3.160 \(\int \frac{(a+b \sin (e+f x))^2}{c+d x} \, dx\)

Optimal. Leaf size=156 \[ \frac{a^2 \log (c+d x)}{d}+\frac{2 a b \text{CosIntegral}\left (\frac{c f}{d}+f x\right ) \sin \left (e-\frac{c f}{d}\right )}{d}+\frac{2 a b \cos \left (e-\frac{c f}{d}\right ) \text{Si}\left (x f+\frac{c f}{d}\right )}{d}-\frac{b^2 \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac{2 c f}{d}\right )}{2 d}+\frac{b^2 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{2 d}+\frac{b^2 \log (c+d x)}{2 d} \]

[Out]

-(b^2*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(2*d) + (a^2*Log[c + d*x])/d + (b^2*Log[c + d*x])/(
2*d) + (2*a*b*CosIntegral[(c*f)/d + f*x]*Sin[e - (c*f)/d])/d + (2*a*b*Cos[e - (c*f)/d]*SinIntegral[(c*f)/d + f
*x])/d + (b^2*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(2*d)

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Rubi [A]  time = 0.324364, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3317, 3303, 3299, 3302, 3312} \[ \frac{a^2 \log (c+d x)}{d}+\frac{2 a b \text{CosIntegral}\left (\frac{c f}{d}+f x\right ) \sin \left (e-\frac{c f}{d}\right )}{d}+\frac{2 a b \cos \left (e-\frac{c f}{d}\right ) \text{Si}\left (x f+\frac{c f}{d}\right )}{d}-\frac{b^2 \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac{2 c f}{d}\right )}{2 d}+\frac{b^2 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{2 d}+\frac{b^2 \log (c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(2*d) + (a^2*Log[c + d*x])/d + (b^2*Log[c + d*x])/(
2*d) + (2*a*b*CosIntegral[(c*f)/d + f*x]*Sin[e - (c*f)/d])/d + (2*a*b*Cos[e - (c*f)/d]*SinIntegral[(c*f)/d + f
*x])/d + (b^2*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(2*d)

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 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]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rubi steps

\begin{align*} \int \frac{(a+b \sin (e+f x))^2}{c+d x} \, dx &=\int \left (\frac{a^2}{c+d x}+\frac{2 a b \sin (e+f x)}{c+d x}+\frac{b^2 \sin ^2(e+f x)}{c+d x}\right ) \, dx\\ &=\frac{a^2 \log (c+d x)}{d}+(2 a b) \int \frac{\sin (e+f x)}{c+d x} \, dx+b^2 \int \frac{\sin ^2(e+f x)}{c+d x} \, dx\\ &=\frac{a^2 \log (c+d x)}{d}+b^2 \int \left (\frac{1}{2 (c+d x)}-\frac{\cos (2 e+2 f x)}{2 (c+d x)}\right ) \, dx+\left (2 a b \cos \left (e-\frac{c f}{d}\right )\right ) \int \frac{\sin \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx+\left (2 a b \sin \left (e-\frac{c f}{d}\right )\right ) \int \frac{\cos \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx\\ &=\frac{a^2 \log (c+d x)}{d}+\frac{b^2 \log (c+d x)}{2 d}+\frac{2 a b \text{Ci}\left (\frac{c f}{d}+f x\right ) \sin \left (e-\frac{c f}{d}\right )}{d}+\frac{2 a b \cos \left (e-\frac{c f}{d}\right ) \text{Si}\left (\frac{c f}{d}+f x\right )}{d}-\frac{1}{2} b^2 \int \frac{\cos (2 e+2 f x)}{c+d x} \, dx\\ &=\frac{a^2 \log (c+d x)}{d}+\frac{b^2 \log (c+d x)}{2 d}+\frac{2 a b \text{Ci}\left (\frac{c f}{d}+f x\right ) \sin \left (e-\frac{c f}{d}\right )}{d}+\frac{2 a b \cos \left (e-\frac{c f}{d}\right ) \text{Si}\left (\frac{c f}{d}+f x\right )}{d}-\frac{1}{2} \left (b^2 \cos \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx+\frac{1}{2} \left (b^2 \sin \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx\\ &=-\frac{b^2 \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Ci}\left (\frac{2 c f}{d}+2 f x\right )}{2 d}+\frac{a^2 \log (c+d x)}{d}+\frac{b^2 \log (c+d x)}{2 d}+\frac{2 a b \text{Ci}\left (\frac{c f}{d}+f x\right ) \sin \left (e-\frac{c f}{d}\right )}{d}+\frac{2 a b \cos \left (e-\frac{c f}{d}\right ) \text{Si}\left (\frac{c f}{d}+f x\right )}{d}+\frac{b^2 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.291788, size = 134, normalized size = 0.86 \[ \frac{2 a^2 \log (c+d x)+4 a b \text{CosIntegral}\left (f \left (\frac{c}{d}+x\right )\right ) \sin \left (e-\frac{c f}{d}\right )+4 a b \cos \left (e-\frac{c f}{d}\right ) \text{Si}\left (f \left (\frac{c}{d}+x\right )\right )-b^2 \text{CosIntegral}\left (\frac{2 f (c+d x)}{d}\right ) \cos \left (2 e-\frac{2 c f}{d}\right )+b^2 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 f (c+d x)}{d}\right )+b^2 \log (c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sin[e + f*x])^2/(c + d*x),x]

