Integrand size = 19, antiderivative size = 875 \[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx=\frac {d \cos (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d \cos (c) \operatorname {CosIntegral}(d x)}{a^3}+\frac {7 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^3}+\frac {7 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^3}-\frac {15 \sqrt {b} \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}}+\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)^{5/2} \sqrt {b}}+\frac {15 \sqrt {b} \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}}-\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)^{5/2} \sqrt {b}}-\frac {\sin (c+d x)}{a^3 x}-\frac {\sqrt {b} \sin (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {7 \sqrt {b} \sin (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} \sin (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}+\sqrt {b} x\right )^2}-\frac {7 \sqrt {b} \sin (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {d \sin (c) \text {Si}(d x)}{a^3}-\frac {15 \sqrt {b} \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}+\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)^{5/2} \sqrt {b}}+\frac {7 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^3}-\frac {15 \sqrt {b} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{7/2}}+\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)^{5/2} \sqrt {b}}-\frac {7 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^3} \]
d*Ci(d*x)*cos(c)/a^3+7/16*d*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*cos(c-d*(-a)^(1/2 )/b^(1/2))/a^3+7/16*d*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*cos(c+d*(-a)^(1/2)/b^( 1/2))/a^3-d*Si(d*x)*sin(c)/a^3-sin(d*x+c)/a^3/x-7/16*d*Si(d*x+d*(-a)^(1/2) /b^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))/a^3-7/16*d*Si(d*x-d*(-a)^(1/2)/b^(1/ 2))*sin(c+d*(-a)^(1/2)/b^(1/2))/a^3-1/16*d^2*cos(c+d*(-a)^(1/2)/b^(1/2))*S i(d*x-d*(-a)^(1/2)/b^(1/2))/(-a)^(5/2)/b^(1/2)+1/16*d^2*cos(c-d*(-a)^(1/2) /b^(1/2))*Si(d*x+d*(-a)^(1/2)/b^(1/2))/(-a)^(5/2)/b^(1/2)+1/16*d^2*Ci(d*x+ d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))/(-a)^(5/2)/b^(1/2)-1/16* d^2*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2))/(-a)^(5/2)/b ^(1/2)+15/16*cos(c+d*(-a)^(1/2)/b^(1/2))*Si(d*x-d*(-a)^(1/2)/b^(1/2))*b^(1 /2)/(-a)^(7/2)-15/16*cos(c-d*(-a)^(1/2)/b^(1/2))*Si(d*x+d*(-a)^(1/2)/b^(1/ 2))*b^(1/2)/(-a)^(7/2)-15/16*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/ 2)/b^(1/2))*b^(1/2)/(-a)^(7/2)+15/16*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*sin(c+d *(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(7/2)-1/16*sin(d*x+c)*b^(1/2)/(-a)^(5/2) /((-a)^(1/2)-x*b^(1/2))^2+1/16*d*cos(d*x+c)/(-a)^(5/2)/((-a)^(1/2)-x*b^(1/ 2))+7/16*sin(d*x+c)*b^(1/2)/a^3/((-a)^(1/2)-x*b^(1/2))+1/16*sin(d*x+c)*b^( 1/2)/(-a)^(5/2)/((-a)^(1/2)+x*b^(1/2))^2+1/16*d*cos(d*x+c)/(-a)^(5/2)/((-a )^(1/2)+x*b^(1/2))-7/16*sin(d*x+c)*b^(1/2)/a^3/((-a)^(1/2)+x*b^(1/2))
Result contains complex when optimal does not.
