Integrand size = 16, antiderivative size = 856 \[ \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {d \cos (c+d x)}{16 (-a)^{3/2} b \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 (-a)^{3/2} b \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^2 b}-\frac {3 d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^2 b}-\frac {3 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {3 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {d^2 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )^2}-\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )^2}+\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{5/2} \sqrt {b}}+\frac {d^2 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {3 d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^2 b}-\frac {3 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{5/2} \sqrt {b}}+\frac {d^2 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {3 d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^2 b} \]
-3/16*d*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*cos(c-d*(-a)^(1/2)/b^(1/2))/a^2/b-3/1 6*d*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*cos(c+d*(-a)^(1/2)/b^(1/2))/a^2/b-1/16*d ^2*cos(c+d*(-a)^(1/2)/b^(1/2))*Si(d*x-d*(-a)^(1/2)/b^(1/2))/(-a)^(3/2)/b^( 3/2)+1/16*d^2*cos(c-d*(-a)^(1/2)/b^(1/2))*Si(d*x+d*(-a)^(1/2)/b^(1/2))/(-a )^(3/2)/b^(3/2)+1/16*d^2*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b ^(1/2))/(-a)^(3/2)/b^(3/2)+3/16*d*Si(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a )^(1/2)/b^(1/2))/a^2/b-1/16*d^2*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a) ^(1/2)/b^(1/2))/(-a)^(3/2)/b^(3/2)+3/16*d*Si(d*x-d*(-a)^(1/2)/b^(1/2))*sin (c+d*(-a)^(1/2)/b^(1/2))/a^2/b+3/16*cos(c+d*(-a)^(1/2)/b^(1/2))*Si(d*x-d*( -a)^(1/2)/b^(1/2))/(-a)^(5/2)/b^(1/2)-3/16*cos(c-d*(-a)^(1/2)/b^(1/2))*Si( d*x+d*(-a)^(1/2)/b^(1/2))/(-a)^(5/2)/b^(1/2)-3/16*Ci(d*x+d*(-a)^(1/2)/b^(1 /2))*sin(c-d*(-a)^(1/2)/b^(1/2))/(-a)^(5/2)/b^(1/2)+3/16*Ci(-d*x+d*(-a)^(1 /2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2))/(-a)^(5/2)/b^(1/2)-1/16*sin(d*x+c )/(-a)^(3/2)/b^(1/2)/((-a)^(1/2)-x*b^(1/2))^2+1/16*d*cos(d*x+c)/(-a)^(3/2) /b/((-a)^(1/2)-x*b^(1/2))-3/16*sin(d*x+c)/a^2/b^(1/2)/((-a)^(1/2)-x*b^(1/2 ))+1/16*sin(d*x+c)/(-a)^(3/2)/b^(1/2)/((-a)^(1/2)+x*b^(1/2))^2+1/16*d*cos( d*x+c)/(-a)^(3/2)/b/((-a)^(1/2)+x*b^(1/2))+3/16*sin(d*x+c)/a^2/b^(1/2)/((- a)^(1/2)+x*b^(1/2))
Result contains complex when optimal does not.
Time = 2.99 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.44 \[ \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (\left (3 b-3 \sqrt {a} \sqrt {b} d+a d^2\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )-\left (3 b+3 \sqrt {a} \sqrt {b} d+a d^2\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )+e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (\left (3 b-3 \sqrt {a} \sqrt {b} d+a d^2\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )-\left (3 b+3 \sqrt {a} \sqrt {b} d+a d^2\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )+\frac {4 \sqrt {a} \sqrt {b} \cos (d x) \left (a d \left (a+b x^2\right ) \cos (c)+b x \left (5 a+3 b x^2\right ) \sin (c)\right )}{\left (a+b x^2\right )^2}-\frac {4 \sqrt {a} \sqrt {b} \left (-b x \left (5 a+3 b x^2\right ) \cos (c)+a d \left (a+b x^2\right ) \sin (c)\right ) \sin (d x)}{\left (a+b x^2\right )^2}}{32 a^{5/2} b^{3/2}} \]
(E^((-I)*c - (Sqrt[a]*d)/Sqrt[b])*((3*b - 3*Sqrt[a]*Sqrt[b]*d + a*d^2)*E^( (2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[-((Sqrt[a]*d)/Sqrt[b]) - I*d*x] - (3* b + 3*Sqrt[a]*Sqrt[b]*d + a*d^2)*ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] - I*d*x ]) + E^(I*c - (Sqrt[a]*d)/Sqrt[b])*((3*b - 3*Sqrt[a]*Sqrt[b]*d + a*d^2)*E^ ((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x] - (3 *b + 3*Sqrt[a]*Sqrt[b]*d + a*d^2)*ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] + I*d* x]) + (4*Sqrt[a]*Sqrt[b]*Cos[d*x]*(a*d*(a + b*x^2)*Cos[c] + b*x*(5*a + 3*b *x^2)*Sin[c]))/(a + b*x^2)^2 - (4*Sqrt[a]*Sqrt[b]*(-(b*x*(5*a + 3*b*x^2)*C os[c]) + a*d*(a + b*x^2)*Sin[c])*Sin[d*x])/(a + b*x^2)^2)/(32*a^(5/2)*b^(3 /2))
Time = 1.59 (sec) , antiderivative size = 856, 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.
