Integrand size = 23, antiderivative size = 113 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {\sqrt {b} (3 a+5 b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{7/2} d}+\frac {(a+2 b) \coth (c+d x)}{a^3 d}-\frac {\coth ^3(c+d x)}{3 a^2 d}+\frac {b (a+b) \tanh (c+d x)}{2 a^3 d \left (a+b \tanh ^2(c+d x)\right )} \] Output:
1/2*b^(1/2)*(3*a+5*b)*arctan(b^(1/2)*tanh(d*x+c)/a^(1/2))/a^(7/2)/d+(a+2*b )*coth(d*x+c)/a^3/d-1/3*coth(d*x+c)^3/a^2/d+1/2*b*(a+b)*tanh(d*x+c)/a^3/d/ (a+b*tanh(d*x+c)^2)
Time = 0.99 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.01 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {3 \sqrt {b} (3 a+5 b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )+2 \sqrt {a} \coth (c+d x) \left (2 a+6 b-a \text {csch}^2(c+d x)\right )+\frac {3 \sqrt {a} b (a+b) \sinh (2 (c+d x))}{a-b+(a+b) \cosh (2 (c+d x))}}{6 a^{7/2} d} \] Input:
Integrate[Csch[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^2,x]
Output:
(3*Sqrt[b]*(3*a + 5*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] + 2*Sqrt[a] *Coth[c + d*x]*(2*a + 6*b - a*Csch[c + d*x]^2) + (3*Sqrt[a]*b*(a + b)*Sinh [2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)]))/(6*a^(7/2)*d)
Time = 0.66 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 4146, 361, 25, 1584, 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.
| \(\displaystyle \int \frac {\text {csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (i c+i d x)^4 \left (a-b \tan (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 4146 |
| \(\displaystyle \frac {\int \frac {\coth ^4(c+d x) \left (1-\tanh ^2(c+d x)\right )}{\left (b \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 361 |
| \(\displaystyle \frac {\frac {b (a+b) \tanh (c+d x)}{2 a^3 \left (a+b \tanh ^2(c+d x)\right )}-\frac {1}{2} b \int -\frac {\coth ^4(c+d x) \left (\frac {(a+b) \tanh ^4(c+d x)}{a^3}-\frac {2 (a+b) \tanh ^2(c+d x)}{a^2 b}+\frac {2}{a b}\right )}{b \tanh ^2(c+d x)+a}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} b \int \frac {\coth ^4(c+d x) \left (\frac {(a+b) \tanh ^4(c+d x)}{a^3}-\frac {2 (a+b) \tanh ^2(c+d x)}{a^2 b}+\frac {2}{a b}\right )}{b \tanh ^2(c+d x)+a}d\tanh (c+d x)+\frac {b (a+b) \tanh (c+d x)}{2 a^3 \left (a+b \tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 1584 |
| \(\displaystyle \frac {\frac {1}{2} b \int \left (\frac {2 \coth ^4(c+d x)}{a^2 b}-\frac {2 (a+2 b) \coth ^2(c+d x)}{a^3 b}+\frac {3 a+5 b}{a^3 \left (b \tanh ^2(c+d x)+a\right )}\right )d\tanh (c+d x)+\frac {b (a+b) \tanh (c+d x)}{2 a^3 \left (a+b \tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {b (a+b) \tanh (c+d x)}{2 a^3 \left (a+b \tanh ^2(c+d x)\right )}+\frac {1}{2} b \left (\frac {(3 a+5 b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{7/2} \sqrt {b}}+\frac {2 (a+2 b) \coth (c+d x)}{a^3 b}-\frac {2 \coth ^3(c+d x)}{3 a^2 b}\right )}{d}\) |
Input:
Int[Csch[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^2,x]
Output:
((b*(((3*a + 5*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^(7/2)*Sqrt[b ]) + (2*(a + 2*b)*Coth[c + d*x])/(a^3*b) - (2*Coth[c + d*x]^3)/(3*a^2*b))) /2 + (b*(a + b)*Tanh[c + d*x])/(2*a^3*(a + b*Tanh[c + d*x]^2)))/d
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[x^m*(a + b*x^2)^(p + 1)*E xpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)] - ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ILtQ[m/ 2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* (a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_ )])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[c*(ff^(m + 1)/f) Subst[Int[x^m*((a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)^(m/ 2 + 1)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x ] && IntegerQ[m/2]
Leaf count of result is larger than twice the leaf count of optimal. \(337\) vs. \(2(99)=198\).
