\(\int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{c-c \sin (e+f x)} \, dx\) [44]
Optimal result
Integrand size = 36, antiderivative size = 156 \[
\int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{c-c \sin (e+f x)} \, dx=-\frac {5 a^3 (3 A+4 B) x}{2 c}+\frac {5 a^3 (3 A+4 B) \cos ^3(e+f x)}{3 c f}-\frac {5 a^3 (3 A+4 B) \cos (e+f x) \sin (e+f x)}{2 c f}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{f (c-c \sin (e+f x))^4}+\frac {2 a^3 (3 A+4 B) c^3 \cos ^5(e+f x)}{f \left (c^2-c^2 \sin (e+f x)\right )^2}
\]
[Out]
-5/2*a^3*(3*A+4*B)*x/c+5/3*a^3*(3*A+4*B)*cos(f*x+e)^3/c/f-5/2*a^3*(3*A+4*B)*cos(f*x+e)*sin(f*x+e)/c/f+a^3*(A+B
)*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^4+2*a^3*(3*A+4*B)*c^3*cos(f*x+e)^5/f/(c^2-c^2*sin(f*x+e))^2
Rubi [A] (verified)
Time = 0.22 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of
steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3046, 2938, 2759, 2761, 2715,
8} \[
\int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{c-c \sin (e+f x)} \, dx=\frac {a^3 c^3 (A+B) \cos ^7(e+f x)}{f (c-c \sin (e+f x))^4}+\frac {2 a^3 c^3 (3 A+4 B) \cos ^5(e+f x)}{f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {5 a^3 (3 A+4 B) \cos ^3(e+f x)}{3 c f}-\frac {5 a^3 (3 A+4 B) \sin (e+f x) \cos (e+f x)}{2 c f}-\frac {5 a^3 x (3 A+4 B)}{2 c}
\]
[In]
Int[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x]),x]
[Out]
(-5*a^3*(3*A + 4*B)*x)/(2*c) + (5*a^3*(3*A + 4*B)*Cos[e + f*x]^3)/(3*c*f) - (5*a^3*(3*A + 4*B)*Cos[e + f*x]*Si
n[e + f*x])/(2*c*f) + (a^3*(A + B)*c^3*Cos[e + f*x]^7)/(f*(c - c*Sin[e + f*x])^4) + (2*a^3*(3*A + 4*B)*c^3*Cos
[e + f*x]^5)/(f*(c^2 - c^2*Sin[e + f*x])^2)
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rule 2715
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] && Integ
erQ[2*n]
Rule 2759
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 2761
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 2938
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 3046
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]))
Rubi steps \begin{align*}
\text {integral}& = \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))^4} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{f (c-c \sin (e+f x))^4}-\left (a^3 (3 A+4 B) c^2\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^3} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{f (c-c \sin (e+f x))^4}+\frac {2 a^3 (3 A+4 B) c \cos ^5(e+f x)}{f (c-c \sin (e+f x))^2}-\left (5 a^3 (3 A+4 B)\right ) \int \frac {\cos ^4(e+f x)}{c-c \sin (e+f x)} \, dx \\ & = \frac {5 a^3 (3 A+4 B) \cos ^3(e+f x)}{3 c f}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{f (c-c \sin (e+f x))^4}+\frac {2 a^3 (3 A+4 B) c \cos ^5(e+f x)}{f (c-c \sin (e+f x))^2}-\frac {\left (5 a^3 (3 A+4 B)\right ) \int \cos ^2(e+f x) \, dx}{c} \\ & = \frac {5 a^3 (3 A+4 B) \cos ^3(e+f x)}{3 c f}-\frac {5 a^3 (3 A+4 B) \cos (e+f x) \sin (e+f x)}{2 c f}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{f (c-c \sin (e+f x))^4}+\frac {2 a^3 (3 A+4 B) c \cos ^5(e+f x)}{f (c-c \sin (e+f x))^2}-\frac {\left (5 a^3 (3 A+4 B)\right ) \int 1 \, dx}{2 c} \\ & = -\frac {5 a^3 (3 A+4 B) x}{2 c}+\frac {5 a^3 (3 A+4 B) \cos ^3(e+f x)}{3 c f}-\frac {5 a^3 (3 A+4 B) \cos (e+f x) \sin (e+f x)}{2 c f}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{f (c-c \sin (e+f x))^4}+\frac {2 a^3 (3 A+4 B) c \cos ^5(e+f x)}{f (c-c \sin (e+f x))^2} \\
\end{align*}
Mathematica [A] (verified)
Time = 11.64 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.43
\[
