\(\int \frac {x^5}{(a+b \csc (c+d x^2))^2} \, dx\) [23]

Optimal result

Integrand size = 18, antiderivative size = 1124 \[ \int \frac {x^5}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx =\text {Too large to display} \] Output:

2*I*b*polylog(3,I*a*exp(I*(d*x^2+c))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^ 
(1/2)/d^3+1/6*x^6/a^2+b^2*x^2*ln(1+a*exp(I*(d*x^2+c))/(I*b-(a^2-b^2)^(1/2) 
))/a^2/(a^2-b^2)/d^2+b^2*x^2*ln(1+a*exp(I*(d*x^2+c))/(I*b+(a^2-b^2)^(1/2)) 
)/a^2/(a^2-b^2)/d^2-I*b^2*polylog(2,-a*exp(I*(d*x^2+c))/(I*b+(a^2-b^2)^(1/ 
2)))/a^2/(a^2-b^2)/d^3+I*b^3*polylog(3,I*a*exp(I*(d*x^2+c))/(b+(-a^2+b^2)^ 
(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^3-1/2*I*b^3*x^4*ln(1-I*a*exp(I*(d*x^2+c))/( 
b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d+I*b*x^4*ln(1-I*a*exp(I*(d*x^2+ 
c))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d-2*I*b*polylog(3,I*a*exp(I 
*(d*x^2+c))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d^3+1/2*I*b^3*x^4*l 
n(1-I*a*exp(I*(d*x^2+c))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d-b^3* 
x^2*polylog(2,I*a*exp(I*(d*x^2+c))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3 
/2)/d^2+2*b*x^2*polylog(2,I*a*exp(I*(d*x^2+c))/(b-(-a^2+b^2)^(1/2)))/a^2/( 
-a^2+b^2)^(1/2)/d^2+b^3*x^2*polylog(2,I*a*exp(I*(d*x^2+c))/(b+(-a^2+b^2)^( 
1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2-2*b*x^2*polylog(2,I*a*exp(I*(d*x^2+c))/(b+ 
(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d^2-I*b*x^4*ln(1-I*a*exp(I*(d*x^2+ 
c))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d-I*b^3*polylog(3,I*a*exp(I 
*(d*x^2+c))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^3-I*b^2*polylog(2 
,-a*exp(I*(d*x^2+c))/(I*b-(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^3-1/2*I*b^2*x^ 
4/a^2/(a^2-b^2)/d-1/2*b^2*x^4*cos(d*x^2+c)/a/(a^2-b^2)/d/(b+a*sin(d*x^2+c) 
)
 

Mathematica [A] (warning: unable to verify)

Time = 7.25 (sec) , antiderivative size = 1956, normalized size of antiderivative = 1.74 \[ \int \frac {x^5}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx =\text {Too large to display} \] Input:

Integrate[x^5/(a + b*Csc[c + d*x^2])^2,x]
 

Output:

(Csc[c + d*x^2]^2*(b + a*Sin[c + d*x^2])*((6*b^2*x^4*Csc[c]*(b*Cos[c] + a* 
Sin[d*x^2]))/((a - b)*(a + b)*d) + 2*x^6*(b + a*Sin[c + d*x^2]) - (6*b*E^( 
(2*I)*c)*((2*I)*b*d^2*E^((2*I)*c)*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^4 + 2*b* 
d*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^2*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E 
^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 2*b*d*E^((2*I)*c)*Sqrt[(a^2 - b 
^2)*E^((2*I)*c)]*x^2*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) - Sqrt[( 
a^2 - b^2)*E^((2*I)*c)])] - 2*a^2*d^2*E^(I*c)*x^4*Log[1 + (a*E^(I*(2*c + d 
*x^2)))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + b^2*d^2*E^(I*c)*x 
^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I 
)*c)])] + 2*a^2*d^2*E^((3*I)*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E 
^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - b^2*d^2*E^((3*I)*c)*x^4*Log[1 + 
 (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 
2*b*d*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^2*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I 
*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 2*b*d*E^((2*I)*c)*Sqrt[(a^2 
 - b^2)*E^((2*I)*c)]*x^2*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sq 
rt[(a^2 - b^2)*E^((2*I)*c)])] + 2*a^2*d^2*E^(I*c)*x^4*Log[1 + (a*E^(I*(2*c 
 + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - b^2*d^2*E^(I* 
c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^( 
(2*I)*c)])] - 2*a^2*d^2*E^((3*I)*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I 
*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + b^2*d^2*E^((3*I)*c)*x^4*...
 

Rubi [A] (verified)

Time = 2.45 (sec) , antiderivative size = 1123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4693, 3042, 4679, 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.

