3.1.34 \(\int \frac {x^2 \sin (c+d x)}{(a+b x)^3} \, dx\) [34]

Optimal. Leaf size=241 \[ -\frac {a^2 d \cos (c+d x)}{2 b^4 (a+b x)}-\frac {2 a d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{b^4}+\frac {\text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^3}-\frac {a^2 d^2 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{2 b^5}-\frac {a^2 \sin (c+d x)}{2 b^3 (a+b x)^2}+\frac {2 a \sin (c+d x)}{b^3 (a+b x)}+\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^3}-\frac {a^2 d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{2 b^5}+\frac {2 a d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^4} \]

[Out]

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

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Rubi [A]
time = 0.37, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6874, 3378, 3384, 3380, 3383} \begin {gather*} -\frac {a^2 d^2 \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{2 b^5}-\frac {a^2 d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{2 b^5}-\frac {a^2 d \cos (c+d x)}{2 b^4 (a+b x)}-\frac {a^2 \sin (c+d x)}{2 b^3 (a+b x)^2}-\frac {2 a d \cos \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^4}+\frac {2 a d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^4}+\frac {\sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {2 a \sin (c+d x)}{b^3 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*Sin[c + d*x])/(a + b*x)^3,x]

[Out]

-1/2*(a^2*d*Cos[c + d*x])/(b^4*(a + b*x)) - (2*a*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^4 + (CosInte
gral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/b^3 - (a^2*d^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/(2*b^5) - (a
^2*Sin[c + d*x])/(2*b^3*(a + b*x)^2) + (2*a*Sin[c + d*x])/(b^3*(a + b*x)) + (Cos[c - (a*d)/b]*SinIntegral[(a*d
)/b + d*x])/b^3 - (a^2*d^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/(2*b^5) + (2*a*d*Sin[c - (a*d)/b]*SinI
ntegral[(a*d)/b + d*x])/b^4

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^2 \sin (c+d x)}{(a+b x)^3} \, dx &=\int \left (\frac {a^2 \sin (c+d x)}{b^2 (a+b x)^3}-\frac {2 a \sin (c+d x)}{b^2 (a+b x)^2}+\frac {\sin (c+d x)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac {\int \frac {\sin (c+d x)}{a+b x} \, dx}{b^2}-\frac {(2 a) \int \frac {\sin (c+d x)}{(a+b x)^2} \, dx}{b^2}+\frac {a^2 \int \frac {\sin (c+d x)}{(a+b x)^3} \, dx}{b^2}\\ &=-\frac {a^2 \sin (c+d x)}{2 b^3 (a+b x)^2}+\frac {2 a \sin (c+d x)}{b^3 (a+b x)}-\frac {(2 a d) \int \frac {\cos (c+d x)}{a+b x} \, dx}{b^3}+\frac {\left (a^2 d\right ) \int \frac {\cos (c+d x)}{(a+b x)^2} \, dx}{2 b^3}+\frac {\cos \left (c-\frac {a d}{b}\right ) \int \frac {\sin \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^2}+\frac {\sin \left (c-\frac {a d}{b}\right ) \int \frac {\cos \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^2}\\ &=-\frac {a^2 d \cos (c+d x)}{2 b^4 (a+b x)}+\frac {\text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^3}-\frac {a^2 \sin (c+d x)}{2 b^3 (a+b x)^2}+\frac {2 a \sin (c+d x)}{b^3 (a+b x)}+\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^3}-\frac {\left (a^2 d^2\right ) \int \frac {\sin (c+d x)}{a+b x} \, dx}{2 b^4}-\frac {\left (2 a 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^3}+\frac {\left (2 a 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^3}\\ &=-\frac {a^2 d \cos (c+d x)}{2 b^4 (a+b x)}-\frac {2 a d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{b^4}+\frac {\text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^3}-\frac {a^2 \sin (c+d x)}{2 b^3 (a+b x)^2}+\frac {2 a \sin (c+d x)}{b^3 (a+b x)}+\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {2 a d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {\left (a^2 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^4}-\frac {\left (a^2 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^4}\\ &=-\frac {a^2 d \cos (c+d x)}{2 b^4 (a+b x)}-\frac {2 a d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{b^4}+\frac {\text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^3}-\frac {a^2 d^2 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{2 b^5}-\frac {a^2 \sin (c+d x)}{2 b^3 (a+b x)^2}+\frac {2 a \sin (c+d x)}{b^3 (a+b x)}+\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^3}-\frac {a^2 d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{2 b^5}+\frac {2 a d \sin \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 = 0.80, size = 154, normalized size = 0.64 \begin {gather*} -\frac {-\text {Ci}\left (d \left (\frac {a}{b}+x\right )\right ) \left (-4 a b d \cos \left (c-\frac {a d}{b}\right )+\left (2 b^2-a^2 d^2\right ) \sin \left (c-\frac {a d}{b}\right )\right )+\frac {a b (a d (a+b x) \cos (c+d x)-b (3 a+4 b x) \sin (c+d x))}{(a+b x)^2}+\left (\left (-2 b^2+a^2 d^2\right ) \cos \left (c-\frac {a d}{b}\right )-4 a b d \sin \left (c-\frac {a d}{b}\right )\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{2 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*Sin[c + d*x])/(a + b*x)^3,x]

