\(\int \frac {x^4 \sin (c+d x)}{a+b x} \, dx\) [18]

Optimal result

Integrand size = 17, antiderivative size = 218 \[ \int \frac {x^4 \sin (c+d x)}{a+b x} \, dx=-\frac {2 a \cos (c+d x)}{b^2 d^3}+\frac {a^3 \cos (c+d x)}{b^4 d}+\frac {6 x \cos (c+d x)}{b d^3}-\frac {a^2 x \cos (c+d x)}{b^3 d}+\frac {a x^2 \cos (c+d x)}{b^2 d}-\frac {x^3 \cos (c+d x)}{b d}+\frac {a^4 \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^5}-\frac {6 \sin (c+d x)}{b d^4}+\frac {a^2 \sin (c+d x)}{b^3 d^2}-\frac {2 a x \sin (c+d x)}{b^2 d^2}+\frac {3 x^2 \sin (c+d x)}{b d^2}+\frac {a^4 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^5} \]

[Out]

-2*a*cos(d*x+c)/b^2/d^3+a^3*cos(d*x+c)/b^4/d+6*x*cos(d*x+c)/b/d^3-a^2*x*cos(d*x+c)/b^3/d+a*x^2*cos(d*x+c)/b^2/
d-x^3*cos(d*x+c)/b/d+a^4*cos(-c+a*d/b)*Si(a*d/b+d*x)/b^5-a^4*Ci(a*d/b+d*x)*sin(-c+a*d/b)/b^5-6*sin(d*x+c)/b/d^
4+a^2*sin(d*x+c)/b^3/d^2-2*a*x*sin(d*x+c)/b^2/d^2+3*x^2*sin(d*x+c)/b/d^2

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {6874, 2718, 3377, 2717, 3384, 3380, 3383} \[ \int \frac {x^4 \sin (c+d x)}{a+b x} \, dx=\frac {a^4 \sin \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{b^5}+\frac {a^4 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^5}+\frac {a^3 \cos (c+d x)}{b^4 d}+\frac {a^2 \sin (c+d x)}{b^3 d^2}-\frac {a^2 x \cos (c+d x)}{b^3 d}-\frac {2 a \cos (c+d x)}{b^2 d^3}-\frac {2 a x \sin (c+d x)}{b^2 d^2}+\frac {a x^2 \cos (c+d x)}{b^2 d}-\frac {6 \sin (c+d x)}{b d^4}+\frac {6 x \cos (c+d x)}{b d^3}+\frac {3 x^2 \sin (c+d x)}{b d^2}-\frac {x^3 \cos (c+d x)}{b d} \]

[In]

Int[(x^4*Sin[c + d*x])/(a + b*x),x]

[Out]

(-2*a*Cos[c + d*x])/(b^2*d^3) + (a^3*Cos[c + d*x])/(b^4*d) + (6*x*Cos[c + d*x])/(b*d^3) - (a^2*x*Cos[c + d*x])
/(b^3*d) + (a*x^2*Cos[c + d*x])/(b^2*d) - (x^3*Cos[c + d*x])/(b*d) + (a^4*CosIntegral[(a*d)/b + d*x]*Sin[c - (
a*d)/b])/b^5 - (6*Sin[c + d*x])/(b*d^4) + (a^2*Sin[c + d*x])/(b^3*d^2) - (2*a*x*Sin[c + d*x])/(b^2*d^2) + (3*x
^2*Sin[c + d*x])/(b*d^2) + (a^4*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^5

