3.51 \(\int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^8} \, dx\)
Optimal. Leaf size=197 \[ \frac{a^3 c^2 (4 A-11 B) \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac{a^3 c^3 (A+B) \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac{2 a^3 (4 A-11 B) \cos ^7(e+f x)}{45045 c f (c-c \sin (e+f x))^7}+\frac{2 a^3 (4 A-11 B) \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac{a^3 c (4 A-11 B) \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9} \]
[Out]
(a^3*(A + B)*c^3*Cos[e + f*x]^7)/(15*f*(c - c*Sin[e + f*x])^11) + (a^3*(4*A - 11*B)*c^2*Cos[e + f*x]^7)/(195*f
*(c - c*Sin[e + f*x])^10) + (a^3*(4*A - 11*B)*c*Cos[e + f*x]^7)/(715*f*(c - c*Sin[e + f*x])^9) + (2*a^3*(4*A -
11*B)*Cos[e + f*x]^7)/(6435*f*(c - c*Sin[e + f*x])^8) + (2*a^3*(4*A - 11*B)*Cos[e + f*x]^7)/(45045*c*f*(c - c
*Sin[e + f*x])^7)
________________________________________________________________________________________
Rubi [A] time = 0.44408, antiderivative size = 197, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 4, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used =
{2967, 2859, 2672, 2671} \[ \frac{a^3 c^2 (4 A-11 B) \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac{a^3 c^3 (A+B) \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac{2 a^3 (4 A-11 B) \cos ^7(e+f x)}{45045 c f (c-c \sin (e+f x))^7}+\frac{2 a^3 (4 A-11 B) \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac{a^3 c (4 A-11 B) \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9} \]
Antiderivative was successfully verified.
[In]
Int[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^8,x]
[Out]
(a^3*(A + B)*c^3*Cos[e + f*x]^7)/(15*f*(c - c*Sin[e + f*x])^11) + (a^3*(4*A - 11*B)*c^2*Cos[e + f*x]^7)/(195*f
*(c - c*Sin[e + f*x])^10) + (a^3*(4*A - 11*B)*c*Cos[e + f*x]^7)/(715*f*(c - c*Sin[e + f*x])^9) + (2*a^3*(4*A -
11*B)*Cos[e + f*x]^7)/(6435*f*(c - c*Sin[e + f*x])^8) + (2*a^3*(4*A - 11*B)*Cos[e + f*x]^7)/(45045*c*f*(c - c
*Sin[e + f*x])^7)
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))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^8} \, dx &=\left (a^3 c^3\right ) \int \frac{\cos ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11}} \, dx\\ &=\frac{a^3 (A+B) c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac{1}{15} \left (a^3 (4 A-11 B) c^2\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^{10}} \, dx\\ &=\frac{a^3 (A+B) c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac{a^3 (4 A-11 B) c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac{1}{65} \left (a^3 (4 A-11 B) c\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^9} \, dx\\ &=\frac{a^3 (A+B) c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac{a^3 (4 A-11 B) c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac{a^3 (4 A-11 B) c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9}+\frac{1}{715} \left (2 a^3 (4 A-11 B)\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^8} \, dx\\ &=\frac{a^3 (A+B) c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac{a^3 (4 A-11 B) c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac{a^3 (4 A-11 B) c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9}+\frac{2 a^3 (4 A-11 B) \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac{\left (2 a^3 (4 A-11 B)\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^7} \, dx}{6435 c}\\ &=\frac{a^3 (A+B) c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac{a^3 (4 A-11 B) c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac{a^3 (4 A-11 B) c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9}+\frac{2 a^3 (4 A-11 B) \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac{2 a^3 (4 A-11 B) \cos ^7(e+f x)}{45045 c f (c-c \sin (e+f x))^7}\\ \end{align*}
