\(\int \frac {(3+3 \sin (e+f x))^3}{(c+d \sin (e+f x))^3} \, dx\) [450]
Optimal result
Integrand size = 25, antiderivative size = 175 \[
\int \frac {(3+3 \sin (e+f x))^3}{(c+d \sin (e+f x))^3} \, dx=\frac {27 x}{d^3}-\frac {27 (c-d) \left (2 c^2+6 c d+7 d^2\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^3 (c+d)^2 \sqrt {c^2-d^2} f}+\frac {(c-d) \cos (e+f x) (27+27 \sin (e+f x))}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {27 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}
\]
[Out]
a^3*x/d^3+1/2*(c-d)*cos(f*x+e)*(a^3+a^3*sin(f*x+e))/d/(c+d)/f/(c+d*sin(f*x+e))^2+1/2*a^3*(c-d)*(2*c+5*d)*cos(f
*x+e)/d^2/(c+d)^2/f/(c+d*sin(f*x+e))-a^3*(c-d)*(2*c^2+6*c*d+7*d^2)*arctan((d+c*tan(1/2*f*x+1/2*e))/(c^2-d^2)^(
1/2))/d^3/(c+d)^2/f/(c^2-d^2)^(1/2)
Rubi [A] (verified)
Time = 0.34 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.07, number of
steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2841, 3047, 3100, 2814, 2739,
632, 210} \[
\int \frac {(3+3 \sin (e+f x))^3}{(c+d \sin (e+f x))^3} \, dx=-\frac {a^3 (c-d) \left (2 c^2+6 c d+7 d^2\right ) \arctan \left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^3 f (c+d)^2 \sqrt {c^2-d^2}}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 f (c+d)^2 (c+d \sin (e+f x))}+\frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{2 d f (c+d) (c+d \sin (e+f x))^2}+\frac {a^3 x}{d^3}
\]
[In]
Int[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^3,x]
[Out]
(a^3*x)/d^3 - (a^3*(c - d)*(2*c^2 + 6*c*d + 7*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^3*(c +
d)^2*Sqrt[c^2 - d^2]*f) + ((c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(2*d*(c + d)*f*(c + d*Sin[e + f*x])
^2) + (a^3*(c - d)*(2*c + 5*d)*Cos[e + f*x])/(2*d^2*(c + d)^2*f*(c + d*Sin[e + f*x]))
Rule 210
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 632
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 2739
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]
Rule 2814
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*(x/d)
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]
Rule 2841
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-b^2)*(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c
+ a*d))), x] + Dist[b^2/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1
)*Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b*c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{
a, b, c, d, e, f}, 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]))
Rule 3047
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Rule 3100
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m
+ 1)*(a^2 - b^2))), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B +
a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b,
e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Rubi steps \begin{align*}
