\(\int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx\) [434]

Optimal result

Integrand size = 25, antiderivative size = 297 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx=\frac {9}{16} \left (24 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right ) x+\frac {3 \left (4 c^5-48 c^4 d-311 c^3 d^2-448 c^2 d^3-288 c d^4-64 d^5\right ) \cos (e+f x)}{20 d f}+\frac {3 \left (8 c^4-96 c^3 d-438 c^2 d^2-464 c d^3-165 d^4\right ) \cos (e+f x) \sin (e+f x)}{80 f}+\frac {3 \left (4 c^3-48 c^2 d-123 c d^2-64 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{40 d f}+\frac {3 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{40 d f}+\frac {3 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{10 d f}-\frac {3 \cos (e+f x) (c+d \sin (e+f x))^5}{2 d f} \]

[Out]

1/16*a^2*(24*c^4+64*c^3*d+84*c^2*d^2+48*c*d^3+11*d^4)*x+1/60*a^2*(4*c^5-48*c^4*d-311*c^3*d^2-448*c^2*d^3-288*c
*d^4-64*d^5)*cos(f*x+e)/d/f+1/240*a^2*(8*c^4-96*c^3*d-438*c^2*d^2-464*c*d^3-165*d^4)*cos(f*x+e)*sin(f*x+e)/f+1
/120*a^2*(4*c^3-48*c^2*d-123*c*d^2-64*d^3)*cos(f*x+e)*(c+d*sin(f*x+e))^2/d/f+1/120*a^2*(4*c^2-48*c*d-55*d^2)*c
os(f*x+e)*(c+d*sin(f*x+e))^3/d/f+1/30*a^2*(c-12*d)*cos(f*x+e)*(c+d*sin(f*x+e))^4/d/f-1/6*a^2*cos(f*x+e)*(c+d*s
in(f*x+e))^5/d/f

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2842, 2832, 2813} \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx=\frac {a^2 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac {a^2 \left (4 c^3-48 c^2 d-123 c d^2-64 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d f}+\frac {a^2 \left (8 c^4-96 c^3 d-438 c^2 d^2-464 c d^3-165 d^4\right ) \sin (e+f x) \cos (e+f x)}{240 f}+\frac {1}{16} a^2 x \left (24 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right )+\frac {a^2 \left (4 c^5-48 c^4 d-311 c^3 d^2-448 c^2 d^3-288 c d^4-64 d^5\right ) \cos (e+f x)}{60 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac {a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f} \]

[In]

Int[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^4,x]

[Out]

(a^2*(24*c^4 + 64*c^3*d + 84*c^2*d^2 + 48*c*d^3 + 11*d^4)*x)/16 + (a^2*(4*c^5 - 48*c^4*d - 311*c^3*d^2 - 448*c
^2*d^3 - 288*c*d^4 - 64*d^5)*Cos[e + f*x])/(60*d*f) + (a^2*(8*c^4 - 96*c^3*d - 438*c^2*d^2 - 464*c*d^3 - 165*d
^4)*Cos[e + f*x]*Sin[e + f*x])/(240*f) + (a^2*(4*c^3 - 48*c^2*d - 123*c*d^2 - 64*d^3)*Cos[e + f*x]*(c + d*Sin[
e + f*x])^2)/(120*d*f) + (a^2*(4*c^2 - 48*c*d - 55*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(120*d*f) + (a^2*
(c - 12*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(30*d*f) - (a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^5)/(6*d*f)

Rule 2813

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*a*c +
b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Cos[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 2832

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d
)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(f*(m + 1))), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Sim
p[b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 2842