\int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{c-c \sin (e+f x)} \, dx=\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3 \left (\cos \left (\frac {1}{2} (e+f x)\right ) (30 (3 A+4 B) (e+f x)-3 (16 A+31 B) \cos (e+f x)+B \cos (3 (e+f x))-3 (A+4 B) \sin (2 (e+f x)))-\sin \left (\frac {1}{2} (e+f x)\right ) (24 B (8+5 e+5 f x)+6 A (32+15 e+15 f x)-3 (16 A+31 B) \cos (e+f x)+B \cos (3 (e+f x))-3 (A+4 B) \sin (2 (e+f x)))\right )}{12 c f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (-1+\sin (e+f x))}
\]
[In]
Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x]),x]
[Out]
(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*(Cos[(e + f*x)/2]*(30*(3*A + 4*B)*(e + f*x) -
3*(16*A + 31*B)*Cos[e + f*x] + B*Cos[3*(e + f*x)] - 3*(A + 4*B)*Sin[2*(e + f*x)]) - Sin[(e + f*x)/2]*(24*B*(8
+ 5*e + 5*f*x) + 6*A*(32 + 15*e + 15*f*x) - 3*(16*A + 31*B)*Cos[e + f*x] + B*Cos[3*(e + f*x)] - 3*(A + 4*B)*Si
n[2*(e + f*x)])))/(12*c*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(-1 + Sin[e + f*x]))
Maple [A] (verified)
Time = 0.66 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.71
| | |
| method | result | size |
| | |
| parallelrisch |
\(\frac {65 a^{3} \left (\frac {4 \left (4 A +\frac {23 B}{3}\right ) \cos \left (2 f x +2 e \right )}{65}+\frac {\left (A +4 B \right ) \sin \left (3 f x +3 e \right )}{65}-\frac {B \cos \left (4 f x +4 e \right )}{195}+\frac {4 \left (-3 f x A -4 f x B +\frac {24}{5} A +\frac {94}{15} B \right ) \cos \left (f x +e \right )}{13}+\left (A +\frac {68 B}{65}\right ) \sin \left (f x +e \right )+\frac {16 A}{13}+\frac {19 B}{13}\right )}{8 c f \cos \left (f x +e \right )}\) |
\(110\) |
| derivativedivides |
\(\frac {2 a^{3} \left (-\frac {\left (2 B +\frac {A}{2}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-4 A -7 B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-8 A -16 B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-2 B -\frac {A}{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-4 A -\frac {23 B}{3}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}-\frac {5 \left (3 A +4 B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {8 A +8 B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}\right )}{f c}\) |
\(152\) |
| default |
\(\frac {2 a^{3} \left (-\frac {\left (2 B +\frac {A}{2}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-4 A -7 B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-8 A -16 B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-2 B -\frac {A}{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-4 A -\frac {23 B}{3}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}-\frac {5 \left (3 A +4 B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {8 A +8 B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}\right )}{f c}\) |
\(152\) |
| risch |
\(-\frac {15 a^{3} x A}{2 c}-\frac {10 a^{3} x B}{c}+\frac {2 a^{3} {\mathrm e}^{i \left (f x +e \right )} A}{c f}+\frac {31 a^{3} {\mathrm e}^{i \left (f x +e \right )} B}{8 c f}+\frac {2 a^{3} {\mathrm e}^{-i \left (f x +e \right )} A}{c f}+\frac {31 a^{3} {\mathrm e}^{-i \left (f x +e \right )} B}{8 c f}+\frac {16 a^{3} A}{f c \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}+\frac {16 a^{3} B}{f c \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}-\frac {B \,a^{3} \cos \left (3 f x +3 e \right )}{12 c f}+\frac {a^{3} \sin \left (2 f x +2 e \right ) A}{4 c f}+\frac {a^{3} \sin \left (2 f x +2 e \right ) B}{c f}\) |
\(220\) |
| norman |