| \(\displaystyle \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 3814 |
| \(\displaystyle \int \left (-\frac {3 b \sin (c+d x)}{8 a^2 \left (-a b-b^2 x^2\right )}-\frac {3 b \sin (c+d x)}{16 a^2 \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {3 b \sin (c+d x)}{16 a^2 \left (\sqrt {-a} \sqrt {b}+b x\right )^2}-\frac {b^{3/2} \sin (c+d x)}{8 (-a)^{3/2} \left (\sqrt {-a} \sqrt {b}-b x\right )^3}-\frac {b^{3/2} \sin (c+d x)}{8 (-a)^{3/2} \left (\sqrt {-a} \sqrt {b}+b x\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{3/2} b^{3/2}}-\frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{3/2} b^{3/2}}+\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d^2}{16 (-a)^{3/2} b^{3/2}}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{3/2} b^{3/2}}+\frac {\cos (c+d x) d}{16 (-a)^{3/2} b \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cos (c+d x) d}{16 (-a)^{3/2} b \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {3 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d}{16 a^2 b}-\frac {3 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^2 b}-\frac {3 \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d}{16 a^2 b}+\frac {3 \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^2 b}-\frac {3 \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}+\frac {3 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {b} x+\sqrt {-a}\right )^2}-\frac {3 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {3 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}\) |
(d*Cos[c + d*x])/(16*(-a)^(3/2)*b*(Sqrt[-a] - Sqrt[b]*x)) + (d*Cos[c + d*x ])/(16*(-a)^(3/2)*b*(Sqrt[-a] + Sqrt[b]*x)) - (3*d*Cos[c + (Sqrt[-a]*d)/Sq rt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^2*b) - (3*d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^2*b) - (3*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]] )/(16*(-a)^(5/2)*Sqrt[b]) + (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*S in[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(3/2)*b^(3/2)) + (3*CosIntegral[(Sq rt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(5/2)*Sqr t[b]) - (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/ Sqrt[b]])/(16*(-a)^(3/2)*b^(3/2)) - Sin[c + d*x]/(16*(-a)^(3/2)*Sqrt[b]*(S qrt[-a] - Sqrt[b]*x)^2) - (3*Sin[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a] - Sqr t[b]*x)) + Sin[c + d*x]/(16*(-a)^(3/2)*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x)^2) + (3*Sin[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x)) - (3*Cos[c + (Sq rt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(5/2) *Sqrt[b]) + (d^2*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sq rt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) - (3*d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]] *SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^2*b) - (3*Cos[c - (Sqrt[-a ]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(5/2)*Sqrt [b]) + (d^2*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(3/2)*b^(3/2)) + (3*d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*S...
