3.26 \(\int \frac {x^4 \sin (c+d x)}{(a+b x)^2} \, dx\)
Optimal. Leaf size=233 \[ \frac {a^4 d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{b^6}-\frac {a^4 d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^6}-\frac {a^4 \sin (c+d x)}{b^5 (a+b x)}-\frac {4 a^3 \sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{b^5}-\frac {4 a^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^5}-\frac {3 a^2 \cos (c+d x)}{b^4 d}-\frac {2 a \sin (c+d x)}{b^3 d^2}+\frac {2 a x \cos (c+d x)}{b^3 d}+\frac {2 \cos (c+d x)}{b^2 d^3}+\frac {2 x \sin (c+d x)}{b^2 d^2}-\frac {x^2 \cos (c+d x)}{b^2 d} \]
[Out]
a^4*d*Ci(a*d/b+d*x)*cos(-c+a*d/b)/b^6+2*cos(d*x+c)/b^2/d^3-3*a^2*cos(d*x+c)/b^4/d+2*a*x*cos(d*x+c)/b^3/d-x^2*c
os(d*x+c)/b^2/d-4*a^3*cos(-c+a*d/b)*Si(a*d/b+d*x)/b^5+4*a^3*Ci(a*d/b+d*x)*sin(-c+a*d/b)/b^5+a^4*d*Si(a*d/b+d*x
)*sin(-c+a*d/b)/b^6-2*a*sin(d*x+c)/b^3/d^2+2*x*sin(d*x+c)/b^2/d^2-a^4*sin(d*x+c)/b^5/(b*x+a)
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Rubi [A] time = 0.51, antiderivative size = 233, normalized size of antiderivative = 1.00,
number of steps used = 15, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used
= {6742, 2638, 3296, 2637, 3297, 3303, 3299, 3302} \[ -\frac {4 a^3 \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^5}+\frac {a^4 d \cos \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^6}-\frac {a^4 d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^6}-\frac {4 a^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^5}-\frac {a^4 \sin (c+d x)}{b^5 (a+b x)}-\frac {3 a^2 \cos (c+d x)}{b^4 d}-\frac {2 a \sin (c+d x)}{b^3 d^2}+\frac {2 a x \cos (c+d x)}{b^3 d}+\frac {2 x \sin (c+d x)}{b^2 d^2}+\frac {2 \cos (c+d x)}{b^2 d^3}-\frac {x^2 \cos (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
[In]
Int[(x^4*Sin[c + d*x])/(a + b*x)^2,x]
[Out]
(2*Cos[c + d*x])/(b^2*d^3) - (3*a^2*Cos[c + d*x])/(b^4*d) + (2*a*x*Cos[c + d*x])/(b^3*d) - (x^2*Cos[c + d*x])/
(b^2*d) + (a^4*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^6 - (4*a^3*CosIntegral[(a*d)/b + d*x]*Sin[c -
(a*d)/b])/b^5 - (2*a*Sin[c + d*x])/(b^3*d^2) + (2*x*Sin[c + d*x])/(b^2*d^2) - (a^4*Sin[c + d*x])/(b^5*(a + b*x
)) - (4*a^3*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^5 - (a^4*d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d
*x])/b^6
Rule 2637
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Rule 2638
Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Rule 3296
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 3297
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 3299
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 3302
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 3303
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 6742
Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]
