\(\int \frac {x^3}{(a+b \csc (c+d x^2))^2} \, dx\) [25]

Optimal result

Integrand size = 18, antiderivative size = 616 \[ \int \frac {x^3}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\frac {x^4}{4 a^2}-\frac {i b^3 x^2 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {i b x^2 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}+\frac {i b^3 x^2 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {i b x^2 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}+\frac {b^2 \log \left (b+a \sin \left (c+d x^2\right )\right )}{2 a^2 \left (a^2-b^2\right ) d^2}-\frac {b^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {b^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {b^2 x^2 \cos \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d x^2\right )\right )} \] Output:

1/4*x^4/a^2-1/2*I*b^3*x^2*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^2*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+1/2*I*b^3*x^2*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^2*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+1/2*b^2*ln(b+a*sin(d*x^2+c))/a 
^2/(a^2-b^2)/d^2-1/2*b^3*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+b*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+1/2*b^3*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-b*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^2-1/2*b^2*x^2*cos(d*x^2 
+c)/a/(a^2-b^2)/d/(b+a*sin(d*x^2+c))
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2566\) vs. \(2(616)=1232\).

Time = 15.11 (sec) , antiderivative size = 2566, normalized size of antiderivative = 4.17 \[ \int \frac {x^3}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\text {Result too large to show} \] Input:

Integrate[x^3/(a + b*Csc[c + d*x^2])^2,x]
 

Output:

((-(b^2*c*Cos[c + d*x^2]) + b^2*(c + d*x^2)*Cos[c + d*x^2])*Csc[c + d*x^2] 
^2*(b + a*Sin[c + d*x^2]))/(2*a*(-a + b)*(a + b)*d^2*(a + b*Csc[c + d*x^2] 
)^2) + ((-c + d*x^2)*(c + d*x^2)*Csc[c + d*x^2]^2*(b + a*Sin[c + d*x^2])^2 
)/(4*a^2*d^2*(a + b*Csc[c + d*x^2])^2) + (Csc[c + d*x^2]^2*(-2*a*b*ArcTanh 
[(a + b*Tan[(c + d*x^2)/2])/Sqrt[a^2 - b^2]] + 2*(a*b + 2*a^2*c - b^2*c)*A 
rcTanh[(a + b*Tan[(c + d*x^2)/2])/Sqrt[a^2 - b^2]] + b*Sqrt[a^2 - b^2]*Log 
[Sec[(c + d*x^2)/2]^2] - b*Sqrt[a^2 - b^2]*Log[Sec[(c + d*x^2)/2]^2*(b + a 
*Sin[c + d*x^2])] + I*(2*a^2 - b^2)*Log[1 - I*Tan[(c + d*x^2)/2]]*Log[(a - 
 Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])/(a - I*b - Sqrt[a^2 - b^2])] - I* 
(2*a^2 - b^2)*Log[1 + I*Tan[(c + d*x^2)/2]]*Log[(a - Sqrt[a^2 - b^2] + b*T 
an[(c + d*x^2)/2])/(a + I*b - Sqrt[a^2 - b^2])] - I*(2*a^2 - b^2)*Log[1 - 
I*Tan[(c + d*x^2)/2]]*Log[(a + Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])/(a 
- I*b + Sqrt[a^2 - b^2])] + I*(2*a^2 - b^2)*Log[1 + I*Tan[(c + d*x^2)/2]]* 
Log[(a + Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])/(a + I*b + Sqrt[a^2 - b^2 
])] - I*(2*a^2 - b^2)*PolyLog[2, (b*(1 + I*Tan[(c + d*x^2)/2]))/((-I)*a + 
b + I*Sqrt[a^2 - b^2])] + I*(2*a^2 - b^2)*PolyLog[2, (b*(1 + I*Tan[(c + d* 
x^2)/2]))/(b - I*(a + Sqrt[a^2 - b^2]))] - I*(2*a^2 - b^2)*PolyLog[2, -((b 
*(I + Tan[(c + d*x^2)/2]))/(a - I*b + Sqrt[a^2 - b^2]))] + I*(2*a^2 - b^2) 
*PolyLog[2, (b*(I + Tan[(c + d*x^2)/2]))/(-a + I*b + Sqrt[a^2 - b^2])])*(b 
 + a*Sin[c + d*x^2])^2*((2*b*c)/((a^2 - b^2)*d*(b + a*Sin[c + d*x^2])) ...
 

Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 607, normalized size of antiderivative = 0.99, 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.

