x) \cos (e+f x)}{30 a^3 f}+\frac{d^3 x \left (-90 c^2 d+40 c^3+78 c d^2-23 d^3\right )}{2 a^3}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{15 a f (a \sin (e+f x)+a)^2} \]
[Out]
(d^3*(40*c^3 - 90*c^2*d + 78*c*d^2 - 23*d^3)*x)/(2*a^3) + (2*d*(2*c^5 + 18*c^4*d + 107*c^3*d^2 - 472*c^2*d^3 +
456*c*d^4 - 136*d^5)*Cos[e + f*x])/(15*a^3*f) + (d^2*(4*c^4 + 36*c^3*d + 216*c^2*d^2 - 626*c*d^3 + 345*d^4)*C
os[e + f*x]*Sin[e + f*x])/(30*a^3*f) + (d*(2*c^3 + 18*c^2*d + 111*c*d^2 - 136*d^3)*Cos[e + f*x]*(c + d*Sin[e +
f*x])^2)/(15*a^3*f) - ((c - d)*(2*c^2 + 18*c*d + 115*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(15*f*(a^3 + a
^3*Sin[e + f*x])) - ((c - d)*(2*c + 13*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(15*a*f*(a + a*Sin[e + f*x])^2)
- ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^5)/(5*f*(a + a*Sin[e + f*x])^3)
________________________________________________________________________________________
Rubi [A] time = 0.789744, antiderivative size = 354, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used =
{2765, 2977, 2753, 2734} \[ \frac{2 d \left (107 c^3 d^2-472 c^2 d^3+18 c^4 d+2 c^5+456 c d^4-136 d^5\right ) \cos (e+f x)}{15 a^3 f}-\frac{(c-d) \left (2 c^2+18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{15 f \left (a^3 \sin (e+f x)+a^3\right )}+\frac{d \left (18 c^2 d+2 c^3+111 c d^2-136 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a^3 f}+\frac{d^2 \left (216 c^2 d^2+36 c^3 d+4 c^4-626 c d^3+345 d^4\right ) \sin (e+f x) \cos (e+f x)}{30 a^3 f}+\frac{d^3 x \left (-90 c^2 d+40 c^3+78 c d^2-23 d^3\right )}{2 a^3}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{15 a f (a \sin (e+f x)+a)^2} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*Sin[e + f*x])^6/(a + a*Sin[e + f*x])^3,x]
[Out]
(d^3*(40*c^3 - 90*c^2*d + 78*c*d^2 - 23*d^3)*x)/(2*a^3) + (2*d*(2*c^5 + 18*c^4*d + 107*c^3*d^2 - 472*c^2*d^3 +
456*c*d^4 - 136*d^5)*Cos[e + f*x])/(15*a^3*f) + (d^2*(4*c^4 + 36*c^3*d + 216*c^2*d^2 - 626*c*d^3 + 345*d^4)*C
os[e + f*x]*Sin[e + f*x])/(30*a^3*f) + (d*(2*c^3 + 18*c^2*d + 111*c*d^2 - 136*d^3)*Cos[e + f*x]*(c + d*Sin[e +
f*x])^2)/(15*a^3*f) - ((c - d)*(2*c^2 + 18*c*d + 115*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(15*f*(a^3 + a
^3*Sin[e + f*x])) - ((c - d)*(2*c + 13*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(15*a*f*(a + a*Sin[e + f*x])^2)
- ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^5)/(5*f*(a + a*Sin[e + f*x])^3)
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))^6}{(a+a \sin (e+f x))^3} \, dx &=-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a+a \sin (e+f x))^3}-\frac{\int \frac{(c+d \sin (e+f x))^4 \left (-a \left (2 c^2+8 c d-5 d^2\right )+a (3 c-8 d) d \sin (e+f x)\right )}{(a+a \sin (e+f x))^2} \, dx}{5 a^2}\\ &=-\frac{(c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{15 a f (a+a \sin (e+f x))^2}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a+a \sin (e+f x))^3}-\frac{\int \frac{(c+d \sin (e+f x))^3 \left (-a^2 \left (2 c^3+10 c^2 d+55 c d^2-52 d^3\right )+3 a^2 d \left (2 c^2+14 c d-21 d^2\right ) \sin (e+f x)\right )}{a+a \sin (e+f x)} \, dx}{15 a^4}\\ &=-\frac{(c-d) \left (2 c^2+18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{15 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac{(c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{15 a f (a+a \sin (e+f x))^2}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a+a \sin (e+f x))^3}-\frac{\int (c+d \sin (e+f x))^2 \left (-3 a^3 d^2 \left (2 c^2+118 c d-115 d^2\right )+3 a^3 d \left (2 c^3+18 c^2 