3.36 \(\int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx\)
Optimal. Leaf size=156 \[ \frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac{2 a^2 (3 A-8 B) \cos ^5(e+f x)}{3465 c f (c-c \sin (e+f x))^5}+\frac{2 a^2 (3 A-8 B) \cos ^5(e+f x)}{693 f (c-c \sin (e+f x))^6}+\frac{a^2 c (3 A-8 B) \cos ^5(e+f x)}{99 f (c-c \sin (e+f x))^7} \]
[Out]
(a^2*(A + B)*c^2*Cos[e + f*x]^5)/(11*f*(c - c*Sin[e + f*x])^8) + (a^2*(3*A - 8*B)*c*Cos[e + f*x]^5)/(99*f*(c -
c*Sin[e + f*x])^7) + (2*a^2*(3*A - 8*B)*Cos[e + f*x]^5)/(693*f*(c - c*Sin[e + f*x])^6) + (2*a^2*(3*A - 8*B)*C
os[e + f*x]^5)/(3465*c*f*(c - c*Sin[e + f*x])^5)
________________________________________________________________________________________
Rubi [A] time = 0.374481, antiderivative size = 156, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used =
{2967, 2859, 2672, 2671} \[ \frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac{2 a^2 (3 A-8 B) \cos ^5(e+f x)}{3465 c f (c-c \sin (e+f x))^5}+\frac{2 a^2 (3 A-8 B) \cos ^5(e+f x)}{693 f (c-c \sin (e+f x))^6}+\frac{a^2 c (3 A-8 B) \cos ^5(e+f x)}{99 f (c-c \sin (e+f x))^7} \]
Antiderivative was successfully verified.
[In]
Int[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^6,x]
[Out]
(a^2*(A + B)*c^2*Cos[e + f*x]^5)/(11*f*(c - c*Sin[e + f*x])^8) + (a^2*(3*A - 8*B)*c*Cos[e + f*x]^5)/(99*f*(c -
c*Sin[e + f*x])^7) + (2*a^2*(3*A - 8*B)*Cos[e + f*x]^5)/(693*f*(c - c*Sin[e + f*x])^6) + (2*a^2*(3*A - 8*B)*C
os[e + f*x]^5)/(3465*c*f*(c - c*Sin[e + f*x])^5)
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 2672
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*Simplify[2*m + p + 1]), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[2*m + p + 1]), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m
, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && !IGtQ[m, 0]
Rule 2671
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*m), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^
2, 0] && EqQ[Simplify[m + p + 1], 0] && !ILtQ[p, 0]
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx &=\left (a^2 c^2\right ) \int \frac{\cos ^4(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^8} \, dx\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac{1}{11} \left (a^2 (3 A-8 B) c\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^7} \, dx\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac{a^2 (3 A-8 B) c \cos ^5(e+f x)}{99 f (c-c \sin (e+f x))^7}+\frac{1}{99} \left (2 a^2 (3 A-8 B)\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^6} \, dx\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac{a^2 (3 A-8 B) c \cos ^5(e+f x)}{99 f (c-c \sin (e+f x))^7}+\frac{2 a^2 (3 A-8 B) \cos ^5(e+f x)}{693 f (c-c \sin (e+f x))^6}+\frac{\left (2 a^2 (3 A-8 B)\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^5} \, dx}{693 c}\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac{a^2 (3 A-8 B) c \cos ^5(e+f x)}{99 f (c-c \sin (e+f x))^7}+\frac{2 a^2 (3 A-8 B) \cos ^5(e+f x)}{693 f (c-c \sin (e+f x))^6}+\frac{2 a^2 (3 A-8 B) \cos ^5(e+f x)}{3465 c f (c-c \sin (e+f x))^5}\\ \end{align*}
Mathematica [A] time = 1.54175, size = 285, normalized size = 1.83 \[ \frac{a^2 (\sin (e+f x)+1)^2 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (231 (27 A+28 B) \cos \left (\frac{1}{2} (e+f x)\right )-2475 (A+2 B) \cos \left (\frac{3}{2} (e+f x)\right )+7623 A \sin \left (\frac{1}{2} (e+f x)\right )+3465 A \sin \left (\frac{3}{2} (e+f x)\right )-495 A \sin \left (\frac{5}{2} (e+f x)\right )+33 A \sin \left (\frac{9}{2} (e+f x)\right )-165 A \cos \left (\frac{7}{2} (e+f x)\right )+3 A \cos \left (\frac{11}{2} (e+f x)\right )+2772 B \sin \left (\frac{1}{2} (e+f x)\right )+2310 B \sin \left (\frac{3}{2} (e+f x)\right )-990 B \sin \left (\frac{5}{2} (e+f x)\right )-88 B \sin \left (\frac{9}{2} (e+f x)\right )-2310 B \cos \left (\frac{5}{2} (e+f x)\right )+440 B \cos \left (\frac{7}{2} (e+f x)\right )-8 B \cos \left (\frac{11}{2} (e+f x)\right )\right )}{27720 c^6 f (\sin (e+f x)-1)^6 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^6,x]