Rubi steps
\begin {align*} \int \frac {x^4 \sin (c+d x)}{(a+b x)^2} \, dx &=\int \left (\frac {3 a^2 \sin (c+d x)}{b^4}-\frac {2 a x \sin (c+d x)}{b^3}+\frac {x^2 \sin (c+d x)}{b^2}+\frac {a^4 \sin (c+d x)}{b^4 (a+b x)^2}-\frac {4 a^3 \sin (c+d x)}{b^4 (a+b x)}\right ) \, dx\\ &=\frac {\left (3 a^2\right ) \int \sin (c+d x) \, dx}{b^4}-\frac {\left (4 a^3\right ) \int \frac {\sin (c+d x)}{a+b x} \, dx}{b^4}+\frac {a^4 \int \frac {\sin (c+d x)}{(a+b x)^2} \, dx}{b^4}-\frac {(2 a) \int x \sin (c+d x) \, dx}{b^3}+\frac {\int x^2 \sin (c+d x) \, dx}{b^2}\\ &=-\frac {3 a^2 \cos (c+d x)}{b^4 d}+\frac {2 a x \cos (c+d x)}{b^3 d}-\frac {x^2 \cos (c+d x)}{b^2 d}-\frac {a^4 \sin (c+d x)}{b^5 (a+b x)}-\frac {(2 a) \int \cos (c+d x) \, dx}{b^3 d}+\frac {2 \int x \cos (c+d x) \, dx}{b^2 d}+\frac {\left (a^4 d\right ) \int \frac {\cos (c+d x)}{a+b x} \, dx}{b^5}-\frac {\left (4 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^4}-\frac {\left (4 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^4}\\ &=-\frac {3 a^2 \cos (c+d x)}{b^4 d}+\frac {2 a x \cos (c+d x)}{b^3 d}-\frac {x^2 \cos (c+d x)}{b^2 d}-\frac {4 a^3 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^5}-\frac {2 a \sin (c+d x)}{b^3 d^2}+\frac {2 x \sin (c+d x)}{b^2 d^2}-\frac {a^4 \sin (c+d x)}{b^5 (a+b x)}-\frac {4 a^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {2 \int \sin (c+d x) \, dx}{b^2 d^2}+\frac {\left (a^4 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^5}-\frac {\left (a^4 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^5}\\ &=\frac {2 \cos (c+d x)}{b^2 d^3}-\frac {3 a^2 \cos (c+d x)}{b^4 d}+\frac {2 a x \cos (c+d x)}{b^3 d}-\frac {x^2 \cos (c+d x)}{b^2 d}+\frac {a^4 d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{b^6}-\frac {4 a^3 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^5}-\frac {2 a \sin (c+d x)}{b^3 d^2}+\frac {2 x \sin (c+d x)}{b^2 d^2}-\frac {a^4 \sin (c+d x)}{b^5 (a+b x)}-\frac {4 a^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {a^4 d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^6}\\ \end {align*}
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Mathematica [A] time = 1.06, size = 177, normalized size = 0.76 \[ \frac {a^3 \text {Ci}\left (d \left (\frac {a}{b}+x\right )\right ) \left (a d \cos \left (c-\frac {a d}{b}\right )-4 b \sin \left (c-\frac {a d}{b}\right )\right )-a^3 \text {Si}\left (d \left (\frac {a}{b}+x\right )\right ) \left (a d \sin \left (c-\frac {a d}{b}\right )+4 b \cos \left (c-\frac {a d}{b}\right )\right )-\frac {b \left (b (a+b x) \left (3 a^2 d^2-2 a b d^2 x+b^2 \left (d^2 x^2-2\right )\right ) \cos (c+d x)+d \left (a^4 d^2+2 a^2 b^2-2 b^4 x^2\right ) \sin (c+d x)\right )}{d^3 (a+b x)}}{b^6} \]
Antiderivative was successfully verified.
