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

3.1.59.1 Optimal result

Integrand size = 19, antiderivative size = 227 \[ \int \frac {x^2 \sin (c+d x)}{a+b x^2} \, dx=-\frac {\cos (c+d x)}{b d}-\frac {\sqrt {-a} \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}+\frac {\sqrt {-a} \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^{3/2}} \]

output

-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)
 

3.1.59.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.65 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.95 \[ \int \frac {x^2 \sin (c+d x)}{a+b x^2} \, dx=-\frac {\cos (c) \cos (d x)}{b d}+\frac {\sqrt {a} e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (-e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )}{4 b^{3/2}}+\frac {\sqrt {a} e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (-e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )+\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )}{4 b^{3/2}}+\frac {\sin (c) \sin (d x)}{b d} \]

input

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

output

-((Cos[c]*Cos[d*x])/(b*d)) + (Sqrt[a]*E^((-I)*c - (Sqrt[a]*d)/Sqrt[b])*(-( 
E^((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[-((Sqrt[a]*d)/Sqrt[b]) - I*d*x]) + 
 ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] - I*d*x]))/(4*b^(3/2)) + (Sqrt[a]*E^(I* 
c - (Sqrt[a]*d)/Sqrt[b])*(-(E^((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[-((Sqr 
t[a]*d)/Sqrt[b]) + I*d*x]) + ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] + I*d*x]))/ 
(4*b^(3/2)) + (Sin[c]*Sin[d*x])/(b*d)
 

3.1.59.3 Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3826, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3826

\(\displaystyle \int \left (\frac {\sin (c+d x)}{b}-\frac {a \sin (c+d x)}{b \left (a+b x^2\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {-a} \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}+\frac {\sqrt {-a} \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}-\frac {\cos (c+d x)}{b d}\)

input

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

output

-(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 - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a] 
*d)/Sqrt[b] + d*x])/(2*b^(3/2))
 

3.1.59.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3826

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

3.1.59.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.16

method	result	size
risch	\(\frac {\sqrt {a b}\, {\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 b^{2}}-\frac {\sqrt {a b}\, {\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 b^{2}}-\frac {{\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) \sqrt {a b}}{4 b^{2}}+\frac {{\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) \sqrt {a b}}{4 b^{2}}-\frac {\cos \left (d x +c \right )}{b d}\)	\(264\)
derivativedivides	\(\frac {d^{2} c^{2} \left (-\frac {\operatorname {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 )+\operatorname {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 b \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\frac {\operatorname {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 )-\operatorname {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 b \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}\right )+\frac {d^{2} \left (d \sqrt {-a b}+c b \right ) c \left (\operatorname {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 )+\operatorname {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 )}{b^{2} \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\frac {d^{2} \left (d \sqrt {-a b}-c b \right ) c \left (\operatorname {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 )-\operatorname {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 )}{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 (a \,d^{2}+c^{2} b -2 c \left (d \sqrt {-a b}+c b \right )\right ) \left (\operatorname {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 )+\operatorname {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 (a \,d^{2}+c^{2} b +2 c \left (d \sqrt {-a b}-c b \right )\right ) \left (\operatorname {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 )-\operatorname {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 )}}{d^{3}}\)	\(782\)
default	\(\frac {d^{2} c^{2} \left (-\frac {\operatorname {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 )+\operatorname {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 b \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\frac {\operatorname {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 )-\operatorname {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 b \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}\right )+\frac {d^{2} \left (d \sqrt {-a b}+c b \right ) c \left (\operatorname {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 )+\operatorname {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 )}{b^{2} \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\frac {d^{2} \left (d \sqrt {-a b}-c b \right ) c \left (\operatorname {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 )-\operatorname {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 )}{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 (a \,d^{2}+c^{2} b -2 c \left (d \sqrt {-a b}+c b \right )\right ) \left (\operatorname {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 )+\operatorname {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 (a \,d^{2}+c^{2} b +2 c \left (d \sqrt {-a b}-c b \right )\right ) \left (\operatorname {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 )-\operatorname {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 )}}{d^{3}}\)	\(782\)

input

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

output

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

3.1.59.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.86 \[ \int \frac {x^2 \sin (c+d x)}{a+b x^2} \, dx=-\frac {\sqrt {\frac {a d^{2}}{b}} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - \sqrt {\frac {a d^{2}}{b}} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + \sqrt {\frac {a d^{2}}{b}} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - \sqrt {\frac {a d^{2}}{b}} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + 4 \, \cos \left (d x + c\right )}{4 \, b d} \]

input

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

output

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

3.1.59.6 Sympy [F]

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

input

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

output

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

3.1.59.7 Maxima [F]

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

input

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

output

-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*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)*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*si 
n(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)*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*sin(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)
 

3.1.59.8 Giac [F]

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

input

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

output

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

3.1.59.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \sin (c+d x)}{a+b x^2} \, dx=\int \frac {x^2\,\sin \left (c+d\,x\right )}{b\,x^2+a} \,d x \]

input

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

output

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