3.1.19 \(\int \frac {x^3 \sin (c+d x)}{a+b x} \, dx\) [19]

Optimal. Leaf size=152 \[ \frac {2 \cos (c+d x)}{b d^3}-\frac {a^2 \cos (c+d x)}{b^3 d}+\frac {a x \cos (c+d x)}{b^2 d}-\frac {x^2 \cos (c+d x)}{b d}-\frac {a^3 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^4}-\frac {a \sin (c+d x)}{b^2 d^2}+\frac {2 x \sin (c+d x)}{b d^2}-\frac {a^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^4} \]

[Out]

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

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Rubi [A]
time = 0.22, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {6874, 2718, 3377, 2717, 3384, 3380, 3383} \begin {gather*} -\frac {a^3 \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {a^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^4}-\frac {a^2 \cos (c+d x)}{b^3 d}-\frac {a \sin (c+d x)}{b^2 d^2}+\frac {a x \cos (c+d x)}{b^2 d}+\frac {2 \cos (c+d x)}{b d^3}+\frac {2 x \sin (c+d x)}{b d^2}-\frac {x^2 \cos (c+d x)}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*Sin[c + d*x])/(a + b*x),x]

[Out]

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

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*} \int \frac {x^3 \sin (c+d x)}{a+b x} \, dx &=\int \left (\frac {a^2 \sin (c+d x)}{b^3}-\frac {a x \sin (c+d x)}{b^2}+\frac {x^2 \sin (c+d x)}{b}-\frac {a^3 \sin (c+d x)}{b^3 (a+b x)}\right ) \, dx\\ &=\frac {a^2 \int \sin (c+d x) \, dx}{b^3}-\frac {a^3 \int \frac {\sin (c+d x)}{a+b x} \, dx}{b^3}-\frac {a \int x \sin (c+d x) \, dx}{b^2}+\frac {\int x^2 \sin (c+d x) \, dx}{b}\\ &=-\frac {a^2 \cos (c+d x)}{b^3 d}+\frac {a x \cos (c+d x)}{b^2 d}-\frac {x^2 \cos (c+d x)}{b d}-\frac {a \int \cos (c+d x) \, dx}{b^2 d}+\frac {2 \int x \cos (c+d x) \, dx}{b d}-\frac {\left (a^3 \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^3}-\frac {\left (a^3 \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^3}\\ &=-\frac {a^2 \cos (c+d x)}{b^3 d}+\frac {a x \cos (c+d x)}{b^2 d}-\frac {x^2 \cos (c+d x)}{b d}-\frac {a^3 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^4}-\frac {a \sin (c+d x)}{b^2 d^2}+\frac {2 x \sin (c+d x)}{b d^2}-\frac {a^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {2 \int \sin (c+d x) \, dx}{b d^2}\\ &=\frac {2 \cos (c+d x)}{b d^3}-\frac {a^2 \cos (c+d x)}{b^3 d}+\frac {a x \cos (c+d x)}{b^2 d}-\frac {x^2 \cos (c+d x)}{b d}-\frac {a^3 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^4}-\frac {a \sin (c+d x)}{b^2 d^2}+\frac {2 x \sin (c+d x)}{b d^2}-\frac {a^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^4}\\ \end {align*}

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Mathematica [A]
time = 0.35, size = 117, normalized size = 0.77 \begin {gather*} -\frac {a^3 d^3 \text {Ci}\left (d \left (\frac {a}{b}+x\right )\right ) \sin \left (c-\frac {a d}{b}\right )+b \left (\left (a^2 d^2-a b d^2 x+b^2 \left (-2+d^2 x^2\right )\right ) \cos (c+d x)+b d (a-2 b x) \sin (c+d x)\right )+a^3 d^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{b^4 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^3*Sin[c + d*x])/(a + b*x),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(515\) vs. \(2(153)=306\).
time = 0.07, size = 516, normalized size = 3.39

