\(\int \frac {\sin (c+d x)}{(a+b x^2)^2} \, dx\) [69]

Optimal result

Integrand size = 16, antiderivative size = 476 \[ \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}-\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a b}+\frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {\sin (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a b} \] Output:

-1/4*d*cos(c+(-a)^(1/2)*d/b^(1/2))*Ci((-a)^(1/2)*d/b^(1/2)-d*x)/a/b-1/4*d* 
cos(c-(-a)^(1/2)*d/b^(1/2))*Ci((-a)^(1/2)*d/b^(1/2)+d*x)/a/b+1/4*Ci((-a)^( 
1/2)*d/b^(1/2)+d*x)*sin(c-(-a)^(1/2)*d/b^(1/2))/(-a)^(3/2)/b^(1/2)-1/4*Ci( 
(-a)^(1/2)*d/b^(1/2)-d*x)*sin(c+(-a)^(1/2)*d/b^(1/2))/(-a)^(3/2)/b^(1/2)-1 
/4*sin(d*x+c)/a/b^(1/2)/((-a)^(1/2)-b^(1/2)*x)+1/4*sin(d*x+c)/a/b^(1/2)/(( 
-a)^(1/2)+b^(1/2)*x)-1/4*cos(c+(-a)^(1/2)*d/b^(1/2))*Si(-(-a)^(1/2)*d/b^(1 
/2)+d*x)/(-a)^(3/2)/b^(1/2)+1/4*d*sin(c+(-a)^(1/2)*d/b^(1/2))*Si(-(-a)^(1/ 
2)*d/b^(1/2)+d*x)/a/b+1/4*cos(c-(-a)^(1/2)*d/b^(1/2))*Si((-a)^(1/2)*d/b^(1 
/2)+d*x)/(-a)^(3/2)/b^(1/2)+1/4*d*sin(c-(-a)^(1/2)*d/b^(1/2))*Si((-a)^(1/2 
)*d/b^(1/2)+d*x)/a/b
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.31 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.59 \[ \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (\left (\sqrt {b}-\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )-\left (\sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )}{b}+\frac {e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (\left (\sqrt {b}-\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )-\left (\sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )}{b}+\frac {4 \sqrt {a} x \cos (d x) \sin (c)}{a+b x^2}+\frac {4 \sqrt {a} x \cos (c) \sin (d x)}{a+b x^2}}{8 a^{3/2}} \] Input:

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

Output:

((E^((-I)*c - (Sqrt[a]*d)/Sqrt[b])*((Sqrt[b] - Sqrt[a]*d)*E^((2*Sqrt[a]*d) 
/Sqrt[b])*ExpIntegralEi[-((Sqrt[a]*d)/Sqrt[b]) - I*d*x] - (Sqrt[b] + Sqrt[ 
a]*d)*ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] - I*d*x]))/b + (E^(I*c - (Sqrt[a]* 
d)/Sqrt[b])*((Sqrt[b] - Sqrt[a]*d)*E^((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi 
[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x] - (Sqrt[b] + Sqrt[a]*d)*ExpIntegralEi[(Sq 
rt[a]*d)/Sqrt[b] + I*d*x]))/b + (4*Sqrt[a]*x*Cos[d*x]*Sin[c])/(a + b*x^2) 
+ (4*Sqrt[a]*x*Cos[c]*Sin[d*x])/(a + b*x^2))/(8*a^(3/2))
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 476, 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.

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

\(\Big \downarrow \) 3814

\(\displaystyle \int \left (-\frac {b \sin (c+d x)}{2 a \left (-a b-b^2 x^2\right )}-\frac {b \sin (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b \sin (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}+b x\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {d \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}-\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a b}-\frac {d \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}+\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a b}+\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {\sin (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}\)

Input:

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

Output:

-1/4*(d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d 
*x])/(a*b) - (d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqr 
t[b] + d*x])/(4*a*b) + (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (S 
qrt[-a]*d)/Sqrt[b]])/(4*(-a)^(3/2)*Sqrt[b]) - (CosIntegral[(Sqrt[-a]*d)/Sq 
rt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*(-a)^(3/2)*Sqrt[b]) - Sin[c 
 + d*x]/(4*a*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)) + Sin[c + d*x]/(4*a*Sqrt[b]*( 
Sqrt[-a] + Sqrt[b]*x)) + (Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[ 
-a]*d)/Sqrt[b] - d*x])/(4*(-a)^(3/2)*Sqrt[b]) - (d*Sin[c + (Sqrt[-a]*d)/Sq 
rt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*a*b) + (Cos[c - (Sqrt[- 
a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(3/2)*Sqrt 
[b]) + (d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + 
 d*x])/(4*a*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 = 1.19 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.03

