Integrand size = 25, antiderivative size = 285 \[ \int \frac {1}{(3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\frac {d^2 \left (12 c^2+16 c d+7 d^2\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{9 (c-d)^4 (c+d)^2 \sqrt {c^2-d^2} f}-\frac {d \left (2 c^2-16 c d-21 d^2\right ) \cos (e+f x)}{54 (c-d)^3 (c+d) f (c+d \sin (e+f x))^2}-\frac {(c-8 d) \cos (e+f x)}{27 (c-d)^2 f (1+\sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\cos (e+f x)}{3 (c-d) f (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac {d \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right ) \cos (e+f x)}{54 (c-d)^4 (c+d)^2 f (c+d \sin (e+f x))} \]
-1/6*d*(2*c^2-16*c*d-21*d^2)*cos(f*x+e)/a^2/(c-d)^3/(c+d)/f/(c+d*sin(f*x+e ))^2-1/3*(c-8*d)*cos(f*x+e)/a^2/(c-d)^2/f/(1+sin(f*x+e))/(c+d*sin(f*x+e))^ 2-1/3*cos(f*x+e)/(c-d)/f/(a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^2-1/6*d*(2*c^ 3-16*c^2*d-59*c*d^2-32*d^3)*cos(f*x+e)/a^2/(c-d)^4/(c+d)^2/f/(c+d*sin(f*x+ e))+d^2*(12*c^2+16*c*d+7*d^2)*arctan((d+c*tan(1/2*f*x+1/2*e))/(c^2-d^2)^(1 /2))/a^2/(c-d)^4/(c+d)^2/f/(c^2-d^2)^(1/2)
Time = 3.26 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.18 \[ \int \frac {1}{(3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (4 (c-d) \sin \left (\frac {1}{2} (e+f x)\right )-2 (c-d) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+4 (c-10 d) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+\frac {6 d^2 \left (12 c^2+16 c d+7 d^2\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}{(c+d)^2 \sqrt {c^2-d^2}}+\frac {3 (c-d) d^3 \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}{(c+d) (c+d \sin (e+f x))^2}+\frac {3 d^3 (7 c+4 d) \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}{(c+d)^2 (c+d \sin (e+f x))}\right )}{54 (c-d)^4 f (1+\sin (e+f x))^2} \]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(4*(c - d)*Sin[(e + f*x)/2] - 2*(c - d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 4*(c - 10*d)*Sin[(e + f*x)/2] *(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (6*d^2*(12*c^2 + 16*c*d + 7*d^2 )*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(Cos[(e + f*x)/2] + Sin [(e + f*x)/2])^3)/((c + d)^2*Sqrt[c^2 - d^2]) + (3*(c - d)*d^3*Cos[e + f*x ]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/((c + d)*(c + d*Sin[e + f*x])^2 ) + (3*d^3*(7*c + 4*d)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^ 3)/((c + d)^2*(c + d*Sin[e + f*x]))))/(54*(c - d)^4*f*(1 + Sin[e + f*x])^2 )
Time = 1.20 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.13, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 3245, 25, 3042, 3457, 25, 3042, 3233, 25, 3042, 3233, 27, 3042, 3139, 1083, 217}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^3}dx\) |
| \(\Big \downarrow \) 3245 |
