3.65 \(\int \frac {x^4 \sin (c+d x)}{(a+b x^2)^2} \, dx\)

Optimal. Leaf size=450 \[ -\frac {3 \sqrt {-a} \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}+\frac {3 \sqrt {-a} \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {3 \sqrt {-a} \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {3 \sqrt {-a} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}-\frac {a d \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^3}-\frac {a d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^3}-\frac {a d \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^3}+\frac {a d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^3}-\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac {x \sin (c+d x)}{2 b^2}-\frac {\cos (c+d x)}{b^2 d} \]

[Out]

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

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Rubi [A]  time = 0.78, antiderivative size = 450, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3343, 3345, 2638, 3333, 3303, 3299, 3302, 3346, 3296} \[ -\frac {3 \sqrt {-a} \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^{5/2}}+\frac {3 \sqrt {-a} \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {a d \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^3}-\frac {a d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^3}-\frac {a d \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^3}+\frac {a d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^3}-\frac {3 \sqrt {-a} \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {3 \sqrt {-a} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}-\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac {x \sin (c+d x)}{2 b^2}-\frac {\cos (c+d x)}{b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*Sin[c + d*x])/(a + b*x^2)^2,x]

[Out]

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

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 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 3333

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sin[c + d*x], (a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])

Rule 3343

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(x^(m - n + 1)*(a + b*
x^n)^(p + 1)*Sin[c + d*x])/(b*n*(p + 1)), x] + (-Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*(a + b*x^n)^(p
+ 1)*Sin[c + d*x], x], x] - Dist[d/(b*n*(p + 1)), Int[x^(m - n + 1)*(a + b*x^n)^(p + 1)*Cos[c + d*x], x], x])
/; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, -1] && IGtQ[n, 0] && (GtQ[m - n + 1, 0] || GtQ[n, 2]) && RationalQ[m]

Rule 3345

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sin[c +
d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ
[p, -1]) && IntegerQ[m]

Rule 3346

Int[Cos[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cos[c +
d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ
[p, -1]) && IntegerQ[m]

Rubi steps

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

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Mathematica [C]  time = 1.18, size = 632, normalized size = 1.40 \[ -\frac {3 i a^{3/2} \sqrt {b} d \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+3 i a^{3/2} \sqrt {b} d \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )-a^2 d^2 \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+a^2 d^2 \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+3 i \sqrt {a} b^{3/2} d x^2 \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+3 i \sqrt {a} b^{3/2} d x^2 \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\sqrt {a} d \left (a+b x^2\right ) \text {Ci}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right ) \left (3 i \sqrt {b} \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )+\sqrt {a} d \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )+\sqrt {a} d \left (a+b x^2\right ) \text {Ci}\left (d \left (x-\frac {i \sqrt {a}}{\sqrt {b}}\right )\right ) \left (\sqrt {a} d \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )-3 i \sqrt {b} \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )-a b d^2 x^2 \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+a b d^2 x^2 \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )-2 a b d x \sin (c+d x)+4 a b \cos (c+d x)+4 b^2 x^2 \cos (c+d x)}{4 b^3 d \left (a+b x^2\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^4*Sin[c + d*x])/(a + b*x^2)^2,x]

[Out]

-1/4*(4*a*b*Cos[c + d*x] + 4*b^2*x^2*Cos[c + d*x] + Sqrt[a]*d*(a + b*x^2)*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] +
x)]*(Sqrt[a]*d*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]] + (3*I)*Sqrt[b]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]) + Sqrt[a]*d*(a
+ b*x^2)*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*(Sqrt[a]*d*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]] - (3*I)*Sqrt[b]
*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]) - 2*a*b*d*x*Sin[c + d*x] + (3*I)*a^(3/2)*Sqrt[b]*d*Cos[c - (I*Sqrt[a]*d)/Sqrt
[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + (3*I)*Sqrt[a]*b^(3/2)*d*x^2*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*Sin
Integral[d*((I*Sqrt[a])/Sqrt[b] + x)] - a^2*d^2*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt
[b] + x)] - a*b*d^2*x^2*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + (3*I)*a^(3/2
)*Sqrt[b]*d*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + (3*I)*Sqrt[a]*b^(3/2)*d*
x^2*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + a^2*d^2*Sin[c + (I*Sqrt[a]*d)/Sq
rt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + a*b*d^2*x^2*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sq
rt[a]*d)/Sqrt[b] - d*x])/(b^3*d*(a + b*x^2))

