3.80 \(\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx\)

Optimal. Leaf size=205 \[ \frac{2 (8 A-3 B) \tan ^5(e+f x)}{165 a^3 c^6 f}+\frac{4 (8 A-3 B) \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac{2 (8 A-3 B) \tan (e+f x)}{33 a^3 c^6 f}+\frac{(8 A-3 B) \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac{(8 A-3 B) \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac{(A+B) \sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3} \]

[Out]

((A + B)*Sec[e + f*x]^5)/(11*a^3*f*(c^2 - c^2*Sin[e + f*x])^3) + ((8*A - 3*B)*Sec[e + f*x]^5)/(99*a^3*f*(c^3 -
c^3*Sin[e + f*x])^2) + ((8*A - 3*B)*Sec[e + f*x]^5)/(99*a^3*f*(c^6 - c^6*Sin[e + f*x])) + (2*(8*A - 3*B)*Tan[
e + f*x])/(33*a^3*c^6*f) + (4*(8*A - 3*B)*Tan[e + f*x]^3)/(99*a^3*c^6*f) + (2*(8*A - 3*B)*Tan[e + f*x]^5)/(165
*a^3*c^6*f)

________________________________________________________________________________________

Rubi [A]  time = 0.34542, antiderivative size = 205, 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, 3767} \[ \frac{2 (8 A-3 B) \tan ^5(e+f x)}{165 a^3 c^6 f}+\frac{4 (8 A-3 B) \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac{2 (8 A-3 B) \tan (e+f x)}{33 a^3 c^6 f}+\frac{(8 A-3 B) \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac{(8 A-3 B) \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac{(A+B) \sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6),x]

[Out]

((A + B)*Sec[e + f*x]^5)/(11*a^3*f*(c^2 - c^2*Sin[e + f*x])^3) + ((8*A - 3*B)*Sec[e + f*x]^5)/(99*a^3*f*(c^3 -
c^3*Sin[e + f*x])^2) + ((8*A - 3*B)*Sec[e + f*x]^5)/(99*a^3*f*(c^6 - c^6*Sin[e + f*x])) + (2*(8*A - 3*B)*Tan[
e + f*x])/(33*a^3*c^6*f) + (4*(8*A - 3*B)*Tan[e + f*x]^3)/(99*a^3*c^6*f) + (2*(8*A - 3*B)*Tan[e + f*x]^5)/(165
*a^3*c^6*f)

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 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin{align*} \int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx &=\frac{\int \frac{\sec ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx}{a^3 c^3}\\ &=\frac{(A+B) \sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac{(8 A-3 B) \int \frac{\sec ^6(e+f x)}{(c-c \sin (e+f x))^2} \, dx}{11 a^3 c^4}\\ &=\frac{(A+B) \sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac{(8 A-3 B) \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac{(7 (8 A-3 B)) \int \frac{\sec ^6(e+f x)}{c-c \sin (e+f x)} \, dx}{99 a^3 c^5}\\ &=\frac{(A+B) \sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac{(8 A-3 B) \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac{(8 A-3 B) \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac{(2 (8 A-3 B)) \int \sec ^6(e+f x) \, dx}{33 a^3 c^6}\\ &=\frac{(A+B) \sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac{(8 A-3 B) \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac{(8 A-3 B) \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}-\frac{(2 (8 A-3 B)) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{33 a^3 c^6 f}\\ &=\frac{(A+B) \sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac{(8 A-3 B) \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac{(8 A-3 B) \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac{2 (8 A-3 B) \tan (e+f x)}{33 a^3 c^6 f}+\frac{4 (8 A-3 B) \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac{2 (8 A-3 B) \tan ^5(e+f x)}{165 a^3 c^6 f}\\ \end{align*}

Mathematica [A]  time = 3.25535, size = 401, normalized size = 1.96 \[ \frac{-3850 (107 A-3 B) \cos (e+f x)+135168 (8 A-3 B) \cos (2 (e+f x))+1802240 A \sin (e+f x)+247170 A \sin (2 (e+f x))+557056 A \sin (3 (e+f x))+187250 A \sin (4 (e+f x))-163840 A \sin (5 (e+f x))+37450 A \sin (6 (e+f x))-98304 A \sin (7 (e+f x))-3745 A \sin (8 (e+f x))-127330 A \cos (3 (e+f x))+819200 A \cos (4 (e+f x))+37450 A \cos (5 (e+f x))+163840 A \cos (6 (e+f x))+22470 A \cos (7 (e+f x))-16384 A \cos (8 (e+f x))-675840 B \sin (e+f x)-6930 B \sin (2 (e+f x))-208896 B \sin (3 (e+f x))-5250 B \sin (4 (e+f x))+61440 B \sin (5 (e+f x))-1050 B \sin (6 (e+f x))+36864 B \sin (7 (e+f x))+105 B \sin (8 (e+f x))+3570 B \cos (3 (e+f x))-307200 B \cos (4 (e+f x))-1050 B \cos (5 (e+f x))-61440 B \cos (6 (e+f x))-630 B \cos (7 (e+f x))+6144 B \cos (8 (e+f x))+1013760 B}{8110080 a^3 c^6 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{11} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6),x]