[Out]
(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2*(231*(27*A + 28*B)*Cos[(e + f*x)/2] - 2475*(A
+ 2*B)*Cos[(3*(e + f*x))/2] - 2310*B*Cos[(5*(e + f*x))/2] - 165*A*Cos[(7*(e + f*x))/2] + 440*B*Cos[(7*(e + f*x
))/2] + 3*A*Cos[(11*(e + f*x))/2] - 8*B*Cos[(11*(e + f*x))/2] + 7623*A*Sin[(e + f*x)/2] + 2772*B*Sin[(e + f*x)
/2] + 3465*A*Sin[(3*(e + f*x))/2] + 2310*B*Sin[(3*(e + f*x))/2] - 495*A*Sin[(5*(e + f*x))/2] - 990*B*Sin[(5*(e
+ f*x))/2] + 33*A*Sin[(9*(e + f*x))/2] - 88*B*Sin[(9*(e + f*x))/2]))/(27720*c^6*f*(Cos[(e + f*x)/2] + Sin[(e
+ f*x)/2])^4*(-1 + Sin[e + f*x])^6)
________________________________________________________________________________________
Maple [A] time = 0.153, size = 249, normalized size = 1.6 \begin{align*} 2\,{\frac{{a}^{2}}{f{c}^{6}} \left ( -1/3\,{\frac{90\,A+26\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{3}}}-1/6\,{\frac{1752\,A+1208\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{6}}}-1/9\,{\frac{1536\,A+1472\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{9}}}-1/4\,{\frac{352\,A+152\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{4}}}-1/8\,{\frac{2304\,A+2048\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{8}}}-1/7\,{\frac{2376\,A+1896\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{7}}}-1/11\,{\frac{128\,A+128\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{11}}}-1/10\,{\frac{640\,A+640\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{10}}}-1/2\,{\frac{14\,A+2\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{2}}}-{\frac{A}{\tan \left ( 1/2\,fx+e/2 \right ) -1}}-1/5\,{\frac{932\,A+528\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{5}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^6,x)
[Out]
2/f*a^2/c^6*(-1/3*(90*A+26*B)/(tan(1/2*f*x+1/2*e)-1)^3-1/6*(1752*A+1208*B)/(tan(1/2*f*x+1/2*e)-1)^6-1/9*(1536*
A+1472*B)/(tan(1/2*f*x+1/2*e)-1)^9-1/4*(352*A+152*B)/(tan(1/2*f*x+1/2*e)-1)^4-1/8*(2304*A+2048*B)/(tan(1/2*f*x
+1/2*e)-1)^8-1/7*(2376*A+1896*B)/(tan(1/2*f*x+1/2*e)-1)^7-1/11*(128*A+128*B)/(tan(1/2*f*x+1/2*e)-1)^11-1/10*(6
40*A+640*B)/(tan(1/2*f*x+1/2*e)-1)^10-1/2*(14*A+2*B)/(tan(1/2*f*x+1/2*e)-1)^2-A/(tan(1/2*f*x+1/2*e)-1)-1/5*(93
2*A+528*B)/(tan(1/2*f*x+1/2*e)-1)^5)
________________________________________________________________________________________
Maxima [B] time = 1.37671, size = 3515, normalized size = 22.53 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^6,x, algorithm="maxima")
[Out]
-2/3465*(5*A*a^2*(913*sin(f*x + e)/(cos(f*x + e) + 1) - 4565*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 12540*sin(f
*x + e)^3/(cos(f*x + e) + 1)^3 - 25080*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 33726*sin(f*x + e)^5/(cos(f*x + e
) + 1)^5 - 33726*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 23100*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 11550*sin(f
*x + e)^8/(cos(f*x + e) + 1)^8 + 3465*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 693*sin(f*x + e)^10/(cos(f*x + e)
+ 1)^10 - 146)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 16
5*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)
^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x + e) +
1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(
f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) - 6*A*a^2*(671*sin(f*x + e)/(co
s(f*x + e) + 1) - 2200*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 6600*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 10890*
sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 15246*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 12936*sin(f*x + e)^6/(cos(f*
x + e) + 1)^6 + 9240*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 3465*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 1155*sin
(f*x + e)^9/(cos(f*x + e) + 1)^9 - 61)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(
cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^
4 - 462*c^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*
