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

Optimal. Leaf size=89 \[ -\frac {a f \text {Ci}\left (x f+\frac {c f}{d}\right ) \sin \left (e-\frac {c f}{d}\right )}{d^2}-\frac {a f \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (x f+\frac {c f}{d}\right )}{d^2}-\frac {a \cos (e+f x)}{d (c+d x)}-\frac {a}{d (c+d x)} \]

[Out]

-a/d/(d*x+c)-a*cos(f*x+e)/d/(d*x+c)-a*f*cos(-e+c*f/d)*Si(c*f/d+f*x)/d^2+a*f*Ci(c*f/d+f*x)*sin(-e+c*f/d)/d^2

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Rubi [A]  time = 0.16, antiderivative size = 89, normalized size of antiderivative = 1.00, 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} \[ -\frac {a f \text {CosIntegral}\left (\frac {c f}{d}+f x\right ) \sin \left (e-\frac {c f}{d}\right )}{d^2}-\frac {a f \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (x f+\frac {c f}{d}\right )}{d^2}-\frac {a \cos (e+f x)}{d (c+d x)}-\frac {a}{d (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 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])

Rubi steps

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

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Mathematica [A]  time = 0.33, size = 78, normalized size = 0.88 \[ -\frac {a \left (f (c+d x) \text {Ci}\left (f \left (\frac {c}{d}+x\right )\right ) \sin \left (e-\frac {c f}{d}\right )+f (c+d x) \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (f \left (\frac {c}{d}+x\right )\right )+d (\cos (e+f x)+1)\right )}{d^2 (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*(d*(1 + Cos[e + f*x]) + f*(c + d*x)*CosIntegral[f*(c/d + x)]*Sin[e - (c*f)/d] + f*(c + d*x)*Cos[e - (c*f)
/d]*SinIntegral[f*(c/d + x)]))/(d^2*(c + d*x)))

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fricas [A]  time = 0.86, size = 135, normalized size = 1.52 \[ -\frac {2 \, a d \cos \left (f x + e\right ) + 2 \, {\left (a d f x + a c f\right )} \cos \left (-\frac {d e - c f}{d}\right ) \operatorname {Si}\left (\frac {d f x + c f}{d}\right ) + 2 \, a d - {\left ({\left (a d f x + a c f\right )} \operatorname {Ci}\left (\frac {d f x + c f}{d}\right ) + {\left (a d f x + a c f\right )} \operatorname {Ci}\left (-\frac {d f x + c f}{d}\right )\right )} \sin \left (-\frac {d e - c f}{d}\right )}{2 \, {\left (d^{3} x + c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a*d*cos(f*x + e) + 2*(a*d*f*x + a*c*f)*cos(-(d*e - c*f)/d)*sin_integral((d*f*x + c*f)/d) + 2*a*d - ((a
*d*f*x + a*c*f)*cos_integral((d*f*x + c*f)/d) + (a*d*f*x + a*c*f)*cos_integral(-(d*f*x + c*f)/d))*sin(-(d*e -
c*f)/d))/(d^3*x + c*d^2)

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giac [B]  time = 0.54, size = 578, normalized size = 6.49 \[ \frac {{\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} f^{2} \operatorname {Ci}\left (-\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e}{d}\right ) \sin \left (\frac {c f - d e}{d}\right ) - c f^{3} \operatorname {Ci}\left (-\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e}{d}\right ) \sin \left (\frac {c f - d e}{d}\right ) + d f^{2} \operatorname {Ci}\left (-\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e}{d}\right ) e \sin \left (\frac {c f - d e}{d}\right ) - {\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} f^{2} \cos \left (\frac {c f - d e}{d}\right ) \operatorname {Si}\left (-\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e}{d}\right ) + c f^{3} \cos \left (\frac {c f - d e}{d}\right ) \operatorname {Si}\left (-\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e}{d}\right ) - d f^{2} \cos \left (\frac {c f - d e}{d}\right ) e \operatorname {Si}\left (-\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e}{d}\right ) + d f^{2} \cos \left (\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )}}{d}\right )\right )} a d^{2}}{{\left ({\left (d x + c\right )} d^{4} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c d^{4} f + d^{5} e\right )} f} - \frac {a}{{\left (d x + c\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c))*f^2*cos_integral(-((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)
) - c*f + d*e)/d)*sin((c*f - d*e)/d) - c*f^3*cos_integral(-((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*
f + d*e)/d)*sin((c*f - d*e)/d) + d*f^2*cos_integral(-((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*
e)/d)*e*sin((c*f - d*e)/d) - (d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c))*f^2*cos((c*f - d*e)/d)*sin_integral
(-((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d) + c*f^3*cos((c*f - d*e)/d)*sin_integral(-((d*
x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d) - d*f^2*cos((c*f - d*e)/d)*e*sin_integral(-((d*x +
c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d) + d*f^2*cos((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c
))/d))*a*d^2/(((d*x + c)*d^4*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*d^4*f + d^5*e)*f) - a/((d*x + c)*d)

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maple [A]  time = 0.04, size = 143, normalized size = 1.61 \[ \frac {f^{2} a \left (-\frac {\cos \left (f x +e \right )}{\left (\left (f x +e \right ) d +c f -d e \right ) d}-\frac {\frac {\Si \left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}-\frac {\Ci \left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}}{d}\right )-\frac {f^{2} a}{\left (\left (f x +e \right ) d +c f -d e \right ) d}}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [C]  time = 1.00, size = 196, normalized size = 2.20 \[ -\frac {\frac {16 \, a f^{2}}{{\left (f x + e\right )} d^{2} - d^{2} e + c d f} + \frac {{\left (8 \, f^{2} {\left (E_{2}\left (\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + E_{2}\left (-\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \cos \left (-\frac {d e - c f}{d}\right ) + f^{2} {\left (8 i \, E_{2}\left (\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) - 8 i \, E_{2}\left (-\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \sin \left (-\frac {d e - c f}{d}\right )\right )} a}{{\left (f x + e\right )} d^{2} - d^{2} e + c d f}}{16 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(16*a*f^2/((f*x + e)*d^2 - d^2*e + c*d*f) + (8*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))*cos(-(d*e - c*f)/d) + f^2*(8*I*exp_integral_e(2, (I*
(f*x + e)*d - I*d*e + I*c*f)/d) - 8*I*exp_integral_e(2, -(I*(f*x + e)*d - I*d*e + I*c*f)/d))*sin(-(d*e - c*f)/
d))*a/((f*x + e)*d^2 - d^2*e + c*d*f))/f

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+a\,\cos \left (e+f\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \frac {\cos {\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {1}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(Integral(cos(e + f*x)/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(1/(c**2 + 2*c*d*x + d**2*x**2), x))

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