method	result	size
derivativedivides	\(\frac {-d \,c^{3} \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )-\frac {3 \left (d a -c b \right ) d \,c^{2} \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b}-\frac {3 d \,c^{2} \cos \left (d x +c \right )}{b}-\frac {3 \left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}\right ) d c \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{2}}+\frac {3 \left (d a -c b -b \right ) d c \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{b^{2}}-\frac {\left (d^{3} a^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) d \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{3}}+\frac {\left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}-a b d +b^{2} c +b^{2}\right ) d \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{b^{3}}}{d^{4}}\)	\(516\)
default	\(\frac {-d \,c^{3} \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )-\frac {3 \left (d a -c b \right ) d \,c^{2} \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b}-\frac {3 d \,c^{2} \cos \left (d x +c \right )}{b}-\frac {3 \left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}\right ) d c \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{2}}+\frac {3 \left (d a -c b -b \right ) d c \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{b^{2}}-\frac {\left (d^{3} a^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) d \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{3}}+\frac {\left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}-a b d +b^{2} c +b^{2}\right ) d \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{b^{3}}}{d^{4}}\)	\(516\)
risch	\(\frac {i \left (2 i b^{4} d^{4} x^{4}-4 i a \,b^{3} d^{4} x^{3}+6 i b^{4} c \,d^{3} x^{3}+6 i a^{2} b^{2} d^{4} x^{2}-12 i a \,b^{3} c \,d^{3} x^{2}+6 i b^{4} c^{2} d^{2} x^{2}-4 i a^{3} b \,d^{4} x +12 i a^{2} b^{2} c \,d^{3} x -6 i a \,b^{3} c^{2} d^{2} x +2 i a^{4} d^{4}-6 i a^{3} b c \,d^{3}+6 i a^{2} b^{2} c^{2} d^{2}-4 i b^{4} d^{2} x^{2}+4 i a \,b^{3} d^{2} x -12 i b^{4} c d x -4 i a^{2} b^{2} d^{2}+12 i a \,b^{3} c d -12 i b^{4} c^{2}\right ) \cos \left (d x +c \right )}{2 d^{3} \left (d^{2} x^{2} b^{2}-a b \,d^{2} x +3 b^{2} c d x +d^{2} a^{2}-3 a b c d +3 b^{2} c^{2}\right ) b^{3}}+\frac {\left (4 d^{3} x^{3} b^{3}-6 a \,b^{2} d^{3} x^{2}+12 b^{3} c \,d^{2} x^{2}+6 a^{2} b \,d^{3} x -18 a \,b^{2} c \,d^{2} x +12 b^{3} c^{2} d x -2 d^{3} a^{3}+6 a^{2} b c \,d^{2}-6 a \,b^{2} c^{2} d \right ) \sin \left (d x +c \right )}{2 d^{3} b^{2} \left (d^{2} x^{2} b^{2}-a b \,d^{2} x +3 b^{2} c d x +d^{2} a^{2}-3 a b c d +3 b^{2} c^{2}\right )}+\frac {i a^{3} \cos \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, \frac {i d \left (b x +a \right )}{b}\right )}{2 b^{4}}-\frac {i a^{3} \cos \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, -\frac {i d \left (b x +a \right )}{b}\right )}{2 b^{4}}-\frac {a^{3} \sin \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, \frac {i d \left (b x +a \right )}{b}\right )}{2 b^{4}}-\frac {a^{3} \sin \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, -\frac {i d \left (b x +a \right )}{b}\right )}{2 b^{4}}\)	\(587\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*sin(d*x+c)/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*sin(d*x+c)/(b*x+a),x, algorithm="maxima")

[Out]