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fricas [C]  time = 0.86, size = 351, normalized size = 0.78 \[ \frac {4 \, a b d x \sin \left (d x + c\right ) - {\left (a b d^{2} x^{2} + a^{2} d^{2} + 3 \, {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (a b d^{2} x^{2} + a^{2} d^{2} - 3 \, {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (a b d^{2} x^{2} + a^{2} d^{2} + 3 \, {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (a b d^{2} x^{2} + a^{2} d^{2} - 3 \, {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 8 \, {\left (b^{2} x^{2} + a b\right )} \cos \left (d x + c\right )}{8 \, {\left (b^{4} d x^{2} + a b^{3} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*sin(d*x+c)/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

1/8*(4*a*b*d*x*sin(d*x + c) - (a*b*d^2*x^2 + a^2*d^2 + 3*(b^2*x^2 + a*b)*sqrt(a*d^2/b))*Ei(I*d*x - sqrt(a*d^2/
b))*e^(I*c + sqrt(a*d^2/b)) - (a*b*d^2*x^2 + a^2*d^2 - 3*(b^2*x^2 + a*b)*sqrt(a*d^2/b))*Ei(I*d*x + sqrt(a*d^2/
b))*e^(I*c - sqrt(a*d^2/b)) - (a*b*d^2*x^2 + a^2*d^2 + 3*(b^2*x^2 + a*b)*sqrt(a*d^2/b))*Ei(-I*d*x - sqrt(a*d^2
/b))*e^(-I*c + sqrt(a*d^2/b)) - (a*b*d^2*x^2 + a^2*d^2 - 3*(b^2*x^2 + a*b)*sqrt(a*d^2/b))*Ei(-I*d*x + sqrt(a*d
^2/b))*e^(-I*c - sqrt(a*d^2/b)) - 8*(b^2*x^2 + a*b)*cos(d*x + c))/(b^4*d*x^2 + a*b^3*d)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*sin(d*x+c)/(b*x^2+a)^2,x, algorithm="giac")

[Out]

integrate(x^4*sin(d*x + c)/(b*x^2 + a)^2, x)

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maple [B]  time = 0.14, size = 3453, normalized size = 7.67 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*sin(d*x+c)/(b*x^2+a)^2,x)

[Out]

