\(\int \frac {a+b \sin (e+f x)}{(c+d x)^3} \, dx\) [156]
Optimal result
Integrand size = 18, antiderivative size = 123 \[
\int \frac {a+b \sin (e+f x)}{(c+d x)^3} \, dx=-\frac {a}{2 d (c+d x)^2}-\frac {b f \cos (e+f x)}{2 d^2 (c+d x)}-\frac {b f^2 \operatorname {CosIntegral}\left (\frac {c f}{d}+f x\right ) \sin \left (e-\frac {c f}{d}\right )}{2 d^3}-\frac {b \sin (e+f x)}{2 d (c+d x)^2}-\frac {b f^2 \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (\frac {c f}{d}+f x\right )}{2 d^3}
\]
[Out]
-1/2*a/d/(d*x+c)^2-1/2*b*f*cos(f*x+e)/d^2/(d*x+c)-1/2*b*f^2*cos(-e+c*f/d)*Si(c*f/d+f*x)/d^3+1/2*b*f^2*Ci(c*f/d
+f*x)*sin(-e+c*f/d)/d^3-1/2*b*sin(f*x+e)/d/(d*x+c)^2
Rubi [A] (verified)
Time = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of
steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3398, 3378, 3384, 3380, 3383}
\[
\int \frac {a+b \sin (e+f x)}{(c+d x)^3} \, dx=-\frac {a}{2 d (c+d x)^2}-\frac {b f^2 \operatorname {CosIntegral}\left (x f+\frac {c f}{d}\right ) \sin \left (e-\frac {c f}{d}\right )}{2 d^3}-\frac {b f^2 \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (x f+\frac {c f}{d}\right )}{2 d^3}-\frac {b f \cos (e+f x)}{2 d^2 (c+d x)}-\frac {b \sin (e+f x)}{2 d (c+d x)^2}
\]
[In]
Int[(a + b*Sin[e + f*x])/(c + d*x)^3,x]
[Out]
-1/2*a/(d*(c + d*x)^2) - (b*f*Cos[e + f*x])/(2*d^2*(c + d*x)) - (b*f^2*CosIntegral[(c*f)/d + f*x]*Sin[e - (c*f
)/d])/(2*d^3) - (b*Sin[e + f*x])/(2*d*(c + d*x)^2) - (b*f^2*Cos[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/(2*d^
3)
Rule 3378
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 3380
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 3383
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 3384
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 3398
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*}
\text {integral}& = \int \left (\frac {a}{(c+d x)^3}+\frac {b \sin (e+f x)}{(c+d x)^3}\right ) \, dx \\ & = -\frac {a}{2 d (c+d x)^2}+b \int \frac {\sin (e+f x)}{(c+d x)^3} \, dx \\ & = -\frac {a}{2 d (c+d x)^2}-\frac {b \sin (e+f x)}{2 d (c+d x)^2}+\frac {(b f) \int \frac {\cos (e+f x)}{(c+d x)^2} \, dx}{2 d} \\ & = -\frac {a}{2 d (c+d x)^2}-\frac {b f \cos (e+f x)}{2 d^2 (c+d x)}-\frac {b \sin (e+f x)}{2 d (c+d x)^2}-\frac {\left (b f^2\right ) \int \frac {\sin (e+f x)}{c+d x} \, dx}{2 d^2} \\ & = -\frac {a}{2 d (c+d x)^2}-\frac {b f \cos (e+f x)}{2 d^2 (c+d x)}-\frac {b \sin (e+f x)}{2 d (c+d x)^2}-\frac {\left (b f^2 \cos \left (e-\frac {c f}{d}\right )\right ) \int \frac {\sin \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx}{2 d^2}-\frac {\left (b f^2 \sin \left (e-\frac {c f}{d}\right )\right ) \int \frac {\cos \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx}{2 d^2} \\ & = -\frac {a}{2 d (c+d x)^2}-\frac {b f \cos (e+f x)}{2 d^2 (c+d x)}-\frac {b f^2 \operatorname {CosIntegral}\left (\frac {c f}{d}+f x\right ) \sin \left (e-\frac {c f}{d}\right )}{2 d^3}-\frac {b \sin (e+f x)}{2 d (c+d x)^2}-\frac {b f^2 \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (\frac {c f}{d}+f x\right )}{2 d^3} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.59 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.76