[In]
Integrate[(x^4*Sin[c + d*x])/(a + b*x)^2,x]
[Out]
(a^3*CosIntegral[d*(a/b + x)]*(a*d*Cos[c - (a*d)/b] - 4*b*Sin[c - (a*d)/b]) - (b*(b*(a + b*x)*(3*a^2*d^2 - 2*a
*b*d^2*x + b^2*(-2 + d^2*x^2))*Cos[c + d*x] + d*(2*a^2*b^2 + a^4*d^2 - 2*b^4*x^2)*Sin[c + d*x]))/(d^3*(a + b*x
)) - a^3*(4*b*Cos[c - (a*d)/b] + a*d*Sin[c - (a*d)/b])*SinIntegral[d*(a/b + x)])/b^6
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fricas [A] time = 0.82, size = 357, normalized size = 1.53 \[ -\frac {2 \, {\left (b^{5} d^{2} x^{3} - a b^{4} d^{2} x^{2} + 3 \, a^{3} b^{2} d^{2} - 2 \, a b^{4} + {\left (a^{2} b^{3} d^{2} - 2 \, b^{5}\right )} x\right )} \cos \left (d x + c\right ) - {\left ({\left (a^{4} b d^{4} x + a^{5} d^{4}\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{4} b d^{4} x + a^{5} d^{4}\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) - 8 \, {\left (a^{3} b^{2} d^{3} x + a^{4} b d^{3}\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) + 2 \, {\left (a^{4} b d^{3} - 2 \, b^{5} d x^{2} + 2 \, a^{2} b^{3} d\right )} \sin \left (d x + c\right ) - 2 \, {\left (2 \, {\left (a^{3} b^{2} d^{3} x + a^{4} b d^{3}\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + 2 \, {\left (a^{3} b^{2} d^{3} x + a^{4} b d^{3}\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) + {\left (a^{4} b d^{4} x + a^{5} d^{4}\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (b^{7} d^{3} x + a b^{6} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*sin(d*x+c)/(b*x+a)^2,x, algorithm="fricas")
[Out]
-1/2*(2*(b^5*d^2*x^3 - a*b^4*d^2*x^2 + 3*a^3*b^2*d^2 - 2*a*b^4 + (a^2*b^3*d^2 - 2*b^5)*x)*cos(d*x + c) - ((a^4
*b*d^4*x + a^5*d^4)*cos_integral((b*d*x + a*d)/b) + (a^4*b*d^4*x + a^5*d^4)*cos_integral(-(b*d*x + a*d)/b) - 8
*(a^3*b^2*d^3*x + a^4*b*d^3)*sin_integral((b*d*x + a*d)/b))*cos(-(b*c - a*d)/b) + 2*(a^4*b*d^3 - 2*b^5*d*x^2 +
2*a^2*b^3*d)*sin(d*x + c) - 2*(2*(a^3*b^2*d^3*x + a^4*b*d^3)*cos_integral((b*d*x + a*d)/b) + 2*(a^3*b^2*d^3*x
+ a^4*b*d^3)*cos_integral(-(b*d*x + a*d)/b) + (a^4*b*d^4*x + a^5*d^4)*sin_integral((b*d*x + a*d)/b))*sin(-(b*
c - a*d)/b))/(b^7*d^3*x + a*b^6*d^3)
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giac [B] time = 1.71, size = 1973, normalized size = 8.47 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*sin(d*x+c)/(b*x+a)^2,x, algorithm="giac")
[Out]
((b*x + a)*a^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^4*cos(-(b*c - a*d)/b)*cos_integral(((b*x + a)*(b*c/(b*x +
a) - a*d/(b*x + a) + d) - b*c + a*d)/b) - a^4*b*c*d^4*cos(-(b*c - a*d)/b)*cos_integral(((b*x + a)*(b*c/(b*x +
a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + a^5*d^5*cos(-(b*c - a*d)/b)*cos_integral(((b*x + a)*(b*c/(b*x + a)
- a*d/(b*x + a) + d) - b*c + a*d)/b) + (b*x + a)*a^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^4*sin(-(b*c - a*d)/
b)*sin_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) - a^4*b*c*d^4*sin(-(b*c - a*d)/
b)*sin_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + a^5*d^5*sin(-(b*c - a*d)/b)*s
in_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + 4*(b*x + a)*a^3*b*(b*c/(b*x + a)
- a*d/(b*x + a) + d)*d^3*cos_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*sin(-(b*c
- a*d)/b) - 4*a^3*b^2*c*d^3*cos_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*sin(-
(b*c - a*d)/b) + 4*a^4*b*d^4*cos_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*sin(-
(b*c - a*d)/b) - 4*(b*x + a)*a^3*b*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^3*cos(-(b*c - a*d)/b)*sin_integral(((
b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + 4*a^3*b^2*c*d^3*cos(-(b*c - a*d)/b)*sin_integra
l(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) - 4*a^4*b*d^4*cos(-(b*c - a*d)/b)*sin_integra
l(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + a^4*b*d^4*sin(-(b*x + a)*(b*c/(b*x + a) - a
*d/(b*x + a) + d)/b) - (b*x + a)^3*b^2*(b*c/(b*x + a) - a*d/(b*x + a) + d)^3*cos(-(b*x + a)*(b*c/(b*x + a) - a
