Integrand size = 23, antiderivative size = 130 \[ \int \frac {\text {csch}^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=-\frac {b^{5/2} \arctan \left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{a^3 \sqrt {a-b} d}-\frac {\left (3 a^2+4 a b+8 b^2\right ) \text {arctanh}(\cosh (c+d x))}{8 a^3 d}+\frac {(3 a+4 b) \coth (c+d x) \text {csch}(c+d x)}{8 a^2 d}-\frac {\coth (c+d x) \text {csch}^3(c+d x)}{4 a d} \]
-1/8*(3*a^2+4*a*b+8*b^2)*arctanh(cosh(d*x+c))/a^3/d+1/8*(3*a+4*b)*coth(d*x +c)*csch(d*x+c)/a^2/d-1/4*coth(d*x+c)*csch(d*x+c)^3/a/d-b^(5/2)*arctan(cos h(d*x+c)*b^(1/2)/(a-b)^(1/2))/a^3/d/(a-b)^(1/2)
Result contains complex when optimal does not.
Time = 8.95 (sec) , antiderivative size = 649, normalized size of antiderivative = 4.99 \[ \int \frac {\text {csch}^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=-\frac {b^{5/2} \arctan \left (\frac {\text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (\sqrt {b} \cosh \left (\frac {1}{2} (c+d x)\right )-i \sqrt {a} \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a-b}}\right ) (2 a-b+b \cosh (2 (c+d x))) \text {csch}^2(c+d x)}{2 a^3 \sqrt {a-b} d \left (b+a \text {csch}^2(c+d x)\right )}-\frac {b^{5/2} \arctan \left (\frac {\text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (\sqrt {b} \cosh \left (\frac {1}{2} (c+d x)\right )+i \sqrt {a} \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a-b}}\right ) (2 a-b+b \cosh (2 (c+d x))) \text {csch}^2(c+d x)}{2 a^3 \sqrt {a-b} d \left (b+a \text {csch}^2(c+d x)\right )}+\frac {(3 a+4 b) (2 a-b+b \cosh (2 (c+d x))) \text {csch}^2\left (\frac {1}{2} (c+d x)\right ) \text {csch}^2(c+d x)}{64 a^2 d \left (b+a \text {csch}^2(c+d x)\right )}-\frac {(2 a-b+b \cosh (2 (c+d x))) \text {csch}^4\left (\frac {1}{2} (c+d x)\right ) \text {csch}^2(c+d x)}{128 a d \left (b+a \text {csch}^2(c+d x)\right )}+\frac {\left (-3 a^2-4 a b-8 b^2\right ) (2 a-b+b \cosh (2 (c+d x))) \text {csch}^2(c+d x) \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )}{16 a^3 d \left (b+a \text {csch}^2(c+d x)\right )}+\frac {\left (3 a^2+4 a b+8 b^2\right ) (2 a-b+b \cosh (2 (c+d x))) \text {csch}^2(c+d x) \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{16 a^3 d \left (b+a \text {csch}^2(c+d x)\right )}+\frac {(3 a+4 b) (2 a-b+b \cosh (2 (c+d x))) \text {csch}^2(c+d x) \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{64 a^2 d \left (b+a \text {csch}^2(c+d x)\right )}+\frac {(2 a-b+b \cosh (2 (c+d x))) \text {csch}^2(c+d x) \text {sech}^4\left (\frac {1}{2} (c+d x)\right )}{128 a d \left (b+a \text {csch}^2(c+d x)\right )} \]