[Out]

-1/2*(-(CosIntegral[d*(a/b + x)]*(-4*a*b*d*Cos[c - (a*d)/b] + (2*b^2 - a^2*d^2)*Sin[c - (a*d)/b])) + (a*b*(a*d
*(a + b*x)*Cos[c + d*x] - b*(3*a + 4*b*x)*Sin[c + d*x]))/(a + b*x)^2 + ((-2*b^2 + a^2*d^2)*Cos[c - (a*d)/b] -
4*a*b*d*Sin[c - (a*d)/b])*SinIntegral[d*(a/b + x)])/b^5

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(778\) vs. \(2(240)=480\).
time = 0.09, size = 779, normalized size = 3.23 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*sin(d*x+c)/(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/d^3*(d^3*c^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)-2*d^3*c/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)+2*d^3*(a*d-b*
c)/b*c*(-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)-2*d^3*(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)+d^3*(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)+d^3/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))

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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^2*sin(d*x+c)/(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/2*((b*cos(c)^2 + b*sin(c)^2)*d*x^2*cos(d*x + c) + ((a*(I*exp_integral_e(4, (I*b*d*x + I*a*d)/b) - I*exp_int
egral_e(4, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a*(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*(exp_integral_e(4, (I*b*d*x + I*a*d)/b) + exp_integ
ral_e(4, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a*(exp_integral_e(4, (I*b*d*x + I*a*d)/b) + exp_integral_e(4, -(I*b
*d*x + I*a*d)/b))*sin(c)^2)*sin(-(b*c - a*d)/b))*cos(d*x + c)^2 + (b*cos(c)^2 + b*sin(c)^2)*x*sin(d*x + c) + (
(a*(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*(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*(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*(exp_inte
gral_e(4, (I*b*d*x + I*a*d)/b) + exp_integral_e(4, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*sin(-(b*c - a*d)/b))*sin(d
*x + c)^2 + ((b*d*x^2*cos(c) - b*x*sin(c))*cos(d*x + c)^2 + (b*d*x^2*cos(c) - b*x*sin(c))*sin(d*x + c)^2)*cos(
d*x + 2*c) - 6*(((a*b^4*cos(c)^2 + a*b^4*sin(c)^2)*d^3*x^3 + 3*(a^2*b^3*cos(c)^2 + a^2*b^3*sin(c)^2)*d^3*x^2 +