\[
\int \frac {a+b \sin (e+f x)}{(c+d x)^3} \, dx=-\frac {b f^2 \operatorname {CosIntegral}\left (f \left (\frac {c}{d}+x\right )\right ) \sin \left (e-\frac {c f}{d}\right )+\frac {d (b f (c+d x) \cos (e+f x)+d (a+b \sin (e+f x)))}{(c+d x)^2}+b f^2 \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (f \left (\frac {c}{d}+x\right )\right )}{2 d^3}
\]
[In]
Integrate[(a + b*Sin[e + f*x])/(c + d*x)^3,x]
[Out]
-1/2*(b*f^2*CosIntegral[f*(c/d + x)]*Sin[e - (c*f)/d] + (d*(b*f*(c + d*x)*Cos[e + f*x] + d*(a + b*Sin[e + f*x]
)))/(c + d*x)^2 + b*f^2*Cos[e - (c*f)/d]*SinIntegral[f*(c/d + x)])/d^3
Maple [A] (verified)
Time = 0.29 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.30
| | |
| method | result | size |
| | |
| parts |
\(-\frac {a}{2 d \left (d x +c \right )^{2}}+b \,f^{2} \left (-\frac {\sin \left (f x +e \right )}{2 \left (c f -d e +d \left (f x +e \right )\right )^{2} d}+\frac {-\frac {\cos \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}-\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}}{d}}{2 d}\right )\) |
\(160\) |
| derivativedivides |
\(\frac {-\frac {f^{3} a}{2 \left (c f -d e +d \left (f x +e \right )\right )^{2} d}+f^{3} b \left (-\frac {\sin \left (f x +e \right )}{2 \left (c f -d e +d \left (f x +e \right )\right )^{2} d}+\frac {-\frac {\cos \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}-\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}}{d}}{2 d}\right )}{f}\) |
\(177\) |
| default |
\(\frac {-\frac {f^{3} a}{2 \left (c f -d e +d \left (f x +e \right )\right )^{2} d}+f^{3} b \left (-\frac {\sin \left (f x +e \right )}{2 \left (c f -d e +d \left (f x +e \right )\right )^{2} d}+\frac {-\frac {\cos \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}-\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}}{d}}{2 d}\right )}{f}\) |
\(177\) |
| risch |
\(-\frac {a}{2 d \left (d x +c \right )^{2}}+\frac {i f^{2} b \,{\mathrm e}^{\frac {i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (i f x +i e +\frac {i \left (c f -d e \right )}{d}\right )}{4 d^{3}}-\frac {i f^{2} b \,{\mathrm e}^{-\frac {i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (-i f x -i e -\frac {i c f -i d e}{d}\right )}{4 d^{3}}+\frac {i b \left (-2 i d^{3} f^{3} x^{3}-6 i c \,d^{2} f^{3} x^{2}-6 i c^{2} d \,f^{3} x -2 i c^{3} f^{3}\right ) \cos \left (f x +e \right )}{4 d^{2} \left (d x +c \right )^{2} \left (-d^{2} x^{2} f^{2}-2 c d \,f^{2} x -c^{2} f^{2}\right )}-\frac {b \left (-2 d^{2} x^{2} f^{2}-4 c d \,f^{2} x -2 c^{2} f^{2}\right ) \sin \left (f x +e \right )}{4 d \left (d x +c \right )^{2} \left (-d^{2} x^{2} f^{2}-2 c d \,f^{2} x -c^{2} f^{2}\right )}\) |