d+111 c d^2-136 d^3\right ) \sin (e+f x)\right ) \, dx}{15 a^6}\\ &=\frac{d \left (2 c^3+18 c^2 d+111 c d^2-136 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a^3 f}-\frac{(c-d) \left (2 c^2+18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{15 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac{(c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{15 a f (a+a \sin (e+f x))^2}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a+a \sin (e+f x))^3}-\frac{\int (c+d \sin (e+f x)) \left (-3 a^3 d^2 \left (2 c^3+318 c^2 d-567 c d^2+272 d^3\right )+3 a^3 d \left (4 c^4+36 c^3 d+216 c^2 d^2-626 c d^3+345 d^4\right ) \sin (e+f x)\right ) \, dx}{45 a^6}\\ &=\frac{d^3 \left (40 c^3-90 c^2 d+78 c d^2-23 d^3\right ) x}{2 a^3}+\frac{2 d \left (2 c^5+18 c^4 d+107 c^3 d^2-472 c^2 d^3+456 c d^4-136 d^5\right ) \cos (e+f x)}{15 a^3 f}+\frac{d^2 \left (4 c^4+36 c^3 d+216 c^2 d^2-626 c d^3+345 d^4\right ) \cos (e+f x) \sin (e+f x)}{30 a^3 f}+\frac{d \left (2 c^3+18 c^2 d+111 c d^2-136 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a^3 f}-\frac{(c-d) \left (2 c^2+18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{15 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac{(c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{15 a f (a+a \sin (e+f x))^2}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a+a \sin (e+f x))^3}\\ \end{align*}
Mathematica [C] time = 2.96154, size = 560, normalized size = 1.58 \[ \frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (16 \left (2 c^2+26 c d+197 d^2\right ) (c-d)^4 \sin \left (\frac{1}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4-60 d^3 \left (90 c^2 d-40 c^3-78 c d^2+23 d^3\right ) (e+f x) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5-45 d^4 \left (20 c^2-24 c d+9 d^2\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5 (\cos (e+f x)-i \sin (e+f x))-45 d^4 \left (20 c^2-24 c d+9 d^2\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5 (\cos (e+f x)+i \sin (e+f x))-45 i d^5 (2 c-d) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5 (\cos (2 (e+f x))-i \sin (2 (e+f x)))+45 i d^5 (2 c-d) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5 (\cos (2 (e+f x))+i \sin (2 (e+f x)))+48 (c-d)^6 \sin \left (\frac{1}{2} (e+f x)\right )-24 (c-d)^6 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )-16 (c+14 d) (c-d)^5 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3+32 (c+14 d) (c-d)^5 \sin \left (\frac{1}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2+10 d^6 \cos (3 (e+f x)) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5\right )}{120 a^3 f (\sin (e+f x)+1)^3} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*Sin[e + f*x])^6/(a + a*Sin[e + f*x])^3,x]
[Out]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(48*(c - d)^6*Sin[(e + f*x)/2] - 24*(c - d)^6*(Cos[(e + f*x)/2] + Sin[(
e + f*x)/2]) + 32*(c - d)^5*(c + 14*d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 16*(c - d)^5
*(c + 14*d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 16*(c - d)^4*(2*c^2 + 26*c*d + 197*d^2)*Sin[(e + f*x)/2]
*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 60*d^3*(-40*c^3 + 90*c^2*d - 78*c*d^2 + 23*d^3)*(e + f*x)*(Cos[(e +
f*x)/2] + Sin[(e + f*x)/2])^5 + 10*d^6*Cos[3*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 - 45*d^4*(20*
c^2 - 24*c*d + 9*d^2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(Cos[e + f*x] - I*Sin[e + f*x]) - 45*d^4*(20*c^2
- 24*c*d + 9*d^2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(Cos[e + f*x] + I*Sin[e + f*x]) - (45*I)*(2*c - d)*
