\(\int \frac {x^{3/2}}{(a+b \csc (c+d \sqrt {x}))^2} \, dx\) [67]

Optimal result

Integrand size = 22, antiderivative size = 1977 \[ \int \frac {x^{3/2}}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \] Output:

-2*b^2*x^2*cos(c+d*x^(1/2))/a/(a^2-b^2)/d/(b+a*sin(c+d*x^(1/2)))-24*I*b^2* 
x*polylog(2,-a*exp(I*(c+d*x^(1/2)))/(I*b+(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d 
^3-24*I*b^2*x*polylog(2,-a*exp(I*(c+d*x^(1/2)))/(I*b-(a^2-b^2)^(1/2)))/a^2 
/(a^2-b^2)/d^3-2*I*b^3*x^2*ln(1-I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/ 
2)))/a^2/(-a^2+b^2)^(3/2)/d-24*I*b^3*x*polylog(3,I*a*exp(I*(c+d*x^(1/2)))/ 
(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^3-4*I*b*x^2*ln(1-I*a*exp(I*(c 
+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d-48*I*b*x*polylog 
(3,I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d^3 
+2*I*b^3*x^2*ln(1-I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2 
+b^2)^(3/2)/d+24*I*b^3*x*polylog(3,I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^ 
(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^3+4*I*b*x^2*ln(1-I*a*exp(I*(c+d*x^(1/2)))/( 
b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d+48*I*b*x*polylog(3,I*a*exp(I*( 
c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d^3-2*I*b^2*x^2/a 
^2/(a^2-b^2)/d-8*b^3*x^(3/2)*polylog(2,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b 
^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2-96*I*b*polylog(5,I*a*exp(I*(c+d*x^(1/ 
2)))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d^5-48*I*b^3*polylog(5,I*a 
*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^5+96*I* 
b*polylog(5,I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^ 
(1/2)/d^5+48*I*b^2*polylog(4,-a*exp(I*(c+d*x^(1/2)))/(I*b+(a^2-b^2)^(1/2)) 
)/a^2/(a^2-b^2)/d^5+48*I*b^2*polylog(4,-a*exp(I*(c+d*x^(1/2)))/(I*b-(a^...
 

Mathematica [A] (warning: unable to verify)

Time = 10.50 (sec) , antiderivative size = 2236, normalized size of antiderivative = 1.13 \[ \int \frac {x^{3/2}}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

(Csc[c + d*Sqrt[x]]^2*(b + a*Sin[c + d*Sqrt[x]])*(2*x^(5/2)*(b + a*Sin[c + 
 d*Sqrt[x]]) - ((10*I)*b*E^(I*c)*(2*b*E^(I*c)*x^2 + ((-1 + E^((2*I)*c))*(( 
4*I)*b*d^3*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^(3/2)*Log[1 + (a*E^(I*(2*c + d* 
Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (2*I)*a^2*d^4* 
E^(I*c)*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - 
 b^2)*E^((2*I)*c)])] + I*b^2*d^4*E^(I*c)*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt 
[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (4*I)*b*d^3*Sqrt[( 
a^2 - b^2)*E^((2*I)*c)]*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E 
^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (2*I)*a^2*d^4*E^(I*c)*x^2*Log[1 
 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c) 
])] - I*b^2*d^4*E^(I*c)*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I* 
c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 4*d^2*(3*b*Sqrt[(a^2 - b^2)*E^((2*I 
)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*x*PolyLog[2, (I*a 
*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 
 4*d^2*(3*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)] + 2*a^2*d*E^(I*c)*Sqrt[x] - b^2* 
d*E^(I*c)*Sqrt[x])*x*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) 
 + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (24*I)*b*d*Sqrt[(a^2 - b^2)*E^((2*I) 
*c)]*Sqrt[x]*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[ 
(a^2 - b^2)*E^((2*I)*c)])] - (24*I)*a^2*d^2*E^(I*c)*x*PolyLog[3, (I*a*E^(I 
*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (...
 

Rubi [A] (verified)

Time = 3.19 (sec) , antiderivative size = 1979, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, 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.

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

\(\Big \downarrow \) 4693

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4679

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {i x^2 \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}+\frac {i x^2 \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}-\frac {4 x^{3/2} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}+\frac {4 x^{3/2} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {12 i x \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac {12 i x \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac {24 \sqrt {x} \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^4}-\frac {24 \sqrt {x} \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^4}+\frac {24 i \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^5}-\frac {24 i \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^5}-\frac {i x^2 b^2}{a^2 \left (a^2-b^2\right ) d}+\frac {4 x^{3/2} \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{i b-\sqrt {a^2-b^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) d^2}+\frac {4 x^{3/2} \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{i b+\sqrt {a^2-b^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac {12 i x \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b-\sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}-\frac {12 i x \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}+\frac {24 \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b-\sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^4}+\frac {24 \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^4}+\frac {24 i \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b-\sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^5}+\frac {24 i \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^5}-\frac {x^2 \cos \left (c+d \sqrt {x}\right ) b^2}{a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d \sqrt {x}\right )\right )}+\frac {2 i x^2 \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d}-\frac {2 i x^2 \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d}+\frac {8 x^{3/2} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^2}-\frac {8 x^{3/2} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^2}+\frac {24 i x \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^3}-\frac {24 i x \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^3}-\frac {48 \sqrt {x} \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^4}+\frac {48 \sqrt {x} \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^4}-\frac {48 i \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^5}+\frac {48 i \operatorname {PolyLog}\left (5,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^5}+\frac {x^{5/2}}{5 a^2}\right )\)

