3.60 \(\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx\)
Optimal. Leaf size=240 \[ \frac{35 c^5 (4 A-7 B) \cos ^3(e+f x)}{4 a^2 f}-\frac{a^5 c^5 (A-B) \cos ^{11}(e+f x)}{3 f (a \sin (e+f x)+a)^7}+\frac{2 a^3 c^5 (4 A-7 B) \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^5}+\frac{6 a^4 c^5 (4 A-7 B) \cos ^7(e+f x)}{f \left (a^2 \sin (e+f x)+a^2\right )^3}+\frac{21 c^5 (4 A-7 B) \cos ^5(e+f x)}{4 f \left (a^2 \sin (e+f x)+a^2\right )}+\frac{105 c^5 (4 A-7 B) \sin (e+f x) \cos (e+f x)}{8 a^2 f}+\frac{105 c^5 x (4 A-7 B)}{8 a^2} \]
[Out]
(105*(4*A - 7*B)*c^5*x)/(8*a^2) + (35*(4*A - 7*B)*c^5*Cos[e + f*x]^3)/(4*a^2*f) + (105*(4*A - 7*B)*c^5*Cos[e +
f*x]*Sin[e + f*x])/(8*a^2*f) - (a^5*(A - B)*c^5*Cos[e + f*x]^11)/(3*f*(a + a*Sin[e + f*x])^7) + (2*a^3*(4*A -
7*B)*c^5*Cos[e + f*x]^9)/(3*f*(a + a*Sin[e + f*x])^5) + (6*a^4*(4*A - 7*B)*c^5*Cos[e + f*x]^7)/(f*(a^2 + a^2*
Sin[e + f*x])^3) + (21*(4*A - 7*B)*c^5*Cos[e + f*x]^5)/(4*f*(a^2 + a^2*Sin[e + f*x]))
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Rubi [A] time = 0.410041, antiderivative size = 240, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 7, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used =
{2967, 2859, 2680, 2679, 2682, 2635, 8} \[ \frac{35 c^5 (4 A-7 B) \cos ^3(e+f x)}{4 a^2 f}-\frac{a^5 c^5 (A-B) \cos ^{11}(e+f x)}{3 f (a \sin (e+f x)+a)^7}+\frac{2 a^3 c^5 (4 A-7 B) \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^5}+\frac{6 a^4 c^5 (4 A-7 B) \cos ^7(e+f x)}{f \left (a^2 \sin (e+f x)+a^2\right )^3}+\frac{21 c^5 (4 A-7 B) \cos ^5(e+f x)}{4 f \left (a^2 \sin (e+f x)+a^2\right )}+\frac{105 c^5 (4 A-7 B) \sin (e+f x) \cos (e+f x)}{8 a^2 f}+\frac{105 c^5 x (4 A-7 B)}{8 a^2} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^5)/(a + a*Sin[e + f*x])^2,x]
[Out]
(105*(4*A - 7*B)*c^5*x)/(8*a^2) + (35*(4*A - 7*B)*c^5*Cos[e + f*x]^3)/(4*a^2*f) + (105*(4*A - 7*B)*c^5*Cos[e +
f*x]*Sin[e + f*x])/(8*a^2*f) - (a^5*(A - B)*c^5*Cos[e + f*x]^11)/(3*f*(a + a*Sin[e + f*x])^7) + (2*a^3*(4*A -
7*B)*c^5*Cos[e + f*x]^9)/(3*f*(a + a*Sin[e + f*x])^5) + (6*a^4*(4*A - 7*B)*c^5*Cos[e + f*x]^7)/(f*(a^2 + a^2*
Sin[e + f*x])^3) + (21*(4*A - 7*B)*c^5*Cos[e + f*x]^5)/(4*f*(a^2 + a^2*Sin[e + f*x]))
Rule 2967
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B
*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I
ntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Rule 2859
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[((b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*(2*m +
p + 1)), x] + Dist[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^
(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[
m + p], 0]) && NeQ[2*m + p + 1, 0]
Rule 2680
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(2*g*(
g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(2*m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(2*m +
p + 1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] && !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]
Rule 2679
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(g*(g*
Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + p)), x] + Dist[(g^2*(p - 1))/(a*(m + p)), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0]
&& LtQ[m, -1] && GtQ[p, 1] && (GtQ[m, -2] || EqQ[2*m + p + 1, 0] || (EqQ[m, -2] && IntegerQ[p])) && NeQ[m + p,
0] && IntegersQ[2*m, 2*p]
Rule 2682
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*(g*Cos[e
+ f*x])^(p - 1))/(b*f*(p - 1)), x] + Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g
}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]