d^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)]) + (45*I)*(2*c - d)*d^5*(Co
s[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])))/(120*a^3*f*(1 + Sin[e + f*x])^3
)
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Maple [B] time = 0.104, size = 1340, 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))^6/(a+a*sin(f*x+e))^3,x)
[Out]
3/f/a^3*d^6/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)-30/f/a^3*d^4/(1+tan(1/2*f*x+1/2*e)^2)^3*c^2+36/f/a^3
*d^5/(1+tan(1/2*f*x+1/2*e)^2)^3*c+40/f/a^3*d^3*arctan(tan(1/2*f*x+1/2*e))*c^3-90/f/a^3*d^4*arctan(tan(1/2*f*x+
1/2*e))*c^2+78/f/a^3*d^5*arctan(tan(1/2*f*x+1/2*e))*c+80/3/f/a^3/(tan(1/2*f*x+1/2*e)+1)^3*c^3*d^3-8/f/a^3/(tan
(1/2*f*x+1/2*e)+1)^3*c*d^5-24/f/a^3/(tan(1/2*f*x+1/2*e)+1)^4*c^5*d+60/f/a^3/(tan(1/2*f*x+1/2*e)+1)^4*c^4*d^2-8
0/f/a^3/(tan(1/2*f*x+1/2*e)+1)^4*c^3*d^3+60/f/a^3/(tan(1/2*f*x+1/2*e)+1)^4*c^2*d^4-28/f/a^3*d^6/(1+tan(1/2*f*x
+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^2-12/f/a^3/(tan(1/2*f*x+1/2*e)+1)^2*c^5*d+40/f/a^3/(tan(1/2*f*x+1/2*e)+1)*c^3*
d^3+40/f/a^3/(tan(1/2*f*x+1/2*e)+1)^2*c^3*d^3-60/f/a^3/(tan(1/2*f*x+1/2*e)+1)^2*c^2*d^4-40/3/f/a^3*d^6/(1+tan(
1/2*f*x+1/2*e)^2)^3-23/f/a^3*d^6*arctan(tan(1/2*f*x+1/2*e))-2/f/a^3/(tan(1/2*f*x+1/2*e)+1)*c^6-20/f/a^3/(tan(1
/2*f*x+1/2*e)+1)*d^6+4/f/a^3/(tan(1/2*f*x+1/2*e)+1)^2*c^6-8/f/a^3/(tan(1/2*f*x+1/2*e)+1)^2*d^6-16/3/f/a^3/(tan
(1/2*f*x+1/2*e)+1)^3*c^6+8/3/f/a^3/(tan(1/2*f*x+1/2*e)+1)^3*d^6-24/f/a^3/(tan(1/2*f*x+1/2*e)+1)^5*c^4*d^2+32/f
/a^3/(tan(1/2*f*x+1/2*e)+1)^5*c^3*d^3+24/f/a^3/(tan(1/2*f*x+1/2*e)+1)^3*c^5*d-40/f/a^3/(tan(1/2*f*x+1/2*e)+1)^
3*c^4*d^2+36/f/a^3/(tan(1/2*f*x+1/2*e)+1)^2*c*d^5-24/f/a^3/(tan(1/2*f*x+1/2*e)+1)^4*c*d^5+48/5/f/a^3/(tan(1/2*
f*x+1/2*e)+1)^5*c^5*d-24/f/a^3/(tan(1/2*f*x+1/2*e)+1)^5*c^2*d^4+48/5/f/a^3/(tan(1/2*f*x+1/2*e)+1)^5*c*d^5-3/f/
a^3*d^6/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^5-12/f/a^3*d^6/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/
2*e)^4-90/f/a^3/(tan(1/2*f*x+1/2*e)+1)*c^2*d^4+72/f/a^3/(tan(1/2*f*x+1/2*e)+1)*c*d^5+36/f/a^3*d^5/(1+tan(1/2*f
*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^4*c-60/f/a^3*d^4/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^2*c^2+72/f/a^
3*d^5/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^2*c-6/f/a^3*d^5/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2
*e)*c+4/f/a^3/(tan(1/2*f*x+1/2*e)+1)^4*c^6+4/f/a^3/(tan(1/2*f*x+1/2*e)+1)^4*d^6-8/5/f/a^3/(tan(1/2*f*x+1/2*e)+
1)^5*c^6-8/5/f/a^3/(tan(1/2*f*x+1/2*e)+1)^5*d^6+6/f/a^3*d^5/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^5*c-
30/f/a^3*d^4/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^4*c^2
________________________________________________________________________________________
Maxima [B] time = 2.14224, size = 2691, normalized size = 7.6 \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))^6/(a+a*sin(f*x+e))^3,x, algorithm="maxima")
[Out]
-1/15*(d^6*((2375*sin(f*x + e)/(cos(f*x + e) + 1) + 5347*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 9230*sin(f*x +
e)^3/(cos(f*x + e) + 1)^3 + 12622*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 13340*sin(f*x + e)^5/(cos(f*x + e) + 1
)^5 + 11684*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 8050*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 4370*sin(f*x + e)
^8/(cos(f*x + e) + 1)^8 + 1725*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 345*sin(f*x + e)^10/(cos(f*x + e) + 1)^10
+ 544)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 13*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 25*a^3*sin
