3.155 \(\int \frac{a+b \sin (e+f x)}{(c+d x)^2} \, dx\)
Optimal. Leaf size=88 \[ -\frac{a}{d (c+d x)}+\frac{b f \text{CosIntegral}\left (\frac{c f}{d}+f x\right ) \cos \left (e-\frac{c f}{d}\right )}{d^2}-\frac{b f \sin \left (e-\frac{c f}{d}\right ) \text{Si}\left (x f+\frac{c f}{d}\right )}{d^2}-\frac{b \sin (e+f x)}{d (c+d x)} \]
[Out]
-(a/(d*(c + d*x))) + (b*f*Cos[e - (c*f)/d]*CosIntegral[(c*f)/d + f*x])/d^2 - (b*Sin[e + f*x])/(d*(c + d*x)) -
(b*f*Sin[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/d^2
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Rubi [A] time = 0.155229, antiderivative size = 88, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used =
{3317, 3297, 3303, 3299, 3302} \[ -\frac{a}{d (c+d x)}+\frac{b f \text{CosIntegral}\left (\frac{c f}{d}+f x\right ) \cos \left (e-\frac{c f}{d}\right )}{d^2}-\frac{b f \sin \left (e-\frac{c f}{d}\right ) \text{Si}\left (x f+\frac{c f}{d}\right )}{d^2}-\frac{b \sin (e+f x)}{d (c+d x)} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Sin[e + f*x])/(c + d*x)^2,x]
[Out]
-(a/(d*(c + d*x))) + (b*f*Cos[e - (c*f)/d]*CosIntegral[(c*f)/d + f*x])/d^2 - (b*Sin[e + f*x])/(d*(c + d*x)) -
(b*f*Sin[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/d^2
Rule 3317
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] ||
IGtQ[m, 0] || NeQ[a^2 - b^2, 0])
Rule 3297
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]
Rule 3303
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]
Rule 3299
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]
Rule 3302
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Rubi steps
\begin{align*} \int \frac{a+b \sin (e+f x)}{(c+d x)^2} \, dx &=\int \left (\frac{a}{(c+d x)^2}+\frac{b \sin (e+f x)}{(c+d x)^2}\right ) \, dx\\ &=-\frac{a}{d (c+d x)}+b \int \frac{\sin (e+f x)}{(c+d x)^2} \, dx\\ &=-\frac{a}{d (c+d x)}-\frac{b \sin (e+f x)}{d (c+d x)}+\frac{(b f) \int \frac{\cos (e+f x)}{c+d x} \, dx}{d}\\ &=-\frac{a}{d (c+d x)}-\frac{b \sin (e+f x)}{d (c+d x)}+\frac{\left (b f \cos \left (e-\frac{c f}{d}\right )\right ) \int \frac{\cos \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx}{d}-\frac{\left (b f \sin \left (e-\frac{c f}{d}\right )\right ) \int \frac{\sin \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx}{d}\\ &=-\frac{a}{d (c+d x)}+\frac{b f \cos \left (e-\frac{c f}{d}\right ) \text{Ci}\left (\frac{c f}{d}+f x\right )}{d^2}-\frac{b \sin (e+f x)}{d (c+d x)}-\frac{b f \sin \left (e-\frac{c f}{d}\right ) \text{Si}\left (\frac{c f}{d}+f x\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 0.34858, size = 72, normalized size = 0.82 \[ \frac{-\frac{d (a+b \sin (e+f x))}{c+d x}+b f \text{CosIntegral}\left (f \left (\frac{c}{d}+x\right )\right ) \cos \left (e-\frac{c f}{d}\right )-b f \sin \left (e-\frac{c f}{d}\right ) \text{Si}\left (f \left (\frac{c}{d}+x\right )\right )}{d^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Sin[e + f*x])/(c + d*x)^2,x]
[Out]
(b*f*Cos[e - (c*f)/d]*CosIntegral[f*(c/d + x)] - (d*(a + b*Sin[e + f*x]))/(c + d*x) - b*f*Sin[e - (c*f)/d]*Sin
Integral[f*(c/d + x)])/d^2
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Maple [A] time = 0.012, size = 141, normalized size = 1.6 \begin{align*}{\frac{1}{f} \left ( -{\frac{{f}^{2}a}{ \left ( \left ( fx+e \right ) d+cf-de \right ) d}}+{f}^{2}b \left ( -{\frac{\sin \left ( fx+e \right ) }{ \left ( \left ( fx+e \right ) d+cf-de \right ) d}}+{\frac{1}{d} \left ({\frac{1}{d}{\it Si} \left ( fx+e+{\frac{cf-de}{d}} \right ) \sin \left ({\frac{cf-de}{d}} \right ) }+{\frac{1}{d}{\it Ci} \left ( fx+e+{\frac{cf-de}{d}} \right ) \cos \left ({\frac{cf-de}{d}} \right ) } \right ) } \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*sin(f*x+e))/(d*x+c)^2,x)
