\(\int \frac {x^{3/2}}{(a+b \sec (c+d \sqrt {x}))^2} \, dx\) [66]

Optimal result

Integrand size = 22, antiderivative size = 1925 \[ \int \frac {x^{3/2}}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \] Output:

-2*I*b^2*x^2/a^2/(a^2-b^2)/d+48*I*b^3*polylog(5,-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+48*I*b^2*polylog(4,-a*exp(I*( 
c+d*x^(1/2)))/(b+I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^5+96*I*b*polylog(5,-a 
*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d^5-96*I* 
b*polylog(5,-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+8*b^2*x^(3/2)*ln(1+a*exp(I*(c+d*x^(1/2)))/(b+I*(a^2-b^2)^(1/2)))/ 
a^2/(a^2-b^2)/d^2+8*b^2*x^(3/2)*ln(1+a*exp(I*(c+d*x^(1/2)))/(b-I*(a^2-b^2) 
^(1/2)))/a^2/(a^2-b^2)/d^2+8*b^3*x^(3/2)*polylog(2,-a*exp(I*(c+d*x^(1/2))) 
/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2-8*b^3*x^(3/2)*polylog(2,-a 
*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2-16*b* 
x^(3/2)*polylog(2,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+ 
b^2)^(1/2)/d^2+16*b*x^(3/2)*polylog(2,-a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2 
)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d^2+48*b^2*x^(1/2)*polylog(3,-a*exp(I*(c+d* 
x^(1/2)))/(b+I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^4+48*b^2*x^(1/2)*polylog( 
3,-a*exp(I*(c+d*x^(1/2)))/(b-I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^4-48*b^3* 
x^(1/2)*polylog(4,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+ 
b^2)^(3/2)/d^4+48*b^3*x^(1/2)*polylog(4,-a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b 
^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^4+96*b*x^(1/2)*polylog(4,-a*exp(I*(c+d* 
x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2)/d^4-96*b*x^(1/2)*poly 
log(4,-a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(1/2...
 

Mathematica [A] (verified)

Time = 11.06 (sec) , antiderivative size = 2161, normalized size of antiderivative = 1.12 \[ \int \frac {x^{3/2}}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

(2*(b + a*Cos[c + d*Sqrt[x]])*Sec[c + d*Sqrt[x]]^2*(x^(5/2)*(b + a*Cos[c + 
 d*Sqrt[x]]) + (5*b*(b + a*Cos[c + d*Sqrt[x]])*(((-2*I)*b*d^4*E^((2*I)*c)* 
x^2)/(1 + E^((2*I)*c)) + (4*b*d^3*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x^(3/2)*L 
og[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(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])))/(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])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] 
 + 4*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])))/(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])))/(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])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 4*d^2*((3*I)*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])))/(b*E^(I*c) - Sqrt[(-a^2 + b^ 
2)*E^((2*I)*c)]))] + 4*d^2*((-3*I)*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])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 24*b*d*Sqrt 
[(-a^2 + b^2)*E^((2*I)*c)]*Sqrt[x]*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])) 
)/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (24*I)*a^2*d^2*E^(I*c)* 
x*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^...
 

Rubi [A] (verified)

Time = 3.32 (sec) , antiderivative size = 1926, 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 = {4692, 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.