\text {integral}& = \frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}-\frac {a \int \frac {(a+a \sin (e+f x)) (a (c-5 d)-2 a (c+d) \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx}{2 d (c+d)} \\ & = \frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}-\frac {a \int \frac {a^2 (c-5 d)+\left (a^2 (c-5 d)-2 a^2 (c+d)\right ) \sin (e+f x)-2 a^2 (c+d) \sin ^2(e+f x)}{(c+d \sin (e+f x))^2} \, dx}{2 d (c+d)} \\ & = \frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}+\frac {a \int \frac {a^2 (c-d) d (c+7 d)+2 a^2 (c-d) (c+d)^2 \sin (e+f x)}{c+d \sin (e+f x)} \, dx}{2 (c-d) d^2 (c+d)^2} \\ & = \frac {a^3 x}{d^3}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}-\frac {\left (a^3 (c-d) \left (2 c^2+6 c d+7 d^2\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{2 d^3 (c+d)^2} \\ & = \frac {a^3 x}{d^3}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}-\frac {\left (a^3 (c-d) \left (2 c^2+6 c d+7 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{c+2 d x+c x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^3 (c+d)^2 f} \\ & = \frac {a^3 x}{d^3}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}+\frac {\left (2 a^3 (c-d) \left (2 c^2+6 c d+7 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (c^2-d^2\right )-x^2} \, dx,x,2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^3 (c+d)^2 f} \\ & = \frac {a^3 x}{d^3}-\frac {a^3 (c-d) \left (2 c^2+6 c d+7 d^2\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^3 (c+d)^2 \sqrt {c^2-d^2} f}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.66 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.91
\[
\int \frac {(3+3 \sin (e+f x))^3}{(c+d \sin (e+f x))^3} \, dx=\frac {27 \left (2 (e+f x)-\frac {2 \left (2 c^3+4 c^2 d+c d^2-7 d^3\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(c+d)^2 \sqrt {c^2-d^2}}-\frac {(c-d)^2 d \cos (e+f x)}{(c+d) (c+d \sin (e+f x))^2}+\frac {3 d \left (c^2+c d-2 d^2\right ) \cos (e+f x)}{(c+d)^2 (c+d \sin (e+f x))}\right )}{2 d^3 f}
\]
[In]
Integrate[(3 + 3*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^3,x]
[Out]
(27*(2*(e + f*x) - (2*(2*c^3 + 4*c^2*d + c*d^2 - 7*d^3)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c
+ d)^2*Sqrt[c^2 - d^2]) - ((c - d)^2*d*Cos[e + f*x])/((c + d)*(c + d*Sin[e + f*x])^2) + (3*d*(c^2 + c*d - 2*d^
2)*Cos[e + f*x])/((c + d)^2*(c + d*Sin[e + f*x]))))/(2*d^3*f)
Maple [A] (verified)
Time = 1.32 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.02
| | |
method | result | size |
| | |
derivativedivides |
\(\frac {2 a^{3} \left (-\frac {\frac {-\frac {d^{2} \left (c^{3}+5 c^{2} d -4 c \,d^{2}-2 d^{3}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c}-\frac {d \left (2 c^{5}+4 c^{4} d -c^{3} d^{2}+7 c^{2} d^{3}-10 c \,d^{4}-2 d^{5}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c^{2}}-\frac {d^{2} \left (7 c^{3}+11 c^{2} d -16 c \,d^{2}-2 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}-\frac {d \left (2 c^{3}+4 c^{2} d -5 c \,d^{2}-d^{3}\right )}{2 \left (c^{2}+2 c d +d^{2}\right )}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{2}}+\frac {\left (2 c^{3}+4 c^{2} d +c \,d^{2}-7 d^{3}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {c^{2}-d^{2}}}}{d^{3}}+\frac {\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d^{3}}\right )}{f}\) |
\(353\) |
default |