Time = 7.43 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.99
| method | result | size |
| derivativedivides | \(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{3}-3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a -8 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{3}}-\frac {2 b \left (\frac {\left (-\frac {a}{2}-\frac {b}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {a}{2}-\frac {b}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}+\frac {\left (3 a +5 b \right ) a \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2}\right )}{a^{3}}-\frac {1}{24 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-3 a -8 b}{8 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(338\) |
| default | \(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{3}-3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a -8 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{3}}-\frac {2 b \left (\frac {\left (-\frac {a}{2}-\frac {b}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {a}{2}-\frac {b}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}+\frac {\left (3 a +5 b \right ) a \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2}\right )}{a^{3}}-\frac {1}{24 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-3 a -8 b}{8 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(338\) |
| risch | \(-\frac {-9 \,{\mathrm e}^{8 d x +8 c} a b -15 \,{\mathrm e}^{8 d x +8 c} b^{2}+12 \,{\mathrm e}^{6 d x +6 c} a^{2}+6 \,{\mathrm e}^{6 d x +6 c} a b +60 \,{\mathrm e}^{6 d x +6 c} b^{2}+20 \,{\mathrm e}^{4 d x +4 c} a^{2}-4 \,{\mathrm e}^{4 d x +4 c} a b -90 \,{\mathrm e}^{4 d x +4 c} b^{2}+4 \,{\mathrm e}^{2 d x +2 c} a^{2}+26 \,{\mathrm e}^{2 d x +2 c} b a +60 b^{2} {\mathrm e}^{2 d x +2 c}-4 a^{2}-19 a b -15 b^{2}}{3 d \,a^{3} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3} \left ({\mathrm e}^{4 d x +4 c} a +b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 \,{\mathrm e}^{2 d x +2 c} b +a +b \right )}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{4 a^{3} d}+\frac {5 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right ) b}{4 a^{4} d}-\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{4 a^{3} d}-\frac {5 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right ) b}{4 a^{4} d}\) | \(419\) |
Input:
int(csch(d*x+c)^4/(a+tanh(d*x+c)^2*b)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/8/a^3*(1/3*tanh(1/2*d*x+1/2*c)^3*a-3*tanh(1/2*d*x+1/2*c)*a-8*b*tan h(1/2*d*x+1/2*c))-2*b/a^3*(((-1/2*a-1/2*b)*tanh(1/2*d*x+1/2*c)^3+(-1/2*a-1 /2*b)*tanh(1/2*d*x+1/2*c))/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^ 2*a+4*b*tanh(1/2*d*x+1/2*c)^2+a)+1/2*(3*a+5*b)*a*(1/2*(a+((a+b)*b)^(1/2)+b )/a/((a+b)*b)^(1/2)/((2*((a+b)*b)^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2* d*x+1/2*c)/((2*((a+b)*b)^(1/2)+a+2*b)*a)^(1/2))-1/2*(-a+((a+b)*b)^(1/2)-b) /a/((a+b)*b)^(1/2)/((2*((a+b)*b)^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2* d*x+1/2*c)/((2*((a+b)*b)^(1/2)-a-2*b)*a)^(1/2))))-1/24/a^2/tanh(1/2*d*x+1/ 2*c)^3-1/8/a^3*(-3*a-8*b)/tanh(1/2*d*x+1/2*c))
Leaf count of result is larger than twice the leaf count of optimal. 2370 vs. \(2 (99) = 198\).
Time = 0.16 (sec) , antiderivative size = 5062, normalized size of antiderivative = 44.80 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^4/(a+b*tanh(d*x+c)^2)^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {\operatorname {csch}^{4}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \] Input:
integrate(csch(d*x+c)**4/(a+b*tanh(d*x+c)**2)**2,x)
Output:
Integral(csch(c + d*x)**4/(a + b*tanh(c + d*x)**2)**2, x)
Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (99) = 198\).