Input:

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

Output:

2*(((-I)*b^2*x^2)/(a^2*(a^2 - b^2)*d) + x^(5/2)/(5*a^2) + (4*b^2*x^(3/2)*L 
og[1 + (a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2 
)*d^2) + (4*b^2*x^(3/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 
- b^2])])/(a^2*(a^2 - b^2)*d^2) - (I*b^3*x^2*Log[1 - (I*a*E^(I*(c + d*Sqrt 
[x])))/(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*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^ 
2 + b^2]*d) + (I*b^3*x^2*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(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*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - ((12*I 
)*b^2*x*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/ 
(a^2*(a^2 - b^2)*d^3) - ((12*I)*b^2*x*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x]) 
))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (4*b^3*x^(3/2)*PolyL 
og[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^ 
2)^(3/2)*d^2) + (8*b*x^(3/2)*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - S 
qrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (4*b^3*x^(3/2)*PolyLog[2, 
(I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/ 
2)*d^2) - (8*b*x^(3/2)*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a 
^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (24*b^2*Sqrt[x]*PolyLog[3, -((a* 
E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) + 
(24*b^2*Sqrt[x]*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(I*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]

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

Input:

int(x^(3/2)/(a+b*csc(c+d*x^(1/2)))^2,x)
 

Output:

int(x^(3/2)/(a+b*csc(c+d*x^(1/2)))^2,x)
 

Fricas [F]

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

integrate(x^(3/2)/(a+b*csc(c+d*x^(1/2)))^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate(x**(3/2)/(a+b*csc(c+d*x**(1/2)))**2,x)
 

Output:

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

Maxima [F(-2)]

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

integrate(x^(3/2)/(a+b*csc(c+d*x^(1/2)))^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]

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

integrate(x^(3/2)/(a+b*csc(c+d*x^(1/2)))^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(x^(3/2)/(a + b/sin(c + d*x^(1/2)))^2,x)
 

Output:

int(x^(3/2)/(a + b/sin(c + d*x^(1/2)))^2, x)
 

Reduce [F]

\[ \int \frac {x^{3/2}}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {too large to display} \] Input:

int(x^(3/2)/(a+b*csc(c+d*x^(1/2)))^2,x)
 

Output:

(4*( - 12960*sqrt( - a**2 + b**2)*atan((tan((sqrt(x)*d + c)/2)*b + a)/sqrt 
( - a**2 + b**2))*sin(sqrt(x)*d + c)*a**6 + 11520*sqrt( - a**2 + b**2)*ata 
n((tan((sqrt(x)*d + c)/2)*b + a)/sqrt( - a**2 + b**2))*sin(sqrt(x)*d + c)* 
a**4*b**2 - 1680*sqrt( - a**2 + b**2)*atan((tan((sqrt(x)*d + c)/2)*b + a)/ 
sqrt( - a**2 + b**2))*sin(sqrt(x)*d + c)*a**2*b**4 - 12960*sqrt( - a**2 + 
b**2)*atan((tan((sqrt(x)*d + c)/2)*b + a)/sqrt( - a**2 + b**2))*a**5*b + 1 
1520*sqrt( - a**2 + b**2)*atan((tan((sqrt(x)*d + c)/2)*b + a)/sqrt( - a**2 
 + b**2))*a**3*b**3 - 1680*sqrt( - a**2 + b**2)*atan((tan((sqrt(x)*d + c)/ 
2)*b + a)/sqrt( - a**2 + b**2))*a*b**5 - 6480*sqrt(x)*cos(sqrt(x)*d + c)*a 
**7*d - 120*sqrt(x)*cos(sqrt(x)*d + c)*a**5*b**2*d**3*x + 9000*sqrt(x)*cos 
(sqrt(x)*d + c)*a**5*b**2*d + 140*sqrt(x)*cos(sqrt(x)*d + c)*a**3*b**4*d** 
3*x - 2640*sqrt(x)*cos(sqrt(x)*d + c)*a**3*b**4*d - 20*sqrt(x)*cos(sqrt(x) 
*d + c)*a*b**6*d**3*x + 120*sqrt(x)*cos(sqrt(x)*d + c)*a*b**6*d + 1080*cos 
(sqrt(x)*d + c)*a**6*b*d**2*x + 10*cos(sqrt(x)*d + c)*a**4*b**3*d**4*x**2 
- 1380*cos(sqrt(x)*d + c)*a**4*b**3*d**2*x - 10*cos(sqrt(x)*d + c)*a**2*b* 
*5*d**4*x**2 + 300*cos(sqrt(x)*d + c)*a**2*b**5*d**2*x + 180*sqrt(x)*sin(s 
qrt(x)*d + c)*a**6*b*d**3*x + sqrt(x)*sin(sqrt(x)*d + c)*a**4*b**3*d**5*x* 
*2 - 260*sqrt(x)*sin(sqrt(x)*d + c)*a**4*b**3*d**3*x - sqrt(x)*sin(sqrt(x) 
*d + c)*a**2*b**5*d**5*x**2 + 80*sqrt(x)*sin(sqrt(x)*d + c)*a**2*b**5*d**3 
*x - 6480*sqrt(x)*a**7*d + 60*sqrt(x)*a**5*b**2*d**3*x + 12240*sqrt(x)*...