\(\displaystyle \int \frac {x^5}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx\)

\(\Big \downarrow \) 4693

\(\displaystyle \frac {1}{2} \int \frac {x^4}{\left (a+b \csc \left (d x^2+c\right )\right )^2}dx^2\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \int \frac {x^4}{\left (a+b \csc \left (d x^2+c\right )\right )^2}dx^2\)

\(\Big \downarrow \) 4679

\(\displaystyle \frac {1}{2} \int \left (-\frac {2 b x^4}{a^2 \left (b+a \sin \left (d x^2+c\right )\right )}+\frac {x^4}{a^2}+\frac {b^2 x^4}{a^2 \left (b+a \sin \left (d x^2+c\right )\right )^2}\right )dx^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {x^6}{3 a^2}+\frac {2 i b \log \left (1-\frac {i a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^4}{a^2 \sqrt {b^2-a^2} d}-\frac {i b^3 \log \left (1-\frac {i a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^4}{a^2 \left (b^2-a^2\right )^{3/2} d}-\frac {2 i b \log \left (1-\frac {i a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^4}{a^2 \sqrt {b^2-a^2} d}+\frac {i b^3 \log \left (1-\frac {i a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^4}{a^2 \left (b^2-a^2\right )^{3/2} d}-\frac {i b^2 x^4}{a^2 \left (a^2-b^2\right ) d}-\frac {b^2 \cos \left (d x^2+c\right ) x^4}{a \left (a^2-b^2\right ) d \left (b+a \sin \left (d x^2+c\right )\right )}+\frac {2 b^2 \log \left (\frac {e^{i \left (d x^2+c\right )} a}{i b-\sqrt {a^2-b^2}}+1\right ) x^2}{a^2 \left (a^2-b^2\right ) d^2}+\frac {2 b^2 \log \left (\frac {e^{i \left (d x^2+c\right )} a}{i b+\sqrt {a^2-b^2}}+1\right ) x^2}{a^2 \left (a^2-b^2\right ) d^2}+\frac {4 b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^2}{a^2 \sqrt {b^2-a^2} d^2}-\frac {2 b^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {4 b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^2}{a^2 \sqrt {b^2-a^2} d^2}+\frac {2 b^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^2+c\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^2+c\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac {4 i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} d^3}-\frac {2 i b^3 \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^3}-\frac {4 i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} d^3}+\frac {2 i b^3 \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^3}\right )\)

Input:

Int[x^5/(a + b*Csc[c + d*x^2])^2,x]
 

Output:

(((-I)*b^2*x^4)/(a^2*(a^2 - b^2)*d) + x^6/(3*a^2) + (2*b^2*x^2*Log[1 + (a* 
E^(I*(c + d*x^2)))/(I*b - Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) + (2*b^ 
2*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(I*b + Sqrt[a^2 - b^2])])/(a^2*(a^2 - 
b^2)*d^2) - (I*b^3*x^4*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^ 
2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + ((2*I)*b*x^4*Log[1 - (I*a*E^(I*(c + d*x 
^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + (I*b^3*x^4*Log[1 
 - (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2 
)*d) - ((2*I)*b*x^4*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2]) 
])/(a^2*Sqrt[-a^2 + b^2]*d) - ((2*I)*b^2*PolyLog[2, -((a*E^(I*(c + d*x^2)) 
)/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - ((2*I)*b^2*PolyLog[2, 
 -((a*E^(I*(c + d*x^2)))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) 
- (2*b^3*x^2*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/( 
a^2*(-a^2 + b^2)^(3/2)*d^2) + (4*b*x^2*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/ 
(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (2*b^3*x^2*PolyLog[2 
, (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2) 
*d^2) - (4*b*x^2*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2]) 
])/(a^2*Sqrt[-a^2 + b^2]*d^2) - ((2*I)*b^3*PolyLog[3, (I*a*E^(I*(c + d*x^2 
)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((4*I)*b*PolyL 
og[3, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^ 
2]*d^3) + ((2*I)*b^3*PolyLog[3, (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 ...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4679

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.) 
, x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Si 
n[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGt 
Q[m, 0]
 

rule 4693

Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Csc[c + d*x])^ 
p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 
 1)/n], 0] && IntegerQ[p]
 

Maple [F]

\[\int \frac {x^{5}}{{\left (a +b \csc \left (d \,x^{2}+c \right )\right )}^{2}}d x\]

Input:

int(x^5/(a+b*csc(d*x^2+c))^2,x)
 

Output:

int(x^5/(a+b*csc(d*x^2+c))^2,x)
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3032 vs. \(2 (966) = 1932\).

