\(\int \frac {\cosh ^2(c+d x)}{(a+b \tanh ^2(c+d x))^3} \, dx\) [125]

Optimal result

Integrand size = 23, antiderivative size = 198 \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {(a+7 b) x}{2 (a+b)^4}+\frac {b^{3/2} \left (35 a^2+14 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} (a+b)^4 d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(2 a-b) b \tanh (c+d x)}{4 a (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(a-3 b) b (4 a+b) \tanh (c+d x)}{8 a^2 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )} \]

[Out]

1/2*(a+7*b)*x/(a+b)^4+1/8*b^(3/2)*(35*a^2+14*a*b+3*b^2)*arctan(b^(1/2)*tanh(d*x+c)/a^(1/2))/a^(5/2)/(a+b)^4/d+
1/2*cosh(d*x+c)*sinh(d*x+c)/(a+b)/d/(a+b*tanh(d*x+c)^2)^2-1/4*(2*a-b)*b*tanh(d*x+c)/a/(a+b)^2/d/(a+b*tanh(d*x+
c)^2)^2-1/8*(a-3*b)*b*(4*a+b)*tanh(d*x+c)/a^2/(a+b)^3/d/(a+b*tanh(d*x+c)^2)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3756, 425, 541, 536, 212, 211} \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {b (a-3 b) (4 a+b) \tanh (c+d x)}{8 a^2 d (a+b)^3 \left (a+b \tanh ^2(c+d x)\right )}+\frac {b^{3/2} \left (35 a^2+14 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} d (a+b)^4}-\frac {b (2 a-b) \tanh (c+d x)}{4 a d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {x (a+7 b)}{2 (a+b)^4} \]

[In]

Int[Cosh[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

((a + 7*b)*x)/(2*(a + b)^4) + (b^(3/2)*(35*a^2 + 14*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a
^(5/2)*(a + b)^4*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) - ((2*a - b)*b*Tan
h[c + d*x])/(4*a*(a + b)^2*d*(a + b*Tanh[c + d*x]^2)^2) - ((a - 3*b)*b*(4*a + b)*Tanh[c + d*x])/(8*a^2*(a + b)
^3*d*(a + b*Tanh[c + d*x]^2))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 3756

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/(c^(m - 1)*f), Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)
^n)^p, x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2] && (IntegersQ[n, p
] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \left (a+b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {a+2 b+5 b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{2 (a+b) d} \\ & = \frac {\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(2 a-b) b \tanh (c+d x)}{4 a (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-2 \left (2 a^2+8 a b+3 b^2\right )-6 (2 a-b) b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{8 a (a+b)^2 d} \\ & = \frac {\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(2 a-b) b \tanh (c+d x)}{4 a (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(a-3 b) b (4 a+b) \tanh (c+d x)}{8 a^2 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {2 \left (4 a^3+24 a^2 b+11 a b^2+3 b^3\right )+2 (a-3 b) b (4 a+b) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{16 a^2 (a+b)^3 d} \\ & = \frac {\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(2 a-b) b \tanh (c+d x)}{4 a (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(a-3 b) b (4 a+b) \tanh (c+d x)}{8 a^2 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )}+\frac {(a+7 b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 (a+b)^4 d}+\frac {\left (b^2 \left (35 a^2+14 a b+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a+b)^4 d} \\ & = \frac {(a+7 b) x}{2 (a+b)^4}+\frac {b^{3/2} \left (35 a^2+14 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} (a+b)^4 d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(2 a-b) b \tanh (c+d x)}{4 a (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(a-3 b) b (4 a+b) \tanh (c+d x)}{8 a^2 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.41 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.83 \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {4 (a+7 b) (c+d x)+\frac {b^{3/2} \left (35 a^2+14 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{5/2}}+2 (a+b) \sinh (2 (c+d x))+\frac {4 b^3 (a+b) \sinh (2 (c+d x))}{a (a-b+(a+b) \cosh (2 (c+d x)))^2}+\frac {b^2 (a+b) (13 a+3 b) \sinh (2 (c+d x))}{a^2 (a-b+(a+b) \cosh (2 (c+d x)))}}{8 (a+b)^4 d} \]