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

\(\Big \downarrow \) 4692

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4679

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {i x^2 \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b-\sqrt {b^2-a^2}}+1\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}+\frac {i x^2 \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b+\sqrt {b^2-a^2}}+1\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}-\frac {4 x^{3/2} \operatorname {PolyLog}\left (2,-\frac {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 {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 {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 {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 {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 {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 {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 {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}{b-i \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}{b+i \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 )}}{b-i \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 )}}{b+i \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 )}}{b-i \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 )}}{b+i \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 )}}{b-i \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 )}}{b+i \sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^5}+\frac {x^2 \sin \left (c+d \sqrt {x}\right ) b^2}{a \left (a^2-b^2\right ) d \left (b+a \cos \left (c+d \sqrt {x}\right )\right )}+\frac {2 i x^2 \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b-\sqrt {b^2-a^2}}+1\right ) b}{a^2 \sqrt {b^2-a^2} d}-\frac {2 i x^2 \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b+\sqrt {b^2-a^2}}+1\right ) b}{a^2 \sqrt {b^2-a^2} d}+\frac {8 x^{3/2} \operatorname {PolyLog}\left (2,-\frac {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 {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 {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 {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 {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 {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 {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 {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*Sec[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])))/(b - I*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])))/(b + I*Sqrt[a^2 
- b^2])])/(a^2*(a^2 - b^2)*d^2) - (I*b^3*x^2*Log[1 + (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*Lo 
g[1 + (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 + (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 + (a*E^(I*(c + d*Sqr 
t[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])))/(b - I*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])))/(b + 
I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (4*b^3*x^(3/2)*PolyLog[2, -( 
(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, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^ 
2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (4*b^3*x^(3/2)*PolyLog[2, -((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, -((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])))/(b - I*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])))/(b + I*Sqrt[a^2 - b^...
 

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 4692

Int[(x_)^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sec[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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^(3/2)/(a+b*sec(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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*( - 960*sqrt( - a**2 + b**2)*atan((tan((sqrt(x)*d + c)/2)*a - tan((sqrt 
(x)*d + c)/2)*b)/sqrt( - a**2 + b**2))*cos(sqrt(x)*d + c)*tan((sqrt(x)*d + 
 c)/2)**2*a**4*b + 2880*sqrt( - a**2 + b**2)*atan((tan((sqrt(x)*d + c)/2)* 
a - tan((sqrt(x)*d + c)/2)*b)/sqrt( - a**2 + b**2))*cos(sqrt(x)*d + c)*tan 
((sqrt(x)*d + c)/2)**2*a**3*b**2 - 2880*sqrt( - a**2 + b**2)*atan((tan((sq 
rt(x)*d + c)/2)*a - tan((sqrt(x)*d + c)/2)*b)/sqrt( - a**2 + b**2))*cos(sq 
rt(x)*d + c)*tan((sqrt(x)*d + c)/2)**2*a**2*b**3 + 960*sqrt( - a**2 + b**2 
)*atan((tan((sqrt(x)*d + c)/2)*a - tan((sqrt(x)*d + c)/2)*b)/sqrt( - a**2 
+ b**2))*cos(sqrt(x)*d + c)*tan((sqrt(x)*d + c)/2)**2*a*b**4 + 960*sqrt( - 
 a**2 + b**2)*atan((tan((sqrt(x)*d + c)/2)*a - tan((sqrt(x)*d + c)/2)*b)/s 
qrt( - a**2 + b**2))*cos(sqrt(x)*d + c)*a**4*b - 960*sqrt( - a**2 + b**2)* 
atan((tan((sqrt(x)*d + c)/2)*a - tan((sqrt(x)*d + c)/2)*b)/sqrt( - a**2 + 
b**2))*cos(sqrt(x)*d + c)*a**3*b**2 - 960*sqrt( - a**2 + b**2)*atan((tan(( 
sqrt(x)*d + c)/2)*a - tan((sqrt(x)*d + c)/2)*b)/sqrt( - a**2 + b**2))*cos( 
sqrt(x)*d + c)*a**2*b**3 + 960*sqrt( - a**2 + b**2)*atan((tan((sqrt(x)*d + 
 c)/2)*a - tan((sqrt(x)*d + c)/2)*b)/sqrt( - a**2 + b**2))*cos(sqrt(x)*d + 
 c)*a*b**4 - 960*sqrt( - a**2 + b**2)*atan((tan((sqrt(x)*d + c)/2)*a - tan 
((sqrt(x)*d + c)/2)*b)/sqrt( - a**2 + b**2))*tan((sqrt(x)*d + c)/2)**2*a** 
3*b**2 + 2880*sqrt( - a**2 + b**2)*atan((tan((sqrt(x)*d + c)/2)*a - tan((s 
qrt(x)*d + c)/2)*b)/sqrt( - a**2 + b**2))*tan((sqrt(x)*d + c)/2)**2*a**...