\(\int \frac {\sin (c+d x)}{a+b x^2} \, dx\) [61]

Optimal result

Integrand size = 16, antiderivative size = 213 \[ \int \frac {\sin (c+d x)}{a+b x^2} \, dx=-\frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 \sqrt {-a} \sqrt {b}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.69 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.77 \[ \int \frac {\sin (c+d x)}{a+b x^2} \, dx=\frac {e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )-\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+e^{2 i c} \left (e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )-\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )\right )}{4 \sqrt {a} \sqrt {b}} \] Input:

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

Output:

(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] + E^((2*I)*c)*(E^((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[-((Sqrt[a]*d)/S 
qrt[b]) + I*d*x] - ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] + I*d*x])))/(4*Sqrt[a 
]*Sqrt[b])
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3814, 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.

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

\(\Big \downarrow \) 3814

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 3814

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])
 

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.06

method	result	size
derivativedivides	\(d \left (-\frac {\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+b c}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+b c}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+b c}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+b c}{b}\right )}{2 b \left (-\frac {d \sqrt {-a b}+b c}{b}+c \right )}-\frac {\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-b c}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-b c}{b}\right )-\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-b c}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-b c}{b}\right )}{2 b \left (\frac {d \sqrt {-a b}-b c}{b}+c \right )}\right )\)	\(225\)
default	\(d \left (-\frac {\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+b c}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+b c}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+b c}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+b c}{b}\right )}{2 b \left (-\frac {d \sqrt {-a b}+b c}{b}+c \right )}-\frac {\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-b c}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-b c}{b}\right )-\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-b c}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-b c}{b}\right )}{2 b \left (\frac {d \sqrt {-a b}-b c}{b}+c \right )}\right )\)	\(225\)
risch	\(\frac {\sqrt {a b}\, {\mathrm e}^{-\frac {i b c +d \sqrt {a b}}{b}} \operatorname {expIntegral}_{1}\left (-\frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a b}-\frac {\sqrt {a b}\, {\mathrm e}^{-\frac {i b c -d \sqrt {a b}}{b}} \operatorname {expIntegral}_{1}\left (-\frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a b}-\frac {\sqrt {a b}\, {\mathrm e}^{\frac {i b c +d \sqrt {a b}}{b}} \operatorname {expIntegral}_{1}\left (\frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a b}+\frac {\sqrt {a b}\, {\mathrm e}^{\frac {i b c -d \sqrt {a b}}{b}} \operatorname {expIntegral}_{1}\left (\frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a b}\)	\(262\)

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.88 \[ \int \frac {\sin (c+d x)}{a+b x^2} \, dx=\frac {\sqrt {\frac {a d^{2}}{b}} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - \sqrt {\frac {a d^{2}}{b}} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + \sqrt {\frac {a d^{2}}{b}} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - \sqrt {\frac {a d^{2}}{b}} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )}}{4 \, a d} \] Input:

integrate(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)) - sqr 
t(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)))/(a*d)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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