-1/2*(b^(5/2)*ArcTan[(Sech[(c + d*x)/2]*(Sqrt[b]*Cosh[(c + d*x)/2] - I*Sqr t[a]*Sinh[(c + d*x)/2]))/Sqrt[a - b]]*(2*a - b + b*Cosh[2*(c + d*x)])*Csch [c + d*x]^2)/(a^3*Sqrt[a - b]*d*(b + a*Csch[c + d*x]^2)) - (b^(5/2)*ArcTan [(Sech[(c + d*x)/2]*(Sqrt[b]*Cosh[(c + d*x)/2] + I*Sqrt[a]*Sinh[(c + d*x)/ 2]))/Sqrt[a - b]]*(2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^2)/(2*a^3* Sqrt[a - b]*d*(b + a*Csch[c + d*x]^2)) + ((3*a + 4*b)*(2*a - b + b*Cosh[2* (c + d*x)])*Csch[(c + d*x)/2]^2*Csch[c + d*x]^2)/(64*a^2*d*(b + a*Csch[c + d*x]^2)) - ((2*a - b + b*Cosh[2*(c + d*x)])*Csch[(c + d*x)/2]^4*Csch[c + d*x]^2)/(128*a*d*(b + a*Csch[c + d*x]^2)) + ((-3*a^2 - 4*a*b - 8*b^2)*(2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^2*Log[Cosh[(c + d*x)/2]])/(16*a^ 3*d*(b + a*Csch[c + d*x]^2)) + ((3*a^2 + 4*a*b + 8*b^2)*(2*a - b + b*Cosh[ 2*(c + d*x)])*Csch[c + d*x]^2*Log[Sinh[(c + d*x)/2]])/(16*a^3*d*(b + a*Csc h[c + d*x]^2)) + ((3*a + 4*b)*(2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x ]^2*Sech[(c + d*x)/2]^2)/(64*a^2*d*(b + a*Csch[c + d*x]^2)) + ((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^2*Sech[(c + d*x)/2]^4)/(128*a*d*(b + a* Csch[c + d*x]^2))
Time = 0.36 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 26, 3665, 316, 402, 397, 218, 219}
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.
| \(\displaystyle \int \frac {\text {csch}^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i}{\sin (i c+i d x)^5 \left (a-b \sin (i c+i d x)^2\right )}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {1}{\sin (i c+i d x)^5 \left (a-b \sin (i c+i d x)^2\right )}dx\) |
| \(\Big \downarrow \) 3665 |
| \(\displaystyle -\frac {\int \frac {1}{\left (1-\cosh ^2(c+d x)\right )^3 \left (b \cosh ^2(c+d x)+a-b\right )}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle -\frac {\frac {\int \frac {3 b \cosh ^2(c+d x)+3 a+b}{\left (1-\cosh ^2(c+d x)\right )^2 \left (b \cosh ^2(c+d x)+a-b\right )}d\cosh (c+d x)}{4 a}+\frac {\cosh (c+d x)}{4 a \left (1-\cosh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {3 a^2+b a+4 b^2+b (3 a+4 b) \cosh ^2(c+d x)}{\left (1-\cosh ^2(c+d x)\right ) \left (b \cosh ^2(c+d x)+a-b\right )}d\cosh (c+d x)}{2 a}+\frac {(3 a+4 b) \cosh (c+d x)}{2 a \left (1-\cosh ^2(c+d x)\right )}}{4 a}+\frac {\cosh (c+d x)}{4 a \left (1-\cosh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\left (3 a^2+4 a b+8 b^2\right ) \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a}+\frac {8 b^3 \int \frac {1}{b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{a}}{2 a}+\frac {(3 a+4 b) \cosh (c+d x)}{2 a \left (1-\cosh ^2(c+d x)\right )}}{4 a}+\frac {\cosh (c+d x)}{4 a \left (1-\cosh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\left (3 a^2+4 a b+8 b^2\right ) \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a}+\frac {8 b^{5/2} \arctan \left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{a \sqrt {a-b}}}{2 a}+\frac {(3 a+4 b) \cosh (c+d x)}{2 a \left (1-\cosh ^2(c+d x)\right )}}{4 a}+\frac {\cosh (c+d x)}{4 a \left (1-\cosh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\left (3 a^2+4 a b+8 b^2\right ) \text {arctanh}(\cosh (c+d x))}{a}+\frac {8 b^{5/2} \arctan \left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{a \sqrt {a-b}}}{2 a}+\frac {(3 a+4 b) \cosh (c+d x)}{2 a \left (1-\cosh ^2(c+d x)\right )}}{4 a}+\frac {\cosh (c+d x)}{4 a \left (1-\cosh ^2(c+d x)\right )^2}}{d}\) |