\(\frac {2 a^{3} \left (-\frac {\frac {-\frac {d^{2} \left (c^{3}+5 c^{2} d -4 c \,d^{2}-2 d^{3}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c}-\frac {d \left (2 c^{5}+4 c^{4} d -c^{3} d^{2}+7 c^{2} d^{3}-10 c \,d^{4}-2 d^{5}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c^{2}}-\frac {d^{2} \left (7 c^{3}+11 c^{2} d -16 c \,d^{2}-2 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}-\frac {d \left (2 c^{3}+4 c^{2} d -5 c \,d^{2}-d^{3}\right )}{2 \left (c^{2}+2 c d +d^{2}\right )}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{2}}+\frac {\left (2 c^{3}+4 c^{2} d +c \,d^{2}-7 d^{3}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {c^{2}-d^{2}}}}{d^{3}}+\frac {\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d^{3}}\right )}{f}\) |
\(353\) |
risch |
\(\frac {a^{3} x}{d^{3}}-\frac {i a^{3} \left (7 i c \,d^{3} {\mathrm e}^{3 i \left (f x +e \right )}+10 i c^{2} d^{2} {\mathrm e}^{i \left (f x +e \right )}-2 i c^{2} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+8 i c^{3} d \,{\mathrm e}^{i \left (f x +e \right )}-i d^{4} {\mathrm e}^{i \left (f x +e \right )}-17 i d^{3} c \,{\mathrm e}^{i \left (f x +e \right )}-4 i c^{3} d \,{\mathrm e}^{3 i \left (f x +e \right )}-i d^{4} {\mathrm e}^{3 i \left (f x +e \right )}+6 c^{4} {\mathrm e}^{2 i \left (f x +e \right )}+6 d \,c^{3} {\mathrm e}^{2 i \left (f x +e \right )}-9 d^{2} c^{2} {\mathrm e}^{2 i \left (f x +e \right )}+3 d^{3} c \,{\mathrm e}^{2 i \left (f x +e \right )}-6 d^{4} {\mathrm e}^{2 i \left (f x +e \right )}-3 c^{2} d^{2}-3 d^{3} c +6 d^{4}\right )}{\left (-i d \,{\mathrm e}^{2 i \left (f x +e \right )}+i d +2 c \,{\mathrm e}^{i \left (f x +e \right )}\right )^{2} \left (c +d \right )^{2} f \,d^{3}}+\frac {\sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c -\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right ) c^{2}}{\left (c +d \right )^{3} f \,d^{3}}+\frac {3 \sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c -\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right ) c}{\left (c +d \right )^{3} f \,d^{2}}+\frac {7 \sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c -\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right )}{2 \left (c +d \right )^{3} f d}-\frac {\sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c +\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right ) c^{2}}{\left (c +d \right )^{3} f \,d^{3}}-\frac {3 \sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c +\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right ) c}{\left (c +d \right )^{3} f \,d^{2}}-\frac {7 \sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c +\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right )}{2 \left (c +d \right )^{3} f d}\) |
\(657\) |
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[In]
int((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x,method=_RETURNVERBOSE)
[Out]
2/f*a^3*(-1/d^3*((-1/2*d^2*(c^3+5*c^2*d-4*c*d^2-2*d^3)/(c^2+2*c*d+d^2)/c*tan(1/2*f*x+1/2*e)^3-1/2*d*(2*c^5+4*c
^4*d-c^3*d^2+7*c^2*d^3-10*c*d^4-2*d^5)/(c^2+2*c*d+d^2)/c^2*tan(1/2*f*x+1/2*e)^2-1/2*d^2*(7*c^3+11*c^2*d-16*c*d
^2-2*d^3)/c/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)-1/2*d*(2*c^3+4*c^2*d-5*c*d^2-d^3)/(c^2+2*c*d+d^2))/(tan(1/2*f*x