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

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 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^3 \sin (c+d x)}{b^4}+\frac {a^2 x \sin (c+d x)}{b^3}-\frac {a x^2 \sin (c+d x)}{b^2}+\frac {x^3 \sin (c+d x)}{b}+\frac {a^4 \sin (c+d x)}{b^4 (a+b x)}\right ) \, dx \\ & = -\frac {a^3 \int \sin (c+d x) \, dx}{b^4}+\frac {a^4 \int \frac {\sin (c+d x)}{a+b x} \, dx}{b^4}+\frac {a^2 \int x \sin (c+d x) \, dx}{b^3}-\frac {a \int x^2 \sin (c+d x) \, dx}{b^2}+\frac {\int x^3 \sin (c+d x) \, dx}{b} \\ & = \frac {a^3 \cos (c+d x)}{b^4 d}-\frac {a^2 x \cos (c+d x)}{b^3 d}+\frac {a x^2 \cos (c+d x)}{b^2 d}-\frac {x^3 \cos (c+d x)}{b d}+\frac {a^2 \int \cos (c+d x) \, dx}{b^3 d}-\frac {(2 a) \int x \cos (c+d x) \, dx}{b^2 d}+\frac {3 \int x^2 \cos (c+d x) \, dx}{b d}+\frac {\left (a^4 \cos \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sin \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^4}+\frac {\left (a^4 \sin \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cos \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^4} \\ & = \frac {a^3 \cos (c+d x)}{b^4 d}-\frac {a^2 x \cos (c+d x)}{b^3 d}+\frac {a x^2 \cos (c+d x)}{b^2 d}-\frac {x^3 \cos (c+d x)}{b d}+\frac {a^4 \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^5}+\frac {a^2 \sin (c+d x)}{b^3 d^2}-\frac {2 a x \sin (c+d x)}{b^2 d^2}+\frac {3 x^2 \sin (c+d x)}{b d^2}+\frac {a^4 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^5}+\frac {(2 a) \int \sin (c+d x) \, dx}{b^2 d^2}-\frac {6 \int x \sin (c+d x) \, dx}{b d^2} \\ & = -\frac {2 a \cos (c+d x)}{b^2 d^3}+\frac {a^3 \cos (c+d x)}{b^4 d}+\frac {6 x \cos (c+d x)}{b d^3}-\frac {a^2 x \cos (c+d x)}{b^3 d}+\frac {a x^2 \cos (c+d x)}{b^2 d}-\frac {x^3 \cos (c+d x)}{b d}+\frac {a^4 \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^5}+\frac {a^2 \sin (c+d x)}{b^3 d^2}-\frac {2 a x \sin (c+d x)}{b^2 d^2}+\frac {3 x^2 \sin (c+d x)}{b d^2}+\frac {a^4 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {6 \int \cos (c+d x) \, dx}{b d^3} \\ & = -\frac {2 a \cos (c+d x)}{b^2 d^3}+\frac {a^3 \cos (c+d x)}{b^4 d}+\frac {6 x \cos (c+d x)}{b d^3}-\frac {a^2 x \cos (c+d x)}{b^3 d}+\frac {a x^2 \cos (c+d x)}{b^2 d}-\frac {x^3 \cos (c+d x)}{b d}+\frac {a^4 \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^5}-\frac {6 \sin (c+d x)}{b d^4}+\frac {a^2 \sin (c+d x)}{b^3 d^2}-\frac {2 a x \sin (c+d x)}{b^2 d^2}+\frac {3 x^2 \sin (c+d x)}{b d^2}+\frac {a^4 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.72 \[ \int \frac {x^4 \sin (c+d x)}{a+b x} \, dx=\frac {a^4 d^4 \operatorname {CosIntegral}\left (d \left (\frac {a}{b}+x\right )\right ) \sin \left (c-\frac {a d}{b}\right )+b \left (d \left (a^3 d^2-a^2 b d^2 x+b^3 x \left (6-d^2 x^2\right )+a b^2 \left (-2+d^2 x^2\right )\right ) \cos (c+d x)+b \left (a^2 d^2-2 a b d^2 x+3 b^2 \left (-2+d^2 x^2\right )\right ) \sin (c+d x)\right )+a^4 d^4 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{b^5 d^4} \]

[In]

Integrate[(x^4*Sin[c + d*x])/(a + b*x),x]

[Out]