[Out]

(1013760*B - 3850*(107*A - 3*B)*Cos[e + f*x] + 135168*(8*A - 3*B)*Cos[2*(e + f*x)] - 127330*A*Cos[3*(e + f*x)]
+ 3570*B*Cos[3*(e + f*x)] + 819200*A*Cos[4*(e + f*x)] - 307200*B*Cos[4*(e + f*x)] + 37450*A*Cos[5*(e + f*x)]
- 1050*B*Cos[5*(e + f*x)] + 163840*A*Cos[6*(e + f*x)] - 61440*B*Cos[6*(e + f*x)] + 22470*A*Cos[7*(e + f*x)] -
630*B*Cos[7*(e + f*x)] - 16384*A*Cos[8*(e + f*x)] + 6144*B*Cos[8*(e + f*x)] + 1802240*A*Sin[e + f*x] - 675840*
B*Sin[e + f*x] + 247170*A*Sin[2*(e + f*x)] - 6930*B*Sin[2*(e + f*x)] + 557056*A*Sin[3*(e + f*x)] - 208896*B*Si
n[3*(e + f*x)] + 187250*A*Sin[4*(e + f*x)] - 5250*B*Sin[4*(e + f*x)] - 163840*A*Sin[5*(e + f*x)] + 61440*B*Sin
[5*(e + f*x)] + 37450*A*Sin[6*(e + f*x)] - 1050*B*Sin[6*(e + f*x)] - 98304*A*Sin[7*(e + f*x)] + 36864*B*Sin[7*
(e + f*x)] - 3745*A*Sin[8*(e + f*x)] + 105*B*Sin[8*(e + f*x)])/(8110080*a^3*c^6*f*(Cos[(e + f*x)/2] - Sin[(e +
f*x)/2])^11*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)

________________________________________________________________________________________

Maple [A]  time = 0.128, size = 365, normalized size = 1.8 \begin{align*} 2\,{\frac{1}{f{a}^{3}{c}^{6}} \left ( -1/11\,{\frac{4\,A+4\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{11}}}-1/10\,{\frac{20\,A+20\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{10}}}-1/9\,{\frac{53\,A+51\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{9}}}-1/8\,{\frac{92\,A+84\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{8}}}-1/4\,{\frac{1}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{4}} \left ({\frac{169\,A}{4}}+{\frac{99\,B}{4}} \right ) }-1/6\,{\frac{1}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{6}} \left ({\frac{217\,A}{2}}+84\,B \right ) }-{\frac{1}{\tan \left ( 1/2\,fx+e/2 \right ) -1} \left ({\frac{219\,A}{256}}+{\frac{21\,B}{256}} \right ) }-1/7\,{\frac{1}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{7}} \left ({\frac{231\,A}{2}}+98\,B \right ) }-1/2\,{\frac{1}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{2}} \left ({\frac{303\,A}{64}}+{\frac{99\,B}{64}} \right ) }-1/5\,{\frac{1}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{5}} \left ({\frac{623\,A}{8}}+{\frac{427\,B}{8}} \right ) }-1/3\,{\frac{1}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{3}} \left ({\frac{1095\,A}{64}}+{\frac{507\,B}{64}} \right ) }-1/2\,{\frac{1}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{2}} \left ( -{\frac{5\,A}{32}}+B/8 \right ) }-1/4\,{\frac{-A/8+B/8}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{4}}}-1/5\,{\frac{A/16-B/16}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{5}}}-1/3\,{\frac{1}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{3}} \left ({\frac{7\,A}{32}}-3/16\,B \right ) }-{\frac{1}{\tan \left ( 1/2\,fx+e/2 \right ) +1} \left ({\frac{37\,A}{256}}-{\frac{21\,B}{256}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x)

[Out]