[Out]

(-(b^2*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*f*(c + d*x))/d]) + 2*a^2*Log[c + d*x] + b^2*Log[c + d*x] + 4*a*b*Co
sIntegral[f*(c/d + x)]*Sin[e - (c*f)/d] + 4*a*b*Cos[e - (c*f)/d]*SinIntegral[f*(c/d + x)] + b^2*Sin[2*e - (2*c
*f)/d]*SinIntegral[(2*f*(c + d*x))/d])/(2*d)

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Maple [A]  time = 0.019, size = 213, normalized size = 1.4 \begin{align*}{\frac{{a}^{2}\ln \left ( \left ( fx+e \right ) d+cf-de \right ) }{d}}+2\,{\frac{ab}{d}{\it Si} \left ( fx+e+{\frac{cf-de}{d}} \right ) \cos \left ({\frac{cf-de}{d}} \right ) }-2\,{\frac{ab}{d}{\it Ci} \left ( fx+e+{\frac{cf-de}{d}} \right ) \sin \left ({\frac{cf-de}{d}} \right ) }+{\frac{{b}^{2}\ln \left ( \left ( fx+e \right ) d+cf-de \right ) }{2\,d}}-{\frac{{b}^{2}}{2\,d}{\it Si} \left ( 2\,fx+2\,e+2\,{\frac{cf-de}{d}} \right ) \sin \left ( 2\,{\frac{cf-de}{d}} \right ) }-{\frac{{b}^{2}}{2\,d}{\it Ci} \left ( 2\,fx+2\,e+2\,{\frac{cf-de}{d}} \right ) \cos \left ( 2\,{\frac{cf-de}{d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*ln((f*x+e)*d+c*f-d*e)/d+2*a*b*Si(f*x+e+(c*f-d*e)/d)*cos((c*f-d*e)/d)/d-2*a*b*Ci(f*x+e+(c*f-d*e)/d)*sin((c*
f-d*e)/d)/d+1/2*b^2*ln((f*x+e)*d+c*f-d*e)/d-1/2*b^2*Si(2*f*x+2*e+2*(c*f-d*e)/d)*sin(2*(c*f-d*e)/d)/d-1/2*b^2*C
i(2*f*x+2*e+2*(c*f-d*e)/d)*cos(2*(c*f-d*e)/d)/d

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Maxima [C]  time = 1.28828, size = 451, normalized size = 2.89 \begin{align*} \frac{\frac{4 \, a^{2} f \log \left (c + \frac{{\left (f x + e\right )} d}{f} - \frac{d e}{f}\right )}{d} + \frac{4 \,{\left (f{\left (-i \, E_{1}\left (\frac{i \,{\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + i \, E_{1}\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{\left (E_{1}\left (\frac{i \,{\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + E_{1}\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 )} a b}{d} + \frac{{\left (f{\left (E_{1}\left (\frac{2 i \,{\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) + E_{1}\left (-\frac{2 i \,{\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right )\right )} \cos \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) + f{\left (i \, E_{1}\left (\frac{2 i \,{\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) - i \, E_{1}\left (-\frac{2 i \,{\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right )\right )} \sin \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) + 2 \, f \log \left ({\left (f x + e\right )} d - d e + c f\right )\right )} b^{2}}{d}}{4 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*a^2*f*log(c + (f*x + e)*d/f - d*e/f)/d + 4*(f*(-I*exp_integral_e(1, (I*(f*x + e)*d - I*d*e + I*c*f)/d)
+ I*exp_integral_e(1, -(I*(f*x + e)*d - I*d*e + I*c*f)/d))*cos(-(d*e - c*f)/d) + f*(exp_integral_e(1, (I*(f*x
+ e)*d - I*d*e + I*c*f)/d) + exp_integral_e(1, -(I*(f*x + e)*d - I*d*e + I*c*f)/d))*sin(-(d*e - c*f)/d))*a*b/d
+ (f*(exp_integral_e(1, (2*I*(f*x + e)*d - 2*I*d*e + 2*I*c*f)/d) + exp_integral_e(1, -(2*I*(f*x + e)*d - 2*I*
d*e + 2*I*c*f)/d))*cos(-2*(d*e - c*f)/d) + f*(I*exp_integral_e(1, (2*I*(f*x + e)*d - 2*I*d*e + 2*I*c*f)/d) - I
*exp_integral_e(1, -(2*I*(f*x + e)*d - 2*I*d*e + 2*I*c*f)/d))*sin(-2*(d*e - c*f)/d) + 2*f*log((f*x + e)*d - d*
e + c*f))*b^2/d)/f