(a^4*d^4*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] + b*(d*(a^3*d^2 - a^2*b*d^2*x + b^3*x*(6 - d^2*x^2) + a*b^2
*(-2 + d^2*x^2))*Cos[c + d*x] + b*(a^2*d^2 - 2*a*b*d^2*x + 3*b^2*(-2 + d^2*x^2))*Sin[c + d*x]) + a^4*d^4*Cos[c
- (a*d)/b]*SinIntegral[d*(a/b + x)])/(b^5*d^4)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.42 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.50

method	result	size
risch	\(-\frac {i \pi \,\operatorname {csgn}\left (\frac {d \left (b x +a \right )}{b}\right ) \sin \left (\frac {d a -c b}{b}\right ) a^{4}}{2 b^{5}}-\frac {x^{3} \cos \left (d x +c \right )}{b d}-\frac {\pi \,\operatorname {csgn}\left (\frac {d \left (b x +a \right )}{b}\right ) \cos \left (\frac {d a -c b}{b}\right ) a^{4}}{2 b^{5}}+\frac {i \operatorname {Si}\left (\frac {d \left (b x +a \right )}{b}\right ) \sin \left (\frac {d a -c b}{b}\right ) a^{4}}{b^{5}}+\frac {a \,x^{2} \cos \left (d x +c \right )}{b^{2} d}+\frac {\operatorname {Si}\left (\frac {d \left (b x +a \right )}{b}\right ) \cos \left (\frac {d a -c b}{b}\right ) a^{4}}{b^{5}}+\frac {\operatorname {Ei}_{1}\left (-\frac {i d \left (b x +a \right )}{b}\right ) \sin \left (\frac {d a -c b}{b}\right ) a^{4}}{b^{5}}+\frac {3 x^{2} \sin \left (d x +c \right )}{b \,d^{2}}-\frac {a^{2} x \cos \left (d x +c \right )}{b^{3} d}-\frac {2 a x \sin \left (d x +c \right )}{b^{2} d^{2}}+\frac {a^{3} \cos \left (d x +c \right )}{b^{4} d}+\frac {a^{2} \sin \left (d x +c \right )}{b^{3} d^{2}}+\frac {6 x \cos \left (d x +c \right )}{b \,d^{3}}-\frac {2 a \cos \left (d x +c \right )}{b^{2} d^{3}}-\frac {6 \sin \left (d x +c \right )}{b \,d^{4}}\)	\(328\)
derivativedivides	\(\frac {d \,c^{4} \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )+\frac {4 \left (d a -c b \right ) d \,c^{3} \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b}+\frac {4 d \,c^{3} \cos \left (d x +c \right )}{b}+\frac {6 \left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}\right ) d \,c^{2} \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{2}}-\frac {6 d \,c^{2} \left (d a -c b -b \right ) \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{b^{2}}+\frac {4 \left (d^{3} a^{3}-3 c \,d^{2} a^{2} b +3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) d c \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{3}}-\frac {4 d c \left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}-a b d +b^{2} c +b^{2}\right ) \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{b^{3}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) d \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{4}}-\frac {d \left (d^{3} a^{3}-3 c \,d^{2} a^{2} b +3 a \,b^{2} c^{2} d -b^{3} c^{3}-d^{2} a^{2} b +2 a \,b^{2} c d -b^{3} c^{2}+a \,b^{2} d -b^{3} c -b^{3}\right ) \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{b^{4}}}{d^{5}}\)	\(784\)
default	\(\frac {d \,c^{4} \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )+\frac {4 \left (d a -c b \right ) d \,c^{3} \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b}+\frac {4 d \,c^{3} \cos \left (d x +c \right )}{b}+\frac {6 \left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}\right ) d \,c^{2} \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{2}}-\frac {6 d \,c^{2} \left (d a -c b -b \right ) \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{b^{2}}+\frac {4 \left (d^{3} a^{3}-3 c \,d^{2} a^{2} b +3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) d c \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{3}}-\frac {4 d c \left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}-a b d +b^{2} c +b^{2}\right ) \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{b^{3}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) d \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{4}}-\frac {d \left (d^{3} a^{3}-3 c \,d^{2} a^{2} b +3 a \,b^{2} c^{2} d -b^{3} c^{3}-d^{2} a^{2} b +2 a \,b^{2} c d -b^{3} c^{2}+a \,b^{2} d -b^{3} c -b^{3}\right ) \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{b^{4}}}{d^{5}}\)	\(784\)