Time = 0.23 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.50 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {4 \, a^{2} + 19 \, a b + 15 \, b^{2} - 2 \, {\left (2 \, a^{2} + 13 \, a b + 30 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, {\left (10 \, a^{2} - 2 \, a b - 45 \, b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 6 \, {\left (2 \, a^{2} + a b + 10 \, b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + 3 \, {\left (3 \, a b + 5 \, b^{2}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{3 \, {\left (a^{4} + a^{3} b - {\left (a^{4} + 5 \, a^{3} b\right )} e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, {\left (a^{4} - 5 \, a^{3} b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 2 \, {\left (a^{4} - 5 \, a^{3} b\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{4} + 5 \, a^{3} b\right )} e^{\left (-8 \, d x - 8 \, c\right )} - {\left (a^{4} + a^{3} b\right )} e^{\left (-10 \, d x - 10 \, c\right )}\right )} d} - \frac {{\left (3 \, a b + 5 \, b^{2}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{3} d} \] Input:
integrate(csch(d*x+c)^4/(a+b*tanh(d*x+c)^2)^2,x, algorithm="maxima")
Output:
1/3*(4*a^2 + 19*a*b + 15*b^2 - 2*(2*a^2 + 13*a*b + 30*b^2)*e^(-2*d*x - 2*c ) - 2*(10*a^2 - 2*a*b - 45*b^2)*e^(-4*d*x - 4*c) - 6*(2*a^2 + a*b + 10*b^2 )*e^(-6*d*x - 6*c) + 3*(3*a*b + 5*b^2)*e^(-8*d*x - 8*c))/((a^4 + a^3*b - ( a^4 + 5*a^3*b)*e^(-2*d*x - 2*c) - 2*(a^4 - 5*a^3*b)*e^(-4*d*x - 4*c) + 2*( a^4 - 5*a^3*b)*e^(-6*d*x - 6*c) + (a^4 + 5*a^3*b)*e^(-8*d*x - 8*c) - (a^4 + a^3*b)*e^(-10*d*x - 10*c))*d) - 1/2*(3*a*b + 5*b^2)*arctan(1/2*((a + b)* e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/(sqrt(a*b)*a^3*d)
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (99) = 198\).
Time = 0.33 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.85 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {\frac {3 \, {\left (3 \, a b + 5 \, b^{2}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} a^{3}} - \frac {6 \, {\left (a b e^{\left (2 \, d x + 2 \, c\right )} - b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a b + b^{2}\right )}}{{\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )} a^{3}} + \frac {8 \, {\left (3 \, b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, a e^{\left (2 \, d x + 2 \, c\right )} - 6 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + 3 \, b\right )}}{a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{6 \, d} \] Input:
integrate(csch(d*x+c)^4/(a+b*tanh(d*x+c)^2)^2,x, algorithm="giac")
Output:
1/6*(3*(3*a*b + 5*b^2)*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/(sqrt(a*b)*a^3) - 6*(a*b*e^(2*d*x + 2*c) - b^2*e^(2*d*x + 2*c) + a*b + b^2)/((a*e^(4*d*x + 4*c) + b*e^(4*d*x + 4*c) + 2*a*e^(2*d* x + 2*c) - 2*b*e^(2*d*x + 2*c) + a + b)*a^3) + 8*(3*b*e^(4*d*x + 4*c) - 3* a*e^(2*d*x + 2*c) - 6*b*e^(2*d*x + 2*c) + a + 3*b)/(a^3*(e^(2*d*x + 2*c) - 1)^3))/d
Timed out. \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^4\,{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \] Input:
int(1/(sinh(c + d*x)^4*(a + b*tanh(c + d*x)^2)^2),x)
Output:
int(1/(sinh(c + d*x)^4*(a + b*tanh(c + d*x)^2)^2), x)
Time = 0.40 (sec) , antiderivative size = 2398, normalized size of antiderivative = 21.22 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:
int(csch(d*x+c)^4/(a+b*tanh(d*x+c)^2)^2,x)
Output:
(9*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqr t(b))/sqrt(a))*a**3 + 69*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**2*b + 135*e**(10*c + 10*d*x)*sqrt( b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a*b**2 + 75* e**(10*c + 10*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b ))/sqrt(a))*b**3 - 9*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*s qrt(a + b) - sqrt(b))/sqrt(a))*a**3 - 105*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a) *atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**2*b - 375*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a)) *a*b**2 - 375*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*b**3 - 18*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*atan((e **(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**3 - 30*e**(6*c + 6*d*x)*sqr t(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**2*b + 4 50*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt( b))/sqrt(a))*a*b**2 + 750*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d *x)*sqrt(a + b) - sqrt(b))/sqrt(a))*b**3 + 18*e**(4*c + 4*d*x)*sqrt(b)*sqr t(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**3 + 30*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a) )*a**2*b - 450*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a*b**2 - 750*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*...