3.1.57 \(\int \frac {x^4 \sin (c+d x)}{a+b x^2} \, dx\) [57]

Optimal. Leaf size=273 \[ \frac {2 \cos (c+d x)}{b d^3}+\frac {a \cos (c+d x)}{b^2 d}-\frac {x^2 \cos (c+d x)}{b d}-\frac {(-a)^{3/2} \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{5/2}}+\frac {(-a)^{3/2} \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{5/2}}+\frac {2 x \sin (c+d x)}{b d^2}-\frac {(-a)^{3/2} \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{5/2}}-\frac {(-a)^{3/2} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.54, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3426, 2718, 3377, 3414, 3384, 3380, 3383} \begin {gather*} -\frac {(-a)^{3/2} \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^{5/2}}+\frac {(-a)^{3/2} \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{5/2}}-\frac {(-a)^{3/2} \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{5/2}}-\frac {(-a)^{3/2} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{5/2}}+\frac {a \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^4*Sin[c + d*x])/(a + b*x^2),x]

[Out]

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

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 3414

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 3426

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]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.31, size = 275, normalized size = 1.01 \begin {gather*} \frac {4 b^{3/2} \cos (c+d x)+2 a \sqrt {b} d^2 \cos (c+d x)-2 b^{3/2} d^2 x^2 \cos (c+d x)+i a^{3/2} d^3 \text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )-i a^{3/2} d^3 \text {Ci}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )+4 b^{3/2} d x \sin (c+d x)+i a^{3/2} d^3 \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+i a^{3/2} d^3 \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )}{2 b^{5/2} d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*b^(3/2)*Cos[c + d*x] + 2*a*Sqrt[b]*d^2*Cos[c + d*x] - 2*b^(3/2)*d^2*x^2*Cos[c + d*x] + I*a^(3/2)*d^3*CosInt
egral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] - I*a^(3/2)*d^3*CosIntegral[d*(((-I)*Sqrt[a]
)/Sqrt[b] + x)]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]] + 4*b^(3/2)*d*x*Sin[c + d*x] + I*a^(3/2)*d^3*Cos[c - (I*Sqrt[a]
*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*a^(3/2)*d^3*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegr
al[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(2*b^(5/2)*d^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1629\) vs. \(2(217)=434\).
time = 0.30, size = 1630, normalized size = 5.97

method	result	size
risch	\(-\frac {a \,{\mathrm e}^{\frac {i b c +d \sqrt {a b}}{b}} \expIntegral \left (1, \frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) \sqrt {a b}}{4 b^{3}}+\frac {a \,{\mathrm e}^{\frac {i b c -d \sqrt {a b}}{b}} \expIntegral \left (1, -\frac {-i b c +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right ) \sqrt {a b}}{4 b^{3}}-\frac {a \sqrt {a b}\, {\mathrm e}^{-\frac {i b c -d \sqrt {a b}}{b}} \expIntegral \left (1, \frac {-i b c +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{4 b^{3}}+\frac {a \sqrt {a b}\, {\mathrm e}^{-\frac {i b c +d \sqrt {a b}}{b}} \expIntegral \left (1, -\frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 b^{3}}-\frac {\left (d^{2} x^{2} b -d^{2} a -2 b \right ) \cos \left (d x +c \right )}{b^{2} d^{3}}-\frac {2 \left (d^{2} x^{2}+3 c d x \right ) \sin \left (d x +c \right )}{d^{3} b \left (-d x -3 c \right )}\)	\(319\)
derivativedivides	\(\text {Expression too large to display}\)	\(1630\)
default	\(\text {Expression too large to display}\)	\(1630\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

-1/2*(((b*d^2*x^4*cos(c) + 2*b*d*x^3*sin(c) - 2*b*x^2*cos(c) + 2*a*d*x*sin(c))*cos(d*x + c)^2 + (b*d^2*x^4*cos
(c) + 2*b*d*x^3*sin(c) - 2*b*x^2*cos(c) + 2*a*d*x*sin(c))*sin(d*x + c)^2)*cos(d*x + 2*c) + ((b*cos(c)^2 + b*si
n(c)^2)*d^2*x^4 - 2*(b*cos(c)^2 + b*sin(c)^2)*x^2)*cos(d*x + c) + 2*(((b^2*cos(c)^2 + b^2*sin(c)^2)*d^3*x^2 +
(a*b*cos(c)^2 + a*b*sin(c)^2)*d^3)*cos(d*x + c)^2 + ((b^2*cos(c)^2 + b^2*sin(c)^2)*d^3*x^2 + (a*b*cos(c)^2 + a
*b*sin(c)^2)*d^3)*sin(d*x + c)^2)*integrate(((a^2*d^2 + 2*a*b)*x*cos(d*x + c) + (a*b*d*x^2 + a^2*d)*sin(d*x +
c))/(b^3*d^3*x^4 + 2*a*b^2*d^3*x^2 + a^2*b*d^3), x) + 2*(((b^2*cos(c)^2 + b^2*sin(c)^2)*d^3*x^2 + (a*b*cos(c)^
2 + a*b*sin(c)^2)*d^3)*cos(d*x + c)^2 + ((b^2*cos(c)^2 + b^2*sin(c)^2)*d^3*x^2 + (a*b*cos(c)^2 + a*b*sin(c)^2)
*d^3)*sin(d*x + c)^2)*integrate(((a^2*d^2 + 2*a*b)*x*cos(d*x + c) + (a*b*d*x^2 + a^2*d)*sin(d*x + c))/((b^3*d^
3*x^4 + 2*a*b^2*d^3*x^2 + a^2*b*d^3)*cos(d*x + c)^2 + (b^3*d^3*x^4 + 2*a*b^2*d^3*x^2 + a^2*b*d^3)*sin(d*x + c)
^2), x) + ((b*d^2*x^4*sin(c) - 2*b*d*x^3*cos(c) - 2*a*d*x*cos(c) - 2*b*x^2*sin(c))*cos(d*x + c)^2 + (b*d^2*x^4
*sin(c) - 2*b*d*x^3*cos(c) - 2*a*d*x*cos(c) - 2*b*x^2*sin(c))*sin(d*x + c)^2)*sin(d*x + 2*c) - 2*((b*cos(c)^2
+ b*sin(c)^2)*d*x^3 + (a*cos(c)^2 + a*sin(c)^2)*d*x)*sin(d*x + c))/(((b^2*cos(c)^2 + b^2*sin(c)^2)*d^3*x^2 + (
a*b*cos(c)^2 + a*b*sin(c)^2)*d^3)*cos(d*x + c)^2 + ((b^2*cos(c)^2 + b^2*sin(c)^2)*d^3*x^2 + (a*b*cos(c)^2 + a*
b*sin(c)^2)*d^3)*sin(d*x + c)^2)

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Fricas [C] Result contains complex when optimal does not.
time = 0.38, size = 240, normalized size = 0.88 \begin {gather*} \frac {\sqrt {\frac {a d^{2}}{b}} a d^{2} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - \sqrt {\frac {a d^{2}}{b}} a d^{2} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + \sqrt {\frac {a d^{2}}{b}} a d^{2} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - \sqrt {\frac {a d^{2}}{b}} a d^{2} {\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 \, b d x \sin \left (d x + c\right ) - 4 \, {\left (b d^{2} x^{2} - a d^{2} - 2 \, b\right )} \cos \left (d x + c\right )}{4 \, b^{2} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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