| \(\displaystyle -\frac {\int -\frac {a (c-5 d)+3 a d \sin (e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^3}dx}{3 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {a (c-5 d)+3 a d \sin (e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^3}dx}{3 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (c-5 d)+3 a d \sin (e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^3}dx}{3 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {-\frac {\int -\frac {21 d^2 a^2+2 (c-8 d) d \sin (e+f x) a^2}{(c+d \sin (e+f x))^3}dx}{a^2 (c-d)}-\frac {(c-8 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {21 d^2 a^2+2 (c-8 d) d \sin (e+f x) a^2}{(c+d \sin (e+f x))^3}dx}{a^2 (c-d)}-\frac {(c-8 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {21 d^2 a^2+2 (c-8 d) d \sin (e+f x) a^2}{(c+d \sin (e+f x))^3}dx}{a^2 (c-d)}-\frac {(c-8 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {2 d^2 (19 c+16 d) a^2+d \left (2 c (c-8 d)-21 d^2\right ) \sin (e+f x) a^2}{(c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right )}-\frac {a^2 d \left (2 c (c-8 d)-21 d^2\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {(c-8 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {2 d^2 (19 c+16 d) a^2+d \left (2 c (c-8 d)-21 d^2\right ) \sin (e+f x) a^2}{(c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right )}-\frac {a^2 d \left (2 c (c-8 d)-21 d^2\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {(c-8 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {2 d^2 (19 c+16 d) a^2+d \left (2 c (c-8 d)-21 d^2\right ) \sin (e+f x) a^2}{(c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right )}-\frac {a^2 d \left (2 c (c-8 d)-21 d^2\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {(c-8 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int -\frac {3 a^2 d^2 \left (12 c^2+16 d c+7 d^2\right )}{c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {a^2 d \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^2 d \left (2 c (c-8 d)-21 d^2\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {(c-8 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 a^2 d^2 \left (12 c^2+16 c d+7 d^2\right ) \int \frac {1}{c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {a^2 d \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^2 d \left (2 c (c-8 d)-21 d^2\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {(c-8 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 a^2 d^2 \left (12 c^2+16 c d+7 d^2\right ) \int \frac {1}{c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {a^2 d \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^2 d \left (2 c (c-8 d)-21 d^2\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {(c-8 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {\frac {\frac {6 a^2 d^2 \left (12 c^2+16 c d+7 d^2\right ) \int \frac {1}{c \tan ^2\left (\frac {1}{2} (e+f x)\right )+2 d \tan \left (\frac {1}{2} (e+f x)\right )+c}d\tan \left (\frac {1}{2} (e+f x)\right )}{f \left (c^2-d^2\right )}-\frac {a^2 d \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^2 d \left (2 c (c-8 d)-21 d^2\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {(c-8 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\frac {-\frac {12 a^2 d^2 \left (12 c^2+16 c d+7 d^2\right ) \int \frac {1}{-\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )^2-4 \left (c^2-d^2\right )}d\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{f \left (c^2-d^2\right )}-\frac {a^2 d \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^2 d \left (2 c (c-8 d)-21 d^2\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {(c-8 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {\frac {6 a^2 d^2 \left (12 c^2+16 c d+7 d^2\right ) \arctan \left (\frac {2 c \tan \left (\frac {1}{2} (e+f x)\right )+2 d}{2 \sqrt {c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{3/2}}-\frac {a^2 d \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^2 d \left (2 c (c-8 d)-21 d^2\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {(c-8 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