+1/2*e)^2*c+2*d*tan(1/2*f*x+1/2*e)+c)^2+1/2*(2*c^3+4*c^2*d+c*d^2-7*d^3)/(c^2+2*c*d+d^2)/(c^2-d^2)^(1/2)*arctan
(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2)))+1/d^3*arctan(tan(1/2*f*x+1/2*e)))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 487 vs. \(2 (178) = 356\).
Time = 0.33 (sec) , antiderivative size = 1064, normalized size of antiderivative = 6.08
\[
\int \frac {(3+3 \sin (e+f x))^3}{(c+d \sin (e+f x))^3} \, dx=\text {Too large to display}
\]
[In]
integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm="fricas")
[Out]
[1/4*(4*(a^3*c^2*d^2 + 2*a^3*c*d^3 + a^3*d^4)*f*x*cos(f*x + e)^2 - 4*(a^3*c^4 + 2*a^3*c^3*d + 2*a^3*c^2*d^2 +
2*a^3*c*d^3 + a^3*d^4)*f*x - (2*a^3*c^4 + 6*a^3*c^3*d + 9*a^3*c^2*d^2 + 6*a^3*c*d^3 + 7*a^3*d^4 - (2*a^3*c^2*d
^2 + 6*a^3*c*d^3 + 7*a^3*d^4)*cos(f*x + e)^2 + 2*(2*a^3*c^3*d + 6*a^3*c^2*d^2 + 7*a^3*c*d^3)*sin(f*x + e))*sqr
t(-(c - d)/(c + d))*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*((c^2 + c*d)*cos(f*
x + e)*sin(f*x + e) + (c*d + d^2)*cos(f*x + e))*sqrt(-(c - d)/(c + d)))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x +
e) - c^2 - d^2)) - 2*(2*a^3*c^3*d + 4*a^3*c^2*d^2 - 5*a^3*c*d^3 - a^3*d^4)*cos(f*x + e) - 2*(4*(a^3*c^3*d + 2*
a^3*c^2*d^2 + a^3*c*d^3)*f*x + 3*(a^3*c^2*d^2 + a^3*c*d^3 - 2*a^3*d^4)*cos(f*x + e))*sin(f*x + e))/((c^2*d^5 +
2*c*d^6 + d^7)*f*cos(f*x + e)^2 - 2*(c^3*d^4 + 2*c^2*d^5 + c*d^6)*f*sin(f*x + e) - (c^4*d^3 + 2*c^3*d^4 + 2*c
^2*d^5 + 2*c*d^6 + d^7)*f), 1/2*(2*(a^3*c^2*d^2 + 2*a^3*c*d^3 + a^3*d^4)*f*x*cos(f*x + e)^2 - 2*(a^3*c^4 + 2*a
^3*c^3*d + 2*a^3*c^2*d^2 + 2*a^3*c*d^3 + a^3*d^4)*f*x - (2*a^3*c^4 + 6*a^3*c^3*d + 9*a^3*c^2*d^2 + 6*a^3*c*d^3
+ 7*a^3*d^4 - (2*a^3*c^2*d^2 + 6*a^3*c*d^3 + 7*a^3*d^4)*cos(f*x + e)^2 + 2*(2*a^3*c^3*d + 6*a^3*c^2*d^2 + 7*a
^3*c*d^3)*sin(f*x + e))*sqrt((c - d)/(c + d))*arctan(-(c*sin(f*x + e) + d)*sqrt((c - d)/(c + d))/((c - d)*cos(
f*x + e))) - (2*a^3*c^3*d + 4*a^3*c^2*d^2 - 5*a^3*c*d^3 - a^3*d^4)*cos(f*x + e) - (4*(a^3*c^3*d + 2*a^3*c^2*d^
2 + a^3*c*d^3)*f*x + 3*(a^3*c^2*d^2 + a^3*c*d^3 - 2*a^3*d^4)*cos(f*x + e))*sin(f*x + e))/((c^2*d^5 + 2*c*d^6 +
d^7)*f*cos(f*x + e)^2 - 2*(c^3*d^4 + 2*c^2*d^5 + c*d^6)*f*sin(f*x + e) - (c^4*d^3 + 2*c^3*d^4 + 2*c^2*d^5 + 2
*c*d^6 + d^7)*f)]
Sympy [F(-1)]
Timed out. \[
\int \frac {(3+3 \sin (e+f x))^3}{(c+d \sin (e+f x))^3} \, dx=\text {Timed out}
\]
[In]
integrate((a+a*sin(f*x+e))**3/(c+d*sin(f*x+e))**3,x)
[Out]
Timed out
Maxima [F(-2)]
Exception generated. \[
\int \frac {(3+3 \sin (e+f x))^3}{(c+d \sin (e+f x))^3} \, dx=\text {Exception raised: ValueError}
\]
[In]
integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*d^2-4*c^2>0)', see `assume?`
for more de
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 503 vs. \(2 (178) = 356\).
Time = 0.35 (sec) , antiderivative size = 503, normalized size of antiderivative = 2.87
\[