[In]

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

[Out]

(4*(a + 7*b)*(c + d*x) + (b^(3/2)*(35*a^2 + 14*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/a^(5/2) +
2*(a + b)*Sinh[2*(c + d*x)] + (4*b^3*(a + b)*Sinh[2*(c + d*x)])/(a*(a - b + (a + b)*Cosh[2*(c + d*x)])^2) + (
b^2*(a + b)*(13*a + 3*b)*Sinh[2*(c + d*x)])/(a^2*(a - b + (a + b)*Cosh[2*(c + d*x)])))/(8*(a + b)^4*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(500\) vs. \(2(180)=360\).

Time = 27.89 (sec) , antiderivative size = 501, normalized size of antiderivative = 2.53

method	result	size
derivativedivides	\(\frac {-\frac {1}{2 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {1}{2 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (a +7 b \right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a +b \right )^{4}}+\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a -7 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 \left (a +b \right )^{4}}-\frac {2 b^{2} \left (\frac {-\frac {\left (13 a^{2}+18 a b +5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 a}-\frac {\left (39 a^{3}+98 a^{2} b +71 a \,b^{2}+12 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 a^{2}}-\frac {\left (39 a^{3}+98 a^{2} b +71 a \,b^{2}+12 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 a^{2}}-\frac {\left (13 a^{2}+18 a b +5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}}{\left (\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 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a \right )^{2}}+\frac {\left (35 a^{2}+14 a b +3 b^{2}\right ) \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 )}{8 a}\right )}{\left (a +b \right )^{4}}}{d}\)	\(501\)
default	\(\frac {-\frac {1}{2 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {1}{2 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (a +7 b \right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a +b \right )^{4}}+\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a -7 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 \left (a +b \right )^{4}}-\frac {2 b^{2} \left (\frac {-\frac {\left (13 a^{2}+18 a b +5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 a}-\frac {\left (39 a^{3}+98 a^{2} b +71 a \,b^{2}+12 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 a^{2}}-\frac {\left (39 a^{3}+98 a^{2} b +71 a \,b^{2}+12 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 a^{2}}-\frac {\left (13 a^{2}+18 a b +5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}}{\left (\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 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a \right )^{2}}+\frac {\left (35 a^{2}+14 a b +3 b^{2}\right ) \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 )}{8 a}\right )}{\left (a +b \right )^{4}}}{d}\)	\(501\)
risch	\(\frac {x a}{2 \left (a +b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {7 x b}{2 \left (a +b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {{\mathrm e}^{2 d x +2 c}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {{\mathrm e}^{-2 d x -2 c}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {b^{2} \left (13 a^{3} {\mathrm e}^{6 d x +6 c}-a^{2} b \,{\mathrm e}^{6 d x +6 c}-17 a \,b^{2} {\mathrm e}^{6 d x +6 c}-3 \,{\mathrm e}^{6 d x +6 c} b^{3}+39 a^{3} {\mathrm e}^{4 d x +4 c}-17 a^{2} b \,{\mathrm e}^{4 d x +4 c}+33 a \,b^{2} {\mathrm e}^{4 d x +4 c}+9 \,{\mathrm e}^{4 d x +4 c} b^{3}+39 a^{3} {\mathrm e}^{2 d x +2 c}+13 a^{2} b \,{\mathrm e}^{2 d x +2 c}-35 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}-9 \,{\mathrm e}^{2 d x +2 c} b^{3}+13 a^{3}+29 a^{2} b +19 a \,b^{2}+3 b^{3}\right )}{4 \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )^{2} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (a +b \right ) d \,a^{2}}+\frac {35 \sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{16 a \left (a +b \right )^{4} d}+\frac {7 \sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{8 a^{2} \left (a +b \right )^{4} d}+\frac {3 \sqrt {-a b}\, b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{16 a^{3} \left (a +b \right )^{4} d}-\frac {35 \sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{16 a \left (a +b \right )^{4} d}-\frac {7 \sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{8 a^{2} \left (a +b \right )^{4} d}-\frac {3 \sqrt {-a b}\, b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{16 a^{3} \left (a +b \right )^{4} d}\)	\(728\)