\(\displaystyle \int \frac {x^3}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx\)

\(\Big \downarrow \) 4693

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4679

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

\(\Big \downarrow \) 2009

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

Input:

Int[x^3/(a + b*Csc[c + d*x^2])^2,x]
 

Output:

(x^4/(2*a^2) - (I*b^3*x^2*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^2*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^2*Lo 
g[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^2*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^ 
2])])/(a^2*Sqrt[-a^2 + b^2]*d) + (b^2*Log[b + a*Sin[c + d*x^2]])/(a^2*(a^2 
 - b^2)*d^2) - (b^3*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) + (2*b*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) + (b^3*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) - (2*b*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) - (b^2*x^2*Cos[c + d*x^2])/(a*(a^2 - b^2)*d*(b + a* 
Sin[c + d*x^2])))/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]

\[\int \frac {x^{3}}{{\left (a +b \csc \left (d \,x^{2}+c \right )\right )}^{2}}d x\]

Input:

int(x^3/(a+b*csc(d*x^2+c))^2,x)
 

Output:

int(x^3/(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. 1906 vs. \(2 (526) = 1052\).

Time = 0.23 (sec) , antiderivative size = 1906, normalized size of antiderivative = 3.09 \[ \int \frac {x^3}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(x^3/(a+b*csc(d*x^2+c))^2,x, algorithm="fricas")
 

Output:

1/4*((a^5 - 2*a^3*b^2 + a*b^4)*d^2*x^4*sin(d*x^2 + c) + (a^4*b - 2*a^2*b^3 
 + b^5)*d^2*x^4 - 2*(a^3*b^2 - a*b^4)*d*x^2*cos(d*x^2 + c) + (2*I*a^3*b^2 
- I*a*b^4 + (2*I*a^4*b - I*a^2*b^3)*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2)* 
dilog((I*b*cos(d*x^2 + c) - b*sin(d*x^2 + c) + (a*cos(d*x^2 + c) + I*a*sin 
(d*x^2 + c))*sqrt((a^2 - b^2)/a^2) - a)/a + 1) + (-2*I*a^3*b^2 + I*a*b^4 + 
 (-2*I*a^4*b + I*a^2*b^3)*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2)*dilog((I*b 
*cos(d*x^2 + c) - b*sin(d*x^2 + c) - (a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c 
))*sqrt((a^2 - b^2)/a^2) - a)/a + 1) + (-2*I*a^3*b^2 + I*a*b^4 + (-2*I*a^4 
*b + I*a^2*b^3)*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2)*dilog((-I*b*cos(d*x^ 
2 + c) - b*sin(d*x^2 + c) + (a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c))*sqrt(( 
a^2 - b^2)/a^2) - a)/a + 1) + (2*I*a^3*b^2 - I*a*b^4 + (2*I*a^4*b - I*a^2* 
b^3)*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2)*dilog((-I*b*cos(d*x^2 + c) - b* 
sin(d*x^2 + c) - (a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/ 
a^2) - a)/a + 1) - ((2*a^3*b^2 - a*b^4)*d*x^2 + (2*a^3*b^2 - a*b^4)*c + (( 
2*a^4*b - a^2*b^3)*d*x^2 + (2*a^4*b - a^2*b^3)*c)*sin(d*x^2 + c))*sqrt((a^ 
2 - b^2)/a^2)*log(-(I*b*cos(d*x^2 + c) - b*sin(d*x^2 + c) + (a*cos(d*x^2 + 
 c) + I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2) - a)/a) + ((2*a^3*b^2 - a* 
b^4)*d*x^2 + (2*a^3*b^2 - a*b^4)*c + ((2*a^4*b - a^2*b^3)*d*x^2 + (2*a^4*b 
 - a^2*b^3)*c)*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2)*log(-(I*b*cos(d*x^2 + 
 c) - b*sin(d*x^2 + c) - (a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*sqrt((...
 

Sympy [F]

\[ \int \frac {x^3}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^{3}}{\left (a + b \csc {\left (c + d x^{2} \right )}\right )^{2}}\, dx \] Input:

integrate(x**3/(a+b*csc(d*x**2+c))**2,x)
                                                                                    
                                                                                    
 

Output:

Integral(x**3/(a + b*csc(c + d*x**2))**2, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^3/(a+b*csc(d*x^2+c))^2,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f 
or more de
 

Giac [F]

\[ \int \frac {x^3}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{3}}{{\left (b \csc \left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \] Input:

integrate(x^3/(a+b*csc(d*x^2+c))^2,x, algorithm="giac")
 

Output:

integrate(x^3/(b*csc(d*x^2 + c) + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^3}{{\left (a+\frac {b}{\sin \left (d\,x^2+c\right )}\right )}^2} \,d x \] Input:

int(x^3/(a + b/sin(c + d*x^2))^2,x)
 

Output:

int(x^3/(a + b/sin(c + d*x^2))^2, x)
 

Reduce [F]

\[ \int \frac {x^3}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^{3}}{\csc \left (d \,x^{2}+c \right )^{2} b^{2}+2 \csc \left (d \,x^{2}+c \right ) a b +a^{2}}d x \] Input:

int(x^3/(a+b*csc(d*x^2+c))^2,x)
 

Output:

int(x**3/(csc(c + d*x**2)**2*b**2 + 2*csc(c + d*x**2)*a*b + a**2),x)