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Fricas [A]  time = 2.19746, size = 482, normalized size = 3.09 \begin{align*} -\frac{2 \, b^{2} \sin \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) \operatorname{Si}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) - 8 \, a b \cos \left (-\frac{d e - c f}{d}\right ) \operatorname{Si}\left (\frac{d f x + c f}{d}\right ) +{\left (b^{2} \operatorname{Ci}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) + b^{2} \operatorname{Ci}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right )\right )} \cos \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) - 2 \,{\left (2 \, a^{2} + b^{2}\right )} \log \left (d x + c\right ) + 4 \,{\left (a b \operatorname{Ci}\left (\frac{d f x + c f}{d}\right ) + a b \operatorname{Ci}\left (-\frac{d f x + c f}{d}\right )\right )} \sin \left (-\frac{d e - c f}{d}\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*b^2*sin(-2*(d*e - c*f)/d)*sin_integral(2*(d*f*x + c*f)/d) - 8*a*b*cos(-(d*e - c*f)/d)*sin_integral((d*
f*x + c*f)/d) + (b^2*cos_integral(2*(d*f*x + c*f)/d) + b^2*cos_integral(-2*(d*f*x + c*f)/d))*cos(-2*(d*e - c*f
)/d) - 2*(2*a^2 + b^2)*log(d*x + c) + 4*(a*b*cos_integral((d*f*x + c*f)/d) + a*b*cos_integral(-(d*f*x + c*f)/d
))*sin(-(d*e - c*f)/d))/d

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \sin{\left (e + f x \right )}\right )^{2}}{c + d x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*sin(e + f*x))**2/(c + d*x), x)

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Giac [C]  time = 1.60154, size = 9986, normalized size = 64.01 \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))^2/(d*x+c),x, algorithm="giac")

[Out]

