3.1.33 \(\int \frac {x^3 \sin (c+d x)}{(a+b x)^3} \, dx\) [33]

Optimal. Leaf size=265 \[ -\frac {\cos (c+d x)}{b^3 d}+\frac {a^3 d \cos (c+d x)}{2 b^5 (a+b x)}+\frac {3 a^2 d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {3 a \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^4}+\frac {a^3 d^2 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{2 b^6}+\frac {a^3 \sin (c+d x)}{2 b^4 (a+b x)^2}-\frac {3 a^2 \sin (c+d x)}{b^4 (a+b x)}-\frac {3 a \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^4}+\frac {a^3 d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{2 b^6}-\frac {3 a^2 d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^5} \]

[Out]

3*a^2*d*Ci(a*d/b+d*x)*cos(-c+a*d/b)/b^5-cos(d*x+c)/b^3/d+1/2*a^3*d*cos(d*x+c)/b^5/(b*x+a)-3*a*cos(-c+a*d/b)*Si
(a*d/b+d*x)/b^4+1/2*a^3*d^2*cos(-c+a*d/b)*Si(a*d/b+d*x)/b^6+3*a*Ci(a*d/b+d*x)*sin(-c+a*d/b)/b^4-1/2*a^3*d^2*Ci
(a*d/b+d*x)*sin(-c+a*d/b)/b^6+3*a^2*d*Si(a*d/b+d*x)*sin(-c+a*d/b)/b^5+1/2*a^3*sin(d*x+c)/b^4/(b*x+a)^2-3*a^2*s
in(d*x+c)/b^4/(b*x+a)

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Rubi [A]
time = 0.42, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {6874, 2718, 3378, 3384, 3380, 3383} \begin {gather*} \frac {a^3 d^2 \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{2 b^6}+\frac {a^3 d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{2 b^6}+\frac {a^3 d \cos (c+d x)}{2 b^5 (a+b x)}+\frac {a^3 \sin (c+d x)}{2 b^4 (a+b x)^2}+\frac {3 a^2 d \cos \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {3 a^2 d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^5}-\frac {3 a^2 \sin (c+d x)}{b^4 (a+b x)}-\frac {3 a \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {3 a \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^4}-\frac {\cos (c+d x)}{b^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*Sin[c + d*x])/(a + b*x)^3,x]

[Out]

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

Rule 2718

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

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

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Mathematica [A]
time = 0.64, size = 235, normalized size = 0.89 \begin {gather*} -\frac {b \cos (d x) \left (-\left ((a+b x) \left (-2 a b^2+a^3 d^2-2 b^3 x\right ) \cos (c)\right )+a^2 b d (5 a+6 b x) \sin (c)\right )+b \left (a^2 b d (5 a+6 b x) \cos (c)+(a+b x) \left (-2 a b^2+a^3 d^2-2 b^3 x\right ) \sin (c)\right ) \sin (d x)-a d (a+b x)^2 \left (\text {Ci}\left (d \left (\frac {a}{b}+x\right )\right ) \left (6 a b d \cos \left (c-\frac {a d}{b}\right )+\left (-6 b^2+a^2 d^2\right ) \sin \left (c-\frac {a d}{b}\right )\right )+\left (\left (-6 b^2+a^2 d^2\right ) \cos \left (c-\frac {a d}{b}\right )-6 a b d \sin \left (c-\frac {a d}{b}\right )\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )\right )}{2 b^6 d (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(b*Cos[d*x]*(-((a + b*x)*(-2*a*b^2 + a^3*d^2 - 2*b^3*x)*Cos[c]) + a^2*b*d*(5*a + 6*b*x)*Sin[c]) + b*(a^2*
b*d*(5*a + 6*b*x)*Cos[c] + (a + b*x)*(-2*a*b^2 + a^3*d^2 - 2*b^3*x)*Sin[c])*Sin[d*x] - a*d*(a + b*x)^2*(CosInt
egral[d*(a/b + x)]*(6*a*b*d*Cos[c - (a*d)/b] + (-6*b^2 + a^2*d^2)*Sin[c - (a*d)/b]) + ((-6*b^2 + a^2*d^2)*Cos[
c - (a*d)/b] - 6*a*b*d*Sin[c - (a*d)/b])*SinIntegral[d*(a/b + x)]))/(b^6*d*(a + b*x)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1207\) vs. \(2(263)=526\).
time = 0.24, size = 1208, normalized size = 4.56