\(\frac {-\frac {17 A \,a^{3}+20 B \,a^{3}}{c f}+\frac {5 a^{3} \left (3 A +4 B \right ) x}{2 c}-\frac {\left (5 A \,a^{3}+2 B \,a^{3}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {\left (17 A \,a^{3}+18 B \,a^{3}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {\left (21 A \,a^{3}+34 B \,a^{3}\right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}-\frac {\left (30 A \,a^{3}+26 B \,a^{3}\right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}-\frac {\left (57 A \,a^{3}+82 B \,a^{3}\right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}-\frac {\left (59 A \,a^{3}+62 B \,a^{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {\left (77 A \,a^{3}+70 B \,a^{3}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {\left (135 A \,a^{3}+110 B \,a^{3}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}-\frac {5 a^{3} \left (3 A +4 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c}+\frac {10 a^{3} \left (3 A +4 B \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {10 a^{3} \left (3 A +4 B \right ) x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}+\frac {15 a^{3} \left (3 A +4 B \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {15 a^{3} \left (3 A +4 B \right ) x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}+\frac {10 a^{3} \left (3 A +4 B \right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {10 a^{3} \left (3 A +4 B \right ) x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}+\frac {5 a^{3} \left (3 A +4 B \right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c}-\frac {5 a^{3} \left (3 A +4 B \right ) x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}\) |
\(565\) |
| | |
|
|
|
|
[In]
int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e)),x,method=_RETURNVERBOSE)
[Out]
65/8*a^3*(4/65*(4*A+23/3*B)*cos(2*f*x+2*e)+1/65*(A+4*B)*sin(3*f*x+3*e)-1/195*B*cos(4*f*x+4*e)+4/13*(-3*f*x*A-4
*f*x*B+24/5*A+94/15*B)*cos(f*x+e)+(A+68/65*B)*sin(f*x+e)+16/13*A+19/13*B)/c/f/cos(f*x+e)
Fricas [A] (verification not implemented)
none
Time = 0.26 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.40
\[
\int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{c-c \sin (e+f x)} \, dx=-\frac {2 \, B a^{3} \cos \left (f x + e\right )^{4} - {\left (3 \, A + 10 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} + 15 \, {\left (3 \, A + 4 \, B\right )} a^{3} f x - 24 \, {\left (A + 2 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 48 \, {\left (A + B\right )} a^{3} + 3 \, {\left (5 \, {\left (3 \, A + 4 \, B\right )} a^{3} f x - {\left (23 \, A + 28 \, B\right )} a^{3}\right )} \cos \left (f x + e\right ) - {\left (2 \, B a^{3} \cos \left (f x + e\right )^{3} + 15 \, {\left (3 \, A + 4 \, B\right )} a^{3} f x + 3 \, {\left (A + 4 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 3 \, {\left (7 \, A + 12 \, B\right )} a^{3} \cos \left (f x + e\right ) + 48 \, {\left (A + B\right )} a^{3}\right )} \sin \left (f x + e\right )}{6 \, {\left (c f \cos \left (f x + e\right ) - c f \sin \left (f x + e\right ) + c f\right )}}
\]
[In]
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e)),x, algorithm="fricas")
[Out]
-1/6*(2*B*a^3*cos(f*x + e)^4 - (3*A + 10*B)*a^3*cos(f*x + e)^3 + 15*(3*A + 4*B)*a^3*f*x - 24*(A + 2*B)*a^3*cos
(f*x + e)^2 - 48*(A + B)*a^3 + 3*(5*(3*A + 4*B)*a^3*f*x - (23*A + 28*B)*a^3)*cos(f*x + e) - (2*B*a^3*cos(f*x +
e)^3 + 15*(3*A + 4*B)*a^3*f*x + 3*(A + 4*B)*a^3*cos(f*x + e)^2 - 3*(7*A + 12*B)*a^3*cos(f*x + e) + 48*(A + B)
*a^3)*sin(f*x + e))/(c*f*cos(f*x + e) - c*f*sin(f*x + e) + c*f)
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 4255 vs. \(2 (144) = 288\).