3.1.75.3.1 Defintions of rubi rules used
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])
Time = 0.52 (sec) , antiderivative size = 598, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(d^{5} \left (-\frac {\sin \left (d x +c \right ) \left (5 a c \,d^{2}-5 a \,d^{2} \left (d x +c \right )+3 c^{3} b -9 b \,c^{2} \left (d x +c \right )+9 b c \left (d x +c \right )^{2}-3 b \left (d x +c \right )^{3}\right )}{8 a^{2} d^{4} \left (a \,d^{2}+c^{2} b -2 b c \left (d x +c \right )+b \left (d x +c \right )^{2}\right )^{2}}+\frac {\cos \left (d x +c \right )}{8 a b \,d^{2} \left (a \,d^{2}+c^{2} b -2 b c \left (d x +c \right )+b \left (d x +c \right )^{2}\right )}-\frac {\left (a \,d^{2}+3 b \right ) \left (\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{16 a^{2} d^{4} b^{2} \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\frac {\left (a \,d^{2}+3 b \right ) \left (\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{16 a^{2} d^{4} b^{2} \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}-\frac {3 \left (-\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{16 a^{2} d^{4} b}-\frac {3 \left (\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )+\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{16 a^{2} d^{4} b}\right )\) | \(598\) |
| default | \(d^{5} \left (-\frac {\sin \left (d x +c \right ) \left (5 a c \,d^{2}-5 a \,d^{2} \left (d x +c \right )+3 c^{3} b -9 b \,c^{2} \left (d x +c \right )+9 b c \left (d x +c \right )^{2}-3 b \left (d x +c \right )^{3}\right )}{8 a^{2} d^{4} \left (a \,d^{2}+c^{2} b -2 b c \left (d x +c \right )+b \left (d x +c \right )^{2}\right )^{2}}+\frac {\cos \left (d x +c \right )}{8 a b \,d^{2} \left (a \,d^{2}+c^{2} b -2 b c \left (d x +c \right )+b \left (d x +c \right )^{2}\right )}-\frac {\left (a \,d^{2}+3 b \right ) \left (\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{16 a^{2} d^{4} b^{2} \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\frac {\left (a \,d^{2}+3 b \right ) \left (\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{16 a^{2} d^{4} b^{2} \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}-\frac {3 \left (-\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{16 a^{2} d^{4} b}-\frac {3 \left (\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )+\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{16 a^{2} d^{4} b}\right )\) | \(598\) |
| risch | \(-\frac {d^{2} \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 )}{32 a^{2} b^{2}}+\frac {d^{2} \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 )}{32 a^{2} b^{2}}+\frac {3 d \,{\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 )}{32 a^{2} b}+\frac {3 d \,{\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 )}{32 a^{2} b}-\frac {3 \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 )}{32 a^{3} b}+\frac {3 \,{\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}}{32 a^{3} b}+\frac {d^{2} {\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}}{32 a^{2} b^{2}}-\frac {d^{2} {\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}}{32 a^{2} b^{2}}+\frac {3 d \,{\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 )}{32 a^{2} b}+\frac {3 d \,{\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 )}{32 a^{2} b}+\frac {3 \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 )}{32 a^{3} b}-\frac {3 \,{\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}}{32 a^{3} b}-\frac {d^{3} \left (d^{2} x^{2} b +a \,d^{2}\right ) \cos \left (d x +c \right )}{8 a b \left (-b^{2} x^{4} d^{4}-2 a b \,d^{4} x^{2}-a^{2} d^{4}\right )}-\frac {d \left (3 d^{3} x^{3} b +5 a \,d^{3} x \right ) \sin \left (d x +c \right )}{8 a^{2} \left (-b^{2} x^{4} d^{4}-2 a b \,d^{4} x^{2}-a^{2} d^{4}\right )}\) | \(902\) |