method	result	size
risch	\(\frac {i \left (2 i a^{3} b^{3} d^{6} x^{3}+6 i a^{4} b^{2} d^{6} x^{2}-4 i b^{6} d^{4} x^{4}+6 i a^{5} b \,d^{6} x -16 i a \,b^{5} d^{4} x^{3}+2 i a^{6} d^{6}-24 i a^{2} b^{4} d^{4} x^{2}-16 i a^{3} b^{3} d^{4} x -4 i a^{4} b^{2} d^{4}\right ) \cos \left (d x +c \right )}{4 b^{5} d^{3} \left (b x +a \right )^{2} \left (-d^{2} x^{2} b^{2}-2 a b \,d^{2} x -d^{2} a^{2}\right )}+\frac {\left (12 a^{2} b^{4} d^{5} x^{3}+34 a^{3} b^{3} d^{5} x^{2}+32 a^{4} b^{2} d^{5} x +10 a^{5} b \,d^{5}\right ) \sin \left (d x +c \right )}{4 b^{5} d^{3} \left (b x +a \right )^{2} \left (-d^{2} x^{2} b^{2}-2 a b \,d^{2} x -d^{2} a^{2}\right )}-\frac {3 d \cos \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, -\frac {i d \left (b x +a \right )}{b}\right ) a^{2}}{2 b^{5}}+\frac {i d^{2} \cos \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, -\frac {i d \left (b x +a \right )}{b}\right ) a^{3}}{4 b^{6}}-\frac {3 d \cos \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, \frac {i d \left (b x +a \right )}{b}\right ) a^{2}}{2 b^{5}}-\frac {i d^{2} \cos \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, \frac {i d \left (b x +a \right )}{b}\right ) a^{3}}{4 b^{6}}-\frac {3 i \cos \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, -\frac {i d \left (b x +a \right )}{b}\right ) a}{2 b^{4}}+\frac {3 i \cos \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, \frac {i d \left (b x +a \right )}{b}\right ) a}{2 b^{4}}+\frac {3 i d \sin \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, -\frac {i d \left (b x +a \right )}{b}\right ) a^{2}}{2 b^{5}}+\frac {d^{2} \sin \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, -\frac {i d \left (b x +a \right )}{b}\right ) a^{3}}{4 b^{6}}-\frac {3 i d \sin \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, \frac {i d \left (b x +a \right )}{b}\right ) a^{2}}{2 b^{5}}+\frac {d^{2} \sin \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, \frac {i d \left (b x +a \right )}{b}\right ) a^{3}}{4 b^{6}}-\frac {3 \sin \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, -\frac {i d \left (b x +a \right )}{b}\right ) a}{2 b^{4}}-\frac {3 \sin \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, \frac {i d \left (b x +a \right )}{b}\right ) a}{2 b^{4}}\)	\(705\)
derivativedivides	\(\text {Expression too large to display}\)	\(1208\)
default	\(\text {Expression too large to display}\)	\(1208\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^4*(-d^3*c^3*(-1/2*sin(d*x+c)/(d*a-c*b+b*(d*x+c))^2/b+1/2*(-cos(d*x+c)/(d*a-c*b+b*(d*x+c))/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)/b)/b)+3*d^3*c^2/b*(-sin(d*x+c)/(d*a-c*b+
b*(d*x+c))/b+(Si(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b+Ci(d*x+c+(a*d-b*c)/b)*cos((a*d-b*c)/b)/b)/b)-3*d^3*c^2*
(a*d-b*c)/b*(-1/2*sin(d*x+c)/(d*a-c*b+b*(d*x+c))^2/b+1/2*(-cos(d*x+c)/(d*a-c*b+b*(d*x+c))/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)/b)/b)+6*d^3*c*(a*d-b*c)/b^2*(-sin(d*x+c)/(d*
a-c*b+b*(d*x+c))/b+(Si(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b+Ci(d*x+c+(a*d-b*c)/b)*cos((a*d-b*c)/b)/b)/b)-3*d^
3*c*(a*d-b*c)^2/b^2*(-1/2*sin(d*x+c)/(d*a-c*b+b*(d*x+c))^2/b+1/2*(-cos(d*x+c)/(d*a-c*b+b*(d*x+c))/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)/b)/b)-3*d^3*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/b^3*(a^2*d^2-2*a*b*c*d+b^2*c^2)*d^3*(-sin(
d*x+c)/(d*a-c*b+b*(d*x+c))/b+(Si(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b+Ci(d*x+c+(a*d-b*c)/b)*cos((a*d-b*c)/b)/
b)/b)-(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*d^3/b^3*(-1/2*sin(d*x+c)/(d*a-c*b+b*(d*x+c))^2/b+1/2*(-cos
(d*x+c)/(d*a-c*b+b*(d*x+c))/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)/b)/b)-3/b^3*(a*d-b*c)*d^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)-d^3/b^3*cos(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)^3,x, algorithm="maxima")

