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

Optimal. Leaf size=72 \[ -\frac{\sin \left (c-\frac{d e}{f}\right ) \text{CosIntegral}\left (\frac{d e}{f}+d x\right )}{a f}-\frac{\cos \left (c-\frac{d e}{f}\right ) \text{Si}\left (\frac{d e}{f}+d x\right )}{a f}+\frac{\log (e+f x)}{a f} \]

[Out]

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

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Rubi [A]  time = 0.20148, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {4523, 31, 3303, 3299, 3302} \[ -\frac{\sin \left (c-\frac{d e}{f}\right ) \text{CosIntegral}\left (\frac{d e}{f}+d x\right )}{a f}-\frac{\cos \left (c-\frac{d e}{f}\right ) \text{Si}\left (\frac{d e}{f}+d x\right )}{a f}+\frac{\log (e+f x)}{a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4523

Int[(Cos[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol
] :> Dist[1/a, Int[(e + f*x)^m*Cos[c + d*x]^(n - 2), x], x] - Dist[1/b, Int[(e + f*x)^m*Cos[c + d*x]^(n - 2)*S
in[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 - b^2, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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

Mathematica [A]  time = 0.275474, size = 58, normalized size = 0.81 \[ \frac{-\sin \left (c-\frac{d e}{f}\right ) \text{CosIntegral}\left (d \left (\frac{e}{f}+x\right )\right )-\cos \left (c-\frac{d e}{f}\right ) \text{Si}\left (d \left (\frac{e}{f}+x\right )\right )+\log (e+f x)}{a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.054, size = 102, normalized size = 1.4 \begin{align*} -{\frac{1}{af}{\it Si} \left ( dx+c+{\frac{-cf+de}{f}} \right ) \cos \left ({\frac{-cf+de}{f}} \right ) }+{\frac{1}{af}{\it Ci} \left ( dx+c+{\frac{-cf+de}{f}} \right ) \sin \left ({\frac{-cf+de}{f}} \right ) }+{\frac{\ln \left ( \left ( dx+c \right ) f-cf+de \right ) }{af}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a*Si(d*x+c+(-c*f+d*e)/f)*cos((-c*f+d*e)/f)/f+1/a*Ci(d*x+c+(-c*f+d*e)/f)*sin((-c*f+d*e)/f)/f+1/a*ln((d*x+c)*
f-c*f+d*e)/f

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Maxima [C]  time = 1.32096, size = 220, normalized size = 3.06 \begin{align*} \frac{d{\left (i \, E_{1}\left (\frac{i \, d e + i \,{\left (d x + c\right )} f - i \, c f}{f}\right ) - i \, E_{1}\left (-\frac{i \, d e + i \,{\left (d x + c\right )} f - i \, c f}{f}\right )\right )} \cos \left (-\frac{d e - c f}{f}\right ) + d{\left (E_{1}\left (\frac{i \, d e + i \,{\left (d x + c\right )} f - i \, c f}{f}\right ) + E_{1}\left (-\frac{i \, d e + i \,{\left (d x + c\right )} f - i \, c f}{f}\right )\right )} \sin \left (-\frac{d e - c f}{f}\right ) + 2 \, d \log \left (d e +{\left (d x + c\right )} f - c f\right )}{2 \, a d f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.73619, size = 230, normalized size = 3.19 \begin{align*} -\frac{{\left (\operatorname{Ci}\left (\frac{d f x + d e}{f}\right ) + \operatorname{Ci}\left (-\frac{d f x + d e}{f}\right )\right )} \sin \left (-\frac{d e - c f}{f}\right ) + 2 \, \cos \left (-\frac{d e - c f}{f}\right ) \operatorname{Si}\left (\frac{d f x + d e}{f}\right ) - 2 \, \log \left (f x + e\right )}{2 \, a f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [C]  time = 1.3586, size = 967, normalized size = 13.43 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(imag_part(cos_integral(d*x + d*e/f))*tan(1/2*c)^2*tan(1/2*d*e/f)^2 - imag_part(cos_integral(-d*x - d*e/f
))*tan(1/2*c)^2*tan(1/2*d*e/f)^2 - 2*log(abs(f*x + e))*tan(1/2*c)^2*tan(1/2*d*e/f)^2 + 2*sin_integral((d*f*x +
d*e)/f)*tan(1/2*c)^2*tan(1/2*d*e/f)^2 + 2*real_part(cos_integral(d*x + d*e/f))*tan(1/2*c)^2*tan(1/2*d*e/f) +
2*real_part(cos_integral(-d*x - d*e/f))*tan(1/2*c)^2*tan(1/2*d*e/f) - 2*real_part(cos_integral(d*x + d*e/f))*t
an(1/2*c)*tan(1/2*d*e/f)^2 - 2*real_part(cos_integral(-d*x - d*e/f))*tan(1/2*c)*tan(1/2*d*e/f)^2 - imag_part(c
os_integral(d*x + d*e/f))*tan(1/2*c)^2 + imag_part(cos_integral(-d*x - d*e/f))*tan(1/2*c)^2 - 2*log(abs(f*x +
e))*tan(1/2*c)^2 - 2*sin_integral((d*f*x + d*e)/f)*tan(1/2*c)^2 + 4*imag_part(cos_integral(d*x + d*e/f))*tan(1
/2*c)*tan(1/2*d*e/f) - 4*imag_part(cos_integral(-d*x - d*e/f))*tan(1/2*c)*tan(1/2*d*e/f) + 8*sin_integral((d*f
*x + d*e)/f)*tan(1/2*c)*tan(1/2*d*e/f) - imag_part(cos_integral(d*x + d*e/f))*tan(1/2*d*e/f)^2 + imag_part(cos
_integral(-d*x - d*e/f))*tan(1/2*d*e/f)^2 - 2*log(abs(f*x + e))*tan(1/2*d*e/f)^2 - 2*sin_integral((d*f*x + d*e
)/f)*tan(1/2*d*e/f)^2 + 2*real_part(cos_integral(d*x + d*e/f))*tan(1/2*c) + 2*real_part(cos_integral(-d*x - d*
e/f))*tan(1/2*c) - 2*real_part(cos_integral(d*x + d*e/f))*tan(1/2*d*e/f) - 2*real_part(cos_integral(-d*x - d*e
/f))*tan(1/2*d*e/f) + imag_part(cos_integral(d*x + d*e/f)) - imag_part(cos_integral(-d*x - d*e/f)) - 2*log(abs
(f*x + e)) + 2*sin_integral((d*f*x + d*e)/f))/(a*f*tan(1/2*c)^2*tan(1/2*d*e/f)^2 + a*f*tan(1/2*c)^2 + a*f*tan(
1/2*d*e/f)^2 + a*f)