Time = 3.79 (sec) , antiderivative size = 4255, normalized size of antiderivative = 27.28
\[
\int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{c-c \sin (e+f x)} \, dx=\text {Too large to display}
\]
[In]
integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e)),x)
[Out]
Piecewise((-45*A*a**3*f*x*tan(e/2 + f*x/2)**7/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*
tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6
*c*f*tan(e/2 + f*x/2) - 6*c*f) + 45*A*a**3*f*x*tan(e/2 + f*x/2)**6/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*tan(e/2
+ f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*c*f*ta
n(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) - 135*A*a**3*f*x*tan(e/2 + f*x/2)**5/(6*c*f*tan(e/2 + f*x/
2)**7 - 6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 +
f*x/2)**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) + 135*A*a**3*f*x*tan(e/2 + f*x/2)**4
/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)
**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) - 135*A*a**3*f
*x*tan(e/2 + f*x/2)**3/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 1
8*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) -
6*c*f) + 135*A*a**3*f*x*tan(e/2 + f*x/2)**2/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*t
an(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6*
c*f*tan(e/2 + f*x/2) - 6*c*f) - 45*A*a**3*f*x*tan(e/2 + f*x/2)/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*tan(e/2 + f*
x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*c*f*tan(e/
2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) + 45*A*a**3*f*x/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*tan(e/2 + f
*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*c*f*tan(e
/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) - 102*A*a**3*tan(e/2 + f*x/2)**6/(6*c*f*tan(e/2 + f*x/2)**7 -
6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)
**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) + 54*A*a**3*tan(e/2 + f*x/2)**5/(6*c*f*tan(
e/2 + f*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f
*tan(e/2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) - 336*A*a**3*tan(e/2 + f*x
/2)**4/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 +
f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) + 96*A*a
**3*tan(e/2 + f*x/2)**3/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 -
18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2)
- 6*c*f) - 378*A*a**3*tan(e/2 + f*x/2)**2/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*tan(
e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6*c*f
*tan(e/2 + f*x/2) - 6*c*f) + 42*A*a**3*tan(e/2 + f*x/2)/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6
+ 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x
/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) - 144*A*a**3/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6 +
18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x/2
)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) - 60*B*a**3*f*x*tan(e/2 + f*x/2)**7/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*
tan(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 1
8*c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) + 60*B*a**3*f*x*tan(e/2 + f*x/2)**6/(6*c*f*tan(e/2
+ f*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*ta
n(e/2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) - 180*B*a**3*f*x*tan(e/2 + f*
x/2)**5/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 +
f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) + 180*B
*a**3*f*x*tan(e/2 + f*x/2)**4/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)
**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f
*x/2) - 6*c*f) - 180*B*a**3*f*x*tan(e/2 + f*x/2)**3/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6 + 1
8*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x/2)*
*2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) + 180*B*a**3*f*x*tan(e/2 + f*x/2)**2/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*t
an(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18
*c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) - 60*B*a**3*f*x*tan(e/2 + f*x/2)/(6*c*f*tan(e/2 + f
*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/
2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) + 60*B*a**3*f*x/(6*c*f*tan(e/2 +
f*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e
/2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) - 120*B*a**3*tan(e/2 + f*x/2)**6
/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)
**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) + 108*B*a**3*t
an(e/2 + f*x/2)**5/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*
f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c
*f) - 372*B*a**3*tan(e/2 + f*x/2)**4/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 +
f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(
e/2 + f*x/2) - 6*c*f) + 192*B*a**3*tan(e/2 + f*x/2)**3/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*tan(e/2 + f*x/2)**6
+ 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*c*f*tan(e/2 + f*x/
2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) - 456*B*a**3*tan(e/2 + f*x/2)**2/(6*c*f*tan(e/2 + f*x/2)**7 - 6*c*f*ta
n(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x/2)**3 - 18*
c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) + 68*B*a**3*tan(e/2 + f*x/2)/(6*c*f*tan(e/2 + f*x/2)
**7 - 6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f
*x/2)**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f) - 188*B*a**3/(6*c*f*tan(e/2 + f*x/2)**
7 - 6*c*f*tan(e/2 + f*x/2)**6 + 18*c*f*tan(e/2 + f*x/2)**5 - 18*c*f*tan(e/2 + f*x/2)**4 + 18*c*f*tan(e/2 + f*x
/2)**3 - 18*c*f*tan(e/2 + f*x/2)**2 + 6*c*f*tan(e/2 + f*x/2) - 6*c*f), Ne(f, 0)), (x*(A + B*sin(e))*(a*sin(e)
+ a)**3/(-c*sin(e) + c), True))
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1139 vs. \(2 (152) = 304\).