[Out]

-1/2*((b^2*cos(c)^2 + b^2*sin(c)^2)*d*x^3*cos(d*x + c) + 3*((a^2*(-I*exp_integral_e(4, (I*b*d*x + I*a*d)/b) +
I*exp_integral_e(4, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a^2*(-I*exp_integral_e(4, (I*b*d*x + I*a*d)/b) + I*exp_i
ntegral_e(4, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*cos(-(b*c - a*d)/b) + (a^2*(exp_integral_e(4, (I*b*d*x + I*a*d)/
b) + exp_integral_e(4, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a^2*(exp_integral_e(4, (I*b*d*x + I*a*d)/b) + exp_int
egral_e(4, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*sin(-(b*c - a*d)/b))*cos(d*x + c)^2 - 3*(a*b*cos(c)^2 + a*b*sin(c)
^2)*x*sin(d*x + c) + 3*((a^2*(-I*exp_integral_e(4, (I*b*d*x + I*a*d)/b) + I*exp_integral_e(4, -(I*b*d*x + I*a*
d)/b))*cos(c)^2 + a^2*(-I*exp_integral_e(4, (I*b*d*x + I*a*d)/b) + I*exp_integral_e(4, -(I*b*d*x + I*a*d)/b))*
sin(c)^2)*cos(-(b*c - a*d)/b) + (a^2*(exp_integral_e(4, (I*b*d*x + I*a*d)/b) + exp_integral_e(4, -(I*b*d*x + I
*a*d)/b))*cos(c)^2 + a^2*(exp_integral_e(4, (I*b*d*x + I*a*d)/b) + exp_integral_e(4, -(I*b*d*x + I*a*d)/b))*si
n(c)^2)*sin(-(b*c - a*d)/b))*sin(d*x + c)^2 + ((b^2*d*x^3*cos(c) + 3*a*b*x*sin(c))*cos(d*x + c)^2 + (b^2*d*x^3
*cos(c) + 3*a*b*x*sin(c))*sin(d*x + c)^2)*cos(d*x + 2*c) + 6*(((a^2*b^5*cos(c)^2 + a^2*b^5*sin(c)^2)*d^3*x^3 +
3*(a^3*b^4*cos(c)^2 + a^3*b^4*sin(c)^2)*d^3*x^2 + 3*(a^4*b^3*cos(c)^2 + a^4*b^3*sin(c)^2)*d^3*x + (a^5*b^2*co
s(c)^2 + a^5*b^2*sin(c)^2)*d^3)*cos(d*x + c)^2 + ((a^2*b^5*cos(c)^2 + a^2*b^5*sin(c)^2)*d^3*x^3 + 3*(a^3*b^4*c
os(c)^2 + a^3*b^4*sin(c)^2)*d^3*x^2 + 3*(a^4*b^3*cos(c)^2 + a^4*b^3*sin(c)^2)*d^3*x + (a^5*b^2*cos(c)^2 + a^5*
b^2*sin(c)^2)*d^3)*sin(d*x + c)^2)*integrate(1/2*x*cos(d*x + c)/(b^5*d^2*x^4 + 4*a*b^4*d^2*x^3 + 6*a^2*b^3*d^2
*x^2 + 4*a^3*b^2*d^2*x + a^4*b*d^2), x) + 6*(((a^2*b^5*cos(c)^2 + a^2*b^5*sin(c)^2)*d^3*x^3 + 3*(a^3*b^4*cos(c
)^2 + a^3*b^4*sin(c)^2)*d^3*x^2 + 3*(a^4*b^3*cos(c)^2 + a^4*b^3*sin(c)^2)*d^3*x + (a^5*b^2*cos(c)^2 + a^5*b^2*
sin(c)^2)*d^3)*cos(d*x + c)^2 + ((a^2*b^5*cos(c)^2 + a^2*b^5*sin(c)^2)*d^3*x^3 + 3*(a^3*b^4*cos(c)^2 + a^3*b^4