[Out]
1/f*(-a*f^2/((f*x+e)*d+c*f-d*e)/d+f^2*b*(-sin(f*x+e)/((f*x+e)*d+c*f-d*e)/d+(Si(f*x+e+(c*f-d*e)/d)*sin((c*f-d*e
)/d)/d+Ci(f*x+e+(c*f-d*e)/d)*cos((c*f-d*e)/d)/d)/d))
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Maxima [C] time = 1.27957, size = 265, normalized size = 3.01 \begin{align*} -\frac{\frac{2 \, a f^{2}}{{\left (f x + e\right )} d^{2} - d^{2} e + c d f} - \frac{{\left (f^{2}{\left (-i \, E_{2}\left (\frac{i \,{\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + i \, E_{2}\left (-\frac{i \,{\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \cos \left (-\frac{d e - c f}{d}\right ) + f^{2}{\left (E_{2}\left (\frac{i \,{\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + E_{2}\left (-\frac{i \,{\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \sin \left (-\frac{d e - c f}{d}\right )\right )} b}{{\left (f x + e\right )} d^{2} - d^{2} e + c d f}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sin(f*x+e))/(d*x+c)^2,x, algorithm="maxima")
[Out]
-1/2*(2*a*f^2/((f*x + e)*d^2 - d^2*e + c*d*f) - (f^2*(-I*exp_integral_e(2, (I*(f*x + e)*d - I*d*e + I*c*f)/d)
+ I*exp_integral_e(2, -(I*(f*x + e)*d - I*d*e + I*c*f)/d))*cos(-(d*e - c*f)/d) + f^2*(exp_integral_e(2, (I*(f*
x + e)*d - I*d*e + I*c*f)/d) + exp_integral_e(2, -(I*(f*x + e)*d - I*d*e + I*c*f)/d))*sin(-(d*e - c*f)/d))*b/(
(f*x + e)*d^2 - d^2*e + c*d*f))/f
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Fricas [A] time = 1.8318, size = 332, normalized size = 3.77 \begin{align*} -\frac{2 \, b d \sin \left (f x + e\right ) - 2 \,{\left (b d f x + b c f\right )} \sin \left (-\frac{d e - c f}{d}\right ) \operatorname{Si}\left (\frac{d f x + c f}{d}\right ) + 2 \, a d -{\left ({\left (b d f x + b c f\right )} \operatorname{Ci}\left (\frac{d f x + c f}{d}\right ) +{\left (b d f x + b c f\right )} \operatorname{Ci}\left (-\frac{d f x + c f}{d}\right )\right )} \cos \left (-\frac{d e - c f}{d}\right )}{2 \,{\left (d^{3} x + c d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sin(f*x+e))/(d*x+c)^2,x, algorithm="fricas")
[Out]
-1/2*(2*b*d*sin(f*x + e) - 2*(b*d*f*x + b*c*f)*sin(-(d*e - c*f)/d)*sin_integral((d*f*x + c*f)/d) + 2*a*d - ((b
*d*f*x + b*c*f)*cos_integral((d*f*x + c*f)/d) + (b*d*f*x + b*c*f)*cos_integral(-(d*f*x + c*f)/d))*cos(-(d*e -
c*f)/d))/(d^3*x + c*d^2)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \sin{\left (e + f x \right )}}{\left (c + d x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sin(f*x+e))/(d*x+c)**2,x)
[Out]
Integral((a + b*sin(e + f*x))/(c + d*x)**2, x)
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Giac [C] time = 1.47167, size = 4251, normalized size = 48.31 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sin(f*x+e))/(d*x+c)^2,x, algorithm="giac")
[Out]
1/2*(d*f*x*real_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2*tan(1/2*e)^2 + d*f*x*real_part
(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2*tan(1/2*e)^2 + 2*d*f*x*imag_part(cos_integral(f*x
+ c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2*tan(1/2*e) - 2*d*f*x*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*
f*x)^2*tan(1/2*c*f/d)^2*tan(1/2*e) + 4*d*f*x*sin_integral((d*f*x + c*f)/d)*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2*tan