(f*x + e)^3/(cos(f*x + e) + 1)^3 + 38*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 46*a^3*sin(f*x + e)^5/(cos(f*x
+ e) + 1)^5 + 46*a^3*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 38*a^3*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 25*a^
3*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 13*a^3*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 5*a^3*sin(f*x + e)^10/(co
s(f*x + e) + 1)^10 + a^3*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) + 345*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/
a^3) - 6*c*d^5*((1325*sin(f*x + e)/(cos(f*x + e) + 1) + 2673*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3805*sin(f*
x + e)^3/(cos(f*x + e) + 1)^3 + 4329*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 3575*sin(f*x + e)^5/(cos(f*x + e) +
1)^5 + 2275*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 975*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 195*sin(f*x + e)^
8/(cos(f*x + e) + 1)^8 + 304)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 12*a^3*sin(f*x + e)^2/(cos(f*x +
e) + 1)^2 + 20*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 26*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 26*a^3*s
in(f*x + e)^5/(cos(f*x + e) + 1)^5 + 20*a^3*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 12*a^3*sin(f*x + e)^7/(cos(f
*x + e) + 1)^7 + 5*a^3*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + a^3*sin(f*x + e)^9/(cos(f*x + e) + 1)^9) + 195*ar
ctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^3) + 90*c^2*d^4*((105*sin(f*x + e)/(cos(f*x + e) + 1) + 189*sin(f*x +
e)^2/(cos(f*x + e) + 1)^2 + 200*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 160*sin(f*x + e)^4/(cos(f*x + e) + 1)^4
+ 75*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 15*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 24)/(a^3 + 5*a^3*sin(f*x +
e)/(cos(f*x + e) + 1) + 11*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 15*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)
^3 + 15*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 11*a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5*a^3*sin(f*x +
e)^6/(cos(f*x + e) + 1)^6 + a^3*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) + 15*arctan(sin(f*x + e)/(cos(f*x + e) +
1))/a^3) - 40*c^3*d^3*((95*sin(f*x + e)/(cos(f*x + e) + 1) + 145*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 75*sin
(f*x + e)^3/(cos(f*x + e) + 1)^3 + 15*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 22)/(a^3 + 5*a^3*sin(f*x + e)/(cos
(f*x + e) + 1) + 10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a
^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 15*arctan(sin(f*x + e)/(co
s(f*x + e) + 1))/a^3) + 2*c^6*(20*sin(f*x + e)/(cos(f*x + e) + 1) + 40*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3
0*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 15*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 7)/(a^3 + 5*a^3*sin(f*x + e)/
(cos(f*x + e) + 1) + 10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 +
5*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 60*c^4*d^2*(5*sin(f*x
+ e)/(cos(f*x + e) + 1) + 10*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e)
+ 1) + 10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x
+ e)^4/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 36*c^5*d*(5*sin(f*x + e)/(cos(f*x +
e) + 1) + 5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1)/(a^3 + 5*a^3*sin(
f*x + e)/(cos(f*x + e) + 1) + 10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e)