Time = 4.04 (sec) , antiderivative size = 593, normalized size of antiderivative = 0.68 \[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx=\frac {8 \sqrt {b} e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (-e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )+\frac {e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (-\left (\left (7 b-7 \sqrt {a} \sqrt {b} d+a d^2\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )+\left (7 b+7 \sqrt {a} \sqrt {b} d+a d^2\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )}{\sqrt {b}}+8 \sqrt {b} e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (-e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )+\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )+\frac {e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (-\left (\left (7 b-7 \sqrt {a} \sqrt {b} d+a d^2\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )+\left (7 b+7 \sqrt {a} \sqrt {b} d+a d^2\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )}{\sqrt {b}}-\frac {4 \sqrt {a} \cos (d x) \left (a d x \left (a+b x^2\right ) \cos (c)+\left (8 a^2+25 a b x^2+15 b^2 x^4\right ) \sin (c)\right )}{x \left (a+b x^2\right )^2}+\frac {4 \sqrt {a} \left (-\left (\left (8 a^2+25 a b x^2+15 b^2 x^4\right ) \cos (c)\right )+a d x \left (a+b x^2\right ) \sin (c)\right ) \sin (d x)}{x \left (a+b x^2\right )^2}+32 \sqrt {a} d (\cos (c) \operatorname {CosIntegral}(d x)-\sin (c) \text {Si}(d x))}{32 a^{7/2}} \]
(8*Sqrt[b]*E^((-I)*c - (Sqrt[a]*d)/Sqrt[b])*(-(E^((2*Sqrt[a]*d)/Sqrt[b])*E xpIntegralEi[-((Sqrt[a]*d)/Sqrt[b]) - I*d*x]) + ExpIntegralEi[(Sqrt[a]*d)/ Sqrt[b] - I*d*x]) + (E^((-I)*c - (Sqrt[a]*d)/Sqrt[b])*(-((7*b - 7*Sqrt[a]* Sqrt[b]*d + a*d^2)*E^((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[-((Sqrt[a]*d)/S qrt[b]) - I*d*x]) + (7*b + 7*Sqrt[a]*Sqrt[b]*d + a*d^2)*ExpIntegralEi[(Sqr t[a]*d)/Sqrt[b] - I*d*x]))/Sqrt[b] + 8*Sqrt[b]*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^(I*c - (Sqrt[a]*d) /Sqrt[b])*(-((7*b - 7*Sqrt[a]*Sqrt[b]*d + a*d^2)*E^((2*Sqrt[a]*d)/Sqrt[b]) *ExpIntegralEi[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x]) + (7*b + 7*Sqrt[a]*Sqrt[b] *d + a*d^2)*ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] + I*d*x]))/Sqrt[b] - (4*Sqrt [a]*Cos[d*x]*(a*d*x*(a + b*x^2)*Cos[c] + (8*a^2 + 25*a*b*x^2 + 15*b^2*x^4) *Sin[c]))/(x*(a + b*x^2)^2) + (4*Sqrt[a]*(-((8*a^2 + 25*a*b*x^2 + 15*b^2*x ^4)*Cos[c]) + a*d*x*(a + b*x^2)*Sin[c])*Sin[d*x])/(x*(a + b*x^2)^2) + 32*S qrt[a]*d*(Cos[c]*CosIntegral[d*x] - Sin[c]*SinIntegral[d*x]))/(32*a^(7/2))
Time = 2.75 (sec) , antiderivative size = 875, 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.
| \(\displaystyle \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 3826 |
| \(\displaystyle \int \left (-\frac {b \sin (c+d x)}{a^3 \left (a+b x^2\right )}+\frac {\sin (c+d x)}{a^3 x^2}-\frac {b \sin (c+d x)}{a^2 \left (a+b x^2\right )^2}-\frac {b \sin (c+d x)}{a \left (a+b x^2\right )^3}\right )dx\) |
| \(\Big \downarrow \) 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)^{5/2} \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 ) d^2}{16 (-a)^{5/2} \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 ) d^2}{16 (-a)^{5/2} \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 ) d^2}{16 (-a)^{5/2} \sqrt {b}}+\frac {\cos (c+d x) d}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cos (c+d x) d}{16 (-a)^{5/2} \left (\sqrt {b} x+\sqrt {-a}\right )}+\frac {\cos (c) \operatorname {CosIntegral}(d x) d}{a^3}+\frac {7 \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^3}+\frac {7 \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^3}-\frac {\sin (c) \text {Si}(d x) d}{a^3}+\frac {7 \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^3}-\frac {7 \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^3}-\frac {15 \sqrt {b} \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}}+\frac {15 \sqrt {b} \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}}-\frac {\sin (c+d x)}{a^3 x}+\frac {7 \sqrt {b} \sin (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {7 \sqrt {b} \sin (c+d x)}{16 a^3 \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {\sqrt {b} \sin (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {\sqrt {b} \sin (c+d x)}{16 (-a)^{5/2} \left (\sqrt {b} x+\sqrt {-a}\right )^2}-\frac {15 \sqrt {b} \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}-\frac {15 \sqrt {b} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}}\) |
(d*Cos[c + d*x])/(16*(-a)^(5/2)*(Sqrt[-a] - Sqrt[b]*x)) + (d*Cos[c + d*x]) /(16*(-a)^(5/2)*(Sqrt[-a] + Sqrt[b]*x)) + (d*Cos[c]*CosIntegral[d*x])/a^3 + (7*d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d* x])/(16*a^3) + (7*d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d) /Sqrt[b] + d*x])/(16*a^3) - (15*Sqrt[b]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(7/2)) + (d^2*CosIntegral[(S qrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(5/2)*Sq rt[b]) + (15*Sqrt[b]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt [-a]*d)/Sqrt[b]])/(16*(-a)^(7/2)) - (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(5/2)*Sqrt[b]) - Sin[c + d* x]/(a^3*x) - (Sqrt[b]*Sin[c + d*x])/(16*(-a)^(5/2)*(Sqrt[-a] - Sqrt[b]*x)^ 2) + (7*Sqrt[b]*Sin[c + d*x])/(16*a^3*(Sqrt[-a] - Sqrt[b]*x)) + (Sqrt[b]*S in[c + d*x])/(16*(-a)^(5/2)*(Sqrt[-a] + Sqrt[b]*x)^2) - (7*Sqrt[b]*Sin[c + d*x])/(16*a^3*(Sqrt[-a] + Sqrt[b]*x)) - (d*Sin[c]*SinIntegral[d*x])/a^3 - (15*Sqrt[b]*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b ] - d*x])/(16*(-a)^(7/2)) + (d^2*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral [(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(5/2)*Sqrt[b]) + (7*d*Sin[c + (Sqrt [-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^3) - (15*S qrt[b]*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d* x])/(16*(-a)^(7/2)) + (d^2*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(S...