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Dist[1/(
d*(m + n)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c*(m - 2) + b^2*d*(n + 1) + a^2*d
*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &
& NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] &&  !LtQ[n, -1] && (IntegersQ[2*m,
2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac {\int \left (11 a^2 d-a^2 (c-12 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^4 \, dx}{6 d} \\ & = \frac {a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac {\int (c+d \sin (e+f x))^3 \left (3 a^2 d (17 c+16 d)-a^2 \left (4 c^2-48 c d-55 d^2\right ) \sin (e+f x)\right ) \, dx}{30 d} \\ & = \frac {a^2 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac {a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac {\int (c+d \sin (e+f x))^2 \left (3 a^2 d \left (64 c^2+112 c d+55 d^2\right )-3 a^2 \left (4 c^3-48 c^2 d-123 c d^2-64 d^3\right ) \sin (e+f x)\right ) \, dx}{120 d} \\ & = \frac {a^2 \left (4 c^3-48 c^2 d-123 c d^2-64 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d f}+\frac {a^2 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac {a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac {\int (c+d \sin (e+f x)) \left (3 a^2 d \left (184 c^3+432 c^2 d+411 c d^2+128 d^3\right )-3 a^2 \left (8 c^4-96 c^3 d-438 c^2 d^2-464 c d^3-165 d^4\right ) \sin (e+f x)\right ) \, dx}{360 d} \\ & = \frac {1}{16} a^2 \left (24 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right ) x+\frac {a^2 \left (4 c^5-48 c^4 d-311 c^3 d^2-448 c^2 d^3-288 c d^4-64 d^5\right ) \cos (e+f x)}{60 d f}+\frac {a^2 \left (8 c^4-96 c^3 d-438 c^2 d^2-464 c d^3-165 d^4\right ) \cos (e+f x) \sin (e+f x)}{240 f}+\frac {a^2 \left (4 c^3-48 c^2 d-123 c d^2-64 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d f}+\frac {a^2 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac {a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.12 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.82 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx=\frac {3 \cos (e+f x) \left (-32 \left (15 c^4+50 c^3 d+60 c^2 d^2+36 c d^3+8 d^4\right )-\frac {30 \left (24 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right ) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(e+f x)}}-15 \left (8 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right ) \sin (e+f x)-64 d \left (5 c^3+15 c^2 d+9 c d^2+2 d^3\right ) \sin ^2(e+f x)-10 d^2 \left (36 c^2+48 c d+11 d^2\right ) \sin ^3(e+f x)-96 d^3 (2 c+d) \sin ^4(e+f x)-40 d^4 \sin ^5(e+f x)\right )}{80 f} \]

[In]

Integrate[(3 + 3*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^4,x]

[Out]

(3*Cos[e + f*x]*(-32*(15*c^4 + 50*c^3*d + 60*c^2*d^2 + 36*c*d^3 + 8*d^4) - (30*(24*c^4 + 64*c^3*d + 84*c^2*d^2
+ 48*c*d^3 + 11*d^4)*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]])/Sqrt[Cos[e + f*x]^2] - 15*(8*c^4 + 64*c^3*d + 84
*c^2*d^2 + 48*c*d^3 + 11*d^4)*Sin[e + f*x] - 64*d*(5*c^3 + 15*c^2*d + 9*c*d^2 + 2*d^3)*Sin[e + f*x]^2 - 10*d^2
*(36*c^2 + 48*c*d + 11*d^2)*Sin[e + f*x]^3 - 96*d^3*(2*c + d)*Sin[e + f*x]^4 - 40*d^4*Sin[e + f*x]^5))/(80*f)

Maple [A] (verified)