Rule 2635
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rubi steps
\begin{align*} \int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx &=\left (a^5 c^5\right ) \int \frac{\cos ^{10}(e+f x) (A+B \sin (e+f x))}{(a+a \sin (e+f x))^7} \, dx\\ &=-\frac{a^5 (A-B) c^5 \cos ^{11}(e+f x)}{3 f (a+a \sin (e+f x))^7}-\frac{1}{3} \left (a^4 (4 A-7 B) c^5\right ) \int \frac{\cos ^{10}(e+f x)}{(a+a \sin (e+f x))^6} \, dx\\ &=-\frac{a^5 (A-B) c^5 \cos ^{11}(e+f x)}{3 f (a+a \sin (e+f x))^7}+\frac{2 a^3 (4 A-7 B) c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^5}+\left (3 a^2 (4 A-7 B) c^5\right ) \int \frac{\cos ^8(e+f x)}{(a+a \sin (e+f x))^4} \, dx\\ &=-\frac{a^5 (A-B) c^5 \cos ^{11}(e+f x)}{3 f (a+a \sin (e+f x))^7}+\frac{2 a^3 (4 A-7 B) c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac{6 a (4 A-7 B) c^5 \cos ^7(e+f x)}{f (a+a \sin (e+f x))^3}+\left (21 (4 A-7 B) c^5\right ) \int \frac{\cos ^6(e+f x)}{(a+a \sin (e+f x))^2} \, dx\\ &=-\frac{a^5 (A-B) c^5 \cos ^{11}(e+f x)}{3 f (a+a \sin (e+f x))^7}+\frac{2 a^3 (4 A-7 B) c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac{6 a (4 A-7 B) c^5 \cos ^7(e+f x)}{f (a+a \sin (e+f x))^3}+\frac{21 (4 A-7 B) c^5 \cos ^5(e+f x)}{4 f \left (a^2+a^2 \sin (e+f x)\right )}+\frac{\left (105 (4 A-7 B) c^5\right ) \int \frac{\cos ^4(e+f x)}{a+a \sin (e+f x)} \, dx}{4 a}\\ &=\frac{35 (4 A-7 B) c^5 \cos ^3(e+f x)}{4 a^2 f}-\frac{a^5 (A-B) c^5 \cos ^{11}(e+f x)}{3 f (a+a \sin (e+f x))^7}+\frac{2 a^3 (4 A-7 B) c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac{6 a (4 A-7 B) c^5 \cos ^7(e+f x)}{f (a+a \sin (e+f x))^3}+\frac{21 (4 A-7 B) c^5 \cos ^5(e+f x)}{4 f \left (a^2+a^2 \sin (e+f x)\right )}+\frac{\left (105 (4 A-7 B) c^5\right ) \int \cos ^2(e+f x) \, dx}{4 a^2}\\ &=\frac{35 (4 A-7 B) c^5 \cos ^3(e+f x)}{4 a^2 f}+\frac{105 (4 A-7 B) c^5 \cos (e+f x) \sin (e+f x)}{8 a^2 f}-\frac{a^5 (A-B) c^5 \cos ^{11}(e+f x)}{3 f (a+a \sin (e+f x))^7}+\frac{2 a^3 (4 A-7 B) c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac{6 a (4 A-7 B) c^5 \cos ^7(e+f x)}{f (a+a \sin (e+f x))^3}+\frac{21 (4 A-7 B) c^5 \cos ^5(e+f x)}{4 f \left (a^2+a^2 \sin (e+f x)\right )}+\frac{\left (105 (4 A-7 B) c^5\right ) \int 1 \, dx}{8 a^2}\\ &=\frac{105 (4 A-7 B) c^5 x}{8 a^2}+\frac{35 (4 A-7 B) c^5 \cos ^3(e+f x)}{4 a^2 f}+\frac{105 (4 A-7 B) c^5 \cos (e+f x) \sin (e+f x)}{8 a^2 f}-\frac{a^5 (A-B) c^5 \cos ^{11}(e+f x)}{3 f (a+a \sin (e+f x))^7}+\frac{2 a^3 (4 A-7 B) c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac{6 a (4 A-7 B) c^5 \cos ^7(e+f x)}{f (a+a \sin (e+f x))^3}+\frac{21 (4 A-7 B) c^5 \cos ^5(e+f x)}{4 f \left (a^2+a^2 \sin (e+f x)\right )}\\ \end{align*}
Mathematica [A] time = 1.98599, size = 354, normalized size = 1.48 \[ \frac{(c-c \sin (e+f x))^5 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (2048 (A-B) \sin \left (\frac{1}{2} (e+f x)\right )+1260 (4 A-7 B) (e+f x) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3+24 (95 A-217 B) \cos (e+f x) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3-8 (A-7 B) \cos (3 (e+f x)) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3-24 (7 A-24 B) \sin (2 (e+f x)) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3-1024 (13 A-19 B) \sin \left (\frac{1}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2-1024 (A-B) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )-3 B \sin (4 (e+f x)) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3\right )}{96 a^2 f (\sin (e+f x)+1)^2 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{10}} \]
Antiderivative was successfully verified.