(1/2*e) - 2*d*f*x*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)*tan(1/2*e)^2 + 2*d*f*x*im
ag_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)*tan(1/2*e)^2 - 4*d*f*x*sin_integral((d*f*x +
c*f)/d)*tan(1/2*f*x)^2*tan(1/2*c*f/d)*tan(1/2*e)^2 + c*f*real_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*
tan(1/2*c*f/d)^2*tan(1/2*e)^2 + c*f*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2*tan(
1/2*e)^2 - d*f*x*real_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2 - d*f*x*real_part(cos_in
tegral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2 + 4*d*f*x*real_part(cos_integral(f*x + c*f/d))*tan(1/2*f
*x)^2*tan(1/2*c*f/d)*tan(1/2*e) + 4*d*f*x*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)*
tan(1/2*e) + 2*c*f*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2*tan(1/2*e) - 2*c*f*ima
g_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2*tan(1/2*e) + 4*c*f*sin_integral((d*f*x + c*
f)/d)*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2*tan(1/2*e) - d*f*x*real_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*t
an(1/2*e)^2 - d*f*x*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)^2 - 2*c*f*imag_part(cos_in
tegral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)*tan(1/2*e)^2 + 2*c*f*imag_part(cos_integral(-f*x - c*f/d))*
tan(1/2*f*x)^2*tan(1/2*c*f/d)*tan(1/2*e)^2 - 4*c*f*sin_integral((d*f*x + c*f)/d)*tan(1/2*f*x)^2*tan(1/2*c*f/d)
*tan(1/2*e)^2 + d*f*x*real_part(cos_integral(f*x + c*f/d))*tan(1/2*c*f/d)^2*tan(1/2*e)^2 + d*f*x*real_part(cos
_integral(-f*x - c*f/d))*tan(1/2*c*f/d)^2*tan(1/2*e)^2 + 2*d*f*x*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*
f*x)^2*tan(1/2*c*f/d) - 2*d*f*x*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d) + 4*d*f*x*
sin_integral((d*f*x + c*f)/d)*tan(1/2*f*x)^2*tan(1/2*c*f/d) - c*f*real_part(cos_integral(f*x + c*f/d))*tan(1/2
*f*x)^2*tan(1/2*c*f/d)^2 - c*f*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2 - 2*d*f*x
*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*e) + 2*d*f*x*imag_part(cos_integral(-f*x - c*f/d)
)*tan(1/2*f*x)^2*tan(1/2*e) - 4*d*f*x*sin_integral((d*f*x + c*f)/d)*tan(1/2*f*x)^2*tan(1/2*e) + 4*c*f*real_par
t(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)*tan(1/2*e) + 4*c*f*real_part(cos_integral(-f*x - c*
f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)*tan(1/2*e) + 2*d*f*x*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*c*f/d)^2
*tan(1/2*e) - 2*d*f*x*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*c*f/d)^2*tan(1/2*e) + 4*d*f*x*sin_integral
((d*f*x + c*f)/d)*tan(1/2*c*f/d)^2*tan(1/2*e) - c*f*real_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/
2*e)^2 - c*f*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)^2 - 2*d*f*x*imag_part(cos_integra
l(f*x + c*f/d))*tan(1/2*c*f/d)*tan(1/2*e)^2 + 2*d*f*x*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*c*f/d)*tan
(1/2*e)^2 - 4*d*f*x*sin_integral((d*f*x + c*f)/d)*tan(1/2*c*f/d)*tan(1/2*e)^2 + c*f*real_part(cos_integral(f*x
+ c*f/d))*tan(1/2*c*f/d)^2*tan(1/2*e)^2 + c*f*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*c*f/d)^2*tan(1/2*
e)^2 + d*f*x*real_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2 + d*f*x*real_part(cos_integral(-f*x - c*f/d))
*tan(1/2*f*x)^2 + 2*c*f*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d) - 2*c*f*imag_part(c
os_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d) + 4*c*f*sin_integral((d*f*x + c*f)/d)*tan(1/2*f*x)^2*