1/4*(4*a*b*imag_part(cos_integral(f*x + c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e)^2 - 4*a*b*im
ag_part(cos_integral(-f*x - c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e)^2 + 4*a^2*log(abs(d*x +
c))*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e)^2 + 2*b^2*log(abs(d*x + c))*tan(c*f/d)^2*tan(1/2*c*f/d)^
2*tan(1/2*e)^2*tan(e)^2 - b^2*real_part(cos_integral(2*f*x + 2*c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(1/2*e
)^2*tan(e)^2 - b^2*real_part(cos_integral(-2*f*x - 2*c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e)
^2 + 8*a*b*sin_integral((d*f*x + c*f)/d)*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e)^2 - 2*b^2*imag_part
(cos_integral(2*f*x + 2*c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e) + 2*b^2*imag_part(cos_integr
al(-2*f*x - 2*c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e) - 4*b^2*sin_integral(2*(d*f*x + c*f)/d
)*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e) - 8*a*b*real_part(cos_integral(f*x + c*f/d))*tan(c*f/d)^2*
tan(1/2*c*f/d)^2*tan(1/2*e)*tan(e)^2 - 8*a*b*real_part(cos_integral(-f*x - c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)
^2*tan(1/2*e)*tan(e)^2 + 8*a*b*real_part(cos_integral(f*x + c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)*tan(1/2*e)^2*t
an(e)^2 + 8*a*b*real_part(cos_integral(-f*x - c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)*tan(1/2*e)^2*tan(e)^2 + 2*b^
2*imag_part(cos_integral(2*f*x + 2*c*f/d))*tan(c*f/d)*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e)^2 - 2*b^2*imag_part
(cos_integral(-2*f*x - 2*c*f/d))*tan(c*f/d)*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e)^2 + 4*b^2*sin_integral(2*(d*f
*x + c*f)/d)*tan(c*f/d)*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e)^2 + 4*a*b*imag_part(cos_integral(f*x + c*f/d))*ta
n(c*f/d)^2*tan(1/2*c*f/d)^2*tan(1/2*e)^2 - 4*a*b*imag_part(cos_integral(-f*x - c*f/d))*tan(c*f/d)^2*tan(1/2*c*
f/d)^2*tan(1/2*e)^2 + 4*a^2*log(abs(d*x + c))*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(1/2*e)^2 + 2*b^2*log(abs(d*x +
c))*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(1/2*e)^2 + b^2*real_part(cos_integral(2*f*x + 2*c*f/d))*tan(c*f/d)^2*ta
n(1/2*c*f/d)^2*tan(1/2*e)^2 + b^2*real_part(cos_integral(-2*f*x - 2*c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(
1/2*e)^2 + 8*a*b*sin_integral((d*f*x + c*f)/d)*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(1/2*e)^2 - 4*b^2*real_part(co
s_integral(2*f*x + 2*c*f/d))*tan(c*f/d)*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e) - 4*b^2*real_part(cos_integral(-2
*f*x - 2*c*f/d))*tan(c*f/d)*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e) - 4*a*b*imag_part(cos_integral(f*x + c*f/d))*
tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(e)^2 + 4*a*b*imag_part(cos_integral(-f*x - c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/
d)^2*tan(e)^2 + 4*a^2*log(abs(d*x + c))*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(e)^2 + 2*b^2*log(abs(d*x + c))*tan(c
*f/d)^2*tan(1/2*c*f/d)^2*tan(e)^2 - b^2*real_part(cos_integral(2*f*x + 2*c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)^2
*tan(e)^2 - b^2*real_part(cos_integral(-2*f*x - 2*c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(e)^2 - 8*a*b*sin_i
ntegral((d*f*x + c*f)/d)*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(e)^2 + 16*a*b*imag_part(cos_integral(f*x + c*f/d))*