\(292\) |
| | |
|
|
|
|
[In]
int((a+b*sin(f*x+e))/(d*x+c)^3,x,method=_RETURNVERBOSE)
[Out]
-1/2*a/d/(d*x+c)^2+b*f^2*(-1/2*sin(f*x+e)/(c*f-d*e+d*(f*x+e))^2/d+1/2*(-cos(f*x+e)/(c*f-d*e+d*(f*x+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)/d)
Fricas [A] (verification not implemented)
none
Time = 0.28 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.45
\[
\int \frac {a+b \sin (e+f x)}{(c+d x)^3} \, dx=-\frac {b d^{2} \sin \left (f x + e\right ) + a d^{2} - {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2}\right )} \operatorname {Ci}\left (\frac {d f x + c f}{d}\right ) \sin \left (-\frac {d e - c f}{d}\right ) + {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2}\right )} \cos \left (-\frac {d e - c f}{d}\right ) \operatorname {Si}\left (\frac {d f x + c f}{d}\right ) + {\left (b d^{2} f x + b c d f\right )} \cos \left (f x + e\right )}{2 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}}
\]
[In]
integrate((a+b*sin(f*x+e))/(d*x+c)^3,x, algorithm="fricas")
[Out]
-1/2*(b*d^2*sin(f*x + e) + a*d^2 - (b*d^2*f^2*x^2 + 2*b*c*d*f^2*x + b*c^2*f^2)*cos_integral((d*f*x + c*f)/d)*s
in(-(d*e - c*f)/d) + (b*d^2*f^2*x^2 + 2*b*c*d*f^2*x + b*c^2*f^2)*cos(-(d*e - c*f)/d)*sin_integral((d*f*x + c*f
)/d) + (b*d^2*f*x + b*c*d*f)*cos(f*x + e))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)
Sympy [F]
\[
\int \frac {a+b \sin (e+f x)}{(c+d x)^3} \, dx=\int \frac {a + b \sin {\left (e + f x \right )}}{\left (c + d x\right )^{3}}\, dx
\]
[In]
integrate((a+b*sin(f*x+e))/(d*x+c)**3,x)
[Out]
Integral((a + b*sin(e + f*x))/(c + d*x)**3, x)
Maxima [C] (verification not implemented)
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.15
\[
\int \frac {a+b \sin (e+f x)}{(c+d x)^3} \, dx=-\frac {\frac {a f^{3}}{{\left (f x + e\right )}^{2} d^{3} + d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2} - 2 \, {\left (d^{3} e - c d^{2} f\right )} {\left (f x + e\right )}} - \frac {{\left (f^{3} {\left (-i \, E_{3}\left (\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + i \, E_{3}\left (-\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \cos \left (-\frac {d e - c f}{d}\right ) + f^{3} {\left (E_{3}\left (\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + E_{3}\left (-\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \sin \left (-\frac {d e - c f}{d}\right )\right )} b}{{\left (f x + e\right )}^{2} d^{3} + d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2} - 2 \, {\left (d^{3} e - c d^{2} f\right )} {\left (f x + e\right )}}}{2 \, f}
\]
[In]
integrate((a+b*sin(f*x+e))/(d*x+c)^3,x, algorithm="maxima")
[Out]
-1/2*(a*f^3/((f*x + e)^2*d^3 + d^3*e^2 - 2*c*d^2*e*f + c^2*d*f^2 - 2*(d^3*e - c*d^2*f)*(f*x + e)) - (f^3*(-I*e