*sin(c)^2)*d^3*x^2 + 3*(a^4*b^3*cos(c)^2 + a^4*b^3*sin(c)^2)*d^3*x + (a^5*b^2*cos(c)^2 + a^5*b^2*sin(c)^2)*d^3
)*sin(d*x + c)^2)*integrate(1/2*x*cos(d*x + c)/((b^5*d^2*x^4 + 4*a*b^4*d^2*x^3 + 6*a^2*b^3*d^2*x^2 + 4*a^3*b^2
*d^2*x + a^4*b*d^2)*cos(d*x + c)^2 + (b^5*d^2*x^4 + 4*a*b^4*d^2*x^3 + 6*a^2*b^3*d^2*x^2 + 4*a^3*b^2*d^2*x + a^
4*b*d^2)*sin(d*x + c)^2), x) - 12*(((a*b^6*cos(c)^2 + a*b^6*sin(c)^2)*d^2*x^3 + 3*(a^2*b^5*cos(c)^2 + a^2*b^5*
sin(c)^2)*d^2*x^2 + 3*(a^3*b^4*cos(c)^2 + a^3*b^4*sin(c)^2)*d^2*x + (a^4*b^3*cos(c)^2 + a^4*b^3*sin(c)^2)*d^2)
*cos(d*x + c)^2 + ((a*b^6*cos(c)^2 + a*b^6*sin(c)^2)*d^2*x^3 + 3*(a^2*b^5*cos(c)^2 + a^2*b^5*sin(c)^2)*d^2*x^2
+ 3*(a^3*b^4*cos(c)^2 + a^3*b^4*sin(c)^2)*d^2*x + (a^4*b^3*cos(c)^2 + a^4*b^3*sin(c)^2)*d^2)*sin(d*x + c)^2)*
integrate(1/2*x*sin(d*x + c)/(b^5*d^2*x^4 + 4*a*b^4*d^2*x^3 + 6*a^2*b^3*d^2*x^2 + 4*a^3*b^2*d^2*x + a^4*b*d^2)
, x) - 12*(((a*b^6*cos(c)^2 + a*b^6*sin(c)^2)*d^2*x^3 + 3*(a^2*b^5*cos(c)^2 + a^2*b^5*sin(c)^2)*d^2*x^2 + 3*(a
^3*b^4*cos(c)^2 + a^3*b^4*sin(c)^2)*d^2*x + (a^4*b^3*cos(c)^2 + a^4*b^3*sin(c)^2)*d^2)*cos(d*x + c)^2 + ((a*b^
6*cos(c)^2 + a*b^6*sin(c)^2)*d^2*x^3 + 3*(a^2*b^5*cos(c)^2 + a^2*b^5*sin(c)^2)*d^2*x^2 + 3*(a^3*b^4*cos(c)^2 +
a^3*b^4*sin(c)^2)*d^2*x + (a^4*b^3*cos(c)^2 + a^4*b^3*sin(c)^2)*d^2)*sin(d*x + c)^2)*integrate(1/2*x*sin(d*x
+ c)/((b^5*d^2*x^4 + 4*a*b^4*d^2*x^3 + 6*a^2*b^3*d^2*x^2 + 4*a^3*b^2*d^2*x + a^4*b*d^2)*cos(d*x + c)^2 + (b^5*
d^2*x^4 + 4*a*b^4*d^2*x^3 + 6*a^2*b^3*d^2*x^2 + 4*a^3*b^2*d^2*x + a^4*b*d^2)*sin(d*x + c)^2), x) + ((b^2*d*x^3
*sin(c) - 3*a*b*x*cos(c))*cos(d*x + c)^2 + (b^2*d*x^3*sin(c) - 3*a*b*x*cos(c))*sin(d*x + c)^2)*sin(d*x + 2*c))
/(((b^5*cos(c)^2 + b^5*sin(c)^2)*d^2*x^3 + 3*(a*b^4*cos(c)^2 + a*b^4*sin(c)^2)*d^2*x^2 + 3*(a^2*b^3*cos(c)^2 +
a^2*b^3*sin(c)^2)*d^2*x + (a^3*b^2*cos(c)^2 + a^3*b^2*sin(c)^2)*d^2)*cos(d*x + c)^2 + ((b^5*cos(c)^2 + b^5*si
n(c)^2)*d^2*x^3 + 3*(a*b^4*cos(c)^2 + a*b^4*sin(c)^2)*d^2*x^2 + 3*(a^2*b^3*cos(c)^2 + a^2*b^3*sin(c)^2)*d^2*x
+ (a^3*b^2*cos(c)^2 + a^3*b^2*sin(c)^2)*d^2)*sin(d*x + c)^2)