tan(c*f/d)^2*tan(1/2*c*f/d)*tan(1/2*e)*tan(e)^2 - 16*a*b*imag_part(cos_integral(-f*x - c*f/d))*tan(c*f/d)^2*ta
n(1/2*c*f/d)*tan(1/2*e)*tan(e)^2 + 32*a*b*sin_integral((d*f*x + c*f)/d)*tan(c*f/d)^2*tan(1/2*c*f/d)*tan(1/2*e)
*tan(e)^2 - 4*a*b*imag_part(cos_integral(f*x + c*f/d))*tan(c*f/d)^2*tan(1/2*e)^2*tan(e)^2 + 4*a*b*imag_part(co
s_integral(-f*x - c*f/d))*tan(c*f/d)^2*tan(1/2*e)^2*tan(e)^2 + 4*a^2*log(abs(d*x + c))*tan(c*f/d)^2*tan(1/2*e)
^2*tan(e)^2 + 2*b^2*log(abs(d*x + c))*tan(c*f/d)^2*tan(1/2*e)^2*tan(e)^2 - b^2*real_part(cos_integral(2*f*x +
2*c*f/d))*tan(c*f/d)^2*tan(1/2*e)^2*tan(e)^2 - b^2*real_part(cos_integral(-2*f*x - 2*c*f/d))*tan(c*f/d)^2*tan(
1/2*e)^2*tan(e)^2 - 8*a*b*sin_integral((d*f*x + c*f)/d)*tan(c*f/d)^2*tan(1/2*e)^2*tan(e)^2 + 4*a*b*imag_part(c
os_integral(f*x + c*f/d))*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e)^2 - 4*a*b*imag_part(cos_integral(-f*x - c*f/d))
*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e)^2 + 4*a^2*log(abs(d*x + c))*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e)^2 + 2*b
^2*log(abs(d*x + c))*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e)^2 + b^2*real_part(cos_integral(2*f*x + 2*c*f/d))*tan
(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e)^2 + b^2*real_part(cos_integral(-2*f*x - 2*c*f/d))*tan(1/2*c*f/d)^2*tan(1/2*e
)^2*tan(e)^2 + 8*a*b*sin_integral((d*f*x + c*f)/d)*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e)^2 - 8*a*b*real_part(co
s_integral(f*x + c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(1/2*e) - 8*a*b*real_part(cos_integral(-f*x - c*f/d)
)*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(1/2*e) + 8*a*b*real_part(cos_integral(f*x + c*f/d))*tan(c*f/d)^2*tan(1/2*c
*f/d)*tan(1/2*e)^2 + 8*a*b*real_part(cos_integral(-f*x - c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)*tan(1/2*e)^2 - 2*
b^2*imag_part(cos_integral(2*f*x + 2*c*f/d))*tan(c*f/d)*tan(1/2*c*f/d)^2*tan(1/2*e)^2 + 2*b^2*imag_part(cos_in
tegral(-2*f*x - 2*c*f/d))*tan(c*f/d)*tan(1/2*c*f/d)^2*tan(1/2*e)^2 - 4*b^2*sin_integral(2*(d*f*x + c*f)/d)*tan
(c*f/d)*tan(1/2*c*f/d)^2*tan(1/2*e)^2 - 2*b^2*imag_part(cos_integral(2*f*x + 2*c*f/d))*tan(c*f/d)^2*tan(1/2*c*
f/d)^2*tan(e) + 2*b^2*imag_part(cos_integral(-2*f*x - 2*c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(e) - 4*b^2*s
in_integral(2*(d*f*x + c*f)/d)*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(e) - 2*b^2*imag_part(cos_integral(2*f*x + 2*c
*f/d))*tan(c*f/d)^2*tan(1/2*e)^2*tan(e) + 2*b^2*imag_part(cos_integral(-2*f*x - 2*c*f/d))*tan(c*f/d)^2*tan(1/2
*e)^2*tan(e) - 4*b^2*sin_integral(2*(d*f*x + c*f)/d)*tan(c*f/d)^2*tan(1/2*e)^2*tan(e) + 2*b^2*imag_part(cos_in
tegral(2*f*x + 2*c*f/d))*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e) - 2*b^2*imag_part(cos_integral(-2*f*x - 2*c*f/d)
)*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e) + 4*b^2*sin_integral(2*(d*f*x + c*f)/d)*tan(1/2*c*f/d)^2*tan(1/2*e)^2*t
an(e) - 8*a*b*real_part(cos_integral(f*x + c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)*tan(e)^2 - 8*a*b*real_part(cos_
integral(-f*x - c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)*tan(e)^2 + 2*b^2*imag_part(cos_integral(2*f*x + 2*c*f/d))*
tan(c*f/d)*tan(1/2*c*f/d)^2*tan(e)^2 - 2*b^2*imag_part(cos_integral(-2*f*x - 2*c*f/d))*tan(c*f/d)*tan(1/2*c*f/
