Integrand size = 12, antiderivative size = 90 \[ \int x \csc ^7\left (a+b x^2\right ) \, dx=-\frac {5 \text {arctanh}\left (\cos \left (a+b x^2\right )\right )}{32 b}-\frac {5 \cot \left (a+b x^2\right ) \csc \left (a+b x^2\right )}{32 b}-\frac {5 \cot \left (a+b x^2\right ) \csc ^3\left (a+b x^2\right )}{48 b}-\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{12 b} \]
-5/32*arctanh(cos(b*x^2+a))/b-5/32*cot(b*x^2+a)*csc(b*x^2+a)/b-5/48*cot(b* x^2+a)*csc(b*x^2+a)^3/b-1/12*cot(b*x^2+a)*csc(b*x^2+a)^5/b
Time = 0.11 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.86 \[ \int x \csc ^7\left (a+b x^2\right ) \, dx=-\frac {5 \csc ^2\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}-\frac {\csc ^4\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}-\frac {\csc ^6\left (\frac {1}{2} \left (a+b x^2\right )\right )}{768 b}-\frac {5 \log \left (\cos \left (\frac {1}{2} \left (a+b x^2\right )\right )\right )}{32 b}+\frac {5 \log \left (\sin \left (\frac {1}{2} \left (a+b x^2\right )\right )\right )}{32 b}+\frac {5 \sec ^2\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}+\frac {\sec ^4\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}+\frac {\sec ^6\left (\frac {1}{2} \left (a+b x^2\right )\right )}{768 b} \]
(-5*Csc[(a + b*x^2)/2]^2)/(128*b) - Csc[(a + b*x^2)/2]^4/(128*b) - Csc[(a + b*x^2)/2]^6/(768*b) - (5*Log[Cos[(a + b*x^2)/2]])/(32*b) + (5*Log[Sin[(a + b*x^2)/2]])/(32*b) + (5*Sec[(a + b*x^2)/2]^2)/(128*b) + Sec[(a + b*x^2) /2]^4/(128*b) + Sec[(a + b*x^2)/2]^6/(768*b)
Time = 0.45 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.16, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {4693, 3042, 4255, 3042, 4255, 3042, 4255, 3042, 4257}
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.
| \(\displaystyle \int x \csc ^7\left (a+b x^2\right ) \, dx\) |
| \(\Big \downarrow \) 4693 |
| \(\displaystyle \frac {1}{2} \int \csc ^7\left (b x^2+a\right )dx^2\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \int \csc \left (b x^2+a\right )^7dx^2\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{2} \left (\frac {5}{6} \int \csc ^5\left (b x^2+a\right )dx^2-\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{6 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\frac {5}{6} \int \csc \left (b x^2+a\right )^5dx^2-\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{6 b}\right )\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{2} \left (\frac {5}{6} \left (\frac {3}{4} \int \csc ^3\left (b x^2+a\right )dx^2-\frac {\cot \left (a+b x^2\right ) \csc ^3\left (a+b x^2\right )}{4 b}\right )-\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{6 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\frac {5}{6} \left (\frac {3}{4} \int \csc \left (b x^2+a\right )^3dx^2-\frac {\cot \left (a+b x^2\right ) \csc ^3\left (a+b x^2\right )}{4 b}\right )-\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{6 b}\right )\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{2} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \csc \left (b x^2+a\right )dx^2-\frac {\cot \left (a+b x^2\right ) \csc \left (a+b x^2\right )}{2 b}\right )-\frac {\cot \left (a+b x^2\right ) \csc ^3\left (a+b x^2\right )}{4 b}\right )-\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{6 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \csc \left (b x^2+a\right )dx^2-\frac {\cot \left (a+b x^2\right ) \csc \left (a+b x^2\right )}{2 b}\right )-\frac {\cot \left (a+b x^2\right ) \csc ^3\left (a+b x^2\right )}{4 b}\right )-\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{6 b}\right )\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{2} \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\text {arctanh}\left (\cos \left (a+b x^2\right )\right )}{2 b}-\frac {\cot \left (a+b x^2\right ) \csc \left (a+b x^2\right )}{2 b}\right )-\frac {\cot \left (a+b x^2\right ) \csc ^3\left (a+b x^2\right )}{4 b}\right )-\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{6 b}\right )\) |
(-1/6*(Cot[a + b*x^2]*Csc[a + b*x^2]^5)/b + (5*(-1/4*(Cot[a + b*x^2]*Csc[a + b*x^2]^3)/b + (3*(-1/2*ArcTanh[Cos[a + b*x^2]]/b - (Cot[a + b*x^2]*Csc[ a + b*x^2])/(2*b)))/4))/6)/2
3.1.15.3.1 Defintions of rubi rules used
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
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]
Time = 0.71 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.81
| method | result | size |
| derivativedivides | \(\frac {\left (-\frac {\csc \left (b \,x^{2}+a \right )^{5}}{6}-\frac {5 \csc \left (b \,x^{2}+a \right )^{3}}{24}-\frac {5 \csc \left (b \,x^{2}+a \right )}{16}\right ) \cot \left (b \,x^{2}+a \right )+\frac {5 \ln \left (\csc \left (b \,x^{2}+a \right )-\cot \left (b \,x^{2}+a \right )\right )}{16}}{2 b}\) | \(73\) |
| default | \(\frac {\left (-\frac {\csc \left (b \,x^{2}+a \right )^{5}}{6}-\frac {5 \csc \left (b \,x^{2}+a \right )^{3}}{24}-\frac {5 \csc \left (b \,x^{2}+a \right )}{16}\right ) \cot \left (b \,x^{2}+a \right )+\frac {5 \ln \left (\csc \left (b \,x^{2}+a \right )-\cot \left (b \,x^{2}+a \right )\right )}{16}}{2 b}\) | \(73\) |
| parallelrisch | \(\frac {\tan \left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )^{6}-\cot \left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )^{6}+9 \tan \left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )^{4}-9 \cot \left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )^{4}+45 \tan \left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )^{2}-45 \cot \left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )^{2}+120 \ln \left (\tan \left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )\right )}{768 b}\) | \(109\) |
| risch | \(\frac {15 \,{\mathrm e}^{11 i \left (b \,x^{2}+a \right )}-85 \,{\mathrm e}^{9 i \left (b \,x^{2}+a \right )}+198 \,{\mathrm e}^{7 i \left (b \,x^{2}+a \right )}+198 \,{\mathrm e}^{5 i \left (b \,x^{2}+a \right )}-85 \,{\mathrm e}^{3 i \left (b \,x^{2}+a \right )}+15 \,{\mathrm e}^{i \left (b \,x^{2}+a \right )}}{48 b \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{6}}-\frac {5 \ln \left ({\mathrm e}^{i \left (b \,x^{2}+a \right )}+1\right )}{32 b}+\frac {5 \ln \left ({\mathrm e}^{i \left (b \,x^{2}+a \right )}-1\right )}{32 b}\) | \(139\) |
1/2/b*((-1/6*csc(b*x^2+a)^5-5/24*csc(b*x^2+a)^3-5/16*csc(b*x^2+a))*cot(b*x ^2+a)+5/16*ln(csc(b*x^2+a)-cot(b*x^2+a)))
Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (82) = 164\).