method	result	size
derivativedivides	\(d^{3} \left (\frac {\sin \left (d x +c \right ) \left (\frac {d x +c}{2 a \,d^{2}}-\frac {c}{2 a \,d^{2}}\right )}{d^{2} a +b \,c^{2}-2 b c \left (d x +c \right )+b \left (d x +c \right )^{2}}-\frac {\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+b c}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+b c}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+b c}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+b c}{b}\right )}{4 a \,d^{2} b \left (-\frac {d \sqrt {-a b}+b c}{b}+c \right )}-\frac {\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-b c}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-b c}{b}\right )-\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-b c}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-b c}{b}\right )}{4 a \,d^{2} b \left (\frac {d \sqrt {-a b}-b c}{b}+c \right )}-\frac {-\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+b c}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+b c}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+b c}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+b c}{b}\right )}{4 a b \,d^{2}}-\frac {\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-b c}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-b c}{b}\right )+\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-b c}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-b c}{b}\right )}{4 a b \,d^{2}}\right )\)	\(491\)
default	\(d^{3} \left (\frac {\sin \left (d x +c \right ) \left (\frac {d x +c}{2 a \,d^{2}}-\frac {c}{2 a \,d^{2}}\right )}{d^{2} a +b \,c^{2}-2 b c \left (d x +c \right )+b \left (d x +c \right )^{2}}-\frac {\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+b c}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+b c}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+b c}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+b c}{b}\right )}{4 a \,d^{2} b \left (-\frac {d \sqrt {-a b}+b c}{b}+c \right )}-\frac {\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-b c}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-b c}{b}\right )-\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-b c}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-b c}{b}\right )}{4 a \,d^{2} b \left (\frac {d \sqrt {-a b}-b c}{b}+c \right )}-\frac {-\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+b c}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+b c}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+b c}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+b c}{b}\right )}{4 a b \,d^{2}}-\frac {\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-b c}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-b c}{b}\right )+\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-b c}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-b c}{b}\right )}{4 a b \,d^{2}}\right )\)	\(491\)
risch	\(\frac {d \,{\mathrm e}^{\frac {i b c +d \sqrt {a b}}{b}} \operatorname {expIntegral}_{1}\left (\frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 a b}+\frac {d \,{\mathrm e}^{\frac {i b c -d \sqrt {a b}}{b}} \operatorname {expIntegral}_{1}\left (\frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 a b}-\frac {\sqrt {a b}\, {\mathrm e}^{\frac {i b c +d \sqrt {a b}}{b}} \operatorname {expIntegral}_{1}\left (\frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 a^{2} b}+\frac {\sqrt {a b}\, {\mathrm e}^{\frac {i b c -d \sqrt {a b}}{b}} \operatorname {expIntegral}_{1}\left (\frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 a^{2} b}+\frac {d \,{\mathrm e}^{-\frac {i b c +d \sqrt {a b}}{b}} \operatorname {expIntegral}_{1}\left (-\frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 a b}+\frac {d \,{\mathrm e}^{-\frac {i b c -d \sqrt {a b}}{b}} \operatorname {expIntegral}_{1}\left (-\frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 a b}+\frac {\sqrt {a b}\, {\mathrm e}^{-\frac {i b c +d \sqrt {a b}}{b}} \operatorname {expIntegral}_{1}\left (-\frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 a^{2} b}-\frac {\sqrt {a b}\, {\mathrm e}^{-\frac {i b c -d \sqrt {a b}}{b}} \operatorname {expIntegral}_{1}\left (-\frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 a^{2} b}-\frac {d^{2} x \sin \left (d x +c \right )}{2 \left (-2 i \left (i d x +i c \right ) b c +\left (i d x +i c \right )^{2} b -d^{2} a -b \,c^{2}\right ) a}\)	\(565\)

Input:

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

Output:

d^3*(sin(d*x+c)*(1/2/a/d^2*(d*x+c)-1/2*c/a/d^2)/(d^2*a+b*c^2-2*b*c*(d*x+c) 
+b*(d*x+c)^2)-1/4/a/d^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))-1/4/a/d^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))-1/4/a/b/d^2*(-Si(d*x+c-(d*(-a 
*b)^(1/2)+b*c)/b)*sin((d*(-a*b)^(1/2)+b*c)/b)+Ci(d*x+c-(d*(-a*b)^(1/2)+b*c 
)/b)*cos((d*(-a*b)^(1/2)+b*c)/b))-1/4/a/b/d^2*(Si(d*x+c+(d*(-a*b)^(1/2)-b* 
c)/b)*sin((d*(-a*b)^(1/2)-b*c)/b)+Ci(d*x+c+(d*(-a*b)^(1/2)-b*c)/b)*cos((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 = 333, normalized size of antiderivative = 0.70 \[ \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {4 \, a b d x \sin \left (d x + c\right ) - {\left (a b d^{2} x^{2} + a^{2} d^{2} - {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (a b d^{2} x^{2} + a^{2} d^{2} + {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (a b d^{2} x^{2} + a^{2} d^{2} - {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (a b d^{2} x^{2} + a^{2} d^{2} + {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )}}{8 \, {\left (a^{2} b^{2} d x^{2} + a^{3} b d\right )}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int \frac {\sin \left (d x +c \right )}{b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \] Input:

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

Output:

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