Integrand size = 19, antiderivative size = 209 \[ \int \frac {x^3 \sin (c+d x)}{a+b x^2} \, dx=-\frac {x \cos (c+d x)}{b d}-\frac {a \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}-\frac {a \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\sin (c+d x)}{b d^2}+\frac {a \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}-\frac {a \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^2} \]
-x*cos(d*x+c)/b/d-1/2*a*cos(c+d*(-a)^(1/2)/b^(1/2))*Si(d*x-d*(-a)^(1/2)/b^ (1/2))/b^2-1/2*a*cos(c-d*(-a)^(1/2)/b^(1/2))*Si(d*x+d*(-a)^(1/2)/b^(1/2))/ b^2+sin(d*x+c)/b/d^2-1/2*a*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2) /b^(1/2))/b^2-1/2*a*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/ 2))/b^2
Result contains complex when optimal does not.
Time = 1.11 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.02 \[ \int \frac {x^3 \sin (c+d x)}{a+b x^2} \, dx=\frac {-i a e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )+i a e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )+\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )-\frac {4 b \cos (d x) (d x \cos (c)-\sin (c))}{d^2}+\frac {4 b (\cos (c)+d x \sin (c)) \sin (d x)}{d^2}}{4 b^2} \]
((-I)*a*E^((-I)*c - (Sqrt[a]*d)/Sqrt[b])*(E^((2*Sqrt[a]*d)/Sqrt[b])*ExpInt egralEi[-((Sqrt[a]*d)/Sqrt[b]) - I*d*x] + ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b ] - I*d*x]) + I*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*Cos[d*x]*(d*x*Cos[c] - Sin[c]))/d^2 + (4*b*(Cos[ c] + d*x*Sin[c])*Sin[d*x])/d^2)/(4*b^2)
Time = 0.54 (sec) , antiderivative size = 209, 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.
\(\displaystyle \int \frac {x^3 \sin (c+d x)}{a+b x^2} \, dx\) |
\(\Big \downarrow \) 3826 |
\(\displaystyle \int \left (\frac {x \sin (c+d x)}{b}-\frac {a x \sin (c+d x)}{b \left (a+b x^2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}-\frac {a \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}+\frac {a \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}-\frac {a \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\sin (c+d x)}{b d^2}-\frac {x \cos (c+d x)}{b d}\) |
-((x*Cos[c + d*x])/(b*d)) - (a*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin [c - (Sqrt[-a]*d)/Sqrt[b]])/(2*b^2) - (a*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*b^2) + Sin[c + d*x]/(b*d^2) + (a* Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2* b^2) - (a*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^2)
3.1.58.3.1 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 = 0.38 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.27
method | result | size |
risch | \(-\frac {i \operatorname {Ei}_{1}\left (\frac {i c b -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) {\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} a}{4 b^{2}}-\frac {i \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) {\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} a}{4 b^{2}}+\frac {i \operatorname {Ei}_{1}\left (-\frac {i c b -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) {\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} a}{4 b^{2}}+\frac {i \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) {\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}} a}{4 b^{2}}-\frac {x \cos \left (d x +c \right )}{b d}+\frac {\sin \left (d x +c \right )}{b \,d^{2}}\) | \(266\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1164\) |