Mathematica [A] time = 6.64529, size = 378, normalized size = 1.92 \[ \frac{(a \sin (e+f x)+a)^3 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (437580 A \sin \left (\frac{1}{2} (e+f x)\right )+240240 A \sin \left (\frac{3}{2} (e+f x)\right )-60060 A \sin \left (\frac{5}{2} (e+f x)\right )-1820 A \sin \left (\frac{9}{2} (e+f x)\right )+60 A \sin \left (\frac{13}{2} (e+f x)\right )+463320 A \cos \left (\frac{1}{2} (e+f x)\right )-260260 A \cos \left (\frac{3}{2} (e+f x)\right )-72072 A \cos \left (\frac{5}{2} (e+f x)\right )+5460 A \cos \left (\frac{7}{2} (e+f x)\right )-420 A \cos \left (\frac{11}{2} (e+f x)\right )+4 A \cos \left (\frac{15}{2} (e+f x)\right )+373230 B \sin \left (\frac{1}{2} (e+f x)\right )+285285 B \sin \left (\frac{3}{2} (e+f x)\right )-150150 B \sin \left (\frac{5}{2} (e+f x)\right )-45045 B \sin \left (\frac{7}{2} (e+f x)\right )+5005 B \sin \left (\frac{9}{2} (e+f x)\right )-165 B \sin \left (\frac{13}{2} (e+f x)\right )+302445 B \cos \left (\frac{1}{2} (e+f x)\right )-230230 B \cos \left (\frac{3}{2} (e+f x)\right )-117117 B \cos \left (\frac{5}{2} (e+f x)\right )+30030 B \cos \left (\frac{7}{2} (e+f x)\right )+1155 B \cos \left (\frac{11}{2} (e+f x)\right )-11 B \cos \left (\frac{15}{2} (e+f x)\right )\right )}{1441440 f (c-c \sin (e+f x))^8 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^6} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^8,x]
[Out]
((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a + a*Sin[e + f*x])^3*(463320*A*Cos[(e + f*x)/2] + 302445*B*Cos[(e + f
*x)/2] - 260260*A*Cos[(3*(e + f*x))/2] - 230230*B*Cos[(3*(e + f*x))/2] - 72072*A*Cos[(5*(e + f*x))/2] - 117117
*B*Cos[(5*(e + f*x))/2] + 5460*A*Cos[(7*(e + f*x))/2] + 30030*B*Cos[(7*(e + f*x))/2] - 420*A*Cos[(11*(e + f*x)
)/2] + 1155*B*Cos[(11*(e + f*x))/2] + 4*A*Cos[(15*(e + f*x))/2] - 11*B*Cos[(15*(e + f*x))/2] + 437580*A*Sin[(e
+ f*x)/2] + 373230*B*Sin[(e + f*x)/2] + 240240*A*Sin[(3*(e + f*x))/2] + 285285*B*Sin[(3*(e + f*x))/2] - 60060
*A*Sin[(5*(e + f*x))/2] - 150150*B*Sin[(5*(e + f*x))/2] - 45045*B*Sin[(7*(e + f*x))/2] - 1820*A*Sin[(9*(e + f*
x))/2] + 5005*B*Sin[(9*(e + f*x))/2] + 60*A*Sin[(13*(e + f*x))/2] - 165*B*Sin[(13*(e + f*x))/2]))/(1441440*f*(
Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c - c*Sin[e + f*x])^8)
________________________________________________________________________________________
Maple [A] time = 0.201, size = 337, normalized size = 1.7 \begin{align*} 2\,{\frac{{a}^{3}}{f{c}^{8}} \left ( -1/10\,{\frac{94144\,A+78144\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{10}}}-1/3\,{\frac{188\,A+38\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{3}}}-1/4\,{\frac{1104\,A+336\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{4}}}-1/14\,{\frac{7168\,A+7168\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{14}}}-1/13\,{\frac{24320\,A+23808\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{13}}}-1/12\,{\frac{52736\,A+49664\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{12}}}-1/7\,{\frac{32288\,A+19176\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{7}}}-1/5\,{\frac{4536\,A+1836\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{5}}}-1/8\,{\frac{58816\,A+40000\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{8}}}-1/11\,{\frac{81344\,A+72512\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{11}}}-1/15\,{\frac{1024\,A+1024\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{15}}}-1/2\,{\frac{20\,A+2\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{2}}}-1/6\,{\frac{13824\,A+6936\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{6}}}-1/9\,{\frac{84112\,A+63856\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{9}}}-{\frac{A}{\tan \left ( 1/2\,fx+e/2 \right ) -1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^8,x)
[Out]
2/f*a^3/c^8*(-1/10*(94144*A+78144*B)/(tan(1/2*f*x+1/2*e)-1)^10-1/3*(188*A+38*B)/(tan(1/2*f*x+1/2*e)-1)^3-1/4*(
1104*A+336*B)/(tan(1/2*f*x+1/2*e)-1)^4-1/14*(7168*A+7168*B)/(tan(1/2*f*x+1/2*e)-1)^14-1/13*(24320*A+23808*B)/(
tan(1/2*f*x+1/2*e)-1)^13-1/12*(52736*A+49664*B)/(tan(1/2*f*x+1/2*e)-1)^12-1/7*(32288*A+19176*B)/(tan(1/2*f*x+1
/2*e)-1)^7-1/5*(4536*A+1836*B)/(tan(1/2*f*x+1/2*e)-1)^5-1/8*(58816*A+40000*B)/(tan(1/2*f*x+1/2*e)-1)^8-1/11*(8