3.1.77.3.1 Defintions of rubi rules used
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]
Result contains complex when optimal does not.
Time = 0.80 (sec) , antiderivative size = 910, normalized size of antiderivative = 1.04
| method | result | size |
| risch | \(\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 )}{32 a^{2} \sqrt {a 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 )}{32 a^{2} \sqrt {a b}}-\frac {7 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^{3}}-\frac {7 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^{3}}+\frac {15 \,{\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 ) b}{32 a^{3} \sqrt {a b}}-\frac {15 \,{\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 ) b}{32 a^{3} \sqrt {a b}}-\frac {d \,\operatorname {Ei}_{1}\left (-i d x \right ) {\mathrm e}^{i c}}{2 a^{3}}-\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 )}{32 a^{2} \sqrt {a 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 )}{32 a^{2} \sqrt {a b}}-\frac {7 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^{3}}-\frac {7 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^{3}}-\frac {15 \,{\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 ) b}{32 a^{3} \sqrt {a b}}+\frac {15 \,{\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 ) b}{32 a^{3} \sqrt {a b}}-\frac {d \,\operatorname {Ei}_{1}\left (i d x \right ) {\mathrm e}^{-i c}}{2 a^{3}}+\frac {d^{2} \left (d^{3} x^{3} b +a \,d^{3} x \right ) \cos \left (d x +c \right )}{8 a^{2} x \left (-b^{2} x^{4} d^{4}-2 a b \,d^{4} x^{2}-a^{2} d^{4}\right )}-\frac {\left (-15 b^{2} x^{4} d^{4}-25 a b \,d^{4} x^{2}-8 a^{2} d^{4}\right ) \sin \left (d x +c \right )}{8 a^{3} x \left (-b^{2} x^{4} d^{4}-2 a b \,d^{4} x^{2}-a^{2} d^{4}\right )}\) | \(910\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1363\) |
| default | \(\text {Expression too large to display}\) | \(1363\) |
1/32/a^2*d^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)-1/32/a^2*d^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)-7/32*d/a^3*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)-7/32*d/a^3*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)+15 /32/a^3/(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)*b-15/32/a^3/(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)*b-1/2*d/a^3*Ei(1,-I*d*x)*exp(I *c)-1/32/a^2*d^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)+1/32/a^2*d^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)-7/32*d/a^3*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)-7/32*d /a^3*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)-15/32/a^3/(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)*b+15/32/a^3/(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)*b-1/2*d/a^3*Ei(1,I*d *x)*exp(-I*c)+1/8/a^2*d^2*(b*d^3*x^3+a*d^3*x)/x/(-b^2*d^4*x^4-2*a*b*d^4*x^ 2-a^2*d^4)*cos(d*x+c)-1/8*(-15*b^2*d^4*x^4-25*a*b*d^4*x^2-8*a^2*d^4)/a^3/x /(-b^2*d^4*x^4-2*a*b*d^4*x^2-a^2*d^4)*sin(d*x+c)