xp_integral_e(3, (I*(f*x + e)*d - I*d*e + I*c*f)/d) + I*exp_integral_e(3, -(I*(f*x + e)*d - I*d*e + I*c*f)/d))
*cos(-(d*e - c*f)/d) + f^3*(exp_integral_e(3, (I*(f*x + e)*d - I*d*e + I*c*f)/d) + exp_integral_e(3, -(I*(f*x
+ e)*d - I*d*e + I*c*f)/d))*sin(-(d*e - c*f)/d))*b/((f*x + e)^2*d^3 + d^3*e^2 - 2*c*d^2*e*f + c^2*d*f^2 - 2*(d
^3*e - c*d^2*f)*(f*x + e)))/f
Giac [C] (verification not implemented)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.43 (sec) , antiderivative size = 6033, normalized size of antiderivative = 49.05
\[
\int \frac {a+b \sin (e+f x)}{(c+d x)^3} \, dx=\text {Too large to display}
\]
[In]
integrate((a+b*sin(f*x+e))/(d*x+c)^3,x, algorithm="giac")
[Out]
-1/4*(b*d^2*f^2*x^2*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)^2*tan(1/2*c*f/d)^2 - b*d^2*
f^2*x^2*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)^2*tan(1/2*c*f/d)^2 + 2*b*d^2*f^2*x^2*s
in_integral((d*f*x + c*f)/d)*tan(1/2*f*x)^2*tan(1/2*e)^2*tan(1/2*c*f/d)^2 + 2*b*d^2*f^2*x^2*real_part(cos_inte
gral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)^2*tan(1/2*c*f/d) + 2*b*d^2*f^2*x^2*real_part(cos_integral(-f*x -
c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)^2*tan(1/2*c*f/d) - 2*b*d^2*f^2*x^2*real_part(cos_integral(f*x + c*f/d))*tan(
1/2*f*x)^2*tan(1/2*e)*tan(1/2*c*f/d)^2 - 2*b*d^2*f^2*x^2*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*
tan(1/2*e)*tan(1/2*c*f/d)^2 + 2*b*c*d*f^2*x*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)^2*t
an(1/2*c*f/d)^2 - 2*b*c*d*f^2*x*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)^2*tan(1/2*c*f/
d)^2 + 4*b*c*d*f^2*x*sin_integral((d*f*x + c*f)/d)*tan(1/2*f*x)^2*tan(1/2*e)^2*tan(1/2*c*f/d)^2 - b*d^2*f^2*x^
2*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)^2 + b*d^2*f^2*x^2*imag_part(cos_integral(-f*x
- c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)^2 - 2*b*d^2*f^2*x^2*sin_integral((d*f*x + c*f)/d)*tan(1/2*f*x)^2*tan(1/2*
e)^2 + 4*b*d^2*f^2*x^2*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)*tan(1/2*c*f/d) - 4*b*d^2
*f^2*x^2*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)*tan(1/2*c*f/d) + 8*b*d^2*f^2*x^2*sin_
integral((d*f*x + c*f)/d)*tan(1/2*f*x)^2*tan(1/2*e)*tan(1/2*c*f/d) + 4*b*c*d*f^2*x*real_part(cos_integral(f*x
+ c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)^2*tan(1/2*c*f/d) + 4*b*c*d*f^2*x*real_part(cos_integral(-f*x - c*f/d))*tan
(1/2*f*x)^2*tan(1/2*e)^2*tan(1/2*c*f/d) - b*d^2*f^2*x^2*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*ta
n(1/2*c*f/d)^2 + b*d^2*f^2*x^2*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2 - 2*b*d^2
*f^2*x^2*sin_integral((d*f*x + c*f)/d)*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2 - 4*b*c*d*f^2*x*real_part(cos_integral(
f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)*tan(1/2*c*f/d)^2 - 4*b*c*d*f^2*x*real_part(cos_integral(-f*x - c*f/d))