\int \frac {(3+3 \sin (e+f x))^3}{(c+d \sin (e+f x))^3} \, dx=\frac {\frac {{\left (f x + e\right )} a^{3}}{d^{3}} - \frac {{\left (2 \, a^{3} c^{3} + 4 \, a^{3} c^{2} d + a^{3} c d^{2} - 7 \, a^{3} d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (c\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )}}{{\left (c^{2} d^{3} + 2 \, c d^{4} + d^{5}\right )} \sqrt {c^{2} - d^{2}}} + \frac {a^{3} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 5 \, a^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a^{3} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, a^{3} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, a^{3} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7 \, a^{3} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 10 \, a^{3} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, a^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7 \, a^{3} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 11 \, a^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 16 \, a^{3} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a^{3} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, a^{3} c^{5} + 4 \, a^{3} c^{4} d - 5 \, a^{3} c^{3} d^{2} - a^{3} c^{2} d^{3}}{{\left (c^{4} d^{2} + 2 \, c^{3} d^{3} + c^{2} d^{4}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c\right )}^{2}}}{f}
\]
[In]
integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm="giac")
[Out]
((f*x + e)*a^3/d^3 - (2*a^3*c^3 + 4*a^3*c^2*d + a^3*c*d^2 - 7*a^3*d^3)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(c
) + arctan((c*tan(1/2*f*x + 1/2*e) + d)/sqrt(c^2 - d^2)))/((c^2*d^3 + 2*c*d^4 + d^5)*sqrt(c^2 - d^2)) + (a^3*c
^4*d*tan(1/2*f*x + 1/2*e)^3 + 5*a^3*c^3*d^2*tan(1/2*f*x + 1/2*e)^3 - 4*a^3*c^2*d^3*tan(1/2*f*x + 1/2*e)^3 - 2*
a^3*c*d^4*tan(1/2*f*x + 1/2*e)^3 + 2*a^3*c^5*tan(1/2*f*x + 1/2*e)^2 + 4*a^3*c^4*d*tan(1/2*f*x + 1/2*e)^2 - a^3
*c^3*d^2*tan(1/2*f*x + 1/2*e)^2 + 7*a^3*c^2*d^3*tan(1/2*f*x + 1/2*e)^2 - 10*a^3*c*d^4*tan(1/2*f*x + 1/2*e)^2 -
2*a^3*d^5*tan(1/2*f*x + 1/2*e)^2 + 7*a^3*c^4*d*tan(1/2*f*x + 1/2*e) + 11*a^3*c^3*d^2*tan(1/2*f*x + 1/2*e) - 1
6*a^3*c^2*d^3*tan(1/2*f*x + 1/2*e) - 2*a^3*c*d^4*tan(1/2*f*x + 1/2*e) + 2*a^3*c^5 + 4*a^3*c^4*d - 5*a^3*c^3*d^
2 - a^3*c^2*d^3)/((c^4*d^2 + 2*c^3*d^3 + c^2*d^4)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2*e) + c)^2)
)/f
Mupad [B] (verification not implemented)
Time = 14.72 (sec) , antiderivative size = 6246, normalized size of antiderivative = 35.69
\[
\int \frac {(3+3 \sin (e+f x))^3}{(c+d \sin (e+f x))^3} \, dx=\text {Too large to display}
\]
[In]
int((a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^3,x)
[Out]
((2*a^3*c^3 - a^3*d^3 - 5*a^3*c*d^2 + 4*a^3*c^2*d)/(d^2*(2*c*d + c^2 + d^2)) + (tan(e/2 + (f*x)/2)^3*(a^3*c^3
- 2*a^3*d^3 - 4*a^3*c*d^2 + 5*a^3*c^2*d))/(c*d*(2*c*d + c^2 + d^2)) + (tan(e/2 + (f*x)/2)*(7*a^3*c^3 - 2*a^3*d
^3 - 16*a^3*c*d^2 + 11*a^3*c^2*d))/(c*d*(2*c*d + c^2 + d^2)) + (tan(e/2 + (f*x)/2)^2*(c^2 + 2*d^2)*(2*a^3*c^3
- a^3*d^3 - 5*a^3*c*d^2 + 4*a^3*c^2*d))/(c^2*d^2*(2*c*d + c^2 + d^2)))/(f*(tan(e/2 + (f*x)/2)^2*(2*c^2 + 4*d^2
) + c^2*tan(e/2 + (f*x)/2)^4 + c^2 + 4*c*d*tan(e/2 + (f*x)/2)^3 + 4*c*d*tan(e/2 + (f*x)/2))) - (2*a^3*atan(-((
(((((8*(4*c^2*d^12 + 16*c^3*d^11 + 24*c^4*d^10 + 16*c^5*d^9 + 4*c^6*d^8))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d
^6 + c^4*d^5) + (8*tan(e/2 + (f*x)/2)*(12*c*d^14 + 48*c^2*d^13 + 64*c^3*d^12 + 16*c^4*d^11 - 36*c^5*d^10 - 32*