1/d^5*(-d^4/b^2*cos(d*x+c)+sin(d*x+c)*(1/2*d^2*(a^2*d^4-6*a*b*c^2*d^2+b^2*c^4)/a*(d*x+c)+1/2*c*d^2*(3*a^2*d^4+
2*a*b*c^2*d^2-b^2*c^4)/a)/b^2/((d*x+c)^2*b-2*(d*x+c)*b*c+a*d^2+b*c^2)+1/4*d^2*(8*(d*(-a*b)^(1/2)+c*b)*a*c*d^2-
3*a^2*d^4-2*a*b*c^2*d^2+b^2*c^4)/a/((d*(-a*b)^(1/2)+c*b)/b-c)/b^3*(Si(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*cos((d*(-a
*b)^(1/2)+c*b)/b)+Ci(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*sin((d*(-a*b)^(1/2)+c*b)/b))+1/4*d^2*(-8*(d*(-a*b)^(1/2)-c*
b)*a*c*d^2-3*a^2*d^4-2*a*b*c^2*d^2+b^2*c^4)/a/(-(d*(-a*b)^(1/2)-c*b)/b-c)/b^3*(Si(d*x+c+(d*(-a*b)^(1/2)-c*b)/b
)*cos((d*(-a*b)^(1/2)-c*b)/b)-Ci(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*sin((d*(-a*b)^(1/2)-c*b)/b))-1/4*d^2*((d*(-a*b)
^(1/2)+c*b)/b*a^2*d^4-6*(d*(-a*b)^(1/2)+c*b)*a*c^2*d^2+(d*(-a*b)^(1/2)+c*b)*b*c^4+3*a^2*c*d^4+2*a*b*c^3*d^2-b^
2*c^5)/a/((d*(-a*b)^(1/2)+c*b)/b-c)/b^3*(-Si(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*sin((d*(-a*b)^(1/2)+c*b)/b)+Ci(d*x+
c-(d*(-a*b)^(1/2)+c*b)/b)*cos((d*(-a*b)^(1/2)+c*b)/b))-1/4*d^2*(-(d*(-a*b)^(1/2)-c*b)/b*a^2*d^4+6*(d*(-a*b)^(1
/2)-c*b)*a*c^2*d^2-(d*(-a*b)^(1/2)-c*b)*b*c^4+3*a^2*c*d^4+2*a*b*c^3*d^2-b^2*c^5)/a/(-(d*(-a*b)^(1/2)-c*b)/b-c)
/b^3*(Si(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*sin((d*(-a*b)^(1/2)-c*b)/b)+Ci(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*cos((d*(-a
*b)^(1/2)-c*b)/b))+sin(d*x+c)*(2*c^2*d^2*(3*a*d^2-b*c^2)/a/b*(d*x+c)-2*c*d^2*(a^2*d^4-b^2*c^4)/a/b^2)/((d*x+c)
^2*b-2*(d*x+c)*b*c+a*d^2+b*c^2)-c*d^2*(2*(d*(-a*b)^(1/2)+c*b)/b*a*d^2+a*c*d^2+b*c^3)/a/b^2/((d*(-a*b)^(1/2)+c*
b)/b-c)*(Si(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*cos((d*(-a*b)^(1/2)+c*b)/b)+Ci(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*sin((d*
(-a*b)^(1/2)+c*b)/b))-c*d^2*(-2*(d*(-a*b)^(1/2)-c*b)/b*a*d^2+a*c*d^2+b*c^3)/a/b^2/(-(d*(-a*b)^(1/2)-c*b)/b-c)*
(Si(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*cos((d*(-a*b)^(1/2)-c*b)/b)-Ci(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*sin((d*(-a*b)^(
1/2)-c*b)/b))-c*d^2*(3*(d*(-a*b)^(1/2)+c*b)*a*c*d^2-(d*(-a*b)^(1/2)+c*b)*b*c^3-a^2*d^4+b^2*c^4)/a/b^3/((d*(-a*
b)^(1/2)+c*b)/b-c)*(-Si(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*sin((d*(-a*b)^(1/2)+c*b)/b)+Ci(d*x+c-(d*(-a*b)^(1/2)+c*b
)/b)*cos((d*(-a*b)^(1/2)+c*b)/b))-c*d^2*(-3*(d*(-a*b)^(1/2)-c*b)*a*c*d^2+(d*(-a*b)^(1/2)-c*b)*b*c^3-a^2*d^4+b^
2*c^4)/a/b^3/(-(d*(-a*b)^(1/2)-c*b)/b-c)*(Si(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*sin((d*(-a*b)^(1/2)-c*b)/b)+Ci(d*x+
c+(d*(-a*b)^(1/2)-c*b)/b)*cos((d*(-a*b)^(1/2)-c*b)/b))+sin(d*x+c)*(-3*c^2*d^2*(a*d^2-b*c^2)/a/b*(d*x+c)-3*c^3*
d^2*(a*d^2+b*c^2)/a/b)/((d*x+c)^2*b-2*(d*x+c)*b*c+a*d^2+b*c^2)+3/2*c^2*d^2*(a*d^2+b*c^2)/a/b^2/((d*(-a*b)^(1/2
)+c*b)/b-c)*(Si(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*cos((d*(-a*b)^(1/2)+c*b)/b)+Ci(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*sin
((d*(-a*b)^(1/2)+c*b)/b))+3/2*c^2*d^2*(a*d^2+b*c^2)/a/b^2/(-(d*(-a*b)^(1/2)-c*b)/b-c)*(Si(d*x+c+(d*(-a*b)^(1/2