3*(a^3*b^2*cos(c)^2 + a^3*b^2*sin(c)^2)*d^3*x + (a^4*b*cos(c)^2 + a^4*b*sin(c)^2)*d^3)*cos(d*x + c)^2 + ((a*b
^4*cos(c)^2 + a*b^4*sin(c)^2)*d^3*x^3 + 3*(a^2*b^3*cos(c)^2 + a^2*b^3*sin(c)^2)*d^3*x^2 + 3*(a^3*b^2*cos(c)^2
+ a^3*b^2*sin(c)^2)*d^3*x + (a^4*b*cos(c)^2 + a^4*b*sin(c)^2)*d^3)*sin(d*x + c)^2)*integrate(1/2*x*cos(d*x + c
)/(b^4*d^2*x^4 + 4*a*b^3*d^2*x^3 + 6*a^2*b^2*d^2*x^2 + 4*a^3*b*d^2*x + a^4*d^2), x) - 6*(((a*b^4*cos(c)^2 + a*
b^4*sin(c)^2)*d^3*x^3 + 3*(a^2*b^3*cos(c)^2 + a^2*b^3*sin(c)^2)*d^3*x^2 + 3*(a^3*b^2*cos(c)^2 + a^3*b^2*sin(c)
^2)*d^3*x + (a^4*b*cos(c)^2 + a^4*b*sin(c)^2)*d^3)*cos(d*x + c)^2 + ((a*b^4*cos(c)^2 + a*b^4*sin(c)^2)*d^3*x^3
+ 3*(a^2*b^3*cos(c)^2 + a^2*b^3*sin(c)^2)*d^3*x^2 + 3*(a^3*b^2*cos(c)^2 + a^3*b^2*sin(c)^2)*d^3*x + (a^4*b*co
s(c)^2 + a^4*b*sin(c)^2)*d^3)*sin(d*x + c)^2)*integrate(1/2*x*cos(d*x + c)/((b^4*d^2*x^4 + 4*a*b^3*d^2*x^3 + 6
*a^2*b^2*d^2*x^2 + 4*a^3*b*d^2*x + a^4*d^2)*cos(d*x + c)^2 + (b^4*d^2*x^4 + 4*a*b^3*d^2*x^3 + 6*a^2*b^2*d^2*x^
2 + 4*a^3*b*d^2*x + a^4*d^2)*sin(d*x + c)^2), x) + 4*(((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*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)*sin(d*x + c)^
2)*integrate(1/2*x*sin(d*x + c)/(b^4*d^2*x^4 + 4*a*b^3*d^2*x^3 + 6*a^2*b^2*d^2*x^2 + 4*a^3*b*d^2*x + a^4*d^2),
x) + 4*(((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*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)*sin(d*x + c)^2)*integrate(1/2*x*sin(d*x + c)/((b^4*d^2*x^4
+ 4*a*b^3*d^2*x^3 + 6*a^2*b^2*d^2*x^2 + 4*a^3*b*d^2*x + a^4*d^2)*cos(d*x + c)^2 + (b^4*d^2*x^4 + 4*a*b^3*d^2*
x^3 + 6*a^2*b^2*d^2*x^2 + 4*a^3*b*d^2*x + a^4*d^2)*sin(d*x + c)^2), x) + ((b*d*x^2*sin(c) + b*x*cos(c))*cos(d*
x + c)^2 + (b*d*x^2*sin(c) + b*x*cos(c))*sin(d*x + c)^2)*sin(d*x + 2*c))/(((b^4*cos(c)^2 + b^4*sin(c)^2)*d^2*x
^3 + 3*(a*b^3*cos(c)^2 + a*b^3*sin(c)^2)*d^2*x^2 + 3*(a^2*b^2*cos(c)^2 + a^2*b^2*sin(c)^2)*d^2*x + (a^3*b*cos(
c)^2 + a^3*b*sin(c)^2)*d^2)*cos(d*x + c)^2 + ((b^4*cos(c)^2 + b^4*sin(c)^2)*d^2*x^3 + 3*(a*b^3*cos(c)^2 + a*b^
3*sin(c)^2)*d^2*x^2 + 3*(a^2*b^2*cos(c)^2 + a^2*b^2*sin(c)^2)*d^2*x + (a^3*b*cos(c)^2 + a^3*b*sin(c)^2)*d^2)*s
in(d*x + c)^2)