Time = 0.35 (sec) , antiderivative size = 1139, normalized size of antiderivative = 7.30
\[
\int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{c-c \sin (e+f x)} \, dx=\text {Too large to display}
\]
[In]
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e)),x, algorithm="maxima")
[Out]
-1/3*(B*a^3*((7*sin(f*x + e)/(cos(f*x + e) + 1) - 39*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 24*sin(f*x + e)^3/(
cos(f*x + e) + 1)^3 - 24*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 9*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 9*sin(f
*x + e)^6/(cos(f*x + e) + 1)^6 - 16)/(c - c*sin(f*x + e)/(cos(f*x + e) + 1) + 3*c*sin(f*x + e)^2/(cos(f*x + e)
+ 1)^2 - 3*c*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*c*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 3*c*sin(f*x + e)
^5/(cos(f*x + e) + 1)^5 + c*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - c*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) + 9*a
rctan(sin(f*x + e)/(cos(f*x + e) + 1))/c) + 18*A*a^3*((sin(f*x + e)/(cos(f*x + e) + 1) - sin(f*x + e)^2/(cos(f
*x + e) + 1)^2 - 2)/(c - c*sin(f*x + e)/(cos(f*x + e) + 1) + c*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - c*sin(f*x
+ e)^3/(cos(f*x + e) + 1)^3) + arctan(sin(f*x + e)/(cos(f*x + e) + 1))/c) + 18*B*a^3*((sin(f*x + e)/(cos(f*x
+ e) + 1) - sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 2)/(c - c*sin(f*x + e)/(cos(f*x + e) + 1) + c*sin(f*x + e)^2
/(cos(f*x + e) + 1)^2 - c*sin(f*x + e)^3/(cos(f*x + e) + 1)^3) + arctan(sin(f*x + e)/(cos(f*x + e) + 1))/c) +
3*A*a^3*((sin(f*x + e)/(cos(f*x + e) + 1) - 5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3*sin(f*x + e)^3/(cos(f*x
+ e) + 1)^3 - 3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 4)/(c - c*sin(f*x + e)/(cos(f*x + e) + 1) + 2*c*sin(f*x
+ e)^2/(cos(f*x + e) + 1)^2 - 2*c*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + c*sin(f*x + e)^4/(cos(f*x + e) + 1)^4
- c*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 3*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/c) + 9*B*a^3*((sin(f*x +
e)/(cos(f*x + e) + 1) - 5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 3*sin(
f*x + e)^4/(cos(f*x + e) + 1)^4 - 4)/(c - c*sin(f*x + e)/(cos(f*x + e) + 1) + 2*c*sin(f*x + e)^2/(cos(f*x + e)
+ 1)^2 - 2*c*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + c*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - c*sin(f*x + e)^5/(
cos(f*x + e) + 1)^5) + 3*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/c) + 18*A*a^3*(arctan(sin(f*x + e)/(cos(f*x +
e) + 1))/c - 1/(c - c*sin(f*x + e)/(cos(f*x + e) + 1))) + 6*B*a^3*(arctan(sin(f*x + e)/(cos(f*x + e) + 1))/c
- 1/(c - c*sin(f*x + e)/(cos(f*x + e) + 1))) - 6*A*a^3/(c - c*sin(f*x + e)/(cos(f*x + e) + 1)))/f
Giac [A] (verification not implemented)
none
Time = 0.38 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.43
\[