d)^2*tan(e)^2 + 4*b^2*sin_integral(2*(d*f*x + c*f)/d)*tan(c*f/d)*tan(1/2*c*f/d)^2*tan(e)^2 + 8*a*b*real_part(c
os_integral(f*x + c*f/d))*tan(c*f/d)^2*tan(1/2*e)*tan(e)^2 + 8*a*b*real_part(cos_integral(-f*x - c*f/d))*tan(c
*f/d)^2*tan(1/2*e)*tan(e)^2 - 8*a*b*real_part(cos_integral(f*x + c*f/d))*tan(1/2*c*f/d)^2*tan(1/2*e)*tan(e)^2
- 8*a*b*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*c*f/d)^2*tan(1/2*e)*tan(e)^2 + 2*b^2*imag_part(cos_integ
ral(2*f*x + 2*c*f/d))*tan(c*f/d)*tan(1/2*e)^2*tan(e)^2 - 2*b^2*imag_part(cos_integral(-2*f*x - 2*c*f/d))*tan(c
*f/d)*tan(1/2*e)^2*tan(e)^2 + 4*b^2*sin_integral(2*(d*f*x + c*f)/d)*tan(c*f/d)*tan(1/2*e)^2*tan(e)^2 + 8*a*b*r
eal_part(cos_integral(f*x + c*f/d))*tan(1/2*c*f/d)*tan(1/2*e)^2*tan(e)^2 + 8*a*b*real_part(cos_integral(-f*x -
c*f/d))*tan(1/2*c*f/d)*tan(1/2*e)^2*tan(e)^2 - 4*a*b*imag_part(cos_integral(f*x + c*f/d))*tan(c*f/d)^2*tan(1/
2*c*f/d)^2 + 4*a*b*imag_part(cos_integral(-f*x - c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)^2 + 4*a^2*log(abs(d*x + c
))*tan(c*f/d)^2*tan(1/2*c*f/d)^2 + 2*b^2*log(abs(d*x + c))*tan(c*f/d)^2*tan(1/2*c*f/d)^2 + b^2*real_part(cos_i
ntegral(2*f*x + 2*c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)^2 + b^2*real_part(cos_integral(-2*f*x - 2*c*f/d))*tan(c*
f/d)^2*tan(1/2*c*f/d)^2 - 8*a*b*sin_integral((d*f*x + c*f)/d)*tan(c*f/d)^2*tan(1/2*c*f/d)^2 + 16*a*b*imag_part
(cos_integral(f*x + c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d)*tan(1/2*e) - 16*a*b*imag_part(cos_integral(-f*x - c*f/
d))*tan(c*f/d)^2*tan(1/2*c*f/d)*tan(1/2*e) + 32*a*b*sin_integral((d*f*x + c*f)/d)*tan(c*f/d)^2*tan(1/2*c*f/d)*
tan(1/2*e) - 4*a*b*imag_part(cos_integral(f*x + c*f/d))*tan(c*f/d)^2*tan(1/2*e)^2 + 4*a*b*imag_part(cos_integr
al(-f*x - c*f/d))*tan(c*f/d)^2*tan(1/2*e)^2 + 4*a^2*log(abs(d*x + c))*tan(c*f/d)^2*tan(1/2*e)^2 + 2*b^2*log(ab
s(d*x + c))*tan(c*f/d)^2*tan(1/2*e)^2 + b^2*real_part(cos_integral(2*f*x + 2*c*f/d))*tan(c*f/d)^2*tan(1/2*e)^2
+ b^2*real_part(cos_integral(-2*f*x - 2*c*f/d))*tan(c*f/d)^2*tan(1/2*e)^2 - 8*a*b*sin_integral((d*f*x + c*f)/
d)*tan(c*f/d)^2*tan(1/2*e)^2 + 4*a*b*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*c*f/d)^2*tan(1/2*e)^2 - 4*a*
b*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*c*f/d)^2*tan(1/2*e)^2 + 4*a^2*log(abs(d*x + c))*tan(1/2*c*f/d)
^2*tan(1/2*e)^2 + 2*b^2*log(abs(d*x + c))*tan(1/2*c*f/d)^2*tan(1/2*e)^2 - b^2*real_part(cos_integral(2*f*x + 2
*c*f/d))*tan(1/2*c*f/d)^2*tan(1/2*e)^2 - b^2*real_part(cos_integral(-2*f*x - 2*c*f/d))*tan(1/2*c*f/d)^2*tan(1/
2*e)^2 + 8*a*b*sin_integral((d*f*x + c*f)/d)*tan(1/2*c*f/d)^2*tan(1/2*e)^2 - 4*b^2*real_part(cos_integral(2*f*
x + 2*c*f/d))*tan(c*f/d)*tan(1/2*c*f/d)^2*tan(e) - 4*b^2*real_part(cos_integral(-2*f*x - 2*c*f/d))*tan(c*f/d)*
tan(1/2*c*f/d)^2*tan(e) - 4*b^2*real_part(cos_integral(2*f*x + 2*c*f/d))*tan(c*f/d)*tan(1/2*e)^2*tan(e) - 4*b^
2*real_part(cos_integral(-2*f*x - 2*c*f/d))*tan(c*f/d)*tan(1/2*e)^2*tan(e) + 4*a*b*imag_part(cos_integral(f*x
+ c*f/d))*tan(c*f/d)^2*tan(e)^2 - 4*a*b*imag_part(cos_integral(-f*x - c*f/d))*tan(c*f/d)^2*tan(e)^2 + 4*a^2*lo
g(abs(d*x + c))*tan(c*f/d)^2*tan(e)^2 + 2*b^2*log(abs(d*x + c))*tan(c*f/d)^2*tan(e)^2 - b^2*real_part(cos_inte
gral(2*f*x + 2*c*f/d))*tan(c*f/d)^2*tan(e)^2 - b^2*real_part(cos_integral(-2*f*x - 2*c*f/d))*tan(c*f/d)^2*tan(
