Integrand size = 23, antiderivative size = 213 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=-\frac {\sqrt {b} \left (15 a^2-10 a b-b^2\right ) \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{8 a^{3/2} (a+b)^4 d}+\frac {(a-5 b) \text {arctanh}(\cosh (c+d x))}{2 (a+b)^4 d}+\frac {(2 a-b) b \cosh (c+d x)}{4 a (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{8 a (a+b)^3 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\cosh (c+d x) \coth ^2(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2} \] Output:
-1/8*b^(1/2)*(15*a^2-10*a*b-b^2)*arctan(a^(1/2)*cosh(d*x+c)/b^(1/2))/a^(3/ 2)/(a+b)^4/d+1/2*(a-5*b)*arctanh(cosh(d*x+c))/(a+b)^4/d+1/4*(2*a-b)*b*cosh (d*x+c)/a/(a+b)^2/d/(b+a*cosh(d*x+c)^2)^2-1/8*(4*a^2-9*a*b-b^2)*cosh(d*x+c )/a/(a+b)^3/d/(b+a*cosh(d*x+c)^2)-1/2*cosh(d*x+c)*coth(d*x+c)^2/(a+b)/d/(b +a*cosh(d*x+c)^2)^2
Result contains complex when optimal does not.
Time = 3.45 (sec) , antiderivative size = 524, normalized size of antiderivative = 2.46 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^5(c+d x) \left (-\frac {8 b^2 (a+b)^2}{a}+\frac {2 b (a+b) (9 a+b) (a+2 b+a \cosh (2 (c+d x)))}{a}+\frac {\sqrt {b} \left (-15 a^2+10 a b+b^2\right ) \arctan \left (\frac {\left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right ) (a+2 b+a \cosh (2 (c+d x)))^2 \text {sech}(c+d x)}{a^{3/2}}+\frac {\sqrt {b} \left (-15 a^2+10 a b+b^2\right ) \arctan \left (\frac {\left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right ) (a+2 b+a \cosh (2 (c+d x)))^2 \text {sech}(c+d x)}{a^{3/2}}-(a+b) (a+2 b+a \cosh (2 (c+d x)))^2 \text {csch}^2\left (\frac {1}{2} (c+d x)\right ) \text {sech}(c+d x)+4 (a-5 b) (a+2 b+a \cosh (2 (c+d x)))^2 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right ) \text {sech}(c+d x)-4 (a-5 b) (a+2 b+a \cosh (2 (c+d x)))^2 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right ) \text {sech}(c+d x)-(a+b) (a+2 b+a \cosh (2 (c+d x)))^2 \text {sech}^2\left (\frac {1}{2} (c+d x)\right ) \text {sech}(c+d x)\right )}{64 (a+b)^4 d \left (a+b \text {sech}^2(c+d x)\right )^3} \] Input:
Integrate[Csch[c + d*x]^3/(a + b*Sech[c + d*x]^2)^3,x]
Output:
((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^5*((-8*b^2*(a + b)^2)/a + ( 2*b*(a + b)*(9*a + b)*(a + 2*b + a*Cosh[2*(c + d*x)]))/a + (Sqrt[b]*(-15*a ^2 + 10*a*b + b^2)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c ])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh [c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)]) ^2*Sech[c + d*x])/a^(3/2) + (Sqrt[b]*(-15*a^2 + 10*a*b + b^2)*ArcTan[((Sqr t[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2] ))/Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x])/a^(3/2) - (a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Csch[(c + d*x)/2]^2*Sech[c + d*x] + 4*(a - 5*b)*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Log[Cosh[(c + d*x)/2]]*Sech [c + d*x] - 4*(a - 5*b)*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Log[Sinh[(c + d* x)/2]]*Sech[c + d*x] - (a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[(c + d*x)/2]^2*Sech[c + d*x]))/(64*(a + b)^4*d*(a + b*Sech[c + d*x]^2)^3)
Time = 0.49 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 26, 4621, 372, 440, 27, 402, 25, 27, 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.