*tan(1/2*f*x)^2*tan(1/2*e)*tan(1/2*c*f/d)^2 + b*d^2*f^2*x^2*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*e)^2*
tan(1/2*c*f/d)^2 - b*d^2*f^2*x^2*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*e)^2*tan(1/2*c*f/d)^2 + 2*b*d^2
*f^2*x^2*sin_integral((d*f*x + c*f)/d)*tan(1/2*e)^2*tan(1/2*c*f/d)^2 + b*c^2*f^2*imag_part(cos_integral(f*x +
c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)^2*tan(1/2*c*f/d)^2 - b*c^2*f^2*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2
*f*x)^2*tan(1/2*e)^2*tan(1/2*c*f/d)^2 + 2*b*c^2*f^2*sin_integral((d*f*x + c*f)/d)*tan(1/2*f*x)^2*tan(1/2*e)^2*
tan(1/2*c*f/d)^2 + 2*b*d^2*f^2*x^2*real_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*e) + 2*b*d^2*f^
2*x^2*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*e) - 2*b*c*d*f^2*x*imag_part(cos_integral(f
*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)^2 + 2*b*c*d*f^2*x*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*
tan(1/2*e)^2 - 4*b*c*d*f^2*x*sin_integral((d*f*x + c*f)/d)*tan(1/2*f*x)^2*tan(1/2*e)^2 - 2*b*d^2*f^2*x^2*real_
part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d) - 2*b*d^2*f^2*x^2*real_part(cos_integral(-f*x -
c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d) + 8*b*c*d*f^2*x*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(
1/2*e)*tan(1/2*c*f/d) - 8*b*c*d*f^2*x*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)*tan(1/2*
c*f/d) + 16*b*c*d*f^2*x*sin_integral((d*f*x + c*f)/d)*tan(1/2*f*x)^2*tan(1/2*e)*tan(1/2*c*f/d) + 2*b*d^2*f^2*x
^2*real_part(cos_integral(f*x + c*f/d))*tan(1/2*e)^2*tan(1/2*c*f/d) + 2*b*d^2*f^2*x^2*real_part(cos_integral(-
f*x - c*f/d))*tan(1/2*e)^2*tan(1/2*c*f/d) + 2*b*c^2*f^2*real_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*ta
n(1/2*e)^2*tan(1/2*c*f/d) + 2*b*c^2*f^2*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)^2*tan(
1/2*c*f/d) - 2*b*c*d*f^2*x*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2 + 2*b*c*d*f^2*
x*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2 - 4*b*c*d*f^2*x*sin_integral((d*f*x +
c*f)/d)*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2 - 2*b*d^2*f^2*x^2*real_part(cos_integral(f*x + c*f/d))*tan(1/2*e)*tan(
1/2*c*f/d)^2 - 2*b*d^2*f^2*x^2*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*e)*tan(1/2*c*f/d)^2 - 2*b*c^2*f^2
*real_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)*tan(1/2*c*f/d)^2 - 2*b*c^2*f^2*real_part(cos_i
ntegral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)*tan(1/2*c*f/d)^2 + 2*b*c*d*f^2*x*imag_part(cos_integral(f*x +
c*f/d))*tan(1/2*e)^2*tan(1/2*c*f/d)^2 - 2*b*c*d*f^2*x*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*e)^2*tan(
1/2*c*f/d)^2 + 4*b*c*d*f^2*x*sin_integral((d*f*x + c*f)/d)*tan(1/2*e)^2*tan(1/2*c*f/d)^2 + 2*b*d^2*f*x*tan(1/2
*f*x)^2*tan(1/2*e)^2*tan(1/2*c*f/d)^2 + b*d^2*f^2*x^2*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2 - b*