-1/3*Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2 ) + (-(((c - 8*d)*Cos[e + f*x])/((c - d)*f*(1 + Sin[e + f*x])*(c + d*Sin[e + f*x])^2)) + (-1/2*(a^2*d*(2*c*(c - 8*d) - 21*d^2)*Cos[e + f*x])/((c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) + ((6*a^2*d^2*(12*c^2 + 16*c*d + 7*d^2)*Ar cTan[(2*d + 2*c*Tan[(e + f*x)/2])/(2*Sqrt[c^2 - d^2])])/((c^2 - d^2)^(3/2) *f) - (a^2*d*(2*c^3 - 16*c^2*d - 59*c*d^2 - 32*d^3)*Cos[e + f*x])/((c^2 - d^2)*f*(c + d*Sin[e + f*x])))/(2*(c^2 - d^2)))/(a^2*(c - d)))/(3*a^2*(c - d))
3.5.69.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[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]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( m + 2)*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] && LtQ[m, -1] && IntegerQ[2*m]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*((c + d*Sin[e + f*x])^(n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/( a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[b*c*(m + 1) - a*d*(2*m + n + 2) + b*d*(m + 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] && LtQ[m, -1] && !GtQ[n, 0] && (Intege rsQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b *d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ [b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Time = 3.58 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.33
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (c -4 d \right )}{\left (c -d \right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {4}{3 \left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {2}{\left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {2 d^{2} \left (\frac {\frac {d^{2} \left (9 c^{2}+4 c d -2 d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (8 c^{4}+4 c^{3} d +15 c^{2} d^{2}+8 d^{3} c -2 d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c^{2} \left (c^{2}+2 c d +d^{2}\right )}+\frac {d^{2} \left (23 c^{2}+12 c d -2 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c}+\frac {d \left (8 c^{2}+4 c d -d^{2}\right )}{2 c^{2}+4 c d +2 d^{2}}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{2}}+\frac {\left (12 c^{2}+16 c d +7 d^{2}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (c -d \right )^{4}}}{a^{2} f}\) | \(378\) |
| default | \(\frac {-\frac {2 \left (c -4 d \right )}{\left (c -d \right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {4}{3 \left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {2}{\left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {2 d^{2} \left (\frac {\frac {d^{2} \left (9 c^{2}+4 c d -2 d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (8 c^{4}+4 c^{3} d +15 c^{2} d^{2}+8 d^{3} c -2 d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c^{2} \left (c^{2}+2 c d +d^{2}\right )}+\frac {d^{2} \left (23 c^{2}+12 c d -2 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c}+\frac {d \left (8 c^{2}+4 c d -d^{2}\right )}{2 c^{2}+4 c d +2 d^{2}}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{2}}+\frac {\left (12 c^{2}+16 c d +7 d^{2}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (c -d \right )^{4}}}{a^{2} f}\) | \(378\) |