[In]

int(x^4*sin(d*x+c)/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

-1/2*I/b^5*Pi*csgn(d*(b*x+a)/b)*sin((a*d-b*c)/b)*a^4-x^3*cos(d*x+c)/b/d-1/2/b^5*Pi*csgn(d*(b*x+a)/b)*cos((a*d-
b*c)/b)*a^4+I/b^5*Si(d*(b*x+a)/b)*sin((a*d-b*c)/b)*a^4+a*x^2*cos(d*x+c)/b^2/d+1/b^5*Si(d*(b*x+a)/b)*cos((a*d-b
*c)/b)*a^4+1/b^5*Ei(1,-I*d*(b*x+a)/b)*sin((a*d-b*c)/b)*a^4+3*x^2*sin(d*x+c)/b/d^2-a^2*x*cos(d*x+c)/b^3/d-2*a*x
*sin(d*x+c)/b^2/d^2+a^3*cos(d*x+c)/b^4/d+a^2*sin(d*x+c)/b^3/d^2+6*x*cos(d*x+c)/b/d^3-2*a*cos(d*x+c)/b^2/d^3-6*
sin(d*x+c)/b/d^4

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.86 \[ \int \frac {x^4 \sin (c+d x)}{a+b x} \, dx=-\frac {a^{4} d^{4} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) \sin \left (-\frac {b c - a d}{b}\right ) - a^{4} d^{4} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) + {\left (b^{4} d^{3} x^{3} - a b^{3} d^{3} x^{2} - a^{3} b d^{3} + 2 \, a b^{3} d + {\left (a^{2} b^{2} d^{3} - 6 \, b^{4} d\right )} x\right )} \cos \left (d x + c\right ) - {\left (3 \, b^{4} d^{2} x^{2} - 2 \, a b^{3} d^{2} x + a^{2} b^{2} d^{2} - 6 \, b^{4}\right )} \sin \left (d x + c\right )}{b^{5} d^{4}} \]

[In]

integrate(x^4*sin(d*x+c)/(b*x+a),x, algorithm="fricas")

[Out]

-(a^4*d^4*cos_integral((b*d*x + a*d)/b)*sin(-(b*c - a*d)/b) - a^4*d^4*cos(-(b*c - a*d)/b)*sin_integral((b*d*x
+ a*d)/b) + (b^4*d^3*x^3 - a*b^3*d^3*x^2 - a^3*b*d^3 + 2*a*b^3*d + (a^2*b^2*d^3 - 6*b^4*d)*x)*cos(d*x + c) - (
3*b^4*d^2*x^2 - 2*a*b^3*d^2*x + a^2*b^2*d^2 - 6*b^4)*sin(d*x + c))/(b^5*d^4)

Sympy [F]

\[ \int \frac {x^4 \sin (c+d x)}{a+b x} \, dx=\int \frac {x^{4} \sin {\left (c + d x \right )}}{a + b x}\, dx \]

[In]

integrate(x**4*sin(d*x+c)/(b*x+a),x)

[Out]

Integral(x**4*sin(c + d*x)/(a + b*x), x)

Maxima [F]

\[ \int \frac {x^4 \sin (c+d x)}{a+b x} \, dx=\int { \frac {x^{4} \sin \left (d x + c\right )}{b x + a} \,d x } \]

[In]

integrate(x^4*sin(d*x+c)/(b*x+a),x, algorithm="maxima")

[Out]