d^2*f^2*x^2*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2 + 2*b*d^2*f^2*x^2*sin_integral((d*f*x + c*f)/
d)*tan(1/2*f*x)^2 + 4*b*c*d*f^2*x*real_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*e) + 4*b*c*d*f^2
*x*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*e) - b*d^2*f^2*x^2*imag_part(cos_integral(f*x
+ c*f/d))*tan(1/2*e)^2 + b*d^2*f^2*x^2*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*e)^2 - 2*b*d^2*f^2*x^2*si
n_integral((d*f*x + c*f)/d)*tan(1/2*e)^2 - b*c^2*f^2*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1
/2*e)^2 + b*c^2*f^2*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)^2 - 2*b*c^2*f^2*sin_integr
al((d*f*x + c*f)/d)*tan(1/2*f*x)^2*tan(1/2*e)^2 - 4*b*c*d*f^2*x*real_part(cos_integral(f*x + c*f/d))*tan(1/2*f
*x)^2*tan(1/2*c*f/d) - 4*b*c*d*f^2*x*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d) + 4*b
*d^2*f^2*x^2*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*e)*tan(1/2*c*f/d) - 4*b*d^2*f^2*x^2*imag_part(cos_in
tegral(-f*x - c*f/d))*tan(1/2*e)*tan(1/2*c*f/d) + 8*b*d^2*f^2*x^2*sin_integral((d*f*x + c*f)/d)*tan(1/2*e)*tan
(1/2*c*f/d) + 4*b*c^2*f^2*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)*tan(1/2*c*f/d) - 4*b*
c^2*f^2*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*e)*tan(1/2*c*f/d) + 8*b*c^2*f^2*sin_integ
ral((d*f*x + c*f)/d)*tan(1/2*f*x)^2*tan(1/2*e)*tan(1/2*c*f/d) + 4*b*c*d*f^2*x*real_part(cos_integral(f*x + c*f
/d))*tan(1/2*e)^2*tan(1/2*c*f/d) + 4*b*c*d*f^2*x*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*e)^2*tan(1/2*c*
f/d) - b*d^2*f^2*x^2*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*c*f/d)^2 + b*d^2*f^2*x^2*imag_part(cos_integ
ral(-f*x - c*f/d))*tan(1/2*c*f/d)^2 - 2*b*d^2*f^2*x^2*sin_integral((d*f*x + c*f)/d)*tan(1/2*c*f/d)^2 - b*c^2*f
^2*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2 + b*c^2*f^2*imag_part(cos_integral(-f*
x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2 - 2*b*c^2*f^2*sin_integral((d*f*x + c*f)/d)*tan(1/2*f*x)^2*tan(1/2
*c*f/d)^2 - 4*b*c*d*f^2*x*real_part(cos_integral(f*x + c*f/d))*tan(1/2*e)*tan(1/2*c*f/d)^2 - 4*b*c*d*f^2*x*rea
l_part(cos_integral(-f*x - c*f/d))*tan(1/2*e)*tan(1/2*c*f/d)^2 + b*c^2*f^2*imag_part(cos_integral(f*x + c*f/d)
)*tan(1/2*e)^2*tan(1/2*c*f/d)^2 - b*c^2*f^2*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*e)^2*tan(1/2*c*f/d)^
2 + 2*b*c^2*f^2*sin_integral((d*f*x + c*f)/d)*tan(1/2*e)^2*tan(1/2*c*f/d)^2 + 2*b*c*d*f*tan(1/2*f*x)^2*tan(1/2
*e)^2*tan(1/2*c*f/d)^2 + 2*b*c*d*f^2*x*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2 - 2*b*c*d*f^2*x*ima
g_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2 + 4*b*c*d*f^2*x*sin_integral((d*f*x + c*f)/d)*tan(1/2*f*x)^2
+ 2*b*d^2*f^2*x^2*real_part(cos_integral(f*x + c*f/d))*tan(1/2*e) + 2*b*d^2*f^2*x^2*real_part(cos_integral(-f