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Fricas [A]
time = 0.38, size = 515, normalized size = 1.94 \begin {gather*} \frac {2 \, {\left (a^{4} b d^{2} - 2 \, b^{5} x^{2} - 2 \, a^{2} b^{3} + {\left (a^{3} b^{2} d^{2} - 4 \, a b^{4}\right )} x\right )} \cos \left (d x + c\right ) + 2 \, {\left (3 \, {\left (a^{2} b^{3} d^{2} x^{2} + 2 \, a^{3} b^{2} d^{2} x + a^{4} b d^{2}\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + 3 \, {\left (a^{2} b^{3} d^{2} x^{2} + 2 \, a^{3} b^{2} d^{2} x + a^{4} b d^{2}\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) + {\left (a^{5} d^{3} - 6 \, a^{3} b^{2} d + {\left (a^{3} b^{2} d^{3} - 6 \, a b^{4} d\right )} x^{2} + 2 \, {\left (a^{4} b d^{3} - 6 \, a^{2} b^{3} d\right )} x\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (6 \, a^{2} b^{3} d x + 5 \, a^{3} b^{2} d\right )} \sin \left (d x + c\right ) - {\left ({\left (a^{5} d^{3} - 6 \, a^{3} b^{2} d + {\left (a^{3} b^{2} d^{3} - 6 \, a b^{4} d\right )} x^{2} + 2 \, {\left (a^{4} b d^{3} - 6 \, a^{2} b^{3} d\right )} x\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{5} d^{3} - 6 \, a^{3} b^{2} d + {\left (a^{3} b^{2} d^{3} - 6 \, a b^{4} d\right )} x^{2} + 2 \, {\left (a^{4} b d^{3} - 6 \, a^{2} b^{3} d\right )} x\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) - 12 \, {\left (a^{2} b^{3} d^{2} x^{2} + 2 \, a^{3} b^{2} d^{2} x + a^{4} b d^{2}\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{4 \, {\left (b^{8} d x^{2} + 2 \, a b^{7} d x + a^{2} b^{6} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*(a^4*b*d^2 - 2*b^5*x^2 - 2*a^2*b^3 + (a^3*b^2*d^2 - 4*a*b^4)*x)*cos(d*x + c) + 2*(3*(a^2*b^3*d^2*x^2 +
2*a^3*b^2*d^2*x + a^4*b*d^2)*cos_integral((b*d*x + a*d)/b) + 3*(a^2*b^3*d^2*x^2 + 2*a^3*b^2*d^2*x + a^4*b*d^2)
*cos_integral(-(b*d*x + a*d)/b) + (a^5*d^3 - 6*a^3*b^2*d + (a^3*b^2*d^3 - 6*a*b^4*d)*x^2 + 2*(a^4*b*d^3 - 6*a^
2*b^3*d)*x)*sin_integral((b*d*x + a*d)/b))*cos(-(b*c - a*d)/b) - 2*(6*a^2*b^3*d*x + 5*a^3*b^2*d)*sin(d*x + c)
- ((a^5*d^3 - 6*a^3*b^2*d + (a^3*b^2*d^3 - 6*a*b^4*d)*x^2 + 2*(a^4*b*d^3 - 6*a^2*b^3*d)*x)*cos_integral((b*d*x
+ a*d)/b) + (a^5*d^3 - 6*a^3*b^2*d + (a^3*b^2*d^3 - 6*a*b^4*d)*x^2 + 2*(a^4*b*d^3 - 6*a^2*b^3*d)*x)*cos_integ
ral(-(b*d*x + a*d)/b) - 12*(a^2*b^3*d^2*x^2 + 2*a^3*b^2*d^2*x + a^4*b*d^2)*sin_integral((b*d*x + a*d)/b))*sin(
-(b*c - a*d)/b))/(b^8*d*x^2 + 2*a*b^7*d*x + a^2*b^6*d)