default | \(\text {Expression too large to display}\) | \(1164\) |
-1/4*I/b^2*Ei(1,(I*c*b-d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)*exp((I*c*b-d*(a*b)^ (1/2))/b)*a-1/4*I/b^2*Ei(1,(I*c*b+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)*exp((I*c *b+d*(a*b)^(1/2))/b)*a+1/4*I/b^2*Ei(1,-(I*c*b-d*(a*b)^(1/2)-b*(I*d*x+I*c)) /b)*exp(-(I*c*b-d*(a*b)^(1/2))/b)*a+1/4*I/b^2*Ei(1,-(I*c*b+d*(a*b)^(1/2)-b *(I*d*x+I*c))/b)*exp(-(I*c*b+d*(a*b)^(1/2))/b)*a-x*cos(d*x+c)/b/d+sin(d*x+ c)/b/d^2
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.89 \[ \int \frac {x^3 \sin (c+d x)}{a+b x^2} \, dx=\frac {i \, a d^{2} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + i \, a d^{2} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - i \, a d^{2} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - i \, a d^{2} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 4 \, b d x \cos \left (d x + c\right ) + 4 \, b \sin \left (d x + c\right )}{4 \, b^{2} d^{2}} \]
1/4*(I*a*d^2*Ei(I*d*x - sqrt(a*d^2/b))*e^(I*c + sqrt(a*d^2/b)) + I*a*d^2*E i(I*d*x + sqrt(a*d^2/b))*e^(I*c - sqrt(a*d^2/b)) - I*a*d^2*Ei(-I*d*x - sqr t(a*d^2/b))*e^(-I*c + sqrt(a*d^2/b)) - I*a*d^2*Ei(-I*d*x + sqrt(a*d^2/b))* e^(-I*c - sqrt(a*d^2/b)) - 4*b*d*x*cos(d*x + c) + 4*b*sin(d*x + c))/(b^2*d ^2)
\[ \int \frac {x^3 \sin (c+d x)}{a+b x^2} \, dx=\int \frac {x^{3} \sin {\left (c + d x \right )}}{a + b x^{2}}\, dx \]
\[ \int \frac {x^3 \sin (c+d x)}{a+b x^2} \, dx=\int { \frac {x^{3} \sin \left (d x + c\right )}{b x^{2} + a} \,d x } \]
-1/2*((cos(c)^2 + sin(c)^2)*d*x^3*cos(d*x + c) - (cos(c)^2 + sin(c)^2)*x^2 *sin(d*x + c) + ((d*x^3*cos(c) + x^2*sin(c))*cos(d*x + c)^2 + (d*x^3*cos(c ) + x^2*sin(c))*sin(d*x + c)^2)*cos(d*x + 2*c) + 2*(((b*cos(c)^2 + b*sin(c )^2)*d^2*x^2 + (a*cos(c)^2 + a*sin(c)^2)*d^2)*cos(d*x + c)^2 + ((b*cos(c)^ 2 + b*sin(c)^2)*d^2*x^2 + (a*cos(c)^2 + a*sin(c)^2)*d^2)*sin(d*x + c)^2)*i ntegrate(-(a*d*x^2*cos(d*x + c) - a*x*sin(d*x + c))/(b^2*d^2*x^4 + 2*a*b*d ^2*x^2 + a^2*d^2), x) + 2*(((b*cos(c)^2 + b*sin(c)^2)*d^2*x^2 + (a*cos(c)^ 2 + a*sin(c)^2)*d^2)*cos(d*x + c)^2 + ((b*cos(c)^2 + b*sin(c)^2)*d^2*x^2 + (a*cos(c)^2 + a*sin(c)^2)*d^2)*sin(d*x + c)^2)*integrate(-(a*d*x^2*cos(d* x + c) - a*x*sin(d*x + c))/((b^2*d^2*x^4 + 2*a*b*d^2*x^2 + a^2*d^2)*cos(d* x + c)^2 + (b^2*d^2*x^4 + 2*a*b*d^2*x^2 + a^2*d^2)*sin(d*x + c)^2), x) + ( (d*x^3*sin(c) - x^2*cos(c))*cos(d*x + c)^2 + (d*x^3*sin(c) - x^2*cos(c))*s in(d*x + c)^2)*sin(d*x + 2*c))/(((b*cos(c)^2 + b*sin(c)^2)*d^2*x^2 + (a*co s(c)^2 + a*sin(c)^2)*d^2)*cos(d*x + c)^2 + ((b*cos(c)^2 + b*sin(c)^2)*d^2* x^2 + (a*cos(c)^2 + a*sin(c)^2)*d^2)*sin(d*x + c)^2)
\[ \int \frac {x^3 \sin (c+d x)}{a+b x^2} \, dx=\int { \frac {x^{3} \sin \left (d x + c\right )}{b x^{2} + a} \,d x } \]
Timed out. \[ \int \frac {x^3 \sin (c+d x)}{a+b x^2} \, dx=\int \frac {x^3\,\sin \left (c+d\,x\right )}{b\,x^2+a} \,d x \]