-((Cosh[c + d*x]/(4*a*(1 - Cosh[c + d*x]^2)^2) + (((8*b^(5/2)*ArcTan[(Sqrt [b]*Cosh[c + d*x])/Sqrt[a - b]])/(a*Sqrt[a - b]) + ((3*a^2 + 4*a*b + 8*b^2 )*ArcTanh[Cosh[c + d*x]])/a)/(2*a) + ((3*a + 4*b)*Cosh[c + d*x])/(2*a*(1 - Cosh[c + d*x]^2)))/(4*a))/d)
3.1.40.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 0.42 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.20
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -4 a -4 b \right )^{2}}{64 a^{3}}-\frac {1}{64 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-4 a -4 b}{32 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (6 a^{2}+8 a b +16 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{3}}-\frac {b^{3} \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{a^{3} \sqrt {a b -b^{2}}}}{d}\) | \(156\) |
| default | \(\frac {\frac {\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -4 a -4 b \right )^{2}}{64 a^{3}}-\frac {1}{64 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-4 a -4 b}{32 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (6 a^{2}+8 a b +16 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{3}}-\frac {b^{3} \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{a^{3} \sqrt {a b -b^{2}}}}{d}\) | \(156\) |
| risch | \(\frac {{\mathrm e}^{d x +c} \left (3 \,{\mathrm e}^{6 d x +6 c} a +4 b \,{\mathrm e}^{6 d x +6 c}-11 \,{\mathrm e}^{4 d x +4 c} a -4 b \,{\mathrm e}^{4 d x +4 c}-11 a \,{\mathrm e}^{2 d x +2 c}-4 b \,{\mathrm e}^{2 d x +2 c}+3 a +4 b \right )}{4 d \,a^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{4}}-\frac {3 \ln \left ({\mathrm e}^{d x +c}+1\right )}{8 a d}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) b}{2 d \,a^{2}}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) b^{2}}{d \,a^{3}}+\frac {3 \ln \left ({\mathrm e}^{d x +c}-1\right )}{8 a d}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) b}{2 d \,a^{2}}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) b^{2}}{d \,a^{3}}+\frac {\sqrt {-b \left (a -b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right )}{2 \left (a -b \right ) d \,a^{3}}-\frac {\sqrt {-b \left (a -b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right )}{2 \left (a -b \right ) d \,a^{3}}\) | \(339\) |
1/d*(1/64*(tanh(1/2*d*x+1/2*c)^2*a-4*a-4*b)^2/a^3-1/64/a/tanh(1/2*d*x+1/2* c)^4-1/32*(-4*a-4*b)/a^2/tanh(1/2*d*x+1/2*c)^2+1/16/a^3*(6*a^2+8*a*b+16*b^ 2)*ln(tanh(1/2*d*x+1/2*c))-b^3/a^3/(a*b-b^2)^(1/2)*arctan(1/4*(2*tanh(1/2* d*x+1/2*c)^2*a-2*a+4*b)/(a*b-b^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 2976 vs. \(2 (116) = 232\).