x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x +
e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) - 3*B*a
^2*(671*sin(f*x + e)/(cos(f*x + e) + 1) - 2200*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 6600*sin(f*x + e)^3/(cos(
f*x + e) + 1)^3 - 10890*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 15246*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 1293
6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 9240*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 3465*sin(f*x + e)^8/(cos(f*
x + e) + 1)^8 + 1155*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 61)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) +
55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x +
e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e)
+ 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*s
in(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f
*x + e) + 1)^11) - 2*B*a^2*(341*sin(f*x + e)/(cos(f*x + e) + 1) - 1705*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 5
115*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 6765*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 9471*sin(f*x + e)^5/(cos(
f*x + e) + 1)^5 - 4851*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 3465*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 31)/(c
^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e
)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)^5/(cos(f*x + e)
+ 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*s
in(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(
f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) + 4*A*a^2*(253*sin(f*x + e)/(cos(f*x + e) + 1) -
1265*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2640*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 5280*sin(f*x + e)^4/(co
s(f*x + e) + 1)^4 + 5313*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 5313*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 2310
*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 1155*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 23)/(c^6 - 11*c^6*sin(f*x +
e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)
^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f
*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x
+ e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6
*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) + 8*B*a^2*(253*sin(f*x + e)/(cos(f*x + e) + 1) - 1265*sin(f*x + e)^2/(
cos(f*x + e) + 1)^2 + 2640*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 5280*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 53
13*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 5313*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 2310*sin(f*x + e)^7/(cos(f
*x + e) + 1)^7 - 1155*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 23)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1)
+ 55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x +
e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e
) + 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*
sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(
f*x + e) + 1)^11))/f
________________________________________________________________________________________
Fricas [B] time = 1.46612, size = 1027, normalized size = 6.58 \begin{align*} -\frac{2 \,{\left (3 \, A - 8 \, B\right )} a^{2} \cos \left (f x + e\right )^{6} + 12 \,{\left (3 \, A - 8 \, B\right )} a^{2} \cos \left (f x + e\right )^{5} - 25 \,{\left (3 \, A - 8 \, B\right )} a^{2} \cos \left (f x + e\right )^{4} - 35 \,{\left (6 \, A + 17 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} - 35 \,{\left (21 \, A + 43 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} + 630 \,{\left (A + B\right )} a^{2} \cos \left (f x + e\right ) + 1260 \,{\left (A + B\right )} a^{2} -{\left (2 \,{\left (3 \, A - 8 \, B\right )} a^{2} \cos \left (f x + e\right )^{5} - 10 \,{\left (3 \, A - 8 \, B\right )} a^{2} \cos \left (f x + e\right )^{4} - 35 \,{\left (3 \, A - 8 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} + 