Time = 0.26 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.03 \[ \int x \csc ^7\left (a+b x^2\right ) \, dx=\frac {30 \, \cos \left (b x^{2} + a\right )^{5} - 80 \, \cos \left (b x^{2} + a\right )^{3} - 15 \, {\left (\cos \left (b x^{2} + a\right )^{6} - 3 \, \cos \left (b x^{2} + a\right )^{4} + 3 \, \cos \left (b x^{2} + a\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x^{2} + a\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (b x^{2} + a\right )^{6} - 3 \, \cos \left (b x^{2} + a\right )^{4} + 3 \, \cos \left (b x^{2} + a\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x^{2} + a\right ) + \frac {1}{2}\right ) + 66 \, \cos \left (b x^{2} + a\right )}{192 \, {\left (b \cos \left (b x^{2} + a\right )^{6} - 3 \, b \cos \left (b x^{2} + a\right )^{4} + 3 \, b \cos \left (b x^{2} + a\right )^{2} - b\right )}} \]
1/192*(30*cos(b*x^2 + a)^5 - 80*cos(b*x^2 + a)^3 - 15*(cos(b*x^2 + a)^6 - 3*cos(b*x^2 + a)^4 + 3*cos(b*x^2 + a)^2 - 1)*log(1/2*cos(b*x^2 + a) + 1/2) + 15*(cos(b*x^2 + a)^6 - 3*cos(b*x^2 + a)^4 + 3*cos(b*x^2 + a)^2 - 1)*log (-1/2*cos(b*x^2 + a) + 1/2) + 66*cos(b*x^2 + a))/(b*cos(b*x^2 + a)^6 - 3*b *cos(b*x^2 + a)^4 + 3*b*cos(b*x^2 + a)^2 - b)
\[ \int x \csc ^7\left (a+b x^2\right ) \, dx=\int x \csc ^{7}{\left (a + b x^{2} \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 3543 vs. \(2 (82) = 164\).
Time = 0.28 (sec) , antiderivative size = 3543, normalized size of antiderivative = 39.37 \[ \int x \csc ^7\left (a+b x^2\right ) \, dx=\text {Too large to display} \]
1/192*(4*(15*cos(11*b*x^2 + 11*a) - 85*cos(9*b*x^2 + 9*a) + 198*cos(7*b*x^ 2 + 7*a) + 198*cos(5*b*x^2 + 5*a) - 85*cos(3*b*x^2 + 3*a) + 15*cos(b*x^2 + a))*cos(12*b*x^2 + 12*a) - 60*(6*cos(10*b*x^2 + 10*a) - 15*cos(8*b*x^2 + 8*a) + 20*cos(6*b*x^2 + 6*a) - 15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a ) - 1)*cos(11*b*x^2 + 11*a) + 24*(85*cos(9*b*x^2 + 9*a) - 198*cos(7*b*x^2 + 7*a) - 198*cos(5*b*x^2 + 5*a) + 85*cos(3*b*x^2 + 3*a) - 15*cos(b*x^2 + a ))*cos(10*b*x^2 + 10*a) - 340*(15*cos(8*b*x^2 + 8*a) - 20*cos(6*b*x^2 + 6* a) + 15*cos(4*b*x^2 + 4*a) - 6*cos(2*b*x^2 + 2*a) + 1)*cos(9*b*x^2 + 9*a) + 60*(198*cos(7*b*x^2 + 7*a) + 198*cos(5*b*x^2 + 5*a) - 85*cos(3*b*x^2 + 3 *a) + 15*cos(b*x^2 + a))*cos(8*b*x^2 + 8*a) - 792*(20*cos(6*b*x^2 + 6*a) - 15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a) - 1)*cos(7*b*x^2 + 7*a) - 80 *(198*cos(5*b*x^2 + 5*a) - 85*cos(3*b*x^2 + 3*a) + 15*cos(b*x^2 + a))*cos( 6*b*x^2 + 6*a) + 792*(15*cos(4*b*x^2 + 4*a) - 6*cos(2*b*x^2 + 2*a) + 1)*co s(5*b*x^2 + 5*a) - 300*(17*cos(3*b*x^2 + 3*a) - 3*cos(b*x^2 + a))*cos(4*b* x^2 + 4*a) + 340*(6*cos(2*b*x^2 + 2*a) - 1)*cos(3*b*x^2 + 3*a) - 360*cos(2 *b*x^2 + 2*a)*cos(b*x^2 + a) + 15*(2*(6*cos(10*b*x^2 + 10*a) - 15*cos(8*b* x^2 + 8*a) + 20*cos(6*b*x^2 + 6*a) - 15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a) - 1)*cos(12*b*x^2 + 12*a) - cos(12*b*x^2 + 12*a)^2 + 12*(15*cos(8* b*x^2 + 8*a) - 20*cos(6*b*x^2 + 6*a) + 15*cos(4*b*x^2 + 4*a) - 6*cos(2*b*x ^2 + 2*a) + 1)*cos(10*b*x^2 + 10*a) - 36*cos(10*b*x^2 + 10*a)^2 + 30*(2...
Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (82) = 164\).
Time = 0.27 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.34 \[ \int x \csc ^7\left (a+b x^2\right ) \, dx=-\frac {\frac {{\left (\frac {9 \, {\left (\cos \left (b x^{2} + a\right ) - 1\right )}}{\cos \left (b x^{2} + a\right ) + 1} - \frac {45 \, {\left (\cos \left (b x^{2} + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x^{2} + a\right ) + 1\right )}^{2}} + \frac {110 \, {\left (\cos \left (b x^{2} + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x^{2} + a\right ) + 1\right )}^{3}} - 1\right )} {\left (\cos \left (b x^{2} + a\right ) + 1\right )}^{3}}{{\left (\cos \left (b x^{2} + a\right ) - 1\right )}^{3}} + \frac {45 \, {\left (\cos \left (b x^{2} + a\right ) - 1\right )}}{\cos \left (b x^{2} + a\right ) + 1} - \frac {9 \, {\left (\cos \left (b x^{2} + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x^{2} + a\right ) + 1\right )}^{2}} + \frac {{\left (\cos \left (b x^{2} + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x^{2} + a\right ) + 1\right )}^{3}} - 60 \, \log \left (-\frac {\cos \left (b x^{2} + a\right ) - 1}{\cos \left (b x^{2} + a\right ) + 1}\right )}{768 \, b} \]
-1/768*((9*(cos(b*x^2 + a) - 1)/(cos(b*x^2 + a) + 1) - 45*(cos(b*x^2 + a) - 1)^2/(cos(b*x^2 + a) + 1)^2 + 110*(cos(b*x^2 + a) - 1)^3/(cos(b*x^2 + a) + 1)^3 - 1)*(cos(b*x^2 + a) + 1)^3/(cos(b*x^2 + a) - 1)^3 + 45*(cos(b*x^2 + a) - 1)/(cos(b*x^2 + a) + 1) - 9*(cos(b*x^2 + a) - 1)^2/(cos(b*x^2 + a) + 1)^2 + (cos(b*x^2 + a) - 1)^3/(cos(b*x^2 + a) + 1)^3 - 60*log(-(cos(b*x ^2 + a) - 1)/(cos(b*x^2 + a) + 1)))/b
Time = 28.55 (sec) , antiderivative size = 491, normalized size of antiderivative = 5.46 \[ \int x \csc ^7\left (a+b x^2\right ) \, dx=-\frac {5\,\ln \left (-\frac {x\,5{}\mathrm {i}}{8}-\frac {x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x^2\,1{}\mathrm {i}}\,5{}\mathrm {i}}{8}\right )}{32\,b}+\frac {5\,\ln \left (\frac {x\,5{}\mathrm {i}}{8}-\frac {x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x^2\,1{}\mathrm {i}}\,5{}\mathrm {i}}{8}\right )}{32\,b}+\frac {8\,{\mathrm {e}}^{3{}\mathrm {i}\,b\,x^2+a\,3{}\mathrm {i}}}{3\,b\,\left (5\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}-10\,{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}+10\,{\mathrm {e}}^{6{}\mathrm {i}\,b\,x^2+a\,6{}\mathrm {i}}-5\,{\mathrm {e}}^{8{}\mathrm {i}\,b\,x^2+a\,8{}\mathrm {i}}+{\mathrm {e}}^{10{}\mathrm {i}\,b\,x^2+a\,10{}\mathrm {i}}-1\right )}+\frac {{\mathrm {e}}^{1{}\mathrm {i}\,b\,x^2+a\,1{}\mathrm {i}}}{6\,b\,\left (3\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}-3\,{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}+{\mathrm {e}}^{6{}\mathrm {i}\,b\,x^2+a\,6{}\mathrm {i}}-1\right )}+\frac {5\,{\mathrm {e}}^{1{}\mathrm {i}\,b\,x^2+a\,1{}\mathrm {i}}}{16\,b\,\left ({\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}-1\right )}+\frac {16\,{\mathrm {e}}^{5{}\mathrm {i}\,b\,x^2+a\,5{}\mathrm {i}}}{3\,b\,\left (1+15\,{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}-20\,{\mathrm {e}}^{6{}\mathrm {i}\,b\,x^2+a\,6{}\mathrm {i}}+15\,{\mathrm {e}}^{8{}\mathrm {i}\,b\,x^2+a\,8{}\mathrm {i}}-6\,{\mathrm {e}}^{10{}\mathrm {i}\,b\,x^2+a\,10{}\mathrm {i}}+{\mathrm {e}}^{12{}\mathrm {i}\,b\,x^2+a\,12{}\mathrm {i}}-6\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}\right )}+\frac {{\mathrm {e}}^{1{}\mathrm {i}\,b\,x^2+a\,1{}\mathrm {i}}}{b\,\left (1+6\,{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}-4\,{\mathrm {e}}^{6{}\mathrm {i}\,b\,x^2+a\,6{}\mathrm {i}}+{\mathrm {e}}^{8{}\mathrm {i}\,b\,x^2+a\,8{}\mathrm {i}}-4\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}\right )}-\frac {5\,{\mathrm {e}}^{1{}\mathrm {i}\,b\,x^2+a\,1{}\mathrm {i}}}{24\,b\,\left (1+{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}-2\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}\right )} \]
(5*log((x*5i)/8 - (x*exp(a*1i)*exp(b*x^2*1i)*5i)/8))/(32*b) - (5*log(- (x* 5i)/8 - (x*exp(a*1i)*exp(b*x^2*1i)*5i)/8))/(32*b) + (8*exp(a*3i + b*x^2*3i ))/(3*b*(5*exp(a*2i + b*x^2*2i) - 10*exp(a*4i + b*x^2*4i) + 10*exp(a*6i + b*x^2*6i) - 5*exp(a*8i + b*x^2*8i) + exp(a*10i + b*x^2*10i) - 1)) + exp(a* 1i + b*x^2*1i)/(6*b*(3*exp(a*2i + b*x^2*2i) - 3*exp(a*4i + b*x^2*4i) + exp (a*6i + b*x^2*6i) - 1)) + (5*exp(a*1i + b*x^2*1i))/(16*b*(exp(a*2i + b*x^2 *2i) - 1)) + (16*exp(a*5i + b*x^2*5i))/(3*b*(15*exp(a*4i + b*x^2*4i) - 6*e xp(a*2i + b*x^2*2i) - 20*exp(a*6i + b*x^2*6i) + 15*exp(a*8i + b*x^2*8i) - 6*exp(a*10i + b*x^2*10i) + exp(a*12i + b*x^2*12i) + 1)) + exp(a*1i + b*x^2 *1i)/(b*(6*exp(a*4i + b*x^2*4i) - 4*exp(a*2i + b*x^2*2i) - 4*exp(a*6i + b* x^2*6i) + exp(a*8i + b*x^2*8i) + 1)) - (5*exp(a*1i + b*x^2*1i))/(24*b*(exp (a*4i + b*x^2*4i) - 2*exp(a*2i + b*x^2*2i) + 1))