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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 )}}{\left (a + b x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 3.25, size = 16724, normalized size = 63.11 \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)^3,x, algorithm="giac")

[Out]

1/4*(a^3*b^2*d^3*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^3*b
^2*d^3*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^3*b^2*d^3*
x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^3*b^2*d^3*x^2*real_part(c
os_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a^3*b^2*d^3*x^2*real_part(cos_integra
l(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - 2*a^3*b^2*d^3*x^2*real_part(cos_integral(d*x + a
*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^3*b^2*d^3*x^2*real_part(cos_integral(-d*x - a*d/b))*ta
n(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*a^4*b*d^3*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*t
an(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a^4*b*d^3*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2
*tan(1/2*a*d/b)^2 + 6*a^2*b^3*d^2*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2
*a*d/b)^2 + 6*a^2*b^3*d^2*x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)
^2 + 4*a^4*b*d^3*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^3*b^2*d^3*x^
2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + a^3*b^2*d^3*x^2*imag_part(cos_integral(-d
*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a^3*b^2*d^3*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(
1/2*c)^2 + 4*a^3*b^2*d^3*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 4
*a^3*b^2*d^3*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) + 8*a^3*b^2*d^
3*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 12*a^2*b^3*d^2*x^2*imag_part(co
s_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 12*a^2*b^3*d^2*x^2*imag_part(cos_integra
l(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 4*a^4*b*d^3*x*real_part(cos_integral(d*x + a*d/b
))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 4*a^4*b*d^3*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*
x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - 24*a^2*b^3*d^2*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^
2*tan(1/2*a*d/b) - a^3*b^2*d^3*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + a^3*
b^2*d^3*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 2*a^3*b^2*d^3*x^2*sin_inte
gral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 12*a^2*b^3*d^2*x^2*imag_part(cos_integral(d*x + a*d/b)
)*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 12*a^2*b^3*d^2*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/
2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 4*a^4*b*d^3*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1
/2*c)*tan(1/2*a*d/b)^2 - 4*a^4*b*d^3*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2
*a*d/b)^2 + 24*a^2*b^3*d^2*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + a^3*
b^2*d^3*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^3*b^2*d^3*x^2*imag_part(cos
_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^3*b^2*d^3*x^2*sin_integral((b*d*x + a*d)/b)*tan(1
/2*c)^2*tan(1/2*a*d/b)^2 + a^5*d^3*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*
d/b)^2 - 6*a*b^4*d*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^5
*d^3*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 6*a*b^4*d*x^2*imag_p
art(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 12*a^3*b^2*d^2*x*real_part(cos_
integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 12*a^3*b^2*d^2*x*real_part(cos_integral(
-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^5*d^3*sin_integral((b*d*x + a*d)/b)*tan(1/2*
d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 12*a*b^4*d*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^
2*tan(1/2*a*d/b)^2 + 2*a^3*b^2*d^3*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a^3*
b^2*d^3*x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a^4*b*d^3*x*imag_part(cos_inte
gral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a^4*b*d^3*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d
*x)^2*tan(1/2*c)^2 - 6*a^2*b^3*d^2*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 6*a^
2*b^3*d^2*x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*a^4*b*d^3*x*sin_integral((
b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a^3*b^2*d^3*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d
*x)^2*tan(1/2*a*d/b) - 2*a^3*b^2*d^3*x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) +
8*a^4*b*d^3*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 8*a^4*b*d^3*x*i
mag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) + 24*a^2*b^3*d^2*x^2*real_part(c
os_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/...

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

[Out]

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

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