*d/(b*x + a) + d)/b) + 3*(b*x + a)^2*b^3*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*c*cos(-(b*x + a)*(b*c/(b*x + a)
- a*d/(b*x + a) + d)/b) - 3*(b*x + a)*b^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c^2*cos(-(b*x + a)*(b*c/(b*x +
a) - a*d/(b*x + a) + d)/b) + b^5*c^3*cos(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + (b*x + a)^2*a*b^2
*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d*cos(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - 2*(b*x + a)*a
*b^3*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c*d*cos(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + a*b^4*c^2
*d*cos(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - (b*x + a)*a^2*b^2*(b*c/(b*x + a) - a*d/(b*x + a) +
d)*d^2*cos(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + a^2*b^3*c*d^2*cos(-(b*x + a)*(b*c/(b*x + a) - a
*d/(b*x + a) + d)/b) - 3*a^3*b^2*d^3*cos(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - 2*(b*x + a)^2*b^3
*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*sin(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + 4*(b*x + a)*b^4
*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c*sin(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - 2*b^5*c^2*sin(-
(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + 2*a^2*b^3*d^2*sin(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a)
+ d)/b) + 2*(b*x + a)*b^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)*cos(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) +
d)/b) - 2*b^5*c*cos(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + 2*a*b^4*d*cos(-(b*x + a)*(b*c/(b*x +
a) - a*d/(b*x + a) + d)/b))*b^2/(((b*x + a)*b^8*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^2 - b^9*c*d^2 + a*b^8*d^
3)*d)
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maple [B] time = 0.04, size = 1214, normalized size = 5.21 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4*sin(d*x+c)/(b*x+a)^2,x)
[Out]
1/d^5*(-4/b^4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*d^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)+(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)*d^2/b^4*
(-sin(d*x+c)/((d*x+c)*b+d*a-c*b)/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*a^2*d^2-6*a*b*c*d+3*b^2*c^2-2*a*b*d+2*b^2*c+b^2)*d^2/b^4*(-(d*x+c)^2*cos(d*x+c)+2*cos(d*x+c)+2*
(d*x+c)*sin(d*x+c))-12/b^3*(a^2*d^2-2*a*b*c*d+b^2*c^2)*d^2*c*(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)+4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*d^2*c/b^3*(-sin(d*x+c)/((d*
x+c)*b+d*a-c*b)/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)-4*d^2
*c*(-2*a*d+2*b*c+b)/b^3*(sin(d*x+c)-(d*x+c)*cos(d*x+c))-12/b^2*(a*d-b*c)*d^2*c^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)+6*(a^2*d^2-2*a*b*c*d+b^2*c^2)*d^2*c^2/b^2*(-sin(d*x+c)/
((d*x+c)*b+d*a-c*b)/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)-6
*d^2*c^2/b^2*cos(d*x+c)-4*d^2*c^3/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)+4*d^2*(a*d-b*c)/b*c^3*(-sin(d*x+c)/((d*x+c)*b+d*a-c*b)/b+(Si(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b+C
i(d*x+c+(a*d-b*c)/b)*cos((a*d-b*c)/b)/b)/b)+d^2*c^4*(-sin(d*x+c)/((d*x+c)*b+d*a-c*b)/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))
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*sin(d*x+c)/(b*x+a)^2,x, algorithm="maxima")
[Out]
Timed out
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,\sin \left (c+d\,x\right )}{{\left (a+b\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^4*sin(c + d*x))/(a + b*x)^2,x)
[Out]
int((x^4*sin(c + d*x))/(a + b*x)^2, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \sin {\left (c + d x \right )}}{\left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**4*sin(d*x+c)/(b*x+a)**2,x)
[Out]
Integral(x**4*sin(c + d*x)/(a + b*x)**2, x)
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