e)^2 + 8*a*b*sin_integral((d*f*x + c*f)/d)*tan(c*f/d)^2*tan(e)^2 - 4*a*b*imag_part(cos_integral(f*x + c*f/d))*
tan(1/2*c*f/d)^2*tan(e)^2 + 4*a*b*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*c*f/d)^2*tan(e)^2 + 4*a^2*log(
abs(d*x + c))*tan(1/2*c*f/d)^2*tan(e)^2 + 2*b^2*log(abs(d*x + c))*tan(1/2*c*f/d)^2*tan(e)^2 + b^2*real_part(co
s_integral(2*f*x + 2*c*f/d))*tan(1/2*c*f/d)^2*tan(e)^2 + b^2*real_part(cos_integral(-2*f*x - 2*c*f/d))*tan(1/2
*c*f/d)^2*tan(e)^2 - 8*a*b*sin_integral((d*f*x + c*f)/d)*tan(1/2*c*f/d)^2*tan(e)^2 + 16*a*b*imag_part(cos_inte
gral(f*x + c*f/d))*tan(1/2*c*f/d)*tan(1/2*e)*tan(e)^2 - 16*a*b*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*c
*f/d)*tan(1/2*e)*tan(e)^2 + 32*a*b*sin_integral((d*f*x + c*f)/d)*tan(1/2*c*f/d)*tan(1/2*e)*tan(e)^2 - 4*a*b*im
ag_part(cos_integral(f*x + c*f/d))*tan(1/2*e)^2*tan(e)^2 + 4*a*b*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2
*e)^2*tan(e)^2 + 4*a^2*log(abs(d*x + c))*tan(1/2*e)^2*tan(e)^2 + 2*b^2*log(abs(d*x + c))*tan(1/2*e)^2*tan(e)^2
+ b^2*real_part(cos_integral(2*f*x + 2*c*f/d))*tan(1/2*e)^2*tan(e)^2 + b^2*real_part(cos_integral(-2*f*x - 2*
c*f/d))*tan(1/2*e)^2*tan(e)^2 - 8*a*b*sin_integral((d*f*x + c*f)/d)*tan(1/2*e)^2*tan(e)^2 - 8*a*b*real_part(co
s_integral(f*x + c*f/d))*tan(c*f/d)^2*tan(1/2*c*f/d) - 8*a*b*real_part(cos_integral(-f*x - c*f/d))*tan(c*f/d)^
2*tan(1/2*c*f/d) - 2*b^2*imag_part(cos_integral(2*f*x + 2*c*f/d))*tan(c*f/d)*tan(1/2*c*f/d)^2 + 2*b^2*imag_par
t(cos_integral(-2*f*x - 2*c*f/d))*tan(c*f/d)*tan(1/2*c*f/d)^2 - 4*b^2*sin_integral(2*(d*f*x + c*f)/d)*tan(c*f/
d)*tan(1/2*c*f/d)^2 + 8*a*b*real_part(cos_integral(f*x + c*f/d))*tan(c*f/d)^2*tan(1/2*e) + 8*a*b*real_part(cos
_integral(-f*x - c*f/d))*tan(c*f/d)^2*tan(1/2*e) - 8*a*b*real_part(cos_integral(f*x + c*f/d))*tan(1/2*c*f/d)^2
*tan(1/2*e) - 8*a*b*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*c*f/d)^2*tan(1/2*e) - 2*b^2*imag_part(cos_in
tegral(2*f*x + 2*c*f/d))*tan(c*f/d)*tan(1/2*e)^2 + 2*b^2*imag_part(cos_integral(-2*f*x - 2*c*f/d))*tan(c*f/d)*
tan(1/2*e)^2 - 4*b^2*sin_integral(2*(d*f*x + c*f)/d)*tan(c*f/d)*tan(1/2*e)^2 + 8*a*b*real_part(cos_integral(f*
x + c*f/d))*tan(1/2*c*f/d)*tan(1/2*e)^2 + 8*a*b*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*c*f/d)*tan(1/2*e
)^2 - 2*b^2*imag_part(cos_integral(2*f*x + 2*c*f/d))*tan(c*f/d)^2*tan(e) + 2*b^2*imag_part(cos_integral(-2*f*x
- 2*c*f/d))*tan(c*f/d)^2*tan(e) - 4*b^2*sin_integral(2*(d*f*x + c*f)/d)*tan(c*f/d)^2*tan(e) + 2*b^2*imag_part
(cos_integral(2*f*x + 2*c*f/d))*tan(1/2*c*f/d)^2*tan(e) - 2*b^2*imag_part(cos_integral(-2*f*x - 2*c*f/d))*tan(
1/2*c*f/d)^2*tan(e) + 4*b^2*sin_integral(2*(d*f*x + c*f)/d)*tan(1/2*c*f/d)^2*tan(e) + 2*b^2*imag_part(cos_inte
gral(2*f*x + 2*c*f/d))*tan(1/2*e)^2*tan(e) - 2*b^2*imag_part(cos_integral(-2*f*x - 2*c*f/d))*tan(1/2*e)^2*tan(
e) + 4*b^2*sin_integral(2*(d*f*x + c*f)/d)*tan(1/2*e)^2*tan(e) + 2*b^2*imag_part(cos_integral(2*f*x + 2*c*f/d)
)*tan(c*f/d)*tan(e)^2 - 2*b^2*imag_part(cos_integral(-2*f*x - 2*c*f/d))*tan(c*f/d)*tan(e)^2 + 4*b^2*sin_integr
al(2*(d*f*x + c*f)/d)*tan(c*f/d)*tan(e)^2 - 8*a*b*real_part(cos_integral(f*x + c*f/d))*tan(1/2*c*f/d)*tan(e)^2
- 8*a*b*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*c*f/d)*tan(e)^2 + 8*a*b*real_part(cos_integral(f*x + c*