tan(1/2*c*f/d) - d*f*x*real_part(cos_integral(f*x + c*f/d))*tan(1/2*c*f/d)^2 - d*f*x*real_part(cos_integral(-f
*x - c*f/d))*tan(1/2*c*f/d)^2 - 2*c*f*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*e) + 2*c*f*i
mag_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*e) - 4*c*f*sin_integral((d*f*x + c*f)/d)*tan(1/2*f
*x)^2*tan(1/2*e) + 4*d*f*x*real_part(cos_integral(f*x + c*f/d))*tan(1/2*c*f/d)*tan(1/2*e) + 4*d*f*x*real_part(
cos_integral(-f*x - c*f/d))*tan(1/2*c*f/d)*tan(1/2*e) + 2*c*f*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*c*f
/d)^2*tan(1/2*e) - 2*c*f*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*c*f/d)^2*tan(1/2*e) + 4*c*f*sin_integra
l((d*f*x + c*f)/d)*tan(1/2*c*f/d)^2*tan(1/2*e) + 4*d*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2*tan(1/2*e) - d*f*x*real_p
art(cos_integral(f*x + c*f/d))*tan(1/2*e)^2 - d*f*x*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*e)^2 - 2*c*f
*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*c*f/d)*tan(1/2*e)^2 + 2*c*f*imag_part(cos_integral(-f*x - c*f/d)
)*tan(1/2*c*f/d)*tan(1/2*e)^2 - 4*c*f*sin_integral((d*f*x + c*f)/d)*tan(1/2*c*f/d)*tan(1/2*e)^2 + 4*d*tan(1/2*
f*x)*tan(1/2*c*f/d)^2*tan(1/2*e)^2 + c*f*real_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2 + c*f*real_part(c
os_integral(-f*x - c*f/d))*tan(1/2*f*x)^2 + 2*d*f*x*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*c*f/d) - 2*d*
f*x*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*c*f/d) + 4*d*f*x*sin_integral((d*f*x + c*f)/d)*tan(1/2*c*f/d
) - c*f*real_part(cos_integral(f*x + c*f/d))*tan(1/2*c*f/d)^2 - c*f*real_part(cos_integral(-f*x - c*f/d))*tan(
1/2*c*f/d)^2 - 2*d*f*x*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*e) + 2*d*f*x*imag_part(cos_integral(-f*x -
c*f/d))*tan(1/2*e) - 4*d*f*x*sin_integral((d*f*x + c*f)/d)*tan(1/2*e) + 4*c*f*real_part(cos_integral(f*x + c*
f/d))*tan(1/2*c*f/d)*tan(1/2*e) + 4*c*f*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*c*f/d)*tan(1/2*e) - c*f*
real_part(cos_integral(f*x + c*f/d))*tan(1/2*e)^2 - c*f*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*e)^2 + d
*f*x*real_part(cos_integral(f*x + c*f/d)) + d*f*x*real_part(cos_integral(-f*x - c*f/d)) + 2*c*f*imag_part(cos_
integral(f*x + c*f/d))*tan(1/2*c*f/d) - 2*c*f*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*c*f/d) + 4*c*f*sin
_integral((d*f*x + c*f)/d)*tan(1/2*c*f/d) - 4*d*tan(1/2*f*x)*tan(1/2*c*f/d)^2 - 2*c*f*imag_part(cos_integral(f
*x + c*f/d))*tan(1/2*e) + 2*c*f*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*e) - 4*c*f*sin_integral((d*f*x +
c*f)/d)*tan(1/2*e) + 4*d*tan(1/2*f*x)^2*tan(1/2*e) - 4*d*tan(1/2*c*f/d)^2*tan(1/2*e) + 4*d*tan(1/2*f*x)*tan(1
/2*e)^2 + c*f*real_part(cos_integral(f*x + c*f/d)) + c*f*real_part(cos_integral(-f*x - c*f/d)) - 4*d*tan(1/2*f
*x) - 4*d*tan(1/2*e))*b/(d^3*x*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2*tan(1/2*e)^2 + c*d^2*tan(1/2*f*x)^2*tan(1/2*c*f
/d)^2*tan(1/2*e)^2 + d^3*x*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2 + d^3*x*tan(1/2*f*x)^2*tan(1/2*e)^2 + d^3*x*tan(1/2
*c*f/d)^2*tan(1/2*e)^2 + c*d^2*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2 + c*d^2*tan(1/2*f*x)^2*tan(1/2*e)^2 + c*d^2*tan
(1/2*c*f/d)^2*tan(1/2*e)^2 + d^3*x*tan(1/2*f*x)^2 + d^3*x*tan(1/2*c*f/d)^2 + d^3*x*tan(1/2*e)^2 + c*d^2*tan(1/
2*f*x)^2 + c*d^2*tan(1/2*c*f/d)^2 + c*d^2*tan(1/2*e)^2 + d^3*x + c*d^2) - a/((d*x + c)*d)