2/f/a^3/c^6*(-1/11*(4*A+4*B)/(tan(1/2*f*x+1/2*e)-1)^11-1/10*(20*A+20*B)/(tan(1/2*f*x+1/2*e)-1)^10-1/9*(53*A+51
*B)/(tan(1/2*f*x+1/2*e)-1)^9-1/8*(92*A+84*B)/(tan(1/2*f*x+1/2*e)-1)^8-1/4*(169/4*A+99/4*B)/(tan(1/2*f*x+1/2*e)
-1)^4-1/6*(217/2*A+84*B)/(tan(1/2*f*x+1/2*e)-1)^6-(219/256*A+21/256*B)/(tan(1/2*f*x+1/2*e)-1)-1/7*(231/2*A+98*
B)/(tan(1/2*f*x+1/2*e)-1)^7-1/2*(303/64*A+99/64*B)/(tan(1/2*f*x+1/2*e)-1)^2-1/5*(623/8*A+427/8*B)/(tan(1/2*f*x
+1/2*e)-1)^5-1/3*(1095/64*A+507/64*B)/(tan(1/2*f*x+1/2*e)-1)^3-1/2*(-5/32*A+1/8*B)/(tan(1/2*f*x+1/2*e)+1)^2-1/
4*(-1/8*A+1/8*B)/(tan(1/2*f*x+1/2*e)+1)^4-1/5*(1/16*A-1/16*B)/(tan(1/2*f*x+1/2*e)+1)^5-1/3*(7/32*A-3/16*B)/(ta
n(1/2*f*x+1/2*e)+1)^3-(37/256*A-21/256*B)/(tan(1/2*f*x+1/2*e)+1))

________________________________________________________________________________________

Maxima [B]  time = 1.22911, size = 1872, normalized size = 9.13 \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))/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algorithm="maxima")

[Out]

-2/495*(A*(255*sin(f*x + e)/(cos(f*x + e) + 1) + 235*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 3065*sin(f*x + e)^3
/(cos(f*x + e) + 1)^3 + 3775*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 667*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 8
217*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 2035*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 8745*sin(f*x + e)^8/(cos(
f*x + e) + 1)^8 - 11715*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 33*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 4917*
sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 2475*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 1815*sin(f*x + e)^13/(cos
(f*x + e) + 1)^13 + 1485*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - 495*sin(f*x + e)^15/(cos(f*x + e) + 1)^15 - 1
25)/(a^3*c^6 - 6*a^3*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10
*a^3*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 50*a^3*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 34*a^3*c^6*sin
(f*x + e)^5/(cos(f*x + e) + 1)^5 + 66*a^3*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 110*a^3*c^6*sin(f*x + e)^7
/(cos(f*x + e) + 1)^7 + 110*a^3*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 66*a^3*c^6*sin(f*x + e)^10/(cos(f*x
+ e) + 1)^10 - 34*a^3*c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 50*a^3*c^6*sin(f*x + e)^12/(cos(f*x + e) + 1
)^12 - 10*a^3*c^6*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - 10*a^3*c^6*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 + 6
*a^3*c^6*sin(f*x + e)^15/(cos(f*x + e) + 1)^15 - a^3*c^6*sin(f*x + e)^16/(cos(f*x + e) + 1)^16) + 3*B*(30*sin(
f*x + e)/(cos(f*x + e) + 1) - 215*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 280*sin(f*x + e)^3/(cos(f*x + e) + 1)^
3 - 245*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 434*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 231*sin(f*x + e)^6/(co
s(f*x + e) + 1)^6 + 880*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 1815*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 330*s
in(f*x + e)^9/(cos(f*x + e) + 1)^9 + 99*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 264*sin(f*x + e)^11/(cos(f*x +
e) + 1)^11 - 495*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 + 330*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - 165*sin(
f*x + e)^14/(cos(f*x + e) + 1)^14 - 5)/(a^3*c^6 - 6*a^3*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*c^6*sin(f
*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 50*a^3*c^6*sin(f*x + e)^4/(c
os(f*x + e) + 1)^4 + 34*a^3*c^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 66*a^3*c^6*sin(f*x + e)^6/(cos(f*x + e)
+ 1)^6 - 110*a^3*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 110*a^3*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 6
6*a^3*c^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 34*a^3*c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 50*a^3*c^
6*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 10*a^3*c^6*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - 10*a^3*c^6*sin(f*
x + e)^14/(cos(f*x + e) + 1)^14 + 6*a^3*c^6*sin(f*x + e)^15/(cos(f*x + e) + 1)^15 - a^3*c^6*sin(f*x + e)^16/(c
os(f*x + e) + 1)^16))/f