| \(\displaystyle \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\sin (i c+i d x)^3 \left (a+b \sec (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\left (b \sec (i c+i d x)^2+a\right )^3 \sin (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 4621 |
| \(\displaystyle \frac {\int \frac {\cosh ^6(c+d x)}{\left (1-\cosh ^2(c+d x)\right )^2 \left (a \cosh ^2(c+d x)+b\right )^3}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {\frac {\cosh ^3(c+d x)}{2 (a+b) \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )^2}-\frac {\int \frac {\cosh ^2(c+d x) \left (3 b-(a-2 b) \cosh ^2(c+d x)\right )}{\left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )^3}d\cosh (c+d x)}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 440 |
| \(\displaystyle \frac {\frac {\cosh ^3(c+d x)}{2 (a+b) \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )^2}-\frac {\frac {\int \frac {2 \left ((2 a-b) b-\left (2 a^2-8 b a-b^2\right ) \cosh ^2(c+d x)\right )}{\left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )^2}d\cosh (c+d x)}{4 a (a+b)}-\frac {b (2 a-b) \cosh (c+d x)}{2 a (a+b) \left (a \cosh ^2(c+d x)+b\right )^2}}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\cosh ^3(c+d x)}{2 (a+b) \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )^2}-\frac {\frac {\int \frac {(2 a-b) b-\left (2 a^2-8 b a-b^2\right ) \cosh ^2(c+d x)}{\left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )^2}d\cosh (c+d x)}{2 a (a+b)}-\frac {b (2 a-b) \cosh (c+d x)}{2 a (a+b) \left (a \cosh ^2(c+d x)+b\right )^2}}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\cosh ^3(c+d x)}{2 (a+b) \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{2 (a+b) \left (a \cosh ^2(c+d x)+b\right )}-\frac {\int -\frac {b \left ((11 a-b) b-\left (4 a^2-9 b a-b^2\right ) \cosh ^2(c+d x)\right )}{\left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )}d\cosh (c+d x)}{2 b (a+b)}}{2 a (a+b)}-\frac {b (2 a-b) \cosh (c+d x)}{2 a (a+b) \left (a \cosh ^2(c+d x)+b\right )^2}}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\cosh ^3(c+d x)}{2 (a+b) \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\int \frac {b \left ((11 a-b) b-\left (4 a^2-9 b a-b^2\right ) \cosh ^2(c+d x)\right )}{\left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )}d\cosh (c+d x)}{2 b (a+b)}+\frac {\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{2 (a+b) \left (a \cosh ^2(c+d x)+b\right )}}{2 a (a+b)}-\frac {b (2 a-b) \cosh (c+d x)}{2 a (a+b) \left (a \cosh ^2(c+d x)+b\right )^2}}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\cosh ^3(c+d x)}{2 (a+b) \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\int \frac {(11 a-b) b-\left (4 a^2-9 b a-b^2\right ) \cosh ^2(c+d x)}{\left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )}d\cosh (c+d x)}{2 (a+b)}+\frac {\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{2 (a+b) \left (a \cosh ^2(c+d x)+b\right )}}{2 a (a+b)}-\frac {b (2 a-b) \cosh (c+d x)}{2 a (a+b) \left (a \cosh ^2(c+d x)+b\right )^2}}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\cosh ^3(c+d x)}{2 (a+b) \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\frac {b \left (15 a^2-10 a b-b^2\right ) \int \frac {1}{a \cosh ^2(c+d x)+b}d\cosh (c+d x)}{a+b}-\frac {4 a (a-5 b) \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a+b}}{2 (a+b)}+\frac {\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{2 (a+b) \left (a \cosh ^2(c+d x)+b\right )}}{2 a (a+b)}-\frac {b (2 a-b) \cosh (c+d x)}{2 a (a+b) \left (a \cosh ^2(c+d x)+b\right )^2}}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\cosh ^3(c+d x)}{2 (a+b) \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\frac {\sqrt {b} \left (15 a^2-10 a b-b^2\right ) \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} (a+b)}-\frac {4 a (a-5 b) \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a+b}}{2 (a+b)}+\frac {\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{2 (a+b) \left (a \cosh ^2(c+d x)+b\right )}}{2 a (a+b)}-\frac {b (2 a-b) \cosh (c+d x)}{2 a (a+b) \left (a \cosh ^2(c+d x)+b\right )^2}}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\cosh ^3(c+d x)}{2 (a+b) \left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\frac {\sqrt {b} \left (15 a^2-10 a b-b^2\right ) \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} (a+b)}-\frac {4 a (a-5 b) \text {arctanh}(\cosh (c+d x))}{a+b}}{2 (a+b)}+\frac {\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{2 (a+b) \left (a \cosh ^2(c+d x)+b\right )}}{2 a (a+b)}-\frac {b (2 a-b) \cosh (c+d x)}{2 a (a+b) \left (a \cosh ^2(c+d x)+b\right )^2}}{2 (a+b)}}{d}\) |
Input:
Int[Csch[c + d*x]^3/(a + b*Sech[c + d*x]^2)^3,x]
Output:
(Cosh[c + d*x]^3/(2*(a + b)*(1 - Cosh[c + d*x]^2)*(b + a*Cosh[c + d*x]^2)^ 2) - (-1/2*((2*a - b)*b*Cosh[c + d*x])/(a*(a + b)*(b + a*Cosh[c + d*x]^2)^ 2) + (((Sqrt[b]*(15*a^2 - 10*a*b - b^2)*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqr t[b]])/(Sqrt[a]*(a + b)) - (4*a*(a - 5*b)*ArcTanh[Cosh[c + d*x]])/(a + b)) /(2*(a + b)) + ((4*a^2 - 9*a*b - b^2)*Cosh[c + d*x])/(2*(a + b)*(b + a*Cos h[c + d*x]^2)))/(2*a*(a + b)))/(2*(a + b)))/d
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_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 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[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ )*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[g*(b*e - a*f)*(g*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] - Simp[ g^2/(2*b*(b*c - a*d)*(p + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f)*(m - 1) + (d*(b*e - a*f)*(m + 2*q + 1) - b*2*(c *f - d*e)*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, q}, x] && LtQ[p, -1] && GtQ[m, 1]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff*x)^n)^p/(ff*x)^(n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/ 2] && IntegerQ[n] && IntegerQ[p]
Time = 0.22 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.64
\[\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a^{3}+24 a^{2} b +24 a \,b^{2}+8 b^{3}}-\frac {1}{8 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a +10 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{4}}-\frac {2 b \left (\frac {-\frac {\left (9 a^{3}-5 a^{2} b -13 a \,b^{2}+b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8 a}-\frac {\left (27 a^{3}-21 a^{2} b +29 a \,b^{2}-3 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 a}-\frac {\left (27 a^{3}+a^{2} b -23 a \,b^{2}+3 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}-\frac {9 a^{3}+17 a^{2} b +7 a \,b^{2}-b^{3}}{8 a}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (15 a^{2}-10 a b -b^{2}\right ) \arctan \left (\frac {2 \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 a -2 b}{4 \sqrt {a b}}\right )}{16 a \sqrt {a b}}\right )}{\left (a +b \right )^{4}}}{d}\]
Input:
int(csch(d*x+c)^3/(a+b*sech(d*x+c)^2)^3,x)
Output:
1/d*(1/8*tanh(1/2*d*x+1/2*c)^2/(a^3+3*a^2*b+3*a*b^2+b^3)-1/8/(a+b)^3/tanh( 1/2*d*x+1/2*c)^2+1/4/(a+b)^4*(-2*a+10*b)*ln(tanh(1/2*d*x+1/2*c))-2*b/(a+b) ^4*((-1/8*(9*a^3-5*a^2*b-13*a*b^2+b^3)/a*tanh(1/2*d*x+1/2*c)^6-1/8*(27*a^3 -21*a^2*b+29*a*b^2-3*b^3)/a*tanh(1/2*d*x+1/2*c)^4-1/8*(27*a^3+a^2*b-23*a*b ^2+3*b^3)/a*tanh(1/2*d*x+1/2*c)^2-1/8*(9*a^3+17*a^2*b+7*a*b^2-b^3)/a)/(tan h(1/2*d*x+1/2*c)^4*a+tanh(1/2*d*x+1/2*c)^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*t anh(1/2*d*x+1/2*c)^2*b+a+b)^2+1/16*(15*a^2-10*a*b-b^2)/a/(a*b)^(1/2)*arcta n(1/4*(2*(a+b)*tanh(1/2*d*x+1/2*c)^2+2*a-2*b)/(a*b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 10990 vs. \(2 (195) = 390\).