Time = 2.65 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.79

method	result	size
parallelrisch	\(-\frac {a^{2} \left (\left (\frac {31}{16} d^{4}+c^{4}+8 c^{3} d +12 c^{2} d^{2}+8 d^{3} c \right ) \sin \left (2 f x +2 e \right )+\left (-\frac {4}{3} c^{3} d -3 d^{3} c -\frac {5}{6} d^{4}-4 c^{2} d^{2}\right ) \cos \left (3 f x +3 e \right )-\frac {3 \left (c +\frac {5 d}{6}\right ) d^{2} \left (c +\frac {d}{2}\right ) \sin \left (4 f x +4 e \right )}{4}+\frac {d^{3} \left (c +\frac {d}{2}\right ) \cos \left (5 f x +5 e \right )}{5}+\frac {d^{4} \sin \left (6 f x +6 e \right )}{48}+\left (8 c^{4}+28 c^{3} d +36 c^{2} d^{2}+22 d^{3} c +5 d^{4}\right ) \cos \left (f x +e \right )+\left (\frac {64}{15}-\frac {11 f x}{4}\right ) d^{4}+c \left (\frac {96}{5}-12 f x \right ) d^{3}+\left (-21 f x +32\right ) c^{2} d^{2}-16 \left (f x -\frac {5}{3}\right ) c^{3} d +\left (-6 f x +8\right ) c^{4}\right )}{4 f}\)	\(235\)
parts	\(a^{2} c^{4} x -\frac {\left (4 a^{2} c \,d^{3}+2 a^{2} d^{4}\right ) \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}-\frac {\left (2 a^{2} c^{4}+4 a^{2} c^{3} d \right ) \cos \left (f x +e \right )}{f}+\frac {\left (6 a^{2} c^{2} d^{2}+8 a^{2} c \,d^{3}+a^{2} d^{4}\right ) \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}-\frac {\left (4 a^{2} c^{3} d +12 a^{2} c^{2} d^{2}+4 a^{2} c \,d^{3}\right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+\frac {\left (a^{2} c^{4}+8 a^{2} c^{3} d +6 a^{2} c^{2} d^{2}\right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {a^{2} d^{4} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}\)	\(311\)
risch	\(-\frac {31 \sin \left (2 f x +2 e \right ) a^{2} d^{4}}{64 f}-\frac {a^{2} d^{3} \cos \left (5 f x +5 e \right ) c}{20 f}-\frac {2 c^{4} a^{2} \cos \left (f x +e \right )}{f}-\frac {c^{4} a^{2} \sin \left (2 f x +2 e \right )}{4 f}+\frac {11 a^{2} d^{4} x}{16}-\frac {5 a^{2} \cos \left (f x +e \right ) d^{4}}{4 f}-\frac {a^{2} d^{4} \cos \left (5 f x +5 e \right )}{40 f}+\frac {5 \sin \left (4 f x +4 e \right ) a^{2} d^{4}}{64 f}+\frac {5 a^{2} d^{4} \cos \left (3 f x +3 e \right )}{24 f}+4 a^{2} c^{3} d x +\frac {21 a^{2} c^{2} d^{2} x}{4}+3 a^{2} c \,d^{3} x -\frac {a^{2} d^{4} \sin \left (6 f x +6 e \right )}{192 f}+\frac {3 \sin \left (4 f x +4 e \right ) a^{2} c^{2} d^{2}}{16 f}+\frac {\sin \left (4 f x +4 e \right ) a^{2} c \,d^{3}}{4 f}+\frac {a^{2} d \cos \left (3 f x +3 e \right ) c^{3}}{3 f}+\frac {a^{2} d^{2} \cos \left (3 f x +3 e \right ) c^{2}}{f}+\frac {3 a^{2} d^{3} \cos \left (3 f x +3 e \right ) c}{4 f}-\frac {7 a^{2} \cos \left (f x +e \right ) c^{3} d}{f}-\frac {9 a^{2} \cos \left (f x +e \right ) c^{2} d^{2}}{f}-\frac {11 a^{2} \cos \left (f x +e \right ) d^{3} c}{2 f}-\frac {2 \sin \left (2 f x +2 e \right ) a^{2} c^{3} d}{f}-\frac {3 \sin \left (2 f x +2 e \right ) a^{2} c^{2} d^{2}}{f}-\frac {2 \sin \left (2 f x +2 e \right ) a^{2} c \,d^{3}}{f}+\frac {3 a^{2} c^{4} x}{2}\)	\(456\)
derivativedivides	\(\frac {a^{2} c^{4} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {4 a^{2} c^{3} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+6 a^{2} c^{2} d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {4 a^{2} c \,d^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+a^{2} d^{4} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )-2 a^{2} c^{4} \cos \left (f x +e \right )+8 a^{2} c^{3} d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-4 a^{2} c^{2} d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+8 a^{2} c \,d^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {2 a^{2} d^{4} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+a^{2} c^{4} \left (f x +e \right )-4 a^{2} c^{3} d \cos \left (f x +e \right )+6 a^{2} c^{2} d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {4 a^{2} c \,d^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+a^{2} d^{4} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\)	\(462\)
default	\(\frac {a^{2} c^{4} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {4 a^{2} c^{3} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+6 a^{2} c^{2} d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {4 a^{2} c \,d^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+a^{2} d^{4} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )-2 a^{2} c^{4} \cos \left (f x +e \right )+8 a^{2} c^{3} d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-4 a^{2} c^{2} d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+8 a^{2} c \,d^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {2 a^{2} d^{4} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+a^{2} c^{4} \left (f x +e \right )-4 a^{2} c^{3} d \cos \left (f x +e \right )+6 a^{2} c^{2} d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {4 a^{2} c \,d^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+a^{2} d^{4} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\)	\(462\)
norman	\(\text {Expression too large to display}\)	\(1043\)