[In]
Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^5)/(a + a*Sin[e + f*x])^2,x]
[Out]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^5*(2048*(A - B)*Sin[(e + f*x)/2] - 1024*(A - B)*(C
os[(e + f*x)/2] + Sin[(e + f*x)/2]) - 1024*(13*A - 19*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]
)^2 + 1260*(4*A - 7*B)*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 24*(95*A - 217*B)*Cos[e + f*x]*(Cos
[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 8*(A - 7*B)*Cos[3*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 2
4*(7*A - 24*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sin[2*(e + f*x)] - 3*B*(Cos[(e + f*x)/2] + Sin[(e + f*x
)/2])^3*Sin[4*(e + f*x)]))/(96*a^2*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10*(1 + Sin[e + f*x])^2)
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Maple [B] time = 0.161, size = 778, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x)
[Out]
7/f*c^5/a^2/(1+tan(1/2*f*x+1/2*e)^2)^4*tan(1/2*f*x+1/2*e)^7*A-95/4/f*c^5/a^2/(1+tan(1/2*f*x+1/2*e)^2)^4*tan(1/
2*f*x+1/2*e)^7*B+46/f*c^5/a^2/(1+tan(1/2*f*x+1/2*e)^2)^4*tan(1/2*f*x+1/2*e)^6*A-98/f*c^5/a^2/(1+tan(1/2*f*x+1/
2*e)^2)^4*tan(1/2*f*x+1/2*e)^6*B+7/f*c^5/a^2/(1+tan(1/2*f*x+1/2*e)^2)^4*tan(1/2*f*x+1/2*e)^5*A-103/4/f*c^5/a^2
/(1+tan(1/2*f*x+1/2*e)^2)^4*tan(1/2*f*x+1/2*e)^5*B+142/f*c^5/a^2/(1+tan(1/2*f*x+1/2*e)^2)^4*tan(1/2*f*x+1/2*e)
^4*A-322/f*c^5/a^2/(1+tan(1/2*f*x+1/2*e)^2)^4*tan(1/2*f*x+1/2*e)^4*B-7/f*c^5/a^2/(1+tan(1/2*f*x+1/2*e)^2)^4*ta
n(1/2*f*x+1/2*e)^3*A+103/4/f*c^5/a^2/(1+tan(1/2*f*x+1/2*e)^2)^4*tan(1/2*f*x+1/2*e)^3*B+430/3/f*c^5/a^2/(1+tan(
1/2*f*x+1/2*e)^2)^4*tan(1/2*f*x+1/2*e)^2*A-994/3/f*c^5/a^2/(1+tan(1/2*f*x+1/2*e)^2)^4*tan(1/2*f*x+1/2*e)^2*B-7
/f*c^5/a^2/(1+tan(1/2*f*x+1/2*e)^2)^4*tan(1/2*f*x+1/2*e)*A+95/4/f*c^5/a^2/(1+tan(1/2*f*x+1/2*e)^2)^4*tan(1/2*f
*x+1/2*e)*B+142/3/f*c^5/a^2/(1+tan(1/2*f*x+1/2*e)^2)^4*A-322/3/f*c^5/a^2/(1+tan(1/2*f*x+1/2*e)^2)^4*B+105/f*c^
5/a^2*arctan(tan(1/2*f*x+1/2*e))*A-735/4/f*c^5/a^2*arctan(tan(1/2*f*x+1/2*e))*B+64/f*c^5/a^2/(tan(1/2*f*x+1/2*
e)+1)^2*A-64/f*c^5/a^2/(tan(1/2*f*x+1/2*e)+1)^2*B+96/f*c^5/a^2/(tan(1/2*f*x+1/2*e)+1)*A-160/f*c^5/a^2/(tan(1/2
*f*x+1/2*e)+1)*B-128/3/f*c^5/a^2/(tan(1/2*f*x+1/2*e)+1)^3*A+128/3/f*c^5/a^2/(tan(1/2*f*x+1/2*e)+1)^3*B
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Maxima [B] time = 1.76675, size = 4026, normalized size = 16.77 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x, algorithm="maxima")
[Out]
-1/12*(B*c^5*((603*sin(f*x + e)/(cos(f*x + e) + 1) + 1297*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2228*sin(f*x +
e)^3/(cos(f*x + e) + 1)^3 + 2628*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 3014*sin(f*x + e)^5/(cos(f*x + e) + 1)
^5 + 2618*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 1980*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 1100*sin(f*x + e)^8
/(cos(f*x + e) + 1)^8 + 495*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 165*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 +
256)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 7*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 13*a^2*sin(f*x
+ e)^3/(cos(f*x + e) + 1)^3 + 18*a^2*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 22*a^2*sin(f*x + e)^5/(cos(f*x + e