c^6*d^9 - 8*c^7*d^8))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*1i)/d^3 - (8*(4*c*d^10 + 2*c^2*d^9 -
6*c^3*d^8 - 2*c^4*d^7 + 2*c^5*d^6))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f*x)/2)
*(28*c*d^11 + 52*c^2*d^10 + 4*c^3*d^9 - 44*c^4*d^8 - 32*c^5*d^7 - 8*c^6*d^6))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*
c^3*d^7 + c^4*d^6))*1i)/d^3 + (8*(4*c^2*d^6 + 16*c^3*d^5 + 24*c^4*d^4 + 16*c^5*d^3 + 4*c^6*d^2))/(4*c*d^8 + d^
9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) - (8*tan(e/2 + (f*x)/2)*(41*c*d^8 - 46*c^2*d^7 - 99*c^3*d^6 - 36*c^4*d^5
+ 36*c^5*d^4 + 32*c^6*d^3 + 8*c^7*d^2))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))/d^3 + ((((8*(4*c*d
^10 + 2*c^2*d^9 - 6*c^3*d^8 - 2*c^4*d^7 + 2*c^5*d^6))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (((8
*(4*c^2*d^12 + 16*c^3*d^11 + 24*c^4*d^10 + 16*c^5*d^9 + 4*c^6*d^8))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c
^4*d^5) + (8*tan(e/2 + (f*x)/2)*(12*c*d^14 + 48*c^2*d^13 + 64*c^3*d^12 + 16*c^4*d^11 - 36*c^5*d^10 - 32*c^6*d^
9 - 8*c^7*d^8))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*1i)/d^3 - (8*tan(e/2 + (f*x)/2)*(28*c*d^11
+ 52*c^2*d^10 + 4*c^3*d^9 - 44*c^4*d^8 - 32*c^5*d^7 - 8*c^6*d^6))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c
^4*d^6))*1i)/d^3 + (8*(4*c^2*d^6 + 16*c^3*d^5 + 24*c^4*d^4 + 16*c^5*d^3 + 4*c^6*d^2))/(4*c*d^8 + d^9 + 6*c^2*d
^7 + 4*c^3*d^6 + c^4*d^5) - (8*tan(e/2 + (f*x)/2)*(41*c*d^8 - 46*c^2*d^7 - 99*c^3*d^6 - 36*c^4*d^5 + 36*c^5*d^
4 + 32*c^6*d^3 + 8*c^7*d^2))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))/d^3)/((16*(18*c^4*d - 49*c*d^
4 + 2*c^5 + 29*c^3*d^2))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) - (((((((8*(4*c^2*d^12 + 16*c^3*d^1
1 + 24*c^4*d^10 + 16*c^5*d^9 + 4*c^6*d^8))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f
*x)/2)*(12*c*d^14 + 48*c^2*d^13 + 64*c^3*d^12 + 16*c^4*d^11 - 36*c^5*d^10 - 32*c^6*d^9 - 8*c^7*d^8))/(4*c*d^9
+ d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*1i)/d^3 - (8*(4*c*d^10 + 2*c^2*d^9 - 6*c^3*d^8 - 2*c^4*d^7 + 2*c^5*
d^6))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f*x)/2)*(28*c*d^11 + 52*c^2*d^10 + 4*c
^3*d^9 - 44*c^4*d^8 - 32*c^5*d^7 - 8*c^6*d^6))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*1i)/d^3 + (
8*(4*c^2*d^6 + 16*c^3*d^5 + 24*c^4*d^4 + 16*c^5*d^3 + 4*c^6*d^2))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4
*d^5) - (8*tan(e/2 + (f*x)/2)*(41*c*d^8 - 46*c^2*d^7 - 99*c^3*d^6 - 36*c^4*d^5 + 36*c^5*d^4 + 32*c^6*d^3 + 8*c
^7*d^2))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*1i)/d^3 + (((((8*(4*c*d^10 + 2*c^2*d^9 - 6*c^3*d^
8 - 2*c^4*d^7 + 2*c^5*d^6))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (((8*(4*c^2*d^12 + 16*c^3*d^11
+ 24*c^4*d^10 + 16*c^5*d^9 + 4*c^6*d^8))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f*
x)/2)*(12*c*d^14 + 48*c^2*d^13 + 64*c^3*d^12 + 16*c^4*d^11 - 36*c^5*d^10 - 32*c^6*d^9 - 8*c^7*d^8))/(4*c*d^9 +
d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*1i)/d^3 - (8*tan(e/2 + (f*x)/2)*(28*c*d^11 + 52*c^2*d^10 + 4*c^3*d^9
- 44*c^4*d^8 - 32*c^5*d^7 - 8*c^6*d^6))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*1i)/d^3 + (8*(4*c