Time = 0.36 (sec) , antiderivative size = 5809, normalized size of antiderivative = 44.68 \[ \int \frac {\text {csch}^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\text {csch}^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\text {csch}^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{5}}{b \sinh \left (d x + c\right )^{2} + a} \,d x } \]
1/4*((3*a*e^(7*c) + 4*b*e^(7*c))*e^(7*d*x) - (11*a*e^(5*c) + 4*b*e^(5*c))* e^(5*d*x) - (11*a*e^(3*c) + 4*b*e^(3*c))*e^(3*d*x) + (3*a*e^c + 4*b*e^c)*e ^(d*x))/(a^2*d*e^(8*d*x + 8*c) - 4*a^2*d*e^(6*d*x + 6*c) + 6*a^2*d*e^(4*d* x + 4*c) - 4*a^2*d*e^(2*d*x + 2*c) + a^2*d) - 1/8*(3*a^2 + 4*a*b + 8*b^2)* log((e^(d*x + c) + 1)*e^(-c))/(a^3*d) + 1/8*(3*a^2 + 4*a*b + 8*b^2)*log((e ^(d*x + c) - 1)*e^(-c))/(a^3*d) - 32*integrate(1/16*(b^3*e^(3*d*x + 3*c) - b^3*e^(d*x + c))/(a^3*b*e^(4*d*x + 4*c) + a^3*b + 2*(2*a^4*e^(2*c) - a^3* b*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\text {csch}^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{5}}{b \sinh \left (d x + c\right )^{2} + a} \,d x } \]
Time = 6.51 (sec) , antiderivative size = 1639, normalized size of antiderivative = 12.61 \[ \int \frac {\text {csch}^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Too large to display} \]
(exp(c + d*x)*(4*a*b + 3*a^2))/(4*a^3*d*(exp(2*c + 2*d*x) - 1)) - (atan((e xp(d*x)*exp(c)*(243*a^12*(-a^6*d^2)^(1/2) + 18432*b^12*(-a^6*d^2)^(1/2) + 6912*a^2*b^10*(-a^6*d^2)^(1/2) - 30720*a^3*b^9*(-a^6*d^2)^(1/2) - 26880*a^ 4*b^8*(-a^6*d^2)^(1/2) - 24192*a^5*b^7*(-a^6*d^2)^(1/2) + 5024*a^6*b^6*(-a ^6*d^2)^(1/2) + 13408*a^7*b^5*(-a^6*d^2)^(1/2) + 17160*a^8*b^4*(-a^6*d^2)^ (1/2) + 9540*a^9*b^3*(-a^6*d^2)^(1/2) + 4563*a^10*b^2*(-a^6*d^2)^(1/2) + 9 216*a*b^11*(-a^6*d^2)^(1/2) + 1134*a^11*b*(-a^6*d^2)^(1/2)))/(81*a^13*d*(6 4*a*b^3 + 24*a^3*b + 9*a^4 + 64*b^4 + 64*a^2*b^2)^(1/2) + 2304*a^3*b^10*d* (64*a*b^3 + 24*a^3*b + 9*a^4 + 64*b^4 + 64*a^2*b^2)^(1/2) - 3840*a^6*b^7*d *(64*a*b^3 + 24*a^3*b + 9*a^4 + 64*b^4 + 64*a^2*b^2)^(1/2) - 1440*a^7*b^6* d*(64*a*b^3 + 24*a^3*b + 9*a^4 + 64*b^4 + 64*a^2*b^2)^(1/2) - 864*a^8*b^5* d*(64*a*b^3 + 24*a^3*b + 9*a^4 + 64*b^4 + 64*a^2*b^2)^(1/2) + 1600*a^9*b^4 *d*(64*a*b^3 + 24*a^3*b + 9*a^4 + 64*b^4 + 64*a^2*b^2)^(1/2) + 1200*a^10*b ^3*d*(64*a*b^3 + 24*a^3*b + 9*a^4 + 64*b^4 + 64*a^2*b^2)^(1/2) + 945*a^11* b^2*d*(64*a*b^3 + 24*a^3*b + 9*a^4 + 64*b^4 + 64*a^2*b^2)^(1/2) + 270*a^12 *b*d*(64*a*b^3 + 24*a^3*b + 9*a^4 + 64*b^4 + 64*a^2*b^2)^(1/2)))*(64*a*b^3 + 24*a^3*b + 9*a^4 + 64*b^4 + 64*a^2*b^2)^(1/2))/(4*(-a^6*d^2)^(1/2)) - ( 6*exp(c + d*x))/(a*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) - (4*exp(c + d*x))/(a*d*(6*exp(4*c + 4*d*x) - 4*exp(2*c + 2*d *x) - 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) - ((2*atan((b^3*exp(d...