Integrand size = 16, antiderivative size = 84 \[ \int x^3 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\frac {a x^4}{4}-\frac {b x^2 \text {arctanh}\left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac {i b \operatorname {PolyLog}\left (2,-e^{i \left (c+d x^2\right )}\right )}{2 d^2}-\frac {i b \operatorname {PolyLog}\left (2,e^{i \left (c+d x^2\right )}\right )}{2 d^2} \] Output:
1/4*a*x^4-b*x^2*arctanh(exp(I*(d*x^2+c)))/d+1/2*I*b*polylog(2,-exp(I*(d*x^ 2+c)))/d^2-1/2*I*b*polylog(2,exp(I*(d*x^2+c)))/d^2
Time = 0.10 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.40 \[ \int x^3 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\frac {a x^4}{4}+\frac {b \left (\left (c+d x^2\right ) \left (\log \left (1-e^{i \left (c+d x^2\right )}\right )-\log \left (1+e^{i \left (c+d x^2\right )}\right )\right )-c \log \left (\tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{i \left (c+d x^2\right )}\right )-\operatorname {PolyLog}\left (2,e^{i \left (c+d x^2\right )}\right )\right )\right )}{2 d^2} \] Input:
Integrate[x^3*(a + b*Csc[c + d*x^2]),x]
Output:
(a*x^4)/4 + (b*((c + d*x^2)*(Log[1 - E^(I*(c + d*x^2))] - Log[1 + E^(I*(c + d*x^2))]) - c*Log[Tan[(c + d*x^2)/2]] + I*(PolyLog[2, -E^(I*(c + d*x^2)) ] - PolyLog[2, E^(I*(c + d*x^2))])))/(2*d^2)
Time = 0.26 (sec) , antiderivative size = 84, 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 = {2010, 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.
\(\displaystyle \int x^3 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (a x^3+b x^3 \csc \left (c+d x^2\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a x^4}{4}-\frac {b x^2 \text {arctanh}\left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac {i b \operatorname {PolyLog}\left (2,-e^{i \left (d x^2+c\right )}\right )}{2 d^2}-\frac {i b \operatorname {PolyLog}\left (2,e^{i \left (d x^2+c\right )}\right )}{2 d^2}\) |
Input:
Int[x^3*(a + b*Csc[c + d*x^2]),x]
Output:
(a*x^4)/4 - (b*x^2*ArcTanh[E^(I*(c + d*x^2))])/d + ((I/2)*b*PolyLog[2, -E^ (I*(c + d*x^2))])/d^2 - ((I/2)*b*PolyLog[2, E^(I*(c + d*x^2))])/d^2
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
\[\int x^{3} \left (a +b \csc \left (d \,x^{2}+c \right )\right )d x\]
Input:
int(x^3*(a+b*csc(d*x^2+c)),x)
Output:
int(x^3*(a+b*csc(d*x^2+c)),x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (66) = 132\).
Time = 0.10 (sec) , antiderivative size = 288, normalized size of antiderivative = 3.43 \[ \int x^3 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\frac {a d^{2} x^{4} - b d x^{2} \log \left (\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right ) + 1\right ) - b d x^{2} \log \left (\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right ) + 1\right ) - b c \log \left (-\frac {1}{2} \, \cos \left (d x^{2} + c\right ) + \frac {1}{2} i \, \sin \left (d x^{2} + c\right ) + \frac {1}{2}\right ) - b c \log \left (-\frac {1}{2} \, \cos \left (d x^{2} + c\right ) - \frac {1}{2} i \, \sin \left (d x^{2} + c\right ) + \frac {1}{2}\right ) - i \, b {\rm Li}_2\left (\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right )\right ) + i \, b {\rm Li}_2\left (\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right )\right ) - i \, b {\rm Li}_2\left (-\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right )\right ) + i \, b {\rm Li}_2\left (-\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right )\right ) + {\left (b d x^{2} + b c\right )} \log \left (-\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right ) + 1\right ) + {\left (b d x^{2} + b c\right )} \log \left (-\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right ) + 1\right )}{4 \, d^{2}} \] Input:
integrate(x^3*(a+b*csc(d*x^2+c)),x, algorithm="fricas")
Output:
1/4*(a*d^2*x^4 - b*d*x^2*log(cos(d*x^2 + c) + I*sin(d*x^2 + c) + 1) - b*d* x^2*log(cos(d*x^2 + c) - I*sin(d*x^2 + c) + 1) - b*c*log(-1/2*cos(d*x^2 + c) + 1/2*I*sin(d*x^2 + c) + 1/2) - b*c*log(-1/2*cos(d*x^2 + c) - 1/2*I*sin (d*x^2 + c) + 1/2) - I*b*dilog(cos(d*x^2 + c) + I*sin(d*x^2 + c)) + I*b*di log(cos(d*x^2 + c) - I*sin(d*x^2 + c)) - I*b*dilog(-cos(d*x^2 + c) + I*sin (d*x^2 + c)) + I*b*dilog(-cos(d*x^2 + c) - I*sin(d*x^2 + c)) + (b*d*x^2 + b*c)*log(-cos(d*x^2 + c) + I*sin(d*x^2 + c) + 1) + (b*d*x^2 + b*c)*log(-co s(d*x^2 + c) - I*sin(d*x^2 + c) + 1))/d^2
\[ \int x^3 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\int x^{3} \left (a + b \csc {\left (c + d x^{2} \right )}\right )\, dx \] Input:
integrate(x**3*(a+b*csc(d*x**2+c)),x)
Output:
Integral(x**3*(a + b*csc(c + d*x**2)), x)
\[ \int x^3 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\int { {\left (b \csc \left (d x^{2} + c\right ) + a\right )} x^{3} \,d x } \] Input:
integrate(x^3*(a+b*csc(d*x^2+c)),x, algorithm="maxima")
Output:
1/4*a*x^4 + b*(integrate(x^3*sin(d*x^2 + c)/(cos(d*x^2 + c)^2 + sin(d*x^2 + c)^2 + 2*cos(d*x^2 + c) + 1), x) + integrate(x^3*sin(d*x^2 + c)/(cos(d*x ^2 + c)^2 + sin(d*x^2 + c)^2 - 2*cos(d*x^2 + c) + 1), x))
\[ \int x^3 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\int { {\left (b \csc \left (d x^{2} + c\right ) + a\right )} x^{3} \,d x } \] Input:
integrate(x^3*(a+b*csc(d*x^2+c)),x, algorithm="giac")
Output:
integrate((b*csc(d*x^2 + c) + a)*x^3, x)
Timed out. \[ \int x^3 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\int x^3\,\left (a+\frac {b}{\sin \left (d\,x^2+c\right )}\right ) \,d x \] Input:
int(x^3*(a + b/sin(c + d*x^2)),x)
Output:
int(x^3*(a + b/sin(c + d*x^2)), x)
\[ \int x^3 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\left (\int \csc \left (d \,x^{2}+c \right ) x^{3}d x \right ) b +\frac {a \,x^{4}}{4} \] Input:
int(x^3*(a+b*csc(d*x^2+c)),x)
Output:
(4*int(csc(c + d*x**2)*x**3,x)*b + a*x**4)/4