d^5*(-1/8*sin(d*x+c)*(5*a*c*d^2-5*a*d^2*(d*x+c)+3*c^3*b-9*b*c^2*(d*x+c)+9* b*c*(d*x+c)^2-3*b*(d*x+c)^3)/a^2/d^4/(a*d^2+c^2*b-2*b*c*(d*x+c)+b*(d*x+c)^ 2)^2+1/8*cos(d*x+c)/a/b/d^2/(a*d^2+c^2*b-2*b*c*(d*x+c)+b*(d*x+c)^2)-1/16*( a*d^2+3*b)/a^2/d^4/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)*si n((d*(-a*b)^(1/2)+c*b)/b))-1/16*(a*d^2+3*b)/a^2/d^4/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))-3/16/a^2/d^4/b* (-Si(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*sin((d*(-a*b)^(1/2)+c*b)/b)+Ci(d*x+c-(d *(-a*b)^(1/2)+c*b)/b)*cos((d*(-a*b)^(1/2)+c*b)/b))-3/16/a^2/d^4/b*(Si(d*x+ c+(d*(-a*b)^(1/2)-c*b)/b)*sin((d*(-a*b)^(1/2)-c*b)/b)+Ci(d*x+c+(d*(-a*b)^( 1/2)-c*b)/b)*cos((d*(-a*b)^(1/2)-c*b)/b)))
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 611, normalized size of antiderivative = 0.71 \[ \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=-\frac {{\left (3 \, a b^{2} d^{2} x^{4} + 6 \, a^{2} b d^{2} x^{2} + 3 \, a^{3} d^{2} - {\left (a^{3} d^{2} + {\left (a b^{2} d^{2} + 3 \, b^{3}\right )} x^{4} + 3 \, a^{2} b + 2 \, {\left (a^{2} b d^{2} + 3 \, a b^{2}\right )} x^{2}\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (3 \, a b^{2} d^{2} x^{4} + 6 \, a^{2} b d^{2} x^{2} + 3 \, a^{3} d^{2} + {\left (a^{3} d^{2} + {\left (a b^{2} d^{2} + 3 \, b^{3}\right )} x^{4} + 3 \, a^{2} b + 2 \, {\left (a^{2} b d^{2} + 3 \, a b^{2}\right )} x^{2}\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (3 \, a b^{2} d^{2} x^{4} + 6 \, a^{2} b d^{2} x^{2} + 3 \, a^{3} d^{2} - {\left (a^{3} d^{2} + {\left (a b^{2} d^{2} + 3 \, b^{3}\right )} x^{4} + 3 \, a^{2} b + 2 \, {\left (a^{2} b d^{2} + 3 \, a b^{2}\right )} x^{2}\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (3 \, a b^{2} d^{2} x^{4} + 6 \, a^{2} b d^{2} x^{2} + 3 \, a^{3} d^{2} + {\left (a^{3} d^{2} + {\left (a b^{2} d^{2} + 3 \, b^{3}\right )} x^{4} + 3 \, a^{2} b + 2 \, {\left (a^{2} b d^{2} + 3 \, a b^{2}\right )} x^{2}\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 4 \, {\left (a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cos \left (d x + c\right ) - 4 \, {\left (3 \, a b^{2} d x^{3} + 5 \, a^{2} b d x\right )} \sin \left (d x + c\right )}{32 \, {\left (a^{3} b^{3} d x^{4} + 2 \, a^{4} b^{2} d x^{2} + a^{5} b d\right )}} \]
-1/32*((3*a*b^2*d^2*x^4 + 6*a^2*b*d^2*x^2 + 3*a^3*d^2 - (a^3*d^2 + (a*b^2* d^2 + 3*b^3)*x^4 + 3*a^2*b + 2*(a^2*b*d^2 + 3*a*b^2)*x^2)*sqrt(a*d^2/b))*E i(I*d*x - sqrt(a*d^2/b))*e^(I*c + sqrt(a*d^2/b)) + (3*a*b^2*d^2*x^4 + 6*a^ 2*b*d^2*x^2 + 3*a^3*d^2 + (a^3*d^2 + (a*b^2*d^2 + 3*b^3)*x^4 + 3*a^2*b + 2 *(a^2*b*d^2 + 3*a*b^2)*x^2)*sqrt(a*d^2/b))*Ei(I*d*x + sqrt(a*d^2/b))*e^(I* c - sqrt(a*d^2/b)) + (3*a*b^2*d^2*x^4 + 6*a^2*b*d^2*x^2 + 3*a^3*d^2 - (a^3 *d^2 + (a*b^2*d^2 + 3*b^3)*x^4 + 3*a^2*b + 2*(a^2*b*d^2 + 3*a*b^2)*x^2)*sq rt(a*d^2/b))*Ei(-I*d*x - sqrt(a*d^2/b))*e^(-I*c + sqrt(a*d^2/b)) + (3*a*b^ 2*d^2*x^4 + 6*a^2*b*d^2*x^2 + 3*a^3*d^2 + (a^3*d^2 + (a*b^2*d^2 + 3*b^3)*x ^4 + 3*a^2*b + 2*(a^2*b*d^2 + 3*a*b^2)*x^2)*sqrt(a*d^2/b))*Ei(-I*d*x + sqr t(a*d^2/b))*e^(-I*c - sqrt(a*d^2/b)) - 4*(a^2*b*d^2*x^2 + a^3*d^2)*cos(d*x + c) - 4*(3*a*b^2*d*x^3 + 5*a^2*b*d*x)*sin(d*x + c))/(a^3*b^3*d*x^4 + 2*a ^4*b^2*d*x^2 + a^5*b*d)
Timed out. \[ \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]
\[ \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int \frac {\sin \left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^3} \,d x \]