| risch | \(\text {Expression too large to display}\) | \(1076\) |
2/f/a^2*(-(c-4*d)/(c-d)^4/(tan(1/2*f*x+1/2*e)+1)-2/3/(c-d)^3/(tan(1/2*f*x+ 1/2*e)+1)^3+1/(c-d)^3/(tan(1/2*f*x+1/2*e)+1)^2+1/(c-d)^4*d^2*((1/2*d^2*(9* c^2+4*c*d-2*d^2)/c/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3+1/2*d*(8*c^4+4*c^3 *d+15*c^2*d^2+8*c*d^3-2*d^4)/c^2/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^2+1/2* d^2*(23*c^2+12*c*d-2*d^2)/(c^2+2*c*d+d^2)/c*tan(1/2*f*x+1/2*e)+1/2*d*(8*c^ 2+4*c*d-d^2)/(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*(12*c^2+16*c*d+7*d^2)/(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))))
Leaf count of result is larger than twice the leaf count of optimal. 1728 vs. \(2 (281) = 562\).
Time = 0.38 (sec) , antiderivative size = 3540, normalized size of antiderivative = 12.42 \[ \int \frac {1}{(3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]
[-1/12*(4*c^7 - 4*c^6*d - 12*c^5*d^2 + 12*c^4*d^3 + 12*c^3*d^4 - 12*c^2*d^ 5 - 4*c*d^6 + 4*d^7 - 2*(2*c^5*d^2 - 16*c^4*d^3 - 61*c^3*d^4 - 16*c^2*d^5 + 59*c*d^6 + 32*d^7)*cos(f*x + e)^4 - 2*(4*c^6*d - 28*c^5*d^2 - 118*c^4*d^ 3 - 106*c^3*d^4 + 71*c^2*d^5 + 134*c*d^6 + 43*d^7)*cos(f*x + e)^3 + 2*(2*c ^7 - 12*c^6*d - 36*c^5*d^2 - 54*c^4*d^3 - 39*c^3*d^4 + 39*c^2*d^5 + 73*c*d ^6 + 27*d^7)*cos(f*x + e)^2 + 3*(24*c^4*d^2 + 80*c^3*d^3 + 102*c^2*d^4 + 6 0*c*d^5 + 14*d^6 + (12*c^2*d^4 + 16*c*d^5 + 7*d^6)*cos(f*x + e)^4 - (24*c^ 3*d^3 + 44*c^2*d^4 + 30*c*d^5 + 7*d^6)*cos(f*x + e)^3 - (12*c^4*d^2 + 64*c ^3*d^3 + 107*c^2*d^4 + 76*c*d^5 + 21*d^6)*cos(f*x + e)^2 + (12*c^4*d^2 + 4 0*c^3*d^3 + 51*c^2*d^4 + 30*c*d^5 + 7*d^6)*cos(f*x + e) + (24*c^4*d^2 + 80 *c^3*d^3 + 102*c^2*d^4 + 60*c*d^5 + 14*d^6 - (12*c^2*d^4 + 16*c*d^5 + 7*d^ 6)*cos(f*x + e)^3 - 2*(12*c^3*d^3 + 28*c^2*d^4 + 23*c*d^5 + 7*d^6)*cos(f*x + e)^2 + (12*c^4*d^2 + 40*c^3*d^3 + 51*c^2*d^4 + 30*c*d^5 + 7*d^6)*cos(f* x + e))*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f *x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 4*(2*c^7 - 5*c^6*d - 36*c^5*d^2 - 75*c^4*d^3 - 39*c^3*d^4 + 60*c ^2*d^5 + 73*c*d^6 + 20*d^7)*cos(f*x + e) - 2*(2*c^7 - 2*c^6*d - 6*c^5*d^2 + 6*c^4*d^3 + 6*c^3*d^4 - 6*c^2*d^5 - 2*c*d^6 + 2*d^7 + (2*c^5*d^2 - 16*c^ 4*d^3 - 61*c^3*d^4 - 16*c^2*d^5 + 59*c*d^6 + 32*d^7)*cos(f*x + e)^3 - (...
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{(3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*d^2-4*c^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (281) = 562\).