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Fricas [A]
time = 0.38, size = 438, normalized size = 1.82 \begin {gather*} -\frac {2 \, {\left (a^{2} b^{2} d x + a^{3} b d\right )} \cos \left (d x + c\right ) + 2 \, {\left (2 \, {\left (a b^{3} d x^{2} + 2 \, a^{2} b^{2} d x + a^{3} b d\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + 2 \, {\left (a b^{3} d x^{2} + 2 \, a^{2} b^{2} d x + a^{3} b d\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) + {\left (a^{4} d^{2} - 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} - 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} - 2 \, a b^{3}\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 (4 \, a b^{3} x + 3 \, a^{2} b^{2}\right )} \sin \left (d x + c\right ) - {\left ({\left (a^{4} d^{2} - 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} - 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} - 2 \, a b^{3}\right )} x\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{4} d^{2} - 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} - 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} - 2 \, a b^{3}\right )} x\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) - 8 \, {\left (a b^{3} d x^{2} + 2 \, a^{2} b^{2} d x + a^{3} b d\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^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*sin(d*x+c)/(b*x+a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \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**2*sin(d*x+c)/(b*x+a)**3,x)

[Out]

Integral(x**2*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 = 5.16, size = 15410, normalized size = 63.94 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*sin(d*x+c)/(b*x+a)^3,x, algorithm="giac")

[Out]

-1/4*(a^2*b^2*d^2*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^2*
b^2*d^2*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^2*b^2*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)^2 + 2*a^2*b^2*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*a^2*b^2*d^2*x^2*real_part(cos_integr
al(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - 2*a^2*b^2*d^2*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^2*b^2*d^2*x^2*real_part(cos_integral(-d*x - a*d/b))*t
an(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*a^3*b*d^2*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 - 2*a^3*b*d^2*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 + 4*a*b^3*d*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*b^3*d*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^3*b*d^2*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^2*b^2*d^2*x^2*imag_
part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + a^2*b^2*d^2*x^2*imag_part(cos_integral(-d*x - a*
d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a^2*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^
2 + 4*a^2*b^2*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) - 4*a^2*b^
2*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) + 8*a^2*b^2*d^2*x^2*s
in_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 8*a*b^3*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) + 8*a*b^3*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) + 4*a^3*b*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) + 4*a^3*b*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) - 16*a*b^3*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) - a^
2*b^2*d^2*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^2*b^2*d^2*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^2*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b)
*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 8*a*b^3*d*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 - 8*a*b^3*d*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^3*b*d^2*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^3*
b*d^2*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 + 16*a*b^3*d*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^2*b^2*d^2*x^2*imag_part(cos_integral(
d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^2*b^2*d^2*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^2*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^4*d^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*b^4*x^2*imag_part(cos_in
tegral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^4*d^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*b^4*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 + 8*a^2*b^2*d*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 + 8*a^2*b^2*d*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^4*d^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 - 4*b^4*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^2*b^2*d^2*x^2*real_part(cos_i
ntegral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a^2*b^2*d^2*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*d^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a^3
*b*d^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*a*b^3*d*x^2*real_part(cos_integ
ral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*a*b^3*d*x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*
x)^2*tan(1/2*c)^2 - 4*a^3*b*d^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a^2*b^2*d^2*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^2*b^2*d^2*x^2*real_part(cos_integra
l(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 8*a^3*b*d^2*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^3*b*d^2*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) + 16*a*b^3*d*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)
+ 16*a*b^3*d*x^2*real_part(cos_integral(-d*x -...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,\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^2*sin(c + d*x))/(a + b*x)^3,x)

[Out]

int((x^2*sin(c + d*x))/(a + b*x)^3, x)

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