[In]

int((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^4,x,method=_RETURNVERBOSE)

[Out]

-1/4*a^2*((31/16*d^4+c^4+8*c^3*d+12*c^2*d^2+8*d^3*c)*sin(2*f*x+2*e)+(-4/3*c^3*d-3*d^3*c-5/6*d^4-4*c^2*d^2)*cos
(3*f*x+3*e)-3/4*(c+5/6*d)*d^2*(c+1/2*d)*sin(4*f*x+4*e)+1/5*d^3*(c+1/2*d)*cos(5*f*x+5*e)+1/48*d^4*sin(6*f*x+6*e
)+(8*c^4+28*c^3*d+36*c^2*d^2+22*c*d^3+5*d^4)*cos(f*x+e)+(64/15-11/4*f*x)*d^4+c*(96/5-12*f*x)*d^3+(-21*f*x+32)*
c^2*d^2-16*(f*x-5/3)*c^3*d+(-6*f*x+8)*c^4)/f

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.01 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx=-\frac {96 \, {\left (2 \, a^{2} c d^{3} + a^{2} d^{4}\right )} \cos \left (f x + e\right )^{5} - 320 \, {\left (a^{2} c^{3} d + 3 \, a^{2} c^{2} d^{2} + 3 \, a^{2} c d^{3} + a^{2} d^{4}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (24 \, a^{2} c^{4} + 64 \, a^{2} c^{3} d + 84 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 11 \, a^{2} d^{4}\right )} f x + 480 \, {\left (a^{2} c^{4} + 4 \, a^{2} c^{3} d + 6 \, a^{2} c^{2} d^{2} + 4 \, a^{2} c d^{3} + a^{2} d^{4}\right )} \cos \left (f x + e\right ) + 5 \, {\left (8 \, a^{2} d^{4} \cos \left (f x + e\right )^{5} - 2 \, {\left (36 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 19 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (8 \, a^{2} c^{4} + 64 \, a^{2} c^{3} d + 108 \, a^{2} c^{2} d^{2} + 80 \, a^{2} c d^{3} + 21 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^4,x, algorithm="fricas")

[Out]

-1/240*(96*(2*a^2*c*d^3 + a^2*d^4)*cos(f*x + e)^5 - 320*(a^2*c^3*d + 3*a^2*c^2*d^2 + 3*a^2*c*d^3 + a^2*d^4)*co
s(f*x + e)^3 - 15*(24*a^2*c^4 + 64*a^2*c^3*d + 84*a^2*c^2*d^2 + 48*a^2*c*d^3 + 11*a^2*d^4)*f*x + 480*(a^2*c^4
+ 4*a^2*c^3*d + 6*a^2*c^2*d^2 + 4*a^2*c*d^3 + a^2*d^4)*cos(f*x + e) + 5*(8*a^2*d^4*cos(f*x + e)^5 - 2*(36*a^2*
c^2*d^2 + 48*a^2*c*d^3 + 19*a^2*d^4)*cos(f*x + e)^3 + 3*(8*a^2*c^4 + 64*a^2*c^3*d + 108*a^2*c^2*d^2 + 80*a^2*c
*d^3 + 21*a^2*d^4)*cos(f*x + e))*sin(f*x + e))/f

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1136 vs. \(2 (299) = 598\).