*x - c*f/d))*tan(1/2*e) + 2*b*c^2*f^2*real_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*e) + 2*b*c^2
*f^2*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*e) - 2*b*c*d*f^2*x*imag_part(cos_integral(f*
x + c*f/d))*tan(1/2*e)^2 + 2*b*c*d*f^2*x*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*e)^2 - 4*b*c*d*f^2*x*si
n_integral((d*f*x + c*f)/d)*tan(1/2*e)^2 + 2*b*d^2*f*x*tan(1/2*f*x)^2*tan(1/2*e)^2 - 2*b*d^2*f^2*x^2*real_part
(cos_integral(f*x + c*f/d))*tan(1/2*c*f/d) - 2*b*d^2*f^2*x^2*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*c*f
/d) - 2*b*c^2*f^2*real_part(cos_integral(f*x + c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d) - 2*b*c^2*f^2*real_part(c
os_integral(-f*x - c*f/d))*tan(1/2*f*x)^2*tan(1/2*c*f/d) + 8*b*c*d*f^2*x*imag_part(cos_integral(f*x + c*f/d))*
tan(1/2*e)*tan(1/2*c*f/d) - 8*b*c*d*f^2*x*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*e)*tan(1/2*c*f/d) + 16
*b*c*d*f^2*x*sin_integral((d*f*x + c*f)/d)*tan(1/2*e)*tan(1/2*c*f/d) + 2*b*c^2*f^2*real_part(cos_integral(f*x
+ c*f/d))*tan(1/2*e)^2*tan(1/2*c*f/d) + 2*b*c^2*f^2*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*e)^2*tan(1/2
*c*f/d) - 2*b*c*d*f^2*x*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*c*f/d)^2 + 2*b*c*d*f^2*x*imag_part(cos_in
tegral(-f*x - c*f/d))*tan(1/2*c*f/d)^2 - 4*b*c*d*f^2*x*sin_integral((d*f*x + c*f)/d)*tan(1/2*c*f/d)^2 - 2*b*d^
2*f*x*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2 - 2*b*c^2*f^2*real_part(cos_integral(f*x + c*f/d))*tan(1/2*e)*tan(1/2*c*
f/d)^2 - 2*b*c^2*f^2*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*e)*tan(1/2*c*f/d)^2 - 8*b*d^2*f*x*tan(1/2*f
*x)*tan(1/2*e)*tan(1/2*c*f/d)^2 - 2*b*d^2*f*x*tan(1/2*e)^2*tan(1/2*c*f/d)^2 + 2*a*d^2*tan(1/2*f*x)^2*tan(1/2*e
)^2*tan(1/2*c*f/d)^2 + b*d^2*f^2*x^2*imag_part(cos_integral(f*x + c*f/d)) - b*d^2*f^2*x^2*imag_part(cos_integr
al(-f*x - c*f/d)) + 2*b*d^2*f^2*x^2*sin_integral((d*f*x + c*f)/d) + b*c^2*f^2*imag_part(cos_integral(f*x + c*f
/d))*tan(1/2*f*x)^2 - b*c^2*f^2*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*f*x)^2 + 2*b*c^2*f^2*sin_integra
l((d*f*x + c*f)/d)*tan(1/2*f*x)^2 + 4*b*c*d*f^2*x*real_part(cos_integral(f*x + c*f/d))*tan(1/2*e) + 4*b*c*d*f^
2*x*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*e) - b*c^2*f^2*imag_part(cos_integral(f*x + c*f/d))*tan(1/2*
e)^2 + b*c^2*f^2*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*e)^2 - 2*b*c^2*f^2*sin_integral((d*f*x + c*f)/d
)*tan(1/2*e)^2 + 2*b*c*d*f*tan(1/2*f*x)^2*tan(1/2*e)^2 - 4*b*c*d*f^2*x*real_part(cos_integral(f*x + c*f/d))*ta
n(1/2*c*f/d) - 4*b*c*d*f^2*x*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*c*f/d) + 4*b*c^2*f^2*imag_part(cos_
integral(f*x + c*f/d))*tan(1/2*e)*tan(1/2*c*f/d) - 4*b*c^2*f^2*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*e
