3.58 \(\int \frac {x^3 \sin (c+d x)}{a+b x^2} \, dx\)

Optimal. Leaf size=209 \[ -\frac {a \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}-\frac {a \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}+\frac {a \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}-\frac {a \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\sin (c+d x)}{b d^2}-\frac {x \cos (c+d x)}{b d} \]

[Out]

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

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Rubi [A]  time = 0.35, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3345, 3296, 2637, 3303, 3299, 3302} \[ -\frac {a \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^2}-\frac {a \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}+\frac {a \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}-\frac {a \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\sin (c+d x)}{b d^2}-\frac {x \cos (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*Sin[c + d*x])/(a + b*x^2),x]

[Out]

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

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 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]

Rubi steps

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

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Mathematica [C]  time = 0.43, size = 202, normalized size = 0.97 \[ -\frac {a d^2 \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+a d^2 \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (x-\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+a d^2 \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )-a d^2 \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )-2 b \sin (c+d x)+2 b d x \cos (c+d x)}{2 b^2 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*(2*b*d*x*Cos[c + d*x] + a*d^2*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] + a
*d^2*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]] - 2*b*Sin[c + d*x] + a*d^2*Cos
[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - a*d^2*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*Si
nIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(b^2*d^2)

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fricas [C]  time = 0.74, size = 185, normalized size = 0.89 \[ \frac {i \, a d^{2} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + i \, a d^{2} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - i \, a d^{2} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - i \, a d^{2} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 4 \, b d x \cos \left (d x + c\right ) + 4 \, b \sin \left (d x + c\right )}{4 \, b^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \sin \left (d x + c\right )}{b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.06, size = 1184, normalized size = 5.67 \[ \frac {\frac {d^{2} \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )-2 c \,d^{2} \cos \left (d x +c \right )}{b}-\frac {d^{2} \left (\frac {\left (d \sqrt {-a b}+c b \right ) a \,d^{2}}{b}-3 \left (d \sqrt {-a b}+c b \right ) c^{2}+2 a c \,d^{2}+2 b \,c^{3}\right ) \left (\Si \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\Ci \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{2 b^{2} \left (\frac {d \sqrt {-a b}+c b}{b}-c \right )}-\frac {d^{2} \left (-\frac {\left (d \sqrt {-a b}-c b \right ) a \,d^{2}}{b}+3 \left (d \sqrt {-a b}-c b \right ) c^{2}+2 a c \,d^{2}+2 b \,c^{3}\right ) \left (\Si \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\Ci \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{2 b^{2} \left (-\frac {d \sqrt {-a b}-c b}{b}-c \right )}+\frac {3 c \,d^{2} \cos \left (d x +c \right )}{b}-\frac {3 c \,d^{2} \left (2 \left (d \sqrt {-a b}+c b \right ) c -a \,d^{2}-b \,c^{2}\right ) \left (\Si \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\Ci \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{2 b^{2} \left (\frac {d \sqrt {-a b}+c b}{b}-c \right )}-\frac {3 c \,d^{2} \left (-2 \left (d \sqrt {-a b}-c b \right ) c -a \,d^{2}-b \,c^{2}\right ) \left (\Si \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\Ci \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{2 b^{2} \left (-\frac {d \sqrt {-a b}-c b}{b}-c \right )}+\frac {3 c^{2} d^{2} \left (d \sqrt {-a b}+c b \right ) \left (\Si \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\Ci \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{2 b^{2} \left (\frac {d \sqrt {-a b}+c b}{b}-c \right )}-\frac {3 c^{2} d^{2} \left (d \sqrt {-a b}-c b \right ) \left (\Si \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\Ci \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{2 b^{2} \left (-\frac {d \sqrt {-a b}-c b}{b}-c \right )}-c^{3} d^{2} \left (\frac {\Si \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\Ci \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )}{2 \left (\frac {d \sqrt {-a b}+c b}{b}-c \right ) b}+\frac {\Si \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\Ci \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )}{2 \left (-\frac {d \sqrt {-a b}-c b}{b}-c \right ) b}\right )}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,\sin \left (c+d\,x\right )}{b\,x^2+a} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*sin(c + d*x))/(a + b*x^2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \sin {\left (c + d x \right )}}{a + b x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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