3.1.59 \(\int \frac {x^2 \sin (c+d x)}{a+b x^2} \, dx\) [59]

Optimal. Leaf size=227 \[ -\frac {\cos (c+d x)}{b d}-\frac {\sqrt {-a} \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}+\frac {\sqrt {-a} \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.30, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3426, 2718, 3414, 3384, 3380, 3383} \begin {gather*} -\frac {\sqrt {-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^{3/2}}+\frac {\sqrt {-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^{3/2}}-\frac {\sqrt {-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^{3/2}}-\frac {\sqrt {-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^{3/2}}-\frac {\cos (c+d x)}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*Sin[c + d*x])/(a + b*x^2),x]

[Out]

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

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.23, size = 216, normalized size = 0.95 \begin {gather*} -\frac {2 \sqrt {b} \cos (c+d x)+i \sqrt {a} d \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 \sqrt {a} d \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 \sqrt {a} d \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 \sqrt {a} 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 )}{2 b^{3/2} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(2*Sqrt[b]*Cos[c + d*x] + I*Sqrt[a]*d*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt
[b]] - I*Sqrt[a]*d*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]] + I*Sqrt[a]*d*Co
s[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*Sqrt[a]*d*Cos[c + (I*Sqrt[a]*d)/Sqrt
[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(b^(3/2)*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(781\) vs. \(2(171)=342\).
time = 0.07, size = 782, normalized size = 3.44

method	result	size
risch	\(-\frac {{\mathrm e}^{\frac {i b c -d \sqrt {a b}}{b}} \expIntegral \left (1, -\frac {-i b c +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right ) \sqrt {a b}}{4 b^{2}}+\frac {{\mathrm e}^{\frac {i b c +d \sqrt {a b}}{b}} \expIntegral \left (1, \frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) \sqrt {a b}}{4 b^{2}}+\frac {\sqrt {a b}\, {\mathrm e}^{-\frac {i b c -d \sqrt {a b}}{b}} \expIntegral \left (1, \frac {-i b c +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{4 b^{2}}-\frac {\sqrt {a b}\, {\mathrm e}^{-\frac {i b c +d \sqrt {a b}}{b}} \expIntegral \left (1, -\frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 b^{2}}-\frac {\cos \left (d x +c \right )}{b d}\)	\(260\)
derivativedivides	\(\frac {d^{2} c^{2} \left (-\frac {\sinIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\cosineIntegral \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 b \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\frac {\sinIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\cosineIntegral \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 b \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}\right )+\frac {d^{2} c \left (d \sqrt {-a b}+c b \right ) \left (\sinIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\cosineIntegral \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 )}{b^{2} \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\frac {d^{2} c \left (d \sqrt {-a b}-c b \right ) \left (\sinIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\cosineIntegral \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 )}{b^{2} \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}-\frac {d^{2} \cos \left (d x +c \right )}{b}+\frac {d^{2} \left (d^{2} a +b \,c^{2}-2 c \left (d \sqrt {-a b}+c b \right )\right ) \left (\sinIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\cosineIntegral \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 (d^{2} a +b \,c^{2}+2 c \left (d \sqrt {-a b}-c b \right )\right ) \left (\sinIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\cosineIntegral \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 )}}{d^{3}}\)	\(782\)
default	\(\frac {d^{2} c^{2} \left (-\frac {\sinIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\cosineIntegral \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 b \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\frac {\sinIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\cosineIntegral \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 b \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}\right )+\frac {d^{2} c \left (d \sqrt {-a b}+c b \right ) \left (\sinIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\cosineIntegral \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 )}{b^{2} \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\frac {d^{2} c \left (d \sqrt {-a b}-c b \right ) \left (\sinIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\cosineIntegral \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 )}{b^{2} \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}-\frac {d^{2} \cos \left (d x +c \right )}{b}+\frac {d^{2} \left (d^{2} a +b \,c^{2}-2 c \left (d \sqrt {-a b}+c b \right )\right ) \left (\sinIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\cosineIntegral \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 (d^{2} a +b \,c^{2}+2 c \left (d \sqrt {-a b}-c b \right )\right ) \left (\sinIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\cosineIntegral \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 )}}{d^{3}}\)	\(782\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^2+a),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains complex when optimal does not.
time = 0.37, size = 195, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {\frac {a d^{2}}{b}} {\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}} {\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}} {\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}} {\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 \, \cos \left (d x + c\right )}{4 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}}{a + b x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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