1344*A+72512*B)/(tan(1/2*f*x+1/2*e)-1)^11-1/15*(1024*A+1024*B)/(tan(1/2*f*x+1/2*e)-1)^15-1/2*(20*A+2*B)/(tan(1
/2*f*x+1/2*e)-1)^2-1/6*(13824*A+6936*B)/(tan(1/2*f*x+1/2*e)-1)^6-1/9*(84112*A+63856*B)/(tan(1/2*f*x+1/2*e)-1)^
9-A/(tan(1/2*f*x+1/2*e)-1))
________________________________________________________________________________________
Maxima [B] time = 1.89065, size = 6433, normalized size = 32.65 \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))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^8,x, algorithm="maxima")
[Out]
2/45045*(3*A*a^3*(17715*sin(f*x + e)/(cos(f*x + e) + 1) - 78960*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 342160*s
in(f*x + e)^3/(cos(f*x + e) + 1)^3 - 891345*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 1960959*sin(f*x + e)^5/(cos(
f*x + e) + 1)^5 - 3043040*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 3912480*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 -
3687255*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 2867865*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 1585584*sin(f*x +
e)^10/(cos(f*x + e) + 1)^10 + 720720*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 195195*sin(f*x + e)^12/(cos(f*x +
e) + 1)^12 + 45045*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - 1181)/(c^8 - 15*c^8*sin(f*x + e)/(cos(f*x + e) + 1
) + 105*c^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1365*c^8*sin(f
*x + e)^4/(cos(f*x + e) + 1)^4 - 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*sin(f*x + e)^6/(cos(f
*x + e) + 1)^6 - 6435*c^8*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 -
5005*c^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 1365*c^8*sin(
f*x + e)^11/(cos(f*x + e) + 1)^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 105*c^8*sin(f*x + e)^13/(c
os(f*x + e) + 1)^13 + 15*c^8*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(cos(f*x + e) + 1)^15
) + B*a^3*(17715*sin(f*x + e)/(cos(f*x + e) + 1) - 78960*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 342160*sin(f*x
+ e)^3/(cos(f*x + e) + 1)^3 - 891345*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 1960959*sin(f*x + e)^5/(cos(f*x + e
) + 1)^5 - 3043040*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 3912480*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 3687255
*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 2867865*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 1585584*sin(f*x + e)^10/(
cos(f*x + e) + 1)^10 + 720720*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 195195*sin(f*x + e)^12/(cos(f*x + e) + 1
)^12 + 45045*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - 1181)/(c^8 - 15*c^8*sin(f*x + e)/(cos(f*x + e) + 1) + 105
*c^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1365*c^8*sin(f*x + e)
^4/(cos(f*x + e) + 1)^4 - 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*sin(f*x + e)^6/(cos(f*x + e)
+ 1)^6 - 6435*c^8*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 5005*c
^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 1365*c^8*sin(f*x + e
)^11/(cos(f*x + e) + 1)^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 105*c^8*sin(f*x + e)^13/(cos(f*x
+ e) + 1)^13 + 15*c^8*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(cos(f*x + e) + 1)^15) - 7*A
*a^3*(7845*sin(f*x + e)/(cos(f*x + e) + 1) - 54915*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 222950*sin(f*x + e)^3
/(cos(f*x + e) + 1)^3 - 668850*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 1444443*sin(f*x + e)^5/(cos(f*x + e) + 1)
^5 - 2407405*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 3063060*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 3063060*sin(f
*x + e)^8/(cos(f*x + e) + 1)^8 + 2357355*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 1414413*sin(f*x + e)^10/(cos(f*
x + e) + 1)^10 + 630630*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 210210*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 +