________________________________________________________________________________________

Fricas [A]  time = 2.02722, size = 543, normalized size = 2.65 \begin{align*} \frac{16 \,{\left (8 \, A - 3 \, B\right )} \cos \left (f x + e\right )^{8} - 72 \,{\left (8 \, A - 3 \, B\right )} \cos \left (f x + e\right )^{6} + 30 \,{\left (8 \, A - 3 \, B\right )} \cos \left (f x + e\right )^{4} + 7 \,{\left (8 \, A - 3 \, B\right )} \cos \left (f x + e\right )^{2} +{\left (48 \,{\left (8 \, A - 3 \, B\right )} \cos \left (f x + e\right )^{6} - 40 \,{\left (8 \, A - 3 \, B\right )} \cos \left (f x + e\right )^{4} - 14 \,{\left (8 \, A - 3 \, B\right )} \cos \left (f x + e\right )^{2} - 72 \, A + 27 \, B\right )} \sin \left (f x + e\right ) + 27 \, A - 72 \, B}{495 \,{\left (3 \, a^{3} c^{6} f \cos \left (f x + e\right )^{7} - 4 \, a^{3} c^{6} f \cos \left (f x + e\right )^{5} -{\left (a^{3} c^{6} f \cos \left (f x + e\right )^{7} - 4 \, a^{3} c^{6} f \cos \left (f x + e\right )^{5}\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))/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algorithm="fricas")

[Out]

1/495*(16*(8*A - 3*B)*cos(f*x + e)^8 - 72*(8*A - 3*B)*cos(f*x + e)^6 + 30*(8*A - 3*B)*cos(f*x + e)^4 + 7*(8*A
- 3*B)*cos(f*x + e)^2 + (48*(8*A - 3*B)*cos(f*x + e)^6 - 40*(8*A - 3*B)*cos(f*x + e)^4 - 14*(8*A - 3*B)*cos(f*
x + e)^2 - 72*A + 27*B)*sin(f*x + e) + 27*A - 72*B)/(3*a^3*c^6*f*cos(f*x + e)^7 - 4*a^3*c^6*f*cos(f*x + e)^5 -
(a^3*c^6*f*cos(f*x + e)^7 - 4*a^3*c^6*f*cos(f*x + e)^5)*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))/(a+a*sin(f*x+e))**3/(c-c*sin(f*x+e))**6,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.27891, size = 641, normalized size = 3.13 \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))/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algorithm="giac")

[Out]

-1/63360*(33*(555*A*tan(1/2*f*x + 1/2*e)^4 - 315*B*tan(1/2*f*x + 1/2*e)^4 + 1920*A*tan(1/2*f*x + 1/2*e)^3 - 10
20*B*tan(1/2*f*x + 1/2*e)^3 + 2710*A*tan(1/2*f*x + 1/2*e)^2 - 1410*B*tan(1/2*f*x + 1/2*e)^2 + 1760*A*tan(1/2*f
*x + 1/2*e) - 900*B*tan(1/2*f*x + 1/2*e) + 463*A - 243*B)/(a^3*c^6*(tan(1/2*f*x + 1/2*e) + 1)^5) + (108405*A*t
an(1/2*f*x + 1/2*e)^10 + 10395*B*tan(1/2*f*x + 1/2*e)^10 - 784080*A*tan(1/2*f*x + 1/2*e)^9 - 5940*B*tan(1/2*f*
x + 1/2*e)^9 + 2901195*A*tan(1/2*f*x + 1/2*e)^8 - 79695*B*tan(1/2*f*x + 1/2*e)^8 - 6652800*A*tan(1/2*f*x + 1/2
*e)^7 + 388080*B*tan(1/2*f*x + 1/2*e)^7 + 10407474*A*tan(1/2*f*x + 1/2*e)^6 - 816354*B*tan(1/2*f*x + 1/2*e)^6
- 11435424*A*tan(1/2*f*x + 1/2*e)^5 + 1114344*B*tan(1/2*f*x + 1/2*e)^5 + 8949270*A*tan(1/2*f*x + 1/2*e)^4 - 99
0990*B*tan(1/2*f*x + 1/2*e)^4 - 4899840*A*tan(1/2*f*x + 1/2*e)^3 + 609840*B*tan(1/2*f*x + 1/2*e)^3 + 1816265*A
*tan(1/2*f*x + 1/2*e)^2 - 235785*B*tan(1/2*f*x + 1/2*e)^2 - 411664*A*tan(1/2*f*x + 1/2*e) + 56364*B*tan(1/2*f*
x + 1/2*e) + 47279*A - 4179*B)/(a^3*c^6*(tan(1/2*f*x + 1/2*e) - 1)^11))/f