35 \,{\left (3 \, A + 25 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} - 630 \,{\left (A + B\right )} a^{2} \cos \left (f x + e\right ) - 1260 \,{\left (A + B\right )} a^{2}\right )} \sin \left (f x + e\right )}{3465 \,{\left (c^{6} f \cos \left (f x + e\right )^{6} - 5 \, c^{6} f \cos \left (f x + e\right )^{5} - 18 \, c^{6} f \cos \left (f x + e\right )^{4} + 20 \, c^{6} f \cos \left (f x + e\right )^{3} + 48 \, c^{6} f \cos \left (f x + e\right )^{2} - 16 \, c^{6} f \cos \left (f x + e\right ) - 32 \, c^{6} f +{\left (c^{6} f \cos \left (f x + e\right )^{5} + 6 \, c^{6} f \cos \left (f x + e\right )^{4} - 12 \, c^{6} f \cos \left (f x + e\right )^{3} - 32 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f \cos \left (f x + e\right ) + 32 \, c^{6} f\right )} \sin \left (f x + e\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^6,x, algorithm="fricas")
[Out]
-1/3465*(2*(3*A - 8*B)*a^2*cos(f*x + e)^6 + 12*(3*A - 8*B)*a^2*cos(f*x + e)^5 - 25*(3*A - 8*B)*a^2*cos(f*x + e
)^4 - 35*(6*A + 17*B)*a^2*cos(f*x + e)^3 - 35*(21*A + 43*B)*a^2*cos(f*x + e)^2 + 630*(A + B)*a^2*cos(f*x + e)
+ 1260*(A + B)*a^2 - (2*(3*A - 8*B)*a^2*cos(f*x + e)^5 - 10*(3*A - 8*B)*a^2*cos(f*x + e)^4 - 35*(3*A - 8*B)*a^
2*cos(f*x + e)^3 + 35*(3*A + 25*B)*a^2*cos(f*x + e)^2 - 630*(A + B)*a^2*cos(f*x + e) - 1260*(A + B)*a^2)*sin(f
*x + e))/(c^6*f*cos(f*x + e)^6 - 5*c^6*f*cos(f*x + e)^5 - 18*c^6*f*cos(f*x + e)^4 + 20*c^6*f*cos(f*x + e)^3 +
48*c^6*f*cos(f*x + e)^2 - 16*c^6*f*cos(f*x + e) - 32*c^6*f + (c^6*f*cos(f*x + e)^5 + 6*c^6*f*cos(f*x + e)^4 -
12*c^6*f*cos(f*x + e)^3 - 32*c^6*f*cos(f*x + e)^2 + 16*c^6*f*cos(f*x + e) + 32*c^6*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+a*sin(f*x+e))**2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**6,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.23176, size = 504, normalized size = 3.23 \begin{align*} -\frac{2 \,{\left (3465 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{10} - 10395 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9} + 3465 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9} + 41580 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 1155 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 69300 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 16170 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 112266 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 6006 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 98406 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 22176 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 81180 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 3960 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 33660 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 8910 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 14685 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 110 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1551 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 671 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 456 \, A a^{2} - 61 \, B a^{2}\right )}}{3465 \, c^{6} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1\right )}^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^6,x, algorithm="giac")
[Out]
-2/3465*(3465*A*a^2*tan(1/2*f*x + 1/2*e)^10 - 10395*A*a^2*tan(1/2*f*x + 1/2*e)^9 + 3465*B*a^2*tan(1/2*f*x + 1/
2*e)^9 + 41580*A*a^2*tan(1/2*f*x + 1/2*e)^8 - 1155*B*a^2*tan(1/2*f*x + 1/2*e)^8 - 69300*A*a^2*tan(1/2*f*x + 1/
2*e)^7 + 16170*B*a^2*tan(1/2*f*x + 1/2*e)^7 + 112266*A*a^2*tan(1/2*f*x + 1/2*e)^6 - 6006*B*a^2*tan(1/2*f*x + 1
/2*e)^6 - 98406*A*a^2*tan(1/2*f*x + 1/2*e)^5 + 22176*B*a^2*tan(1/2*f*x + 1/2*e)^5 + 81180*A*a^2*tan(1/2*f*x +
1/2*e)^4 - 3960*B*a^2*tan(1/2*f*x + 1/2*e)^4 - 33660*A*a^2*tan(1/2*f*x + 1/2*e)^3 + 8910*B*a^2*tan(1/2*f*x + 1
/2*e)^3 + 14685*A*a^2*tan(1/2*f*x + 1/2*e)^2 + 110*B*a^2*tan(1/2*f*x + 1/2*e)^2 - 1551*A*a^2*tan(1/2*f*x + 1/2
*e) + 671*B*a^2*tan(1/2*f*x + 1/2*e) + 456*A*a^2 - 61*B*a^2)/(c^6*f*(tan(1/2*f*x + 1/2*e) - 1)^11)