Time = 0.27 (sec) , antiderivative size = 3032, normalized size of antiderivative = 2.70 \[ \int \frac {x^5}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(x^5/(a+b*csc(d*x^2+c))^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {x^5}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^{5}}{\left (a + b \csc {\left (c + d x^{2} \right )}\right )^{2}}\, dx \] Input:

integrate(x**5/(a+b*csc(d*x**2+c))**2,x)
                                                                                    
                                                                                    
 

Output:

Integral(x**5/(a + b*csc(c + d*x**2))**2, x)
 

Maxima [F]

\[ \int \frac {x^5}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{5}}{{\left (b \csc \left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \] Input:

integrate(x^5/(a+b*csc(d*x^2+c))^2,x, algorithm="maxima")
 

Output:

1/6*((a^4 - a^2*b^2)*d*x^6*cos(2*d*x^2 + 2*c)^2 + 4*(a^2*b^2 - b^4)*d*x^6* 
cos(d*x^2 + c)^2 + (a^4 - a^2*b^2)*d*x^6*sin(2*d*x^2 + 2*c)^2 + 4*(a^2*b^2 
 - b^4)*d*x^6*sin(d*x^2 + c)^2 - 6*a*b^3*x^4*cos(d*x^2 + c) + 4*(a^3*b - a 
*b^3)*d*x^6*sin(d*x^2 + c) + (a^4 - a^2*b^2)*d*x^6 - 2*(3*a*b^3*x^4*cos(d* 
x^2 + c) + 2*(a^3*b - a*b^3)*d*x^6*sin(d*x^2 + c) + (a^4 - a^2*b^2)*d*x^6) 
*cos(2*d*x^2 + 2*c) - 6*((a^6 - a^4*b^2)*d*cos(2*d*x^2 + 2*c)^2 + 4*(a^4*b 
^2 - a^2*b^4)*d*cos(d*x^2 + c)^2 + 4*(a^5*b - a^3*b^3)*d*cos(d*x^2 + c)*si 
n(2*d*x^2 + 2*c) + (a^6 - a^4*b^2)*d*sin(2*d*x^2 + 2*c)^2 + 4*(a^4*b^2 - a 
^2*b^4)*d*sin(d*x^2 + c)^2 + 4*(a^5*b - a^3*b^3)*d*sin(d*x^2 + c) + (a^6 - 
 a^4*b^2)*d - 2*(2*(a^5*b - a^3*b^3)*d*sin(d*x^2 + c) + (a^6 - a^4*b^2)*d) 
*cos(2*d*x^2 + 2*c))*integrate(2*(2*(2*a^2*b^2 - b^4)*d*x^5*cos(d*x^2 + c) 
^2 + 2*(2*a^2*b^2 - b^4)*d*x^5*sin(d*x^2 + c)^2 - 2*a*b^3*x^3*cos(d*x^2 + 
c) + (2*a^3*b - a*b^3)*d*x^5*sin(d*x^2 + c) - (2*a*b^3*x^3*cos(d*x^2 + c) 
+ (2*a^3*b - a*b^3)*d*x^5*sin(d*x^2 + c))*cos(2*d*x^2 + 2*c) + ((2*a^3*b - 
 a*b^3)*d*x^5*cos(d*x^2 + c) - 2*a*b^3*x^3*sin(d*x^2 + c) - 2*a^2*b^2*x^3) 
*sin(2*d*x^2 + 2*c))/((a^6 - a^4*b^2)*d*cos(2*d*x^2 + 2*c)^2 + 4*(a^4*b^2 
- a^2*b^4)*d*cos(d*x^2 + c)^2 + 4*(a^5*b - a^3*b^3)*d*cos(d*x^2 + c)*sin(2 
*d*x^2 + 2*c) + (a^6 - a^4*b^2)*d*sin(2*d*x^2 + 2*c)^2 + 4*(a^4*b^2 - a^2* 
b^4)*d*sin(d*x^2 + c)^2 + 4*(a^5*b - a^3*b^3)*d*sin(d*x^2 + c) + (a^6 - a^ 
4*b^2)*d - 2*(2*(a^5*b - a^3*b^3)*d*sin(d*x^2 + c) + (a^6 - a^4*b^2)*d)...
 

Giac [F]

\[ \int \frac {x^5}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{5}}{{\left (b \csc \left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \] Input:

integrate(x^5/(a+b*csc(d*x^2+c))^2,x, algorithm="giac")
 

Output:

integrate(x^5/(b*csc(d*x^2 + c) + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^5}{{\left (a+\frac {b}{\sin \left (d\,x^2+c\right )}\right )}^2} \,d x \] Input:

int(x^5/(a + b/sin(c + d*x^2))^2,x)
 

Output:

int(x^5/(a + b/sin(c + d*x^2))^2, x)
 

Reduce [F]

\[ \int \frac {x^5}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^{5}}{\csc \left (d \,x^{2}+c \right )^{2} b^{2}+2 \csc \left (d \,x^{2}+c \right ) a b +a^{2}}d x \] Input:

int(x^5/(a+b*csc(d*x^2+c))^2,x)
 

Output:

int(x**5/(csc(c + d*x**2)**2*b**2 + 2*csc(c + d*x**2)*a*b + a**2),x)