^2*d^6 + 16*c^3*d^5 + 24*c^4*d^4 + 16*c^5*d^3 + 4*c^6*d^2))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5)
- (8*tan(e/2 + (f*x)/2)*(41*c*d^8 - 46*c^2*d^7 - 99*c^3*d^6 - 36*c^4*d^5 + 36*c^5*d^4 + 32*c^6*d^3 + 8*c^7*d^2
))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*1i)/d^3 - (16*tan(e/2 + (f*x)/2)*(28*c*d^5 - 32*c^5*d -
8*c^6 + 52*c^2*d^4 + 4*c^3*d^3 - 44*c^4*d^2))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))))/(d^3*f) +
(a^3*atan(((a^3*(-(c + d)^5*(c - d))^(1/2)*(3*c*d + c^2 + (7*d^2)/2)*((8*(4*a^6*c^2*d^6 + 16*a^6*c^3*d^5 + 24
*a^6*c^4*d^4 + 16*a^6*c^5*d^3 + 4*a^6*c^6*d^2))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) - (8*tan(e/2
+ (f*x)/2)*(41*a^6*c*d^8 - 46*a^6*c^2*d^7 - 99*a^6*c^3*d^6 - 36*a^6*c^4*d^5 + 36*a^6*c^5*d^4 + 32*a^6*c^6*d^3
+ 8*a^6*c^7*d^2))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6) + (a^3*(-(c + d)^5*(c - d))^(1/2)*(3*c*d
+ c^2 + (7*d^2)/2)*((8*tan(e/2 + (f*x)/2)*(28*a^3*c*d^11 + 52*a^3*c^2*d^10 + 4*a^3*c^3*d^9 - 44*a^3*c^4*d^8 -
32*a^3*c^5*d^7 - 8*a^3*c^6*d^6))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6) - (8*(4*a^3*c*d^10 + 2*a^
3*c^2*d^9 - 6*a^3*c^3*d^8 - 2*a^3*c^4*d^7 + 2*a^3*c^5*d^6))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5)
+ (a^3*(-(c + d)^5*(c - d))^(1/2)*((8*(4*c^2*d^12 + 16*c^3*d^11 + 24*c^4*d^10 + 16*c^5*d^9 + 4*c^6*d^8))/(4*c*
d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f*x)/2)*(12*c*d^14 + 48*c^2*d^13 + 64*c^3*d^12 +
16*c^4*d^11 - 36*c^5*d^10 - 32*c^6*d^9 - 8*c^7*d^8))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*(3*c*
d + c^2 + (7*d^2)/2))/(5*c*d^7 + d^8 + 10*c^2*d^6 + 10*c^3*d^5 + 5*c^4*d^4 + c^5*d^3)))/(5*c*d^7 + d^8 + 10*c^
2*d^6 + 10*c^3*d^5 + 5*c^4*d^4 + c^5*d^3))*1i)/(5*c*d^7 + d^8 + 10*c^2*d^6 + 10*c^3*d^5 + 5*c^4*d^4 + c^5*d^3)
+ (a^3*(-(c + d)^5*(c - d))^(1/2)*(3*c*d + c^2 + (7*d^2)/2)*((8*(4*a^6*c^2*d^6 + 16*a^6*c^3*d^5 + 24*a^6*c^4*
d^4 + 16*a^6*c^5*d^3 + 4*a^6*c^6*d^2))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) - (8*tan(e/2 + (f*x)/
2)*(41*a^6*c*d^8 - 46*a^6*c^2*d^7 - 99*a^6*c^3*d^6 - 36*a^6*c^4*d^5 + 36*a^6*c^5*d^4 + 32*a^6*c^6*d^3 + 8*a^6*
c^7*d^2))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6) + (a^3*(-(c + d)^5*(c - d))^(1/2)*(3*c*d + c^2 +
(7*d^2)/2)*((8*(4*a^3*c*d^10 + 2*a^3*c^2*d^9 - 6*a^3*c^3*d^8 - 2*a^3*c^4*d^7 + 2*a^3*c^5*d^6))/(4*c*d^8 + d^9
+ 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) - (8*tan(e/2 + (f*x)/2)*(28*a^3*c*d^11 + 52*a^3*c^2*d^10 + 4*a^3*c^3*d^9 -
44*a^3*c^4*d^8 - 32*a^3*c^5*d^7 - 8*a^3*c^6*d^6))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6) + (a^3*(-
(c + d)^5*(c - d))^(1/2)*((8*(4*c^2*d^12 + 16*c^3*d^11 + 24*c^4*d^10 + 16*c^5*d^9 + 4*c^6*d^8))/(4*c*d^8 + d^9
+ 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f*x)/2)*(12*c*d^14 + 48*c^2*d^13 + 64*c^3*d^12 + 16*c^4*d^
11 - 36*c^5*d^10 - 32*c^6*d^9 - 8*c^7*d^8))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*(3*c*d + c^2 +
(7*d^2)/2))/(5*c*d^7 + d^8 + 10*c^2*d^6 + 10*c^3*d^5 + 5*c^4*d^4 + c^5*d^3)))/(5*c*d^7 + d^8 + 10*c^2*d^6 + 1
0*c^3*d^5 + 5*c^4*d^4 + c^5*d^3))*1i)/(5*c*d^7 + d^8 + 10*c^2*d^6 + 10*c^3*d^5 + 5*c^4*d^4 + c^5*d^3))/((16*(2