)*tan(1/2*c*f/d) + 8*b*c^2*f^2*sin_integral((d*f*x + c*f)/d)*tan(1/2*e)*tan(1/2*c*f/d) - b*c^2*f^2*imag_part(c
os_integral(f*x + c*f/d))*tan(1/2*c*f/d)^2 + b*c^2*f^2*imag_part(cos_integral(-f*x - c*f/d))*tan(1/2*c*f/d)^2
- 2*b*c^2*f^2*sin_integral((d*f*x + c*f)/d)*tan(1/2*c*f/d)^2 - 2*b*c*d*f*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2 - 8*b
*c*d*f*tan(1/2*f*x)*tan(1/2*e)*tan(1/2*c*f/d)^2 - 4*b*d^2*tan(1/2*f*x)^2*tan(1/2*e)*tan(1/2*c*f/d)^2 - 2*b*c*d
*f*tan(1/2*e)^2*tan(1/2*c*f/d)^2 - 4*b*d^2*tan(1/2*f*x)*tan(1/2*e)^2*tan(1/2*c*f/d)^2 + 2*b*c*d*f^2*x*imag_par
t(cos_integral(f*x + c*f/d)) - 2*b*c*d*f^2*x*imag_part(cos_integral(-f*x - c*f/d)) + 4*b*c*d*f^2*x*sin_integra
l((d*f*x + c*f)/d) - 2*b*d^2*f*x*tan(1/2*f*x)^2 + 2*b*c^2*f^2*real_part(cos_integral(f*x + c*f/d))*tan(1/2*e)
+ 2*b*c^2*f^2*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*e) - 8*b*d^2*f*x*tan(1/2*f*x)*tan(1/2*e) - 2*b*d^2
*f*x*tan(1/2*e)^2 + 2*a*d^2*tan(1/2*f*x)^2*tan(1/2*e)^2 - 2*b*c^2*f^2*real_part(cos_integral(f*x + c*f/d))*tan
(1/2*c*f/d) - 2*b*c^2*f^2*real_part(cos_integral(-f*x - c*f/d))*tan(1/2*c*f/d) + 2*b*d^2*f*x*tan(1/2*c*f/d)^2
+ 2*a*d^2*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2 + 2*a*d^2*tan(1/2*e)^2*tan(1/2*c*f/d)^2 + b*c^2*f^2*imag_part(cos_in
tegral(f*x + c*f/d)) - b*c^2*f^2*imag_part(cos_integral(-f*x - c*f/d)) + 2*b*c^2*f^2*sin_integral((d*f*x + c*f
)/d) - 2*b*c*d*f*tan(1/2*f*x)^2 - 8*b*c*d*f*tan(1/2*f*x)*tan(1/2*e) - 4*b*d^2*tan(1/2*f*x)^2*tan(1/2*e) - 2*b*
c*d*f*tan(1/2*e)^2 - 4*b*d^2*tan(1/2*f*x)*tan(1/2*e)^2 + 2*b*c*d*f*tan(1/2*c*f/d)^2 + 4*b*d^2*tan(1/2*f*x)*tan
(1/2*c*f/d)^2 + 4*b*d^2*tan(1/2*e)*tan(1/2*c*f/d)^2 + 2*b*d^2*f*x + 2*a*d^2*tan(1/2*f*x)^2 + 2*a*d^2*tan(1/2*e
)^2 + 2*a*d^2*tan(1/2*c*f/d)^2 + 2*b*c*d*f + 4*b*d^2*tan(1/2*f*x) + 4*b*d^2*tan(1/2*e) + 2*a*d^2)/(d^5*x^2*tan
(1/2*f*x)^2*tan(1/2*e)^2*tan(1/2*c*f/d)^2 + 2*c*d^4*x*tan(1/2*f*x)^2*tan(1/2*e)^2*tan(1/2*c*f/d)^2 + d^5*x^2*t
an(1/2*f*x)^2*tan(1/2*e)^2 + d^5*x^2*tan(1/2*f*x)^2*tan(1/2*c*f/d)^2 + d^5*x^2*tan(1/2*e)^2*tan(1/2*c*f/d)^2 +
c^2*d^3*tan(1/2*f*x)^2*tan(1/2*e)^2*tan(1/2*c*f/d)^2 + 2*c*d^4*x*tan(1/2*f*x)^2*tan(1/2*e)^2 + 2*c*d^4*x*tan(
1/2*f*x)^2*tan(1/2*c*f/d)^2 + 2*c*d^4*x*tan(1/2*e)^2*tan(1/2*c*f/d)^2 + d^5*x^2*tan(1/2*f*x)^2 + d^5*x^2*tan(1
/2*e)^2 + c^2*d^3*tan(1/2*f*x)^2*tan(1/2*e)^2 + d^5*x^2*tan(1/2*c*f/d)^2 + c^2*d^3*tan(1/2*f*x)^2*tan(1/2*c*f/
d)^2 + c^2*d^3*tan(1/2*e)^2*tan(1/2*c*f/d)^2 + 2*c*d^4*x*tan(1/2*f*x)^2 + 2*c*d^4*x*tan(1/2*e)^2 + 2*c*d^4*x*t
an(1/2*c*f/d)^2 + d^5*x^2 + c^2*d^3*tan(1/2*f*x)^2 + c^2*d^3*tan(1/2*e)^2 + c^2*d^3*tan(1/2*c*f/d)^2 + 2*c*d^4
*x + c^2*d^3)
Mupad [F(-1)]
Timed out. \[
\int \frac {a+b \sin (e+f x)}{(c+d x)^3} \, dx=\int \frac {a+b\,\sin \left (e+f\,x\right )}{{\left (c+d\,x\right )}^3} \,d x
\]
[In]
int((a + b*sin(e + f*x))/(c + d*x)^3,x)
[Out]
int((a + b*sin(e + f*x))/(c + d*x)^3, x)