)-c*b)/b)*cos((d*(-a*b)^(1/2)-c*b)/b)-Ci(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*sin((d*(-a*b)^(1/2)-c*b)/b))+3/2*c^2*d^
2*((d*(-a*b)^(1/2)+c*b)/b*a*d^2-(d*(-a*b)^(1/2)+c*b)*c^2+a*c*d^2+b*c^3)/a/b^2/((d*(-a*b)^(1/2)+c*b)/b-c)*(-Si(
d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*sin((d*(-a*b)^(1/2)+c*b)/b)+Ci(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*cos((d*(-a*b)^(1/2)
+c*b)/b))+3/2*c^2*d^2*(-(d*(-a*b)^(1/2)-c*b)/b*a*d^2+(d*(-a*b)^(1/2)-c*b)*c^2+a*c*d^2+b*c^3)/a/b^2/(-(d*(-a*b)
^(1/2)-c*b)/b-c)*(Si(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*sin((d*(-a*b)^(1/2)-c*b)/b)+Ci(d*x+c+(d*(-a*b)^(1/2)-c*b)/b
)*cos((d*(-a*b)^(1/2)-c*b)/b))+sin(d*x+c)*(-2*c^4*d^2/a*(d*x+c)+2*c^3*d^2*(a*d^2+b*c^2)/a/b)/((d*x+c)^2*b-2*(d
*x+c)*b*c+a*d^2+b*c^2)-c^4*d^2/a/((d*(-a*b)^(1/2)+c*b)/b-c)/b*(Si(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*cos((d*(-a*b)^
(1/2)+c*b)/b)+Ci(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*sin((d*(-a*b)^(1/2)+c*b)/b))-c^4*d^2/a/(-(d*(-a*b)^(1/2)-c*b)/b
-c)/b*(Si(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*cos((d*(-a*b)^(1/2)-c*b)/b)-Ci(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*sin((d*(-
a*b)^(1/2)-c*b)/b))+c^3*d^2*((d*(-a*b)^(1/2)+c*b)*c-a*d^2-b*c^2)/a/b^2/((d*(-a*b)^(1/2)+c*b)/b-c)*(-Si(d*x+c-(
d*(-a*b)^(1/2)+c*b)/b)*sin((d*(-a*b)^(1/2)+c*b)/b)+Ci(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*cos((d*(-a*b)^(1/2)+c*b)/b
))+c^3*d^2*(-(d*(-a*b)^(1/2)-c*b)*c-a*d^2-b*c^2)/a/b^2/(-(d*(-a*b)^(1/2)-c*b)/b-c)*(Si(d*x+c+(d*(-a*b)^(1/2)-c
*b)/b)*sin((d*(-a*b)^(1/2)-c*b)/b)+Ci(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*cos((d*(-a*b)^(1/2)-c*b)/b))+c^4*d^4*(sin(
d*x+c)*(1/2/a/d^2*(d*x+c)-1/2*c/a/d^2)/((d*x+c)^2*b-2*(d*x+c)*b*c+a*d^2+b*c^2)+1/4/a/d^2/((d*(-a*b)^(1/2)+c*b)
/b-c)/b*(Si(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*cos((d*(-a*b)^(1/2)+c*b)/b)+Ci(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*sin((d*
(-a*b)^(1/2)+c*b)/b))+1/4/a/d^2/(-(d*(-a*b)^(1/2)-c*b)/b-c)/b*(Si(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*cos((d*(-a*b)^
(1/2)-c*b)/b)-Ci(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*sin((d*(-a*b)^(1/2)-c*b)/b))-1/4/a/b/d^2*(-Si(d*x+c-(d*(-a*b)^(
1/2)+c*b)/b)*sin((d*(-a*b)^(1/2)+c*b)/b)+Ci(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*cos((d*(-a*b)^(1/2)+c*b)/b))-1/4/a/b
/d^2*(Si(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*sin((d*(-a*b)^(1/2)-c*b)/b)+Ci(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*cos((d*(-a
*b)^(1/2)-c*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^2+a)^2,x, algorithm="maxima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,\sin \left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*sin(c + d*x))/(a + b*x^2)^2,x)

[Out]

int((x^4*sin(c + d*x))/(a + b*x^2)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \sin {\left (c + d x \right )}}{\left (a + b x^{2}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*sin(d*x+c)/(b*x**2+a)**2,x)

[Out]

Integral(x**4*sin(c + d*x)/(a + b*x**2)**2, x)

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