Time = 0.50 (sec) , antiderivative size = 1136, normalized size of antiderivative = 3.82 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))**2*(c+d*sin(f*x+e))**4,x)

[Out]

Piecewise((a**2*c**4*x*sin(e + f*x)**2/2 + a**2*c**4*x*cos(e + f*x)**2/2 + a**2*c**4*x - a**2*c**4*sin(e + f*x
)*cos(e + f*x)/(2*f) - 2*a**2*c**4*cos(e + f*x)/f + 4*a**2*c**3*d*x*sin(e + f*x)**2 + 4*a**2*c**3*d*x*cos(e +
f*x)**2 - 4*a**2*c**3*d*sin(e + f*x)**2*cos(e + f*x)/f - 4*a**2*c**3*d*sin(e + f*x)*cos(e + f*x)/f - 8*a**2*c*
*3*d*cos(e + f*x)**3/(3*f) - 4*a**2*c**3*d*cos(e + f*x)/f + 9*a**2*c**2*d**2*x*sin(e + f*x)**4/4 + 9*a**2*c**2
*d**2*x*sin(e + f*x)**2*cos(e + f*x)**2/2 + 3*a**2*c**2*d**2*x*sin(e + f*x)**2 + 9*a**2*c**2*d**2*x*cos(e + f*
x)**4/4 + 3*a**2*c**2*d**2*x*cos(e + f*x)**2 - 15*a**2*c**2*d**2*sin(e + f*x)**3*cos(e + f*x)/(4*f) - 12*a**2*
c**2*d**2*sin(e + f*x)**2*cos(e + f*x)/f - 9*a**2*c**2*d**2*sin(e + f*x)*cos(e + f*x)**3/(4*f) - 3*a**2*c**2*d
**2*sin(e + f*x)*cos(e + f*x)/f - 8*a**2*c**2*d**2*cos(e + f*x)**3/f + 3*a**2*c*d**3*x*sin(e + f*x)**4 + 6*a**
2*c*d**3*x*sin(e + f*x)**2*cos(e + f*x)**2 + 3*a**2*c*d**3*x*cos(e + f*x)**4 - 4*a**2*c*d**3*sin(e + f*x)**4*c
os(e + f*x)/f - 5*a**2*c*d**3*sin(e + f*x)**3*cos(e + f*x)/f - 16*a**2*c*d**3*sin(e + f*x)**2*cos(e + f*x)**3/
(3*f) - 4*a**2*c*d**3*sin(e + f*x)**2*cos(e + f*x)/f - 3*a**2*c*d**3*sin(e + f*x)*cos(e + f*x)**3/f - 32*a**2*
c*d**3*cos(e + f*x)**5/(15*f) - 8*a**2*c*d**3*cos(e + f*x)**3/(3*f) + 5*a**2*d**4*x*sin(e + f*x)**6/16 + 15*a*
*2*d**4*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 3*a**2*d**4*x*sin(e + f*x)**4/8 + 15*a**2*d**4*x*sin(e + f*x)**
2*cos(e + f*x)**4/16 + 3*a**2*d**4*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 5*a**2*d**4*x*cos(e + f*x)**6/16 + 3*
a**2*d**4*x*cos(e + f*x)**4/8 - 11*a**2*d**4*sin(e + f*x)**5*cos(e + f*x)/(16*f) - 2*a**2*d**4*sin(e + f*x)**4
*cos(e + f*x)/f - 5*a**2*d**4*sin(e + f*x)**3*cos(e + f*x)**3/(6*f) - 5*a**2*d**4*sin(e + f*x)**3*cos(e + f*x)
/(8*f) - 8*a**2*d**4*sin(e + f*x)**2*cos(e + f*x)**3/(3*f) - 5*a**2*d**4*sin(e + f*x)*cos(e + f*x)**5/(16*f) -
3*a**2*d**4*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 16*a**2*d**4*cos(e + f*x)**5/(15*f), Ne(f, 0)), (x*(c + d*si
n(e))**4*(a*sin(e) + a)**2, True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.52 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx=\frac {240 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{4} + 960 \, {\left (f x + e\right )} a^{2} c^{4} + 1280 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c^{3} d + 1920 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{3} d + 3840 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c^{2} d^{2} + 180 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{2} d^{2} + 1440 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{2} d^{2} - 256 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{2} c d^{3} + 1280 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c d^{3} + 240 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c d^{3} - 128 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{2} d^{4} + 5 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{4} + 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{4} - 1920 \, a^{2} c^{4} \cos \left (f x + e\right ) - 3840 \, a^{2} c^{3} d \cos \left (f x + e\right )}{960 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^4,x, algorithm="maxima")