) + 1)^5 + 22*a^2*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 18*a^2*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 13*a^2*si
n(f*x + e)^8/(cos(f*x + e) + 1)^8 + 7*a^2*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3*a^2*sin(f*x + e)^10/(cos(f*x
+ e) + 1)^10 + a^2*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) + 165*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2)
- 20*A*c^5*((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*si
n(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/(c
os(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/(cos
(f*x + e) + 1)^7) + 21*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) + 40*B*c^5*((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/(cos(f*x + e) + 1)^7) + 21*arctan(sin(f*x + e)
/(cos(f*x + e) + 1))/a^2) - 8*A*c^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*sin(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)/(cos(f*x + e) + 1) + 6*a^2*sin(f*x + e)^2/(co
s(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*B*c^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*sin(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)/(cos(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) - 160*A*c^5*((12*sin(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) + 160*B*c^5*((12*sin(f*x + e)/(co
s(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) - 80*A*c^5*((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*arctan(sin(f*x + e)/(
cos(f*x + e) + 1))/a^2) + 40*B*c^5*((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*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) + 8*A*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) - 40*A*c^5*(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) + 8*B*c^5*(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 [A] time = 1.52659, size = 922, normalized size = 3.84 \begin{align*} -\frac{6 \, B c^{5} \cos \left (f x + e\right )^{6} + 4 \,{\left (2 \, A - 11 \, B\right )} c^{5} \cos \left (f x + e\right )^{5} +{\left (76 \, A - 241 \, B\right )} c^{5} \cos \left (f x + e\right )^{4} - 2 \,{\left (212 \, A - 431 \, B\right )} c^{5} \cos \left (f x + e\right )^{3} + 630 \,{\left (4 \, A - 7 \, B\right )} c^{5} f x - 256 \,{\left (A - B\right )} c^{5} -{\left (315 \,{\left (4 \, A - 7 \, B\right )} c^{5} f x -{\left (2156 \, A - 3485 \, B\right )} c^{5}\right )} \cos \left (f x + e\right )^{2} +{\left (315 \,{\left (4 \, A - 7 \, B\right )} c^{5} f x + 2 \,{\left (1196 \, A - 2141 \, B\right )} c^{5}\right )} \cos \left (f x + e\right ) +{\left (6 \, B c^{5} \cos \left (f x + e\right )^{5} - 2 \,{\left (4 \, A - 25 \, B\right )} c^{5} \cos \left (f x + e\right )^{4} +{\left (68 \, A - 191 \, B\right )} c^{5} \cos \left (f x + e\right )^{3} + 630 \,{\left (4 \, A - 7 \, B\right )} c^{5} f x + 3 \,{\left (164 \, A - 351 \, B\right )} c^{5} \cos \left (f x + e\right )^{2} + 256 \,{\left (A - B\right )} c^{5} +{\left (315 \,{\left (4 \, A - 7 \, B\right )} c^{5} f x + 2 \,{\left (1324 \, A - 2269 \, B\right )} c^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \,{\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((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x, algorithm="fricas")