1/2*(((2*a*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 + 2*a
*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 - (a^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 + a^2*(-I*exp_integra
l_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)*cos(-(b*c - a*d)/b) + (2
*a*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 + 2*a*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 - (a^2*(exp_int
egral_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*(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)*sin(-(b*c - a*d)/b))*cos(d*x + c)^
2 + ((2*a*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 + 2*a*
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 - (a^2*(-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))*cos(c)^2 + a^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)*d)*cos(-(b*c - a*d)/b) + (2*
a*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 + 2*a*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 - (a^2*(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))*cos(c)^2 + a^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)*d)*sin(-(b*c - a*d)/b))*sin(d*x + c)^2
- ((b^2*d^2*x^3*cos(c) + 2*b^2*d*x^2*sin(c) + (a*b*d*sin(c) - 2*b^2*cos(c))*x)*cos(d*x + c)^2 + (b^2*d^2*x^3*
cos(c) + 2*b^2*d*x^2*sin(c) + (a*b*d*sin(c) - 2*b^2*cos(c))*x)*sin(d*x + c)^2)*cos(d*x + 2*c) - ((b^2*cos(c)^2
+ b^2*sin(c)^2)*d^2*x^3 - 2*(b^2*cos(c)^2 + b^2*sin(c)^2)*x)*cos(d*x + c) - 2*(((a^2*b^3*cos(c)^2 + a^2*b^3*s
in(c)^2)*d^5*x + (a^3*b^2*cos(c)^2 + a^3*b^2*sin(c)^2)*d^5)*cos(d*x + c)^2 + ((a^2*b^3*cos(c)^2 + a^2*b^3*sin(
c)^2)*d^5*x + (a^3*b^2*cos(c)^2 + a^3*b^2*sin(c)^2)*d^5)*sin(d*x + c)^2)*integrate(1/2*x*cos(d*x + c)/(b^3*d^3
*x^2 + 2*a*b^2*d^3*x + a^2*b*d^3), x) - 2*(((a^2*b^3*cos(c)^2 + a^2*b^3*sin(c)^2)*d^5*x + (a^3*b^2*cos(c)^2 +
a^3*b^2*sin(c)^2)*d^5)*cos(d*x + c)^2 + ((a^2*b^3*cos(c)^2 + a^2*b^3*sin(c)^2)*d^5*x + (a^3*b^2*cos(c)^2 + a^3
*b^2*sin(c)^2)*d^5)*sin(d*x + c)^2)*integrate(1/2*x*cos(d*x + c)/((b^3*d^3*x^2 + 2*a*b^2*d^3*x + a^2*b*d^3)*co
s(d*x + c)^2 + (b^3*d^3*x^2 + 2*a*b^2*d^3*x + a^2*b*d^3)*sin(d*x + c)^2), x) - 4*(((a*b^4*cos(c)^2 + a*b^4*sin
(c)^2)*d^4*x + (a^2*b^3*cos(c)^2 + a^2*b^3*sin(c)^2)*d^4)*cos(d*x + c)^2 + ((a*b^4*cos(c)^2 + a*b^4*sin(c)^2)*
d^4*x + (a^2*b^3*cos(c)^2 + a^2*b^3*sin(c)^2)*d^4)*sin(d*x + c)^2)*integrate(1/2*x*sin(d*x + c)/(b^3*d^3*x^2 +
2*a*b^2*d^3*x + a^2*b*d^3), x) - 4*(((a*b^4*cos(c)^2 + a*b^4*sin(c)^2)*d^4*x + (a^2*b^3*cos(c)^2 + a^2*b^3*si
n(c)^2)*d^4)*cos(d*x + c)^2 + ((a*b^4*cos(c)^2 + a*b^4*sin(c)^2)*d^4*x + (a^2*b^3*cos(c)^2 + a^2*b^3*sin(c)^2)
*d^4)*sin(d*x + c)^2)*integrate(1/2*x*sin(d*x + c)/((b^3*d^3*x^2 + 2*a*b^2*d^3*x + a^2*b*d^3)*cos(d*x + c)^2 +
(b^3*d^3*x^2 + 2*a*b^2*d^3*x + a^2*b*d^3)*sin(d*x + c)^2), x) - ((b^2*d^2*x^3*sin(c) - 2*b^2*d*x^2*cos(c) - (
a*b*d*cos(c) + 2*b^2*sin(c))*x)*cos(d*x + c)^2 + (b^2*d^2*x^3*sin(c) - 2*b^2*d*x^2*cos(c) - (a*b*d*cos(c) + 2*
b^2*sin(c))*x)*sin(d*x + c)^2)*sin(d*x + 2*c) + (2*(b^2*cos(c)^2 + b^2*sin(c)^2)*d*x^2 + (a*b*cos(c)^2 + a*b*s
in(c)^2)*d*x)*sin(d*x + c))/(((b^3*cos(c)^2 + b^3*sin(c)^2)*d^3*x + (a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^3)*cos
(d*x + c)^2 + ((b^3*cos(c)^2 + b^3*sin(c)^2)*d^3*x + (a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^3)*sin(d*x + c)^2)

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Fricas [A]
time = 0.35, size = 167, normalized size = 1.10 \begin {gather*} -\frac {2 \, a^{3} d^{3} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) + 2 \, {\left (b^{3} d^{2} x^{2} - a b^{2} d^{2} x + a^{2} b d^{2} - 2 \, b^{3}\right )} \cos \left (d x + c\right ) - 2 \, {\left (2 \, b^{3} d x - a b^{2} d\right )} \sin \left (d x + c\right ) - {\left (a^{3} d^{3} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + a^{3} d^{3} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{2 \, b^{4} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*sin(d*x+c)/(b*x+a),x, algorithm="fricas")

[Out]

-1/2*(2*a^3*d^3*cos(-(b*c - a*d)/b)*sin_integral((b*d*x + a*d)/b) + 2*(b^3*d^2*x^2 - a*b^2*d^2*x + a^2*b*d^2 -
2*b^3)*cos(d*x + c) - 2*(2*b^3*d*x - a*b^2*d)*sin(d*x + c) - (a^3*d^3*cos_integral((b*d*x + a*d)/b) + a^3*d^3
*cos_integral(-(b*d*x + a*d)/b))*sin(-(b*c - a*d)/b))/(b^4*d^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \sin {\left (c + d x \right )}}{a + b x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*sin(d*x+c)/(b*x+a),x)

[Out]

Integral(x**3*sin(c + d*x)/(a + b*x), x)