[Out]

1/960*(240*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^2*c^4 + 960*(f*x + e)*a^2*c^4 + 1280*(cos(f*x + e)^3 - 3*cos(f*x
+ e))*a^2*c^3*d + 1920*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^2*c^3*d + 3840*(cos(f*x + e)^3 - 3*cos(f*x + e))*a^
2*c^2*d^2 + 180*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a^2*c^2*d^2 + 1440*(2*f*x + 2*e - sin(
2*f*x + 2*e))*a^2*c^2*d^2 - 256*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*a^2*c*d^3 + 1280*(cos
(f*x + e)^3 - 3*cos(f*x + e))*a^2*c*d^3 + 240*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a^2*c*d^
3 - 128*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*a^2*d^4 + 5*(4*sin(2*f*x + 2*e)^3 + 60*f*x +
60*e + 9*sin(4*f*x + 4*e) - 48*sin(2*f*x + 2*e))*a^2*d^4 + 30*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x
+ 2*e))*a^2*d^4 - 1920*a^2*c^4*cos(f*x + e) - 3840*a^2*c^3*d*cos(f*x + e))/f

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.51 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx=\frac {a^{2} c d^{3} \cos \left (3 \, f x + 3 \, e\right )}{3 \, f} - \frac {a^{2} d^{4} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {a^{2} d^{4} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac {1}{16} \, {\left (8 \, a^{2} c^{4} + 64 \, a^{2} c^{3} d + 36 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 5 \, a^{2} d^{4}\right )} x + \frac {1}{8} \, {\left (8 \, a^{2} c^{4} + 24 \, a^{2} c^{2} d^{2} + 3 \, a^{2} d^{4}\right )} x - \frac {{\left (2 \, a^{2} c d^{3} + a^{2} d^{4}\right )} \cos \left (5 \, f x + 5 \, e\right )}{40 \, f} + \frac {{\left (8 \, a^{2} c^{3} d + 24 \, a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 5 \, a^{2} d^{4}\right )} \cos \left (3 \, f x + 3 \, e\right )}{24 \, f} - \frac {{\left (8 \, a^{2} c^{4} + 12 \, a^{2} c^{3} d + 36 \, a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 5 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )}{4 \, f} - \frac {{\left (4 \, a^{2} c^{3} d + 3 \, a^{2} c d^{3}\right )} \cos \left (f x + e\right )}{f} + \frac {{\left (12 \, a^{2} c^{2} d^{2} + 16 \, a^{2} c d^{3} + 3 \, a^{2} d^{4}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac {{\left (16 \, a^{2} c^{4} + 128 \, a^{2} c^{3} d + 96 \, a^{2} c^{2} d^{2} + 128 \, a^{2} c d^{3} + 15 \, a^{2} d^{4}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} - \frac {{\left (6 \, a^{2} c^{2} d^{2} + a^{2} d^{4}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^4,x, algorithm="giac")

[Out]