Time = 0.38 (sec) , antiderivative size = 594, normalized size of antiderivative = 2.08 \[ \int \frac {1}{(3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\frac {\frac {3 \, {\left (12 \, c^{2} d^{2} + 16 \, c d^{3} + 7 \, d^{4}\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 (a^{2} c^{6} - 2 \, a^{2} c^{5} d - a^{2} c^{4} d^{2} + 4 \, a^{2} c^{3} d^{3} - a^{2} c^{2} d^{4} - 2 \, a^{2} c d^{5} + a^{2} d^{6}\right )} \sqrt {c^{2} - d^{2}}} + \frac {3 \, {\left (9 \, c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 4 \, c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, c d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 8 \, c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 15 \, c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 8 \, c d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 23 \, c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, c d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 8 \, c^{4} d^{3} + 4 \, c^{3} d^{4} - c^{2} d^{5}\right )}}{{\left (a^{2} c^{8} - 2 \, a^{2} c^{7} d - a^{2} c^{6} d^{2} + 4 \, a^{2} c^{5} d^{3} - a^{2} c^{4} d^{4} - 2 \, a^{2} c^{3} d^{5} + a^{2} c^{2} d^{6}\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}} - \frac {2 \, {\left (3 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 21 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, c - 11 \, d\right )}}{{\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 )} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}}}{3 \, f} \]
1/3*(3*(12*c^2*d^2 + 16*c*d^3 + 7*d^4)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*s gn(c) + arctan((c*tan(1/2*f*x + 1/2*e) + d)/sqrt(c^2 - d^2)))/((a^2*c^6 - 2*a^2*c^5*d - a^2*c^4*d^2 + 4*a^2*c^3*d^3 - a^2*c^2*d^4 - 2*a^2*c*d^5 + a^ 2*d^6)*sqrt(c^2 - d^2)) + 3*(9*c^3*d^4*tan(1/2*f*x + 1/2*e)^3 + 4*c^2*d^5* tan(1/2*f*x + 1/2*e)^3 - 2*c*d^6*tan(1/2*f*x + 1/2*e)^3 + 8*c^4*d^3*tan(1/ 2*f*x + 1/2*e)^2 + 4*c^3*d^4*tan(1/2*f*x + 1/2*e)^2 + 15*c^2*d^5*tan(1/2*f *x + 1/2*e)^2 + 8*c*d^6*tan(1/2*f*x + 1/2*e)^2 - 2*d^7*tan(1/2*f*x + 1/2*e )^2 + 23*c^3*d^4*tan(1/2*f*x + 1/2*e) + 12*c^2*d^5*tan(1/2*f*x + 1/2*e) - 2*c*d^6*tan(1/2*f*x + 1/2*e) + 8*c^4*d^3 + 4*c^3*d^4 - c^2*d^5)/((a^2*c^8 - 2*a^2*c^7*d - a^2*c^6*d^2 + 4*a^2*c^5*d^3 - a^2*c^4*d^4 - 2*a^2*c^3*d^5 + a^2*c^2*d^6)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2*e) + c)^2 ) - 2*(3*c*tan(1/2*f*x + 1/2*e)^2 - 12*d*tan(1/2*f*x + 1/2*e)^2 + 3*c*tan( 1/2*f*x + 1/2*e) - 21*d*tan(1/2*f*x + 1/2*e) + 2*c - 11*d)/((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)*(tan(1/2*f*x + 1/2*e) + 1)^3))/f
Time = 11.05 (sec) , antiderivative size = 1199, normalized size of antiderivative = 4.21 \[ \int \frac {1}{(3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\frac {\frac {-4\,c^5+14\,c^4\,d+40\,c^3\,d^2+46\,c^2\,d^3+12\,c\,d^4-3\,d^5}{3\,\left (c+d\right )\,\left (c^2-d^2\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (-2\,c^6+4\,c^5\,d+38\,c^4\,d^2+40\,c^3\,d^3+23\,c^2\,d^4+4\,c\,d^5-2\,d^6\right )}{c^2\,\left (c^5-3\,c^4\,d+2\,c^3\,d^2+2\,c^2\,d^3-3\,c\,d^4+d^5\right )}+\frac {2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (-6\,c^6+16\,c^5\,d+102\,c^4\,d^2+212\,c^3\,d^3+177\,c^2\,d^4+33\,c\,d^5-9\,d^6\right )}{3\,c^2\,\left (c^2-d^2\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}+\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (-6\,c^5+20\,c^4\,d+114\,c^3\,d^2+160\,c^2\,d^3+33\,c\,d^4-6\,d^5\right )}{3\,c\,\left (c^2-d^2\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (-14\,c^7+16\,c^6\,d+226\,c^5\,d^2+532\,c^4\,d^3+583\,c^3\,d^4+232\,c^2\,d^5+6\,c\,d^6-6\,d^7\right )}{3\,c^2\,\left (c+d\right )\,\left (c^2-d^2\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (-16\,c^7+14\,c^6\,d+220\,c^5\,d^2+502\,c^4\,d^3+522\,c^3\,d^4+303\,c^2\,d^5+48\,c\,d^6-18\,d^7\right )}{3\,c^2\,\left (c+d\right )\,\left (c^2-d^2\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (-2\,c^6+4\,c^5\,d+14\,c^4\,d^2+8\,c^3\,d^3+9\,c^2\,d^4+4\,c\,d^5-2\,d^6\right )}{c\,\left (c-d\right )\,\left (c^2+2\,c\,d+d^2\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}}{f\,\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (3\,a^2\,c^2+4\,d\,a^2\,c\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (5\,a^2\,c^2+12\,a^2\,c\,d+4\,a^2\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (5\,a^2\,c^2+12\,a^2\,c\,d+4\,a^2\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (7\,a^2\,c^2+16\,a^2\,c\,d+12\,a^2\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (7\,a^2\,c^2+16\,a^2\,c\,d+12\,a^2\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (3\,a^2\,c^2+4\,d\,a^2\,c\right )+a^2\,c^2+a^2\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\right )}-\frac {d^2\,\mathrm {atan}\left (\frac {\frac {d^2\,\left (12\,c^2+16\,c\,d+7\,d^2\right )\,\left (-2\,a^2\,c^6\,d+4\,a^2\,c^5\,d^2+2\,a^2\,c^4\,d^3-8\,a^2\,c^3\,d^4+2\,a^2\,c^2\,d^5+4\,a^2\,c\,d^6-2\,a^2\,d^7\right )}{2\,a^2\,{\left (c+d\right )}^{5/2}\,{\left (c-d\right )}^{9/2}}+\frac {c\,d^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (12\,c^2+16\,c\,d+7\,d^2\right )\,\left (-a^2\,c^6+2\,a^2\,c^5\,d+a^2\,c^4\,d^2-4\,a^2\,c^3\,d^3+a^2\,c^2\,d^4+2\,a^2\,c\,d^5-a^2\,d^6\right )}{a^2\,{\left (c+d\right )}^{5/2}\,{\left (c-d\right )}^{9/2}}}{12\,c^2\,d^2+16\,c\,d^3+7\,d^4}\right )\,\left (12\,c^2+16\,c\,d+7\,d^2\right )}{a^2\,f\,{\left (c+d\right )}^{5/2}\,{\left (c-d\right )}^{9/2}} \]
((12*c*d^4 + 14*c^4*d - 4*c^5 - 3*d^5 + 46*c^2*d^3 + 40*c^3*d^2)/(3*(c + d )*(c^2 - d^2)*(3*c*d^2 - 3*c^2*d + c^3 - d^3)) + (tan(e/2 + (f*x)/2)^5*(4* c*d^5 + 4*c^5*d - 2*c^6 - 2*d^6 + 23*c^2*d^4 + 40*c^3*d^3 + 38*c^4*d^2))/( c^2*(c^5 - 3*c^4*d - 3*c*d^4 + d^5 + 2*c^2*d^3 + 2*c^3*d^2)) + (2*tan(e/2 + (f*x)/2)^3*(33*c*d^5 + 16*c^5*d - 6*c^6 - 9*d^6 + 177*c^2*d^4 + 212*c^3* d^3 + 102*c^4*d^2))/(3*c^2*(c^2 - d^2)*(3*c*d^2 - 3*c^2*d + c^3 - d^3)) + (tan(e/2 + (f*x)/2)*(33*c*d^4 + 20*c^4*d - 6*c^5 - 6*d^5 + 160*c^2*d^3 + 1 14*c^3*d^2))/(3*c*(c^2 - d^2)*(3*c*d^2 - 3*c^2*d + c^3 - d^3)) + (tan(e/2 + (f*x)/2)^2*(6*c*d^6 + 16*c^6*d - 14*c^7 - 6*d^7 + 232*c^2*d^5 + 583*c^3* d^4 + 532*c^4*d^3 + 226*c^5*d^2))/(3*c^2*(c + d)*(c^2 - d^2)*(3*c*d^2 - 3* c^2*d + c^3 - d^3)) + (tan(e/2 + (f*x)/2)^4*(48*c*d^6 + 14*c^6*d - 16*c^7 - 18*d^7 + 303*c^2*d^5 + 522*c^3*d^4 + 502*c^4*d^3 + 220*c^5*d^2))/(3*c^2* (c + d)*(c^2 - d^2)*(3*c*d^2 - 3*c^2*d + c^3 - d^3)) + (tan(e/2 + (f*x)/2) ^6*(4*c*d^5 + 4*c^5*d - 2*c^6 - 2*d^6 + 9*c^2*d^4 + 8*c^3*d^3 + 14*c^4*d^2 ))/(c*(c - d)*(2*c*d + c^2 + d^2)*(3*c*d^2 - 3*c^2*d + c^3 - d^3)))/(f*(ta n(e/2 + (f*x)/2)*(3*a^2*c^2 + 4*a^2*c*d) + tan(e/2 + (f*x)/2)^2*(5*a^2*c^2 + 4*a^2*d^2 + 12*a^2*c*d) + tan(e/2 + (f*x)/2)^5*(5*a^2*c^2 + 4*a^2*d^2 + 12*a^2*c*d) + tan(e/2 + (f*x)/2)^3*(7*a^2*c^2 + 12*a^2*d^2 + 16*a^2*c*d) + tan(e/2 + (f*x)/2)^4*(7*a^2*c^2 + 12*a^2*d^2 + 16*a^2*c*d) + tan(e/2 + ( f*x)/2)^6*(3*a^2*c^2 + 4*a^2*c*d) + a^2*c^2 + a^2*c^2*tan(e/2 + (f*x)/2...