+ 1)^3 + 5*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5))/f
________________________________________________________________________________________
Fricas [B] time = 1.9088, size = 1949, normalized size = 5.51 \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))^6/(a+a*sin(f*x+e))^3,x, algorithm="fricas")
[Out]
1/30*(10*d^6*cos(f*x + e)^6 + 6*c^6 - 36*c^5*d + 90*c^4*d^2 - 120*c^3*d^3 + 90*c^2*d^4 - 36*c*d^5 + 6*d^6 + 15
*(6*c*d^5 - d^6)*cos(f*x + e)^5 - 10*(45*c^2*d^4 - 36*c*d^5 + 14*d^6)*cos(f*x + e)^4 - (4*c^6 + 36*c^5*d + 210
*c^4*d^2 - 1280*c^3*d^3 + 3510*c^2*d^4 - 2694*c*d^5 + 839*d^6 - 15*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^
6)*f*x)*cos(f*x + e)^3 - 60*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x + (8*c^6 + 72*c^5*d - 30*c^4*d^2
- 760*c^3*d^3 + 2520*c^2*d^4 - 2148*c*d^5 + 668*d^6 + 45*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x)*c
os(f*x + e)^2 + 6*(3*c^6 + 12*c^5*d + 45*c^4*d^2 - 360*c^3*d^3 + 945*c^2*d^4 - 768*c*d^5 + 233*d^6 - 5*(40*c^3
*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x)*cos(f*x + e) + (10*d^6*cos(f*x + e)^5 - 6*c^6 + 36*c^5*d - 90*c^4*
d^2 + 120*c^3*d^3 - 90*c^2*d^4 + 36*c*d^5 - 6*d^6 - 5*(18*c*d^5 - 5*d^6)*cos(f*x + e)^4 - 5*(90*c^2*d^4 - 54*c
*d^5 + 23*d^6)*cos(f*x + e)^3 - 60*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x + (4*c^6 + 36*c^5*d + 210
*c^4*d^2 - 1280*c^3*d^3 + 3060*c^2*d^4 - 2424*c*d^5 + 724*d^6 + 15*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^
6)*f*x)*cos(f*x + e)^2 + 6*(2*c^6 + 18*c^5*d + 30*c^4*d^2 - 340*c^3*d^3 + 930*c^2*d^4 - 762*c*d^5 + 232*d^6 -
5*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x)*cos(f*x + e))*sin(f*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3
*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*s
in(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))**6/(a+a*sin(f*x+e))**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.32225, size = 1048, normalized size = 2.96 \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))^6/(a+a*sin(f*x+e))^3,x, algorithm="giac")
[Out]
1/30*(15*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*(f*x + e)/a^3 + 10*(18*c*d^5*tan(1/2*f*x + 1/2*e)^5 - 9
*d^6*tan(1/2*f*x + 1/2*e)^5 - 90*c^2*d^4*tan(1/2*f*x + 1/2*e)^4 + 108*c*d^5*tan(1/2*f*x + 1/2*e)^4 - 36*d^6*ta
n(1/2*f*x + 1/2*e)^4 - 180*c^2*d^4*tan(1/2*f*x + 1/2*e)^2 + 216*c*d^5*tan(1/2*f*x + 1/2*e)^2 - 84*d^6*tan(1/2*
f*x + 1/2*e)^2 - 18*c*d^5*tan(1/2*f*x + 1/2*e) + 9*d^6*tan(1/2*f*x + 1/2*e) - 90*c^2*d^4 + 108*c*d^5 - 40*d^6)
/((tan(1/2*f*x + 1/2*e)^2 + 1)^3*a^3) - 4*(15*c^6*tan(1/2*f*x + 1/2*e)^4 - 300*c^3*d^3*tan(1/2*f*x + 1/2*e)^4
+ 675*c^2*d^4*tan(1/2*f*x + 1/2*e)^4 - 540*c*d^5*tan(1/2*f*x + 1/2*e)^4 + 150*d^6*tan(1/2*f*x + 1/2*e)^4 + 30*
c^6*tan(1/2*f*x + 1/2*e)^3 + 90*c^5*d*tan(1/2*f*x + 1/2*e)^3 - 1500*c^3*d^3*tan(1/2*f*x + 1/2*e)^3 + 3150*c^2*
d^4*tan(1/2*f*x + 1/2*e)^3 - 2430*c*d^5*tan(1/2*f*x + 1/2*e)^3 + 660*d^6*tan(1/2*f*x + 1/2*e)^3 + 40*c^6*tan(1
/2*f*x + 1/2*e)^2 + 90*c^5*d*tan(1/2*f*x + 1/2*e)^2 + 300*c^4*d^2*tan(1/2*f*x + 1/2*e)^2 - 2900*c^3*d^3*tan(1/
2*f*x + 1/2*e)^2 + 5400*c^2*d^4*tan(1/2*f*x + 1/2*e)^2 - 3990*c*d^5*tan(1/2*f*x + 1/2*e)^2 + 1060*d^6*tan(1/2*
f*x + 1/2*e)^2 + 20*c^6*tan(1/2*f*x + 1/2*e) + 90*c^5*d*tan(1/2*f*x + 1/2*e) + 150*c^4*d^2*tan(1/2*f*x + 1/2*e
) - 1900*c^3*d^3*tan(1/2*f*x + 1/2*e) + 3600*c^2*d^4*tan(1/2*f*x + 1/2*e) - 2670*c*d^5*tan(1/2*f*x + 1/2*e) +
710*d^6*tan(1/2*f*x + 1/2*e) + 7*c^6 + 18*c^5*d + 30*c^4*d^2 - 440*c^3*d^3 + 855*c^2*d^4 - 642*c*d^5 + 172*d^6
)/(a^3*(tan(1/2*f*x + 1/2*e) + 1)^5))/f