1/3*a^2*c*d^3*cos(3*f*x + 3*e)/f - 1/192*a^2*d^4*sin(6*f*x + 6*e)/f + 1/32*a^2*d^4*sin(4*f*x + 4*e)/f + 1/16*(
8*a^2*c^4 + 64*a^2*c^3*d + 36*a^2*c^2*d^2 + 48*a^2*c*d^3 + 5*a^2*d^4)*x + 1/8*(8*a^2*c^4 + 24*a^2*c^2*d^2 + 3*
a^2*d^4)*x - 1/40*(2*a^2*c*d^3 + a^2*d^4)*cos(5*f*x + 5*e)/f + 1/24*(8*a^2*c^3*d + 24*a^2*c^2*d^2 + 10*a^2*c*d
^3 + 5*a^2*d^4)*cos(3*f*x + 3*e)/f - 1/4*(8*a^2*c^4 + 12*a^2*c^3*d + 36*a^2*c^2*d^2 + 10*a^2*c*d^3 + 5*a^2*d^4
)*cos(f*x + e)/f - (4*a^2*c^3*d + 3*a^2*c*d^3)*cos(f*x + e)/f + 1/64*(12*a^2*c^2*d^2 + 16*a^2*c*d^3 + 3*a^2*d^
4)*sin(4*f*x + 4*e)/f - 1/64*(16*a^2*c^4 + 128*a^2*c^3*d + 96*a^2*c^2*d^2 + 128*a^2*c*d^3 + 15*a^2*d^4)*sin(2*
f*x + 2*e)/f - 1/4*(6*a^2*c^2*d^2 + a^2*d^4)*sin(2*f*x + 2*e)/f

Mupad [B] (verification not implemented)

Time = 10.79 (sec) , antiderivative size = 865, normalized size of antiderivative = 2.91 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx=\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (24\,c^4+64\,c^3\,d+84\,c^2\,d^2+48\,c\,d^3+11\,d^4\right )}{8\,\left (3\,a^2\,c^4+8\,a^2\,c^3\,d+\frac {21\,a^2\,c^2\,d^2}{2}+6\,a^2\,c\,d^3+\frac {11\,a^2\,d^4}{8}\right )}\right )\,\left (24\,c^4+64\,c^3\,d+84\,c^2\,d^2+48\,c\,d^3+11\,d^4\right )}{8\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (20\,a^2\,c^4+56\,a^2\,c^3\,d+48\,a^2\,c^2\,d^2+16\,a^2\,c\,d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (4\,a^2\,c^4+8\,d\,a^2\,c^3\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (a^2\,c^4+8\,a^2\,c^3\,d+\frac {21\,a^2\,c^2\,d^2}{2}+6\,a^2\,c\,d^3+\frac {11\,a^2\,d^4}{8}\right )+4\,a^2\,c^4+\frac {32\,a^2\,d^4}{15}-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}\,\left (a^2\,c^4+8\,a^2\,c^3\,d+\frac {21\,a^2\,c^2\,d^2}{2}+6\,a^2\,c\,d^3+\frac {11\,a^2\,d^4}{8}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (2\,a^2\,c^4+16\,a^2\,c^3\,d+33\,a^2\,c^2\,d^2+28\,a^2\,c\,d^3+\frac {47\,a^2\,d^4}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (2\,a^2\,c^4+16\,a^2\,c^3\,d+33\,a^2\,c^2\,d^2+28\,a^2\,c\,d^3+\frac {47\,a^2\,d^4}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (3\,a^2\,c^4+24\,a^2\,c^3\,d+\frac {87\,a^2\,c^2\,d^2}{2}+34\,a^2\,c\,d^3+\frac {187\,a^2\,d^4}{24}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (3\,a^2\,c^4+24\,a^2\,c^3\,d+\frac {87\,a^2\,c^2\,d^2}{2}+34\,a^2\,c\,d^3+\frac {187\,a^2\,d^4}{24}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (40\,a^2\,c^4+144\,a^2\,c^3\,d+192\,a^2\,c^2\,d^2+128\,a^2\,c\,d^3+32\,a^2\,d^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (20\,a^2\,c^4+72\,a^2\,c^3\,d+96\,a^2\,c^2\,d^2+\frac {288\,a^2\,c\,d^3}{5}+\frac {64\,a^2\,d^4}{5}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (40\,a^2\,c^4+\frac {400\,a^2\,c^3\,d}{3}+160\,a^2\,c^2\,d^2+96\,a^2\,c\,d^3+\frac {64\,a^2\,d^4}{3}\right )+\frac {48\,a^2\,c\,d^3}{5}+\frac {40\,a^2\,c^3\,d}{3}+16\,a^2\,c^2\,d^2}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]