-1/2*(((6*a*b^2*(I*exp_integral_e(2, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^
2 + 6*a*b^2*(I*exp_integral_e(2, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*sin(c)^2 -
(a^3*(-I*exp_integral_e(2, (I*b*d*x + I*a*d)/b) + I*exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a^3*(-
I*exp_integral_e(2, (I*b*d*x + I*a*d)/b) + I*exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*d^2 - 4*(a^2*b
*(exp_integral_e(2, (I*b*d*x + I*a*d)/b) + exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a^2*b*(exp_inte
gral_e(2, (I*b*d*x + I*a*d)/b) + exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*d)*cos(-(b*c - a*d)/b) - (
6*a*b^2*(exp_integral_e(2, (I*b*d*x + I*a*d)/b) + exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + 6*a*b^2*
(exp_integral_e(2, (I*b*d*x + I*a*d)/b) + exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*sin(c)^2 + (a^3*(exp_integr
al_e(2, (I*b*d*x + I*a*d)/b) + exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a^3*(exp_integral_e(2, (I*b
*d*x + I*a*d)/b) + exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*d^2 - 4*(a^2*b*(-I*exp_integral_e(2, (I*
b*d*x + I*a*d)/b) + I*exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a^2*b*(-I*exp_integral_e(2, (I*b*d*x
+ I*a*d)/b) + I*exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*d)*sin(-(b*c - a*d)/b))*cos(d*x + c)^2 + (
(6*a*b^2*(I*exp_integral_e(2, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + 6*a
*b^2*(I*exp_integral_e(2, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*sin(c)^2 - (a^3*(-
I*exp_integral_e(2, (I*b*d*x + I*a*d)/b) + I*exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a^3*(-I*exp_i
ntegral_e(2, (I*b*d*x + I*a*d)/b) + I*exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*d^2 - 4*(a^2*b*(exp_i
ntegral_e(2, (I*b*d*x + I*a*d)/b) + exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a^2*b*(exp_integral_e(
2, (I*b*d*x + I*a*d)/b) + exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*d)*cos(-(b*c - a*d)/b) - (6*a*b^2
*(exp_integral_e(2, (I*b*d*x + I*a*d)/b) + exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + 6*a*b^2*(exp_in
tegral_e(2, (I*b*d*x + I*a*d)/b) + exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*sin(c)^2 + (a^3*(exp_integral_e(2,
(I*b*d*x + I*a*d)/b) + exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a^3*(exp_integral_e(2, (I*b*d*x +
I*a*d)/b) + exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*d^2 - 4*(a^2*b*(-I*exp_integral_e(2, (I*b*d*x +
I*a*d)/b) + I*exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a^2*b*(-I*exp_integral_e(2, (I*b*d*x + I*a*
d)/b) + I*exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*d)*sin(-(b*c - a*d)/b))*sin(d*x + c)^2 + ((b^3*d^
3*x^4*cos(c) + 3*b^3*d^2*x^3*sin(c) + (a*b^2*d^2*sin(c) - 6*b^3*d*cos(c))*x^2 - (a^2*b*d^2*sin(c) + 4*a*b^2*d*