*a^9*c^5 - 49*a^9*c*d^4 + 18*a^9*c^4*d + 29*a^9*c^3*d^2))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) +
(16*tan(e/2 + (f*x)/2)*(8*a^9*c^6 - 28*a^9*c*d^5 + 32*a^9*c^5*d - 52*a^9*c^2*d^4 - 4*a^9*c^3*d^3 + 44*a^9*c^4*
d^2))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6) - (a^3*(-(c + d)^5*(c - d))^(1/2)*(3*c*d + c^2 + (7*d
^2)/2)*((8*(4*a^6*c^2*d^6 + 16*a^6*c^3*d^5 + 24*a^6*c^4*d^4 + 16*a^6*c^5*d^3 + 4*a^6*c^6*d^2))/(4*c*d^8 + d^9
+ 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) - (8*tan(e/2 + (f*x)/2)*(41*a^6*c*d^8 - 46*a^6*c^2*d^7 - 99*a^6*c^3*d^6 - 3
6*a^6*c^4*d^5 + 36*a^6*c^5*d^4 + 32*a^6*c^6*d^3 + 8*a^6*c^7*d^2))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^
4*d^6) + (a^3*(-(c + d)^5*(c - d))^(1/2)*(3*c*d + c^2 + (7*d^2)/2)*((8*tan(e/2 + (f*x)/2)*(28*a^3*c*d^11 + 52*
a^3*c^2*d^10 + 4*a^3*c^3*d^9 - 44*a^3*c^4*d^8 - 32*a^3*c^5*d^7 - 8*a^3*c^6*d^6))/(4*c*d^9 + d^10 + 6*c^2*d^8 +
4*c^3*d^7 + c^4*d^6) - (8*(4*a^3*c*d^10 + 2*a^3*c^2*d^9 - 6*a^3*c^3*d^8 - 2*a^3*c^4*d^7 + 2*a^3*c^5*d^6))/(4*
c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (a^3*(-(c + d)^5*(c - d))^(1/2)*((8*(4*c^2*d^12 + 16*c^3*d^11
+ 24*c^4*d^10 + 16*c^5*d^9 + 4*c^6*d^8))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f*
x)/2)*(12*c*d^14 + 48*c^2*d^13 + 64*c^3*d^12 + 16*c^4*d^11 - 36*c^5*d^10 - 32*c^6*d^9 - 8*c^7*d^8))/(4*c*d^9 +
d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*(3*c*d + c^2 + (7*d^2)/2))/(5*c*d^7 + d^8 + 10*c^2*d^6 + 10*c^3*d^5
+ 5*c^4*d^4 + c^5*d^3)))/(5*c*d^7 + d^8 + 10*c^2*d^6 + 10*c^3*d^5 + 5*c^4*d^4 + c^5*d^3)))/(5*c*d^7 + d^8 + 10
*c^2*d^6 + 10*c^3*d^5 + 5*c^4*d^4 + c^5*d^3) + (a^3*(-(c + d)^5*(c - d))^(1/2)*(3*c*d + c^2 + (7*d^2)/2)*((8*(
4*a^6*c^2*d^6 + 16*a^6*c^3*d^5 + 24*a^6*c^4*d^4 + 16*a^6*c^5*d^3 + 4*a^6*c^6*d^2))/(4*c*d^8 + d^9 + 6*c^2*d^7
+ 4*c^3*d^6 + c^4*d^5) - (8*tan(e/2 + (f*x)/2)*(41*a^6*c*d^8 - 46*a^6*c^2*d^7 - 99*a^6*c^3*d^6 - 36*a^6*c^4*d^
5 + 36*a^6*c^5*d^4 + 32*a^6*c^6*d^3 + 8*a^6*c^7*d^2))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6) + (a^
3*(-(c + d)^5*(c - d))^(1/2)*(3*c*d + c^2 + (7*d^2)/2)*((8*(4*a^3*c*d^10 + 2*a^3*c^2*d^9 - 6*a^3*c^3*d^8 - 2*a
^3*c^4*d^7 + 2*a^3*c^5*d^6))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) - (8*tan(e/2 + (f*x)/2)*(28*a^3
*c*d^11 + 52*a^3*c^2*d^10 + 4*a^3*c^3*d^9 - 44*a^3*c^4*d^8 - 32*a^3*c^5*d^7 - 8*a^3*c^6*d^6))/(4*c*d^9 + d^10
+ 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6) + (a^3*(-(c + d)^5*(c - d))^(1/2)*((8*(4*c^2*d^12 + 16*c^3*d^11 + 24*c^4*d^
10 + 16*c^5*d^9 + 4*c^6*d^8))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f*x)/2)*(12*c*
d^14 + 48*c^2*d^13 + 64*c^3*d^12 + 16*c^4*d^11 - 36*c^5*d^10 - 32*c^6*d^9 - 8*c^7*d^8))/(4*c*d^9 + d^10 + 6*c^
2*d^8 + 4*c^3*d^7 + c^4*d^6))*(3*c*d + c^2 + (7*d^2)/2))/(5*c*d^7 + d^8 + 10*c^2*d^6 + 10*c^3*d^5 + 5*c^4*d^4
+ c^5*d^3)))/(5*c*d^7 + d^8 + 10*c^2*d^6 + 10*c^3*d^5 + 5*c^4*d^4 + c^5*d^3)))/(5*c*d^7 + d^8 + 10*c^2*d^6 + 1
0*c^3*d^5 + 5*c^4*d^4 + c^5*d^3)))*(-(c + d)^5*(c - d))^(1/2)*(3*c*d + c^2 + (7*d^2)/2)*2i)/(f*(5*c*d^7 + d^8
+ 10*c^2*d^6 + 10*c^3*d^5 + 5*c^4*d^4 + c^5*d^3))