[In]

int((a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^4,x)

[Out]

(a^2*atan((a^2*tan(e/2 + (f*x)/2)*(48*c*d^3 + 64*c^3*d + 24*c^4 + 11*d^4 + 84*c^2*d^2))/(8*(3*a^2*c^4 + (11*a^
2*d^4)/8 + 6*a^2*c*d^3 + 8*a^2*c^3*d + (21*a^2*c^2*d^2)/2)))*(48*c*d^3 + 64*c^3*d + 24*c^4 + 11*d^4 + 84*c^2*d
^2))/(8*f) - (tan(e/2 + (f*x)/2)^8*(20*a^2*c^4 + 16*a^2*c*d^3 + 56*a^2*c^3*d + 48*a^2*c^2*d^2) + tan(e/2 + (f*
x)/2)^10*(4*a^2*c^4 + 8*a^2*c^3*d) + tan(e/2 + (f*x)/2)*(a^2*c^4 + (11*a^2*d^4)/8 + 6*a^2*c*d^3 + 8*a^2*c^3*d
+ (21*a^2*c^2*d^2)/2) + 4*a^2*c^4 + (32*a^2*d^4)/15 - tan(e/2 + (f*x)/2)^11*(a^2*c^4 + (11*a^2*d^4)/8 + 6*a^2*
c*d^3 + 8*a^2*c^3*d + (21*a^2*c^2*d^2)/2) + tan(e/2 + (f*x)/2)^5*(2*a^2*c^4 + (47*a^2*d^4)/4 + 28*a^2*c*d^3 +
16*a^2*c^3*d + 33*a^2*c^2*d^2) - tan(e/2 + (f*x)/2)^7*(2*a^2*c^4 + (47*a^2*d^4)/4 + 28*a^2*c*d^3 + 16*a^2*c^3*
d + 33*a^2*c^2*d^2) + tan(e/2 + (f*x)/2)^3*(3*a^2*c^4 + (187*a^2*d^4)/24 + 34*a^2*c*d^3 + 24*a^2*c^3*d + (87*a
^2*c^2*d^2)/2) - tan(e/2 + (f*x)/2)^9*(3*a^2*c^4 + (187*a^2*d^4)/24 + 34*a^2*c*d^3 + 24*a^2*c^3*d + (87*a^2*c^
2*d^2)/2) + tan(e/2 + (f*x)/2)^4*(40*a^2*c^4 + 32*a^2*d^4 + 128*a^2*c*d^3 + 144*a^2*c^3*d + 192*a^2*c^2*d^2) +
tan(e/2 + (f*x)/2)^2*(20*a^2*c^4 + (64*a^2*d^4)/5 + (288*a^2*c*d^3)/5 + 72*a^2*c^3*d + 96*a^2*c^2*d^2) + tan(
e/2 + (f*x)/2)^6*(40*a^2*c^4 + (64*a^2*d^4)/3 + 96*a^2*c*d^3 + (400*a^2*c^3*d)/3 + 160*a^2*c^2*d^2) + (48*a^2*
c*d^3)/5 + (40*a^2*c^3*d)/3 + 16*a^2*c^2*d^2)/(f*(6*tan(e/2 + (f*x)/2)^2 + 15*tan(e/2 + (f*x)/2)^4 + 20*tan(e/
2 + (f*x)/2)^6 + 15*tan(e/2 + (f*x)/2)^8 + 6*tan(e/2 + (f*x)/2)^10 + tan(e/2 + (f*x)/2)^12 + 1))