45045*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - 6435*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - 952)/(c^8 - 15*c^8
*sin(f*x + e)/(cos(f*x + e) + 1) + 105*c^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f
*x + e) + 1)^3 + 1365*c^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 +
5005*c^8*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 6435*c^8*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*
x + e)^8/(cos(f*x + e) + 1)^8 - 5005*c^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f
*x + e) + 1)^10 - 1365*c^8*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^
12 - 105*c^8*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 + 15*c^8*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin(f*
x + e)^15/(cos(f*x + e) + 1)^15) - 12*A*a^3*(1740*sin(f*x + e)/(cos(f*x + e) + 1) - 12180*sin(f*x + e)^2/(cos(
f*x + e) + 1)^2 + 37765*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 113295*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 204
204*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 340340*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 373230*sin(f*x + e)^7/(
cos(f*x + e) + 1)^7 - 373230*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 240240*sin(f*x + e)^9/(cos(f*x + e) + 1)^9
- 144144*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 45045*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 15015*sin(f*x +
e)^12/(cos(f*x + e) + 1)^12 - 116)/(c^8 - 15*c^8*sin(f*x + e)/(cos(f*x + e) + 1) + 105*c^8*sin(f*x + e)^2/(co
s(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1365*c^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^4
- 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 6435*c^8*sin(
f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 5005*c^8*sin(f*x + e)^9/(cos(
f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 1365*c^8*sin(f*x + e)^11/(cos(f*x + e) + 1)
^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 105*c^8*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 + 15*c^8*s
in(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(cos(f*x + e) + 1)^15) - 12*B*a^3*(1740*sin(f*x + e
)/(cos(f*x + e) + 1) - 12180*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 37765*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 -
113295*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 204204*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 340340*sin(f*x + e)
^6/(cos(f*x + e) + 1)^6 + 373230*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 373230*sin(f*x + e)^8/(cos(f*x + e) + 1
)^8 + 240240*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 144144*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 45045*sin(f*
x + e)^11/(cos(f*x + e) + 1)^11 - 15015*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 116)/(c^8 - 15*c^8*sin(f*x + e
)/(cos(f*x + e) + 1) + 105*c^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)
^3 + 1365*c^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*si
n(f*x + e)^6/(cos(f*x + e) + 1)^6 - 6435*c^8*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(co
s(f*x + e) + 1)^8 - 5005*c^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)
^10 - 1365*c^8*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 105*c^8
*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 + 15*c^8*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(c
os(f*x + e) + 1)^15) + 6*A*a^3*(675*sin(f*x + e)/(cos(f*x + e) + 1) - 4725*sin(f*x + e)^2/(cos(f*x + e) + 1)^2
+ 20475*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 46410*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 102102*sin(f*x + e)
^5/(cos(f*x + e) + 1)^5 - 130130*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 167310*sin(f*x + e)^7/(cos(f*x + e) + 1
)^7 - 122265*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 95095*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 33033*sin(f*x +
e)^10/(cos(f*x + e) + 1)^10 + 15015*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 45)/(c^8 - 15*c^8*sin(f*x + e)/(c