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 714, normalized size of antiderivative = 0.82 \[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx=\frac {32 \, {\left (a b^{2} d^{2} x^{5} + 2 \, a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \cos \left (c\right ) \operatorname {Ci}\left (d x\right ) + {\left (7 \, a b^{2} d^{2} x^{5} + 14 \, a^{2} b d^{2} x^{3} + 7 \, a^{3} d^{2} x - {\left ({\left (a b^{2} d^{2} + 15 \, b^{3}\right )} x^{5} + 2 \, {\left (a^{2} b d^{2} + 15 \, a b^{2}\right )} x^{3} + {\left (a^{3} d^{2} + 15 \, a^{2} b\right )} x\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 (7 \, a b^{2} d^{2} x^{5} + 14 \, a^{2} b d^{2} x^{3} + 7 \, a^{3} d^{2} x + {\left ({\left (a b^{2} d^{2} + 15 \, b^{3}\right )} x^{5} + 2 \, {\left (a^{2} b d^{2} + 15 \, a b^{2}\right )} x^{3} + {\left (a^{3} d^{2} + 15 \, a^{2} b\right )} x\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 (7 \, a b^{2} d^{2} x^{5} + 14 \, a^{2} b d^{2} x^{3} + 7 \, a^{3} d^{2} x - {\left ({\left (a b^{2} d^{2} + 15 \, b^{3}\right )} x^{5} + 2 \, {\left (a^{2} b d^{2} + 15 \, a b^{2}\right )} x^{3} + {\left (a^{3} d^{2} + 15 \, a^{2} b\right )} x\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 (7 \, a b^{2} d^{2} x^{5} + 14 \, a^{2} b d^{2} x^{3} + 7 \, a^{3} d^{2} x + {\left ({\left (a b^{2} d^{2} + 15 \, b^{3}\right )} x^{5} + 2 \, {\left (a^{2} b d^{2} + 15 \, a b^{2}\right )} x^{3} + {\left (a^{3} d^{2} + 15 \, a^{2} b\right )} x\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 )} - 32 \, {\left (a b^{2} d^{2} x^{5} + 2 \, a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \sin \left (c\right ) \operatorname {Si}\left (d x\right ) - 4 \, {\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \cos \left (d x + c\right ) - 4 \, {\left (15 \, a b^{2} d x^{4} + 25 \, a^{2} b d x^{2} + 8 \, a^{3} d\right )} \sin \left (d x + c\right )}{32 \, {\left (a^{4} b^{2} d x^{5} + 2 \, a^{5} b d x^{3} + a^{6} d x\right )}} \]
1/32*(32*(a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2*x)*cos(c)*cos_integral (d*x) + (7*a*b^2*d^2*x^5 + 14*a^2*b*d^2*x^3 + 7*a^3*d^2*x - ((a*b^2*d^2 + 15*b^3)*x^5 + 2*(a^2*b*d^2 + 15*a*b^2)*x^3 + (a^3*d^2 + 15*a^2*b)*x)*sqrt( a*d^2/b))*Ei(I*d*x - sqrt(a*d^2/b))*e^(I*c + sqrt(a*d^2/b)) + (7*a*b^2*d^2 *x^5 + 14*a^2*b*d^2*x^3 + 7*a^3*d^2*x + ((a*b^2*d^2 + 15*b^3)*x^5 + 2*(a^2 *b*d^2 + 15*a*b^2)*x^3 + (a^3*d^2 + 15*a^2*b)*x)*sqrt(a*d^2/b))*Ei(I*d*x + sqrt(a*d^2/b))*e^(I*c - sqrt(a*d^2/b)) + (7*a*b^2*d^2*x^5 + 14*a^2*b*d^2* x^3 + 7*a^3*d^2*x - ((a*b^2*d^2 + 15*b^3)*x^5 + 2*(a^2*b*d^2 + 15*a*b^2)*x ^3 + (a^3*d^2 + 15*a^2*b)*x)*sqrt(a*d^2/b))*Ei(-I*d*x - sqrt(a*d^2/b))*e^( -I*c + sqrt(a*d^2/b)) + (7*a*b^2*d^2*x^5 + 14*a^2*b*d^2*x^3 + 7*a^3*d^2*x + ((a*b^2*d^2 + 15*b^3)*x^5 + 2*(a^2*b*d^2 + 15*a*b^2)*x^3 + (a^3*d^2 + 15 *a^2*b)*x)*sqrt(a*d^2/b))*Ei(-I*d*x + sqrt(a*d^2/b))*e^(-I*c - sqrt(a*d^2/ b)) - 32*(a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2*x)*sin(c)*sin_integral (d*x) - 4*(a^2*b*d^2*x^3 + a^3*d^2*x)*cos(d*x + c) - 4*(15*a*b^2*d*x^4 + 2 5*a^2*b*d*x^2 + 8*a^3*d)*sin(d*x + c))/(a^4*b^2*d*x^5 + 2*a^5*b*d*x^3 + a^ 6*d*x)
Timed out. \[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3} x^{2}} \,d x } \]
\[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx=\int \frac {\sin \left (c+d\,x\right )}{x^2\,{\left (b\,x^2+a\right )}^3} \,d x \]