f/d))*tan(1/2*e)*tan(e)^2 + 8*a*b*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*e)*tan(e)^2 + 4*a*b*imag_part(
cos_integral(f*x + c*f/d))*tan(c*f/d)^2 - 4*a*b*imag_part(cos_integral(-f*x - c*f/d))*tan(c*f/d)^2 + 4*a^2*log
(abs(d*x + c))*tan(c*f/d)^2 + 2*b^2*log(abs(d*x + c))*tan(c*f/d)^2 + b^2*real_part(cos_integral(2*f*x + 2*c*f/
d))*tan(c*f/d)^2 + b^2*real_part(cos_integral(-2*f*x - 2*c*f/d))*tan(c*f/d)^2 + 8*a*b*sin_integral((d*f*x + c*
f)/d)*tan(c*f/d)^2 - 4*a*b*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*c*f/d)^2 + 4*a*b*imag_part(cos_integra
l(-f*x - c*f/d))*tan(1/2*c*f/d)^2 + 4*a^2*log(abs(d*x + c))*tan(1/2*c*f/d)^2 + 2*b^2*log(abs(d*x + c))*tan(1/2
*c*f/d)^2 - b^2*real_part(cos_integral(2*f*x + 2*c*f/d))*tan(1/2*c*f/d)^2 - b^2*real_part(cos_integral(-2*f*x
- 2*c*f/d))*tan(1/2*c*f/d)^2 - 8*a*b*sin_integral((d*f*x + c*f)/d)*tan(1/2*c*f/d)^2 + 16*a*b*imag_part(cos_int
egral(f*x + c*f/d))*tan(1/2*c*f/d)*tan(1/2*e) - 16*a*b*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*c*f/d)*ta
n(1/2*e) + 32*a*b*sin_integral((d*f*x + c*f)/d)*tan(1/2*c*f/d)*tan(1/2*e) - 4*a*b*imag_part(cos_integral(f*x +
c*f/d))*tan(1/2*e)^2 + 4*a*b*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*e)^2 + 4*a^2*log(abs(d*x + c))*tan
(1/2*e)^2 + 2*b^2*log(abs(d*x + c))*tan(1/2*e)^2 - b^2*real_part(cos_integral(2*f*x + 2*c*f/d))*tan(1/2*e)^2 -
b^2*real_part(cos_integral(-2*f*x - 2*c*f/d))*tan(1/2*e)^2 - 8*a*b*sin_integral((d*f*x + c*f)/d)*tan(1/2*e)^2
- 4*b^2*real_part(cos_integral(2*f*x + 2*c*f/d))*tan(c*f/d)*tan(e) - 4*b^2*real_part(cos_integral(-2*f*x - 2*
c*f/d))*tan(c*f/d)*tan(e) + 4*a*b*imag_part(cos_integral(f*x + c*f/d))*tan(e)^2 - 4*a*b*imag_part(cos_integral
(-f*x - c*f/d))*tan(e)^2 + 4*a^2*log(abs(d*x + c))*tan(e)^2 + 2*b^2*log(abs(d*x + c))*tan(e)^2 + b^2*real_part
(cos_integral(2*f*x + 2*c*f/d))*tan(e)^2 + b^2*real_part(cos_integral(-2*f*x - 2*c*f/d))*tan(e)^2 + 8*a*b*sin_
integral((d*f*x + c*f)/d)*tan(e)^2 - 2*b^2*imag_part(cos_integral(2*f*x + 2*c*f/d))*tan(c*f/d) + 2*b^2*imag_pa
rt(cos_integral(-2*f*x - 2*c*f/d))*tan(c*f/d) - 4*b^2*sin_integral(2*(d*f*x + c*f)/d)*tan(c*f/d) - 8*a*b*real_
part(cos_integral(f*x + c*f/d))*tan(1/2*c*f/d) - 8*a*b*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*c*f/d) +
8*a*b*real_part(cos_integral(f*x + c*f/d))*tan(1/2*e) + 8*a*b*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*e)
+ 2*b^2*imag_part(cos_integral(2*f*x + 2*c*f/d))*tan(e) - 2*b^2*imag_part(cos_integral(-2*f*x - 2*c*f/d))*tan
(e) + 4*b^2*sin_integral(2*(d*f*x + c*f)/d)*tan(e) + 4*a*b*imag_part(cos_integral(f*x + c*f/d)) - 4*a*b*imag_p
art(cos_integral(-f*x - c*f/d)) + 4*a^2*log(abs(d*x + c)) + 2*b^2*log(abs(d*x + c)) - b^2*real_part(cos_integr
al(2*f*x + 2*c*f/d)) - b^2*real_part(cos_integral(-2*f*x - 2*c*f/d)) + 8*a*b*sin_integral((d*f*x + c*f)/d))/(d
*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(e)^2 + d*tan(c*f/d)^2*tan(1/2*c*f/d)^2*tan(1/2*e)^2 + d*tan(c*
f/d)^2*tan(1/2*c*f/d)^2*tan(e)^2 + d*tan(c*f/d)^2*tan(1/2*e)^2*tan(e)^2 + d*tan(1/2*c*f/d)^2*tan(1/2*e)^2*tan(
e)^2 + d*tan(c*f/d)^2*tan(1/2*c*f/d)^2 + d*tan(c*f/d)^2*tan(1/2*e)^2 + d*tan(1/2*c*f/d)^2*tan(1/2*e)^2 + d*tan
(c*f/d)^2*tan(e)^2 + d*tan(1/2*c*f/d)^2*tan(e)^2 + d*tan(1/2*e)^2*tan(e)^2 + d*tan(c*f/d)^2 + d*tan(1/2*c*f/d)
^2 + d*tan(1/2*e)^2 + d*tan(e)^2 + d)