cos(c) + 6*b^3*sin(c))*x)*cos(d*x + c)^2 + (b^3*d^3*x^4*cos(c) + 3*b^3*d^2*x^3*sin(c) + (a*b^2*d^2*sin(c) - 6*
b^3*d*cos(c))*x^2 - (a^2*b*d^2*sin(c) + 4*a*b^2*d*cos(c) + 6*b^3*sin(c))*x)*sin(d*x + c)^2)*cos(d*x + 2*c) + (
(b^3*cos(c)^2 + b^3*sin(c)^2)*d^3*x^4 - 6*(b^3*cos(c)^2 + b^3*sin(c)^2)*d*x^2 - 4*(a*b^2*cos(c)^2 + a*b^2*sin(
c)^2)*d*x)*cos(d*x + c) - 2*(((a^4*b^3*cos(c)^2 + a^4*b^3*sin(c)^2)*d^7 - 6*(a^2*b^5*cos(c)^2 + a^2*b^5*sin(c)
^2)*d^5 + ((a^3*b^4*cos(c)^2 + a^3*b^4*sin(c)^2)*d^7 - 6*(a*b^6*cos(c)^2 + a*b^6*sin(c)^2)*d^5)*x)*cos(d*x + c
)^2 + ((a^4*b^3*cos(c)^2 + a^4*b^3*sin(c)^2)*d^7 - 6*(a^2*b^5*cos(c)^2 + a^2*b^5*sin(c)^2)*d^5 + ((a^3*b^4*cos
(c)^2 + a^3*b^4*sin(c)^2)*d^7 - 6*(a*b^6*cos(c)^2 + a*b^6*sin(c)^2)*d^5)*x)*sin(d*x + c)^2)*integrate(1/2*x*co
s(d*x + c)/(b^4*d^4*x^2 + 2*a*b^3*d^4*x + a^2*b^2*d^4), x) - 2*(((a^4*b^3*cos(c)^2 + a^4*b^3*sin(c)^2)*d^7 - 6
*(a^2*b^5*cos(c)^2 + a^2*b^5*sin(c)^2)*d^5 + ((a^3*b^4*cos(c)^2 + a^3*b^4*sin(c)^2)*d^7 - 6*(a*b^6*cos(c)^2 +
a*b^6*sin(c)^2)*d^5)*x)*cos(d*x + c)^2 + ((a^4*b^3*cos(c)^2 + a^4*b^3*sin(c)^2)*d^7 - 6*(a^2*b^5*cos(c)^2 + a^
2*b^5*sin(c)^2)*d^5 + ((a^3*b^4*cos(c)^2 + a^3*b^4*sin(c)^2)*d^7 - 6*(a*b^6*cos(c)^2 + a*b^6*sin(c)^2)*d^5)*x)
*sin(d*x + c)^2)*integrate(1/2*x*cos(d*x + c)/((b^4*d^4*x^2 + 2*a*b^3*d^4*x + a^2*b^2*d^4)*cos(d*x + c)^2 + (b
^4*d^4*x^2 + 2*a*b^3*d^4*x + a^2*b^2*d^4)*sin(d*x + c)^2), x) - 4*(((a^2*b^5*cos(c)^2 + a^2*b^5*sin(c)^2)*d^6*
x + (a^3*b^4*cos(c)^2 + a^3*b^4*sin(c)^2)*d^6)*cos(d*x + c)^2 + ((a^2*b^5*cos(c)^2 + a^2*b^5*sin(c)^2)*d^6*x +
(a^3*b^4*cos(c)^2 + a^3*b^4*sin(c)^2)*d^6)*sin(d*x + c)^2)*integrate(1/2*x*sin(d*x + c)/(b^4*d^4*x^2 + 2*a*b^
3*d^4*x + a^2*b^2*d^4), x) - 4*(((a^2*b^5*cos(c)^2 + a^2*b^5*sin(c)^2)*d^6*x + (a^3*b^4*cos(c)^2 + a^3*b^4*sin
(c)^2)*d^6)*cos(d*x + c)^2 + ((a^2*b^5*cos(c)^2 + a^2*b^5*sin(c)^2)*d^6*x + (a^3*b^4*cos(c)^2 + a^3*b^4*sin(c)
^2)*d^6)*sin(d*x + c)^2)*integrate(1/2*x*sin(d*x + c)/((b^4*d^4*x^2 + 2*a*b^3*d^4*x + a^2*b^2*d^4)*cos(d*x + c
)^2 + (b^4*d^4*x^2 + 2*a*b^3*d^4*x + a^2*b^2*d^4)*sin(d*x + c)^2), x) + ((b^3*d^3*x^4*sin(c) - 3*b^3*d^2*x^3*c
os(c) - (a*b^2*d^2*cos(c) + 6*b^3*d*sin(c))*x^2 + (a^2*b*d^2*cos(c) - 4*a*b^2*d*sin(c) + 6*b^3*cos(c))*x)*cos(
d*x + c)^2 + (b^3*d^3*x^4*sin(c) - 3*b^3*d^2*x^3*cos(c) - (a*b^2*d^2*cos(c) + 6*b^3*d*sin(c))*x^2 + (a^2*b*d^2
*cos(c) - 4*a*b^2*d*sin(c) + 6*b^3*cos(c))*x)*sin(d*x + c)^2)*sin(d*x + 2*c) - (3*(b^3*cos(c)^2 + b^3*sin(c)^2
)*d^2*x^3 + (a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^2*x^2 - (6*b^3*cos(c)^2 + 6*b^3*sin(c)^2 + (a^2*b*cos(c)^2 + a
^2*b*sin(c)^2)*d^2)*x)*sin(d*x + c))/(((b^4*cos(c)^2 + b^4*sin(c)^2)*d^4*x + (a*b^3*cos(c)^2 + a*b^3*sin(c)^2)
*d^4)*cos(d*x + c)^2 + ((b^4*cos(c)^2 + b^4*sin(c)^2)*d^4*x + (a*b^3*cos(c)^2 + a*b^3*sin(c)^2)*d^4)*sin(d*x +
c)^2)