Time = 0.48 (sec) , antiderivative size = 20341, normalized size of antiderivative = 95.50 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^3/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {\operatorname {csch}^{3}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \] Input:
integrate(csch(d*x+c)**3/(a+b*sech(d*x+c)**2)**3,x)
Output:
Integral(csch(c + d*x)**3/(a + b*sech(c + d*x)**2)**3, x)
\[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{3}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \] Input:
integrate(csch(d*x+c)^3/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")
Output:
1/2*(a - 5*b)*log((e^(d*x + c) + 1)*e^(-c))/(a^4*d + 4*a^3*b*d + 6*a^2*b^2 *d + 4*a*b^3*d + b^4*d) - 1/2*(a - 5*b)*log((e^(d*x + c) - 1)*e^(-c))/(a^4 *d + 4*a^3*b*d + 6*a^2*b^2*d + 4*a*b^3*d + b^4*d) - 1/4*((4*a^3*e^(11*c) - 9*a^2*b*e^(11*c) - a*b^2*e^(11*c))*e^(11*d*x) + (20*a^3*e^(9*c) + 23*a^2* b*e^(9*c) - 29*a*b^2*e^(9*c) + 4*b^3*e^(9*c))*e^(9*d*x) + 2*(20*a^3*e^(7*c ) + 57*a^2*b*e^(7*c) + 47*a*b^2*e^(7*c) - 2*b^3*e^(7*c))*e^(7*d*x) + 2*(20 *a^3*e^(5*c) + 57*a^2*b*e^(5*c) + 47*a*b^2*e^(5*c) - 2*b^3*e^(5*c))*e^(5*d *x) + (20*a^3*e^(3*c) + 23*a^2*b*e^(3*c) - 29*a*b^2*e^(3*c) + 4*b^3*e^(3*c ))*e^(3*d*x) + (4*a^3*e^c - 9*a^2*b*e^c - a*b^2*e^c)*e^(d*x))/(a^6*d + 3*a ^5*b*d + 3*a^4*b^2*d + a^3*b^3*d + (a^6*d*e^(12*c) + 3*a^5*b*d*e^(12*c) + 3*a^4*b^2*d*e^(12*c) + a^3*b^3*d*e^(12*c))*e^(12*d*x) + 2*(a^6*d*e^(10*c) + 7*a^5*b*d*e^(10*c) + 15*a^4*b^2*d*e^(10*c) + 13*a^3*b^3*d*e^(10*c) + 4*a ^2*b^4*d*e^(10*c))*e^(10*d*x) - (a^6*d*e^(8*c) + 3*a^5*b*d*e^(8*c) - 13*a^ 4*b^2*d*e^(8*c) - 47*a^3*b^3*d*e^(8*c) - 48*a^2*b^4*d*e^(8*c) - 16*a*b^5*d *e^(8*c))*e^(8*d*x) - 4*(a^6*d*e^(6*c) + 7*a^5*b*d*e^(6*c) + 23*a^4*b^2*d* e^(6*c) + 37*a^3*b^3*d*e^(6*c) + 28*a^2*b^4*d*e^(6*c) + 8*a*b^5*d*e^(6*c)) *e^(6*d*x) - (a^6*d*e^(4*c) + 3*a^5*b*d*e^(4*c) - 13*a^4*b^2*d*e^(4*c) - 4 7*a^3*b^3*d*e^(4*c) - 48*a^2*b^4*d*e^(4*c) - 16*a*b^5*d*e^(4*c))*e^(4*d*x) + 2*(a^6*d*e^(2*c) + 7*a^5*b*d*e^(2*c) + 15*a^4*b^2*d*e^(2*c) + 13*a^3*b^ 3*d*e^(2*c) + 4*a^2*b^4*d*e^(2*c))*e^(2*d*x)) - 8*integrate(1/32*((15*a...
Exception generated. \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(csch(d*x+c)^3/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Timed out. \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \] Input:
int(1/(sinh(c + d*x)^3*(a + b/cosh(c + d*x)^2)^3),x)
Output:
int(cosh(c + d*x)^6/(sinh(c + d*x)^3*(b + a*cosh(c + d*x)^2)^3), x)
Time = 2.04 (sec) , antiderivative size = 13590, normalized size of antiderivative = 63.80 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:
int(csch(d*x+c)^3/(a+b*sech(d*x+c)^2)^3,x)
Output:
( - 30*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt( a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b ) + a + 2*b)))*a**4 + 20*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sq rt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2* sqrt(b)*sqrt(a + b) + a + 2*b)))*a**3*b + 2*e**(12*c + 12*d*x)*sqrt(b)*sqr t(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)* a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a**2*b**2 - 60*e**(10* c + 10*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2 *b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b))) *a**4 - 200*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)* sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt( a + b) + a + 2*b)))*a**3*b + 164*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a) *sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a**2*b**2 + 16*e**(10*c + 10*d*x) *sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e **(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a*b**3 + 3 0*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a**4 - 20*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt (b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b...