\int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{c-c \sin (e+f x)} \, dx=-\frac {\frac {15 \, {\left (3 \, A a^{3} + 4 \, B a^{3}\right )} {\left (f x + e\right )}}{c} + \frac {96 \, {\left (A a^{3} + B a^{3}\right )}}{c {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}} + \frac {2 \, {\left (3 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 12 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 24 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 42 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 48 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 96 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 12 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 24 \, A a^{3} - 46 \, B a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{3} c}}{6 \, f}
\]
[In]
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e)),x, algorithm="giac")
[Out]
-1/6*(15*(3*A*a^3 + 4*B*a^3)*(f*x + e)/c + 96*(A*a^3 + B*a^3)/(c*(tan(1/2*f*x + 1/2*e) - 1)) + 2*(3*A*a^3*tan(
1/2*f*x + 1/2*e)^5 + 12*B*a^3*tan(1/2*f*x + 1/2*e)^5 - 24*A*a^3*tan(1/2*f*x + 1/2*e)^4 - 42*B*a^3*tan(1/2*f*x
+ 1/2*e)^4 - 48*A*a^3*tan(1/2*f*x + 1/2*e)^2 - 96*B*a^3*tan(1/2*f*x + 1/2*e)^2 - 3*A*a^3*tan(1/2*f*x + 1/2*e)
- 12*B*a^3*tan(1/2*f*x + 1/2*e) - 24*A*a^3 - 46*B*a^3)/((tan(1/2*f*x + 1/2*e)^2 + 1)^3*c))/f
Mupad [B] (verification not implemented)
Time = 14.85 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.07
\[
\int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{c-c \sin (e+f x)} \, dx=\frac {24\,A\,a^3-\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (7\,A\,a^3+\frac {34\,B\,a^3}{3}\right )+\frac {94\,B\,a^3}{3}-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (9\,A\,a^3+18\,B\,a^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (17\,A\,a^3+20\,B\,a^3\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (16\,A\,a^3+32\,B\,a^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (56\,A\,a^3+62\,B\,a^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (63\,A\,a^3+76\,B\,a^3\right )}{f\,\left (-c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-3\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+3\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-3\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+3\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+c\right )}-\frac {5\,a^3\,\mathrm {atan}\left (\frac {5\,a^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (3\,A+4\,B\right )}{15\,A\,a^3+20\,B\,a^3}\right )\,\left (3\,A+4\,B\right )}{c\,f}
\]
[In]
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3)/(c - c*sin(e + f*x)),x)
[Out]
(24*A*a^3 - tan(e/2 + (f*x)/2)*(7*A*a^3 + (34*B*a^3)/3) + (94*B*a^3)/3 - tan(e/2 + (f*x)/2)^5*(9*A*a^3 + 18*B*
a^3) + tan(e/2 + (f*x)/2)^6*(17*A*a^3 + 20*B*a^3) - tan(e/2 + (f*x)/2)^3*(16*A*a^3 + 32*B*a^3) + tan(e/2 + (f*
x)/2)^4*(56*A*a^3 + 62*B*a^3) + tan(e/2 + (f*x)/2)^2*(63*A*a^3 + 76*B*a^3))/(f*(c - c*tan(e/2 + (f*x)/2) + 3*c
*tan(e/2 + (f*x)/2)^2 - 3*c*tan(e/2 + (f*x)/2)^3 + 3*c*tan(e/2 + (f*x)/2)^4 - 3*c*tan(e/2 + (f*x)/2)^5 + c*tan
(e/2 + (f*x)/2)^6 - c*tan(e/2 + (f*x)/2)^7)) - (5*a^3*atan((5*a^3*tan(e/2 + (f*x)/2)*(3*A + 4*B))/(15*A*a^3 +
20*B*a^3))*(3*A + 4*B))/(c*f)