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.36 (sec) , antiderivative size = 3337, normalized size of antiderivative = 15.31 \[ \int \frac {x^4 \sin (c+d x)}{a+b x} \, dx=\text {Too large to display} \]

[In]

integrate(x^4*sin(d*x+c)/(b*x+a),x, algorithm="giac")

[Out]

1/2*(2*b^4*d^3*x^3*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a*b^3*d^3*x^2*tan(1/2*d*x + 1/2*c)
^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^4*d^4*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2
*c)^2*tan(1/2*a*d/b)^2 - a^4*d^4*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan
(1/2*a*d/b)^2 + 2*a^4*d^4*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 +
2*b^4*d^3*x^3*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + 2*a^4*d^4*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*
x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a^4*d^4*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c
)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*b^4*d^3*x^3*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a^4*d^4*real_part(
cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^4*d^4*real_part(cos_integr
al(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*b^4*d^3*x^3*tan(1/2*c)^2*tan(1/2*a*d/
b)^2 + 2*a^2*b^2*d^3*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a*b^3*d^3*x^2*tan(1/2*d*x + 1/
2*c)^2*tan(1/2*c)^2 - a^4*d^4*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + a^4*d
^4*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 - 2*a^4*d^4*sin_integral((b*d*x +
a*d)/b)*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + 4*a^4*d^4*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/
2*c)^2*tan(1/2*c)*tan(1/2*a*d/b) - 4*a^4*d^4*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(
1/2*c)*tan(1/2*a*d/b) + 8*a^4*d^4*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)*tan(1/2*a*d/
b) - 2*a*b^3*d^3*x^2*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 - a^4*d^4*imag_part(cos_integral(d*x + a*d/b))*ta
n(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 + a^4*d^4*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*t
an(1/2*a*d/b)^2 - 2*a^4*d^4*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*b^3*d^
3*x^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^4*d^4*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)
^2 - a^4*d^4*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^4*d^4*sin_integral((b*d
*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 12*b^4*d^2*x^2*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2*tan(1/2*a*d/b)^2
- 2*a^3*b*d^3*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^4*d^3*x^3*tan(1/2*d*x + 1/2*c)^2 + 2
*a^4*d^4*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c) + 2*a^4*d^4*real_part(cos_inte
gral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c) - 2*b^4*d^3*x^3*tan(1/2*c)^2 + 2*a^2*b^2*d^3*x*tan(1/2*d
*x + 1/2*c)^2*tan(1/2*c)^2 - 2*a^4*d^4*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d
/b) - 2*a^4*d^4*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b) + 2*a^4*d^4*real_p
art(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a^4*d^4*real_part(cos_integral(-d*x - a*d/b))*t
an(1/2*c)^2*tan(1/2*a*d/b) - 2*b^4*d^3*x^3*tan(1/2*a*d/b)^2 + 2*a^2*b^2*d^3*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a
*d/b)^2 - 2*a^4*d^4*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^4*d^4*real_part(cos
_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^2*b^2*d^3*x*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 8*a*b^3
*d^2*x*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 12*b^4*d*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan
(1/2*a*d/b)^2 - 2*a*b^3*d^3*x^2*tan(1/2*d*x + 1/2*c)^2 + a^4*d^4*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*
d*x + 1/2*c)^2 - a^4*d^4*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2 + 2*a^4*d^4*sin_integral
((b*d*x + a*d)/b)*tan(1/2*d*x + 1/2*c)^2 + 2*a*b^3*d^3*x^2*tan(1/2*c)^2 - a^4*d^4*imag_part(cos_integral(d*x +
a*d/b))*tan(1/2*c)^2 + a^4*d^4*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 - 2*a^4*d^4*sin_integral((b
*d*x + a*d)/b)*tan(1/2*c)^2 + 12*b^4*d^2*x^2*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2 - 2*a^3*b*d^3*tan(1/2*d*x + 1/2
*c)^2*tan(1/2*c)^2 + 4*a^4*d^4*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 4*a^4*d^4*imag
_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 8*a^4*d^4*sin_integral((b*d*x + a*d)/b)*tan(1/2*
c)*tan(1/2*a*d/b) + 2*a*b^3*d^3*x^2*tan(1/2*a*d/b)^2 - a^4*d^4*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*
d/b)^2 + a^4*d^4*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 - 2*a^4*d^4*sin_integral((b*d*x + a*d)
/b)*tan(1/2*a*d/b)^2 + 12*b^4*d^2*x^2*tan(1/2*d*x + 1/2*c)*tan(1/2*a*d/b)^2 - 2*a^3*b*d^3*tan(1/2*d*x + 1/2*c)
^2*tan(1/2*a*d/b)^2 + 2*a^3*b*d^3*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 4*a^2*b^2*d^2*tan(1/2*d*x + 1/2*c)*tan(1/2*c
)^2*tan(1/2*a*d/b)^2 + 4*a*b^3*d*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*b^4*d^3*x^3 + 2*a^2*
b^2*d^3*x*tan(1/2*d*x + 1/2*c)^2 + 2*a^4*d^4*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c) + 2*a^4*d^4*real_
part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 2*a^2*b^2*d^3*x*tan(1/2*c)^2 - 8*a*b^3*d^2*x*tan(1/2*d*x + 1/2*c
)*tan(1/2*c)^2 - 12*b^4*d*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 - 2*a^4*d^4*real_part(cos_integral(d*x + a*d/b
))*tan(1/2*a*d/b) - 2*a^4*d^4*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) - 2*a^2*b^2*d^3*x*tan(1/2*a
*d/b)^2 - 8*a*b^3*d^2*x*tan(1/2*d*x + 1/2*c)*tan(1/2*a*d/b)^2 - 12*b^4*d*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/
b)^2 + 12*b^4*d*x*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*b^3*d^3*x^2 + a^4*d^4*imag_part(cos_integral(d*x + a*d/b
)) - a^4*d^4*imag_part(cos_integral(-d*x - a*d/b)) + 2*a^4*d^4*sin_integral((b*d*x + a*d)/b) + 12*b^4*d^2*x^2*
tan(1/2*d*x + 1/2*c) - 2*a^3*b*d^3*tan(1/2*d*x + 1/2*c)^2 + 2*a^3*b*d^3*tan(1/2*c)^2 + 4*a^2*b^2*d^2*tan(1/2*d
*x + 1/2*c)*tan(1/2*c)^2 + 4*a*b^3*d*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + 2*a^3*b*d^3*tan(1/2*a*d/b)^2 + 4*a^
2*b^2*d^2*tan(1/2*d*x + 1/2*c)*tan(1/2*a*d/b)^2 + 4*a*b^3*d*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 - 4*a*b^3*
d*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 24*b^4*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a^2*b^2*d^3*x
- 8*a*b^3*d^2*x*tan(1/2*d*x + 1/2*c) - 12*b^4*d*x*tan(1/2*d*x + 1/2*c)^2 + 12*b^4*d*x*tan(1/2*c)^2 + 12*b^4*d*
x*tan(1/2*a*d/b)^2 + 2*a^3*b*d^3 + 4*a^2*b^2*d^2*tan(1/2*d*x + 1/2*c) + 4*a*b^3*d*tan(1/2*d*x + 1/2*c)^2 - 4*a
*b^3*d*tan(1/2*c)^2 - 24*b^4*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2 - 4*a*b^3*d*tan(1/2*a*d/b)^2 - 24*b^4*tan(1/2*d
*x + 1/2*c)*tan(1/2*a*d/b)^2 + 12*b^4*d*x - 4*a*b^3*d - 24*b^4*tan(1/2*d*x + 1/2*c))/(b^5*d^4*tan(1/2*d*x + 1/
2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^5*d^4*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + b^5*d^4*tan(1/2*d*x + 1/2
*c)^2*tan(1/2*a*d/b)^2 + b^5*d^4*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^5*d^4*tan(1/2*d*x + 1/2*c)^2 + b^5*d^4*tan(
1/2*c)^2 + b^5*d^4*tan(1/2*a*d/b)^2 + b^5*d^4)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \sin (c+d x)}{a+b x} \, dx=\int \frac {x^4\,\sin \left (c+d\,x\right )}{a+b\,x} \,d x \]

[In]

int((x^4*sin(c + d*x))/(a + b*x),x)

[Out]

int((x^4*sin(c + d*x))/(a + b*x), x)