[Out]
-1/24*(6*B*c^5*cos(f*x + e)^6 + 4*(2*A - 11*B)*c^5*cos(f*x + e)^5 + (76*A - 241*B)*c^5*cos(f*x + e)^4 - 2*(212
*A - 431*B)*c^5*cos(f*x + e)^3 + 630*(4*A - 7*B)*c^5*f*x - 256*(A - B)*c^5 - (315*(4*A - 7*B)*c^5*f*x - (2156*
A - 3485*B)*c^5)*cos(f*x + e)^2 + (315*(4*A - 7*B)*c^5*f*x + 2*(1196*A - 2141*B)*c^5)*cos(f*x + e) + (6*B*c^5*
cos(f*x + e)^5 - 2*(4*A - 25*B)*c^5*cos(f*x + e)^4 + (68*A - 191*B)*c^5*cos(f*x + e)^3 + 630*(4*A - 7*B)*c^5*f
*x + 3*(164*A - 351*B)*c^5*cos(f*x + e)^2 + 256*(A - B)*c^5 + (315*(4*A - 7*B)*c^5*f*x + 2*(1324*A - 2269*B)*c
^5)*cos(f*x + e))*sin(f*x + e))/(a^2*f*cos(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((A+B*sin(f*x+e))*(c-c*sin(f*x+e))**5/(a+a*sin(f*x+e))**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [A] time = 1.22966, size = 556, normalized size = 2.32 \begin{align*} \frac{\frac{315 \,{\left (4 \, A c^{5} - 7 \, B c^{5}\right )}{\left (f x + e\right )}}{a^{2}} + \frac{256 \,{\left (9 \, A c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 15 \, B c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 24 \, A c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 36 \, B c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 11 \, A c^{5} - 17 \, B c^{5}\right )}}{a^{2}{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )}^{3}} + \frac{2 \,{\left (84 \, A c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} - 285 \, B c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 552 \, A c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 1176 \, B c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} + 84 \, A c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 309 \, B c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 1704 \, A c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 3864 \, B c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 84 \, A c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 309 \, B c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 1720 \, A c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 3976 \, B c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 84 \, A c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 285 \, B c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 568 \, A c^{5} - 1288 \, B c^{5}\right )}}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1\right )}^{4} a^{2}}}{24 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x, algorithm="giac")
[Out]
1/24*(315*(4*A*c^5 - 7*B*c^5)*(f*x + e)/a^2 + 256*(9*A*c^5*tan(1/2*f*x + 1/2*e)^2 - 15*B*c^5*tan(1/2*f*x + 1/2
*e)^2 + 24*A*c^5*tan(1/2*f*x + 1/2*e) - 36*B*c^5*tan(1/2*f*x + 1/2*e) + 11*A*c^5 - 17*B*c^5)/(a^2*(tan(1/2*f*x
+ 1/2*e) + 1)^3) + 2*(84*A*c^5*tan(1/2*f*x + 1/2*e)^7 - 285*B*c^5*tan(1/2*f*x + 1/2*e)^7 + 552*A*c^5*tan(1/2*
f*x + 1/2*e)^6 - 1176*B*c^5*tan(1/2*f*x + 1/2*e)^6 + 84*A*c^5*tan(1/2*f*x + 1/2*e)^5 - 309*B*c^5*tan(1/2*f*x +
1/2*e)^5 + 1704*A*c^5*tan(1/2*f*x + 1/2*e)^4 - 3864*B*c^5*tan(1/2*f*x + 1/2*e)^4 - 84*A*c^5*tan(1/2*f*x + 1/2
*e)^3 + 309*B*c^5*tan(1/2*f*x + 1/2*e)^3 + 1720*A*c^5*tan(1/2*f*x + 1/2*e)^2 - 3976*B*c^5*tan(1/2*f*x + 1/2*e)
^2 - 84*A*c^5*tan(1/2*f*x + 1/2*e) + 285*B*c^5*tan(1/2*f*x + 1/2*e) + 568*A*c^5 - 1288*B*c^5)/((tan(1/2*f*x +
1/2*e)^2 + 1)^4*a^2))/f