os(f*x + e) + 1) + 105*c^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 +
1365*c^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*sin(f*
x + e)^6/(cos(f*x + e) + 1)^6 - 6435*c^8*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(cos(f*
x + e) + 1)^8 - 5005*c^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10
- 1365*c^8*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 105*c^8*sin
(f*x + e)^13/(cos(f*x + e) + 1)^13 + 15*c^8*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(cos(f
*x + e) + 1)^15) + 18*B*a^3*(675*sin(f*x + e)/(cos(f*x + e) + 1) - 4725*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 +
20475*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 46410*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 102102*sin(f*x + e)^5/
(cos(f*x + e) + 1)^5 - 130130*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 167310*sin(f*x + e)^7/(cos(f*x + e) + 1)^7
- 122265*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 95095*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 33033*sin(f*x + e)
^10/(cos(f*x + e) + 1)^10 + 15015*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 45)/(c^8 - 15*c^8*sin(f*x + e)/(cos(
f*x + e) + 1) + 105*c^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 13
65*c^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*sin(f*x +
e)^6/(cos(f*x + e) + 1)^6 - 6435*c^8*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(cos(f*x +
e) + 1)^8 - 5005*c^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 1
365*c^8*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 105*c^8*sin(f*
x + e)^13/(cos(f*x + e) + 1)^13 + 15*c^8*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(cos(f*x
+ e) + 1)^15) - 48*B*a^3*(60*sin(f*x + e)/(cos(f*x + e) + 1) - 420*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1820*
sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 5460*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 9009*sin(f*x + e)^5/(cos(f*x
+ e) + 1)^5 - 15015*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 12870*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 12870*si
n(f*x + e)^8/(cos(f*x + e) + 1)^8 + 5005*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 3003*sin(f*x + e)^10/(cos(f*x +
e) + 1)^10 - 4)/(c^8 - 15*c^8*sin(f*x + e)/(cos(f*x + e) + 1) + 105*c^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 -
455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1365*c^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 3003*c^8*sin(f*x
+ e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 6435*c^8*sin(f*x + e)^7/(cos(f*x
+ e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 5005*c^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3
003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 1365*c^8*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 455*c^8*sin(f
*x + e)^12/(cos(f*x + e) + 1)^12 - 105*c^8*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 + 15*c^8*sin(f*x + e)^14/(cos
(f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(cos(f*x + e) + 1)^15))/f
________________________________________________________________________________________
Fricas [B] time = 1.6306, size = 1400, normalized size = 7.11 \begin{align*} \frac{2 \,{\left (4 \, A - 11 \, B\right )} a^{3} \cos \left (f x + e\right )^{8} + 16 \,{\left (4 \, A - 11 \, B\right )} a^{3} \cos \left (f x + e\right )^{7} - 49 \,{\left (4 \, A - 11 \, B\right )} a^{3} \cos \left (f x + e\right )^{6} - 168 \,{\left (4 \, A - 11 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} + 105 \,{\left (7 \, A + 88 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} - 231 \,{\left (31 \, A + 61 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - 924 \,{\left (22 \, A + 37 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 12012 \,{\left (A + B\right )} a^{3} \cos \left (f x + e\right ) + 24024 \,{\left (A + B\right )} a^{3} -{\left (2 \,{\left (4 \, A - 11 \, B\right )} a^{3} \cos \left (f x + e\right )^{7} - 14 \,{\left (4 \, A - 11 \, B\right )} a^{3} \cos \left (f x + e\right )^{6} - 63 \,{\left (4 \, A - 11 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} + 105 \,{\left (4 \, A - 11 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} + 1155 \,{\left (A + 7 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} + 2772 \,{\left (3 \, A + 8 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 12012 \,{\left (A + B\right )} a^{3} \cos \left (f x + e\right ) - 24024 \,{\left (A + B\right )} a^{3}\right )} \sin \left (f x + e\right )}{45045 \,{\left (c^{8} f \cos \left (f x + e\right )^{8} - 7 \, c^{8} f \cos \left (f x + e\right )^{7} - 32 \, c^{8} f \cos \left (f x + e\right )^{6} + 56 \, c^{8} f \cos \left (f x + e\right )^{5} + 160 \, c^{8} f \cos \left (f x + e\right )^{4} - 112 \, c^{8} f \cos \left (f x + e\right )^{3} - 256 \, c^{8} f \cos \left (f x + e\right )^{2} + 64 \, c^{8} f \cos \left (f x + e\right ) + 128 \, c^{8} f +{\left (c^{8} f \cos \left (f x + e\right )^{7} + 8 \, c^{8} f \cos \left (f x + e\right )^{6} - 24 \, c^{8} f \cos \left (f x + e\right )^{5} - 80 \, c^{8} f \cos \left (f x + e\right )^{4} + 80 \, c^{8} f \cos \left (f x + e\right )^{3} + 192 \, c^{8} f \cos \left (f x + e\right )^{2} - 64 \, c^{8} f \cos \left (f x + e\right ) - 128 \, c^{8} 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))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^8,x, algorithm="fricas")
[Out]
1/45045*(2*(4*A - 11*B)*a^3*cos(f*x + e)^8 + 16*(4*A - 11*B)*a^3*cos(f*x + e)^7 - 49*(4*A - 11*B)*a^3*cos(f*x
+ e)^6 - 168*(4*A - 11*B)*a^3*cos(f*x + e)^5 + 105*(7*A + 88*B)*a^3*cos(f*x + e)^4 - 231*(31*A + 61*B)*a^3*cos
(f*x + e)^3 - 924*(22*A + 37*B)*a^3*cos(f*x + e)^2 + 12012*(A + B)*a^3*cos(f*x + e) + 24024*(A + B)*a^3 - (2*(
4*A - 11*B)*a^3*cos(f*x + e)^7 - 14*(4*A - 11*B)*a^3*cos(f*x + e)^6 - 63*(4*A - 11*B)*a^3*cos(f*x + e)^5 + 105
*(4*A - 11*B)*a^3*cos(f*x + e)^4 + 1155*(A + 7*B)*a^3*cos(f*x + e)^3 + 2772*(3*A + 8*B)*a^3*cos(f*x + e)^2 - 1
2012*(A + B)*a^3*cos(f*x + e) - 24024*(A + B)*a^3)*sin(f*x + e))/(c^8*f*cos(f*x + e)^8 - 7*c^8*f*cos(f*x + e)^
7 - 32*c^8*f*cos(f*x + e)^6 + 56*c^8*f*cos(f*x + e)^5 + 160*c^8*f*cos(f*x + e)^4 - 112*c^8*f*cos(f*x + e)^3 -
256*c^8*f*cos(f*x + e)^2 + 64*c^8*f*cos(f*x + e) + 128*c^8*f + (c^8*f*cos(f*x + e)^7 + 8*c^8*f*cos(f*x + e)^6
- 24*c^8*f*cos(f*x + e)^5 - 80*c^8*f*cos(f*x + e)^4 + 80*c^8*f*cos(f*x + e)^3 + 192*c^8*f*cos(f*x + e)^2 - 64*
c^8*f*cos(f*x + e) - 128*c^8*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))**3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**8,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.26361, size = 698, normalized size = 3.54 \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))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^8,x, algorithm="giac")
[Out]
-2/45045*(45045*A*a^3*tan(1/2*f*x + 1/2*e)^14 - 180180*A*a^3*tan(1/2*f*x + 1/2*e)^13 + 45045*B*a^3*tan(1/2*f*x
+ 1/2*e)^13 + 1066065*A*a^3*tan(1/2*f*x + 1/2*e)^12 - 15015*B*a^3*tan(1/2*f*x + 1/2*e)^12 - 2702700*A*a^3*tan
(1/2*f*x + 1/2*e)^11 + 450450*B*a^3*tan(1/2*f*x + 1/2*e)^11 + 6675669*A*a^3*tan(1/2*f*x + 1/2*e)^10 - 306306*B
*a^3*tan(1/2*f*x + 1/2*e)^10 - 10210200*A*a^3*tan(1/2*f*x + 1/2*e)^9 + 1456455*B*a^3*tan(1/2*f*x + 1/2*e)^9 +
14124825*A*a^3*tan(1/2*f*x + 1/2*e)^8 - 791505*B*a^3*tan(1/2*f*x + 1/2*e)^8 - 13178880*A*a^3*tan(1/2*f*x + 1/2
*e)^7 + 1827540*B*a^3*tan(1/2*f*x + 1/2*e)^7 + 11026015*A*a^3*tan(1/2*f*x + 1/2*e)^6 - 580580*B*a^3*tan(1/2*f*
x + 1/2*e)^6 - 6066060*A*a^3*tan(1/2*f*x + 1/2*e)^5 + 915915*B*a^3*tan(1/2*f*x + 1/2*e)^5 + 3088995*A*a^3*tan(
1/2*f*x + 1/2*e)^4 - 105105*B*a^3*tan(1/2*f*x + 1/2*e)^4 - 864500*A*a^3*tan(1/2*f*x + 1/2*e)^3 + 170170*B*a^3*
tan(1/2*f*x + 1/2*e)^3 + 265335*A*a^3*tan(1/2*f*x + 1/2*e)^2 + 2310*B*a^3*tan(1/2*f*x + 1/2*e)^2 - 18600*A*a^3
*tan(1/2*f*x + 1/2*e) + 6105*B*a^3*tan(1/2*f*x + 1/2*e) + 4243*A*a^3 - 407*B*a^3)/(c^8*f*(tan(1/2*f*x + 1/2*e)
- 1)^15)