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Giac [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 3.55, size = 2709, normalized size = 17.82 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*sin(d*x+c)/(b*x+a),x, algorithm="giac")

[Out]

1/2*(2*b^3*d^2*x^2*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^3*d^3*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^3*d^3*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^3*d^3*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*a^3*d^3*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^3*d^3*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^3*d^3*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^3*d^3*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*b^2*d^2*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^3*d^2*x^2*tan(1/2*d*x + 1/2*c)^2*ta
n(1/2*c)^2 + a^3*d^3*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 - a^3*d^3*imag_p
art(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + 2*a^3*d^3*sin_integral((b*d*x + a*d)/b)*
tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 - 4*a^3*d^3*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*ta
n(1/2*c)*tan(1/2*a*d/b) + 4*a^3*d^3*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)*ta
n(1/2*a*d/b) - 8*a^3*d^3*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*b^
3*d^2*x^2*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 + a^3*d^3*imag_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^3*d^3*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/
b)^2 + 2*a^3*d^3*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*b^3*d^2*x^2*tan(1/2
*c)^2*tan(1/2*a*d/b)^2 - a^3*d^3*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^3*d^3*
imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a^3*d^3*sin_integral((b*d*x + a*d)/b)*
tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*b*d^2*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a^3*d^3*r
eal_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c) - 2*a^3*d^3*real_part(cos_integral(-d*x
- a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c) - 2*a*b^2*d^2*x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + 2*a^3*d^3*re
al_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^3*d^3*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^3*d^3*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)
^2*tan(1/2*a*d/b) - 2*a^3*d^3*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) - 2*a*b^2*d^2*
x*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^3*d^3*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*
a*d/b)^2 + 2*a^3*d^3*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*a*b^2*d^2*x*tan(1/2
*c)^2*tan(1/2*a*d/b)^2 + 8*b^3*d*x*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^3*d^2*x^2*tan(1/2*
d*x + 1/2*c)^2 - a^3*d^3*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2 + a^3*d^3*imag_part(cos_i
ntegral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2 - 2*a^3*d^3*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x + 1/2*c)^2
- 2*b^3*d^2*x^2*tan(1/2*c)^2 + a^3*d^3*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 - a^3*d^3*imag_part(
cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 + 2*a^3*d^3*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2 + 2*a^2*b*d^2*
tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 - 4*a^3*d^3*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)
+ 4*a^3*d^3*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 8*a^3*d^3*sin_integral((b*d*x +
a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b) - 2*b^3*d^2*x^2*tan(1/2*a*d/b)^2 + a^3*d^3*imag_part(cos_integral(d*x + a*d
/b))*tan(1/2*a*d/b)^2 - a^3*d^3*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 + 2*a^3*d^3*sin_integra
l((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 + 2*a^2*b*d^2*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a^2*b*d^2*tan(1/
2*c)^2*tan(1/2*a*d/b)^2 - 4*a*b^2*d*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 4*b^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^2*d^2*x*tan(1/2*d*x + 1/2*c)^2 - 2*a^3*d^3*real_part(cos_integra
l(d*x + a*d/b))*tan(1/2*c) - 2*a^3*d^3*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) + 2*a*b^2*d^2*x*tan(1/
2*c)^2 + 8*b^3*d*x*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2 + 2*a^3*d^3*real_part(cos_integral(d*x + a*d/b))*tan(1/2*
a*d/b) + 2*a^3*d^3*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) + 2*a*b^2*d^2*x*tan(1/2*a*d/b)^2 + 8*b
^3*d*x*tan(1/2*d*x + 1/2*c)*tan(1/2*a*d/b)^2 - 2*b^3*d^2*x^2 - a^3*d^3*imag_part(cos_integral(d*x + a*d/b)) +
a^3*d^3*imag_part(cos_integral(-d*x - a*d/b)) - 2*a^3*d^3*sin_integral((b*d*x + a*d)/b) + 2*a^2*b*d^2*tan(1/2*
d*x + 1/2*c)^2 - 2*a^2*b*d^2*tan(1/2*c)^2 - 4*a*b^2*d*tan(1/2*d*x + 1/2*c)*tan(1/2*c)^2 - 4*b^3*tan(1/2*d*x +
1/2*c)^2*tan(1/2*c)^2 - 2*a^2*b*d^2*tan(1/2*a*d/b)^2 - 4*a*b^2*d*tan(1/2*d*x + 1/2*c)*tan(1/2*a*d/b)^2 - 4*b^3
*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a*d/b)^2 + 4*b^...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3\,\sin \left (c+d\,x\right )}{a+b\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*sin(c + d*x))/(a + b*x),x)

[Out]

int((x^3*sin(c + d*x))/(a + b*x), x)

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