[In]

int(cosh(d*x+c)^2/(a+b*tanh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/2/(a+b)^3/(1+tanh(1/2*d*x+1/2*c))^2+1/2/(a+b)^3/(1+tanh(1/2*d*x+1/2*c))+1/2*(a+7*b)/(a+b)^4*ln(1+tanh(
1/2*d*x+1/2*c))+1/2/(a+b)^3/(tanh(1/2*d*x+1/2*c)-1)^2+1/2/(a+b)^3/(tanh(1/2*d*x+1/2*c)-1)+1/2/(a+b)^4*(-a-7*b)
*ln(tanh(1/2*d*x+1/2*c)-1)-2*b^2/(a+b)^4*((-1/8*(13*a^2+18*a*b+5*b^2)/a*tanh(1/2*d*x+1/2*c)^7-1/8*(39*a^3+98*a
^2*b+71*a*b^2+12*b^3)/a^2*tanh(1/2*d*x+1/2*c)^5-1/8*(39*a^3+98*a^2*b+71*a*b^2+12*b^3)/a^2*tanh(1/2*d*x+1/2*c)^
3-1/8*(13*a^2+18*a*b+5*b^2)/a*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*tanh(1
/2*d*x+1/2*c)^2*b+a)^2+1/8/a*(35*a^2+14*a*b+3*b^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)*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)))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6780 vs. \(2 (180) = 360\).

Time = 0.46 (sec) , antiderivative size = 13887, normalized size of antiderivative = 70.14 \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(cosh(d*x+c)**2/(a+b*tanh(d*x+c)**2)**3,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1806 vs. \(2 (180) = 360\).

Time = 0.57 (sec) , antiderivative size = 1806, normalized size of antiderivative = 9.12 \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(cosh(d*x+c)^2/(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

3/4*b*log((a + b)*e^(4*d*x + 4*c) + 2*(a - b)*e^(2*d*x + 2*c) + a + b)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 +
b^4)*d) - 3/4*b*log(2*(a - b)*e^(-2*d*x - 2*c) + (a + b)*e^(-4*d*x - 4*c) + a + b)/((a^4 + 4*a^3*b + 6*a^2*b^
2 + 4*a*b^3 + b^4)*d) - 3/32*(5*a^3*b - 15*a^2*b^2 - 5*a*b^3 - b^4)*arctan(1/2*((a + b)*e^(2*d*x + 2*c) + a -
b)/sqrt(a*b))/((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*sqrt(a*b)*d) + 3/32*(5*a^3*b - 15*a^2*b^2 - 5
*a*b^3 - b^4)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3
+ a^2*b^4)*sqrt(a*b)*d) - 1/16*(15*a^2*b + 10*a*b^2 + 3*b^3)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sq
rt(a*b))/((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sqrt(a*b)*d) + 1/16*(9*a^4*b + 4*a^3*b^2 - 22*a^2*b^3 - 20*a*b
^4 - 3*b^5 + 3*(3*a^4*b - 22*a^3*b^2 - 20*a^2*b^3 + 6*a*b^4 + b^5)*e^(6*d*x + 6*c) + (27*a^4*b - 156*a^3*b^2 +
110*a^2*b^3 - 36*a*b^4 - 9*b^5)*e^(4*d*x + 4*c) + (27*a^4*b - 86*a^3*b^2 - 84*a^2*b^3 + 38*a*b^4 + 9*b^5)*e^(
2*d*x + 2*c))/((a^8 + 6*a^7*b + 15*a^6*b^2 + 20*a^5*b^3 + 15*a^4*b^4 + 6*a^3*b^5 + a^2*b^6 + (a^8 + 6*a^7*b +
15*a^6*b^2 + 20*a^5*b^3 + 15*a^4*b^4 + 6*a^3*b^5 + a^2*b^6)*e^(8*d*x + 8*c) + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 - 5
*a^4*b^4 - 4*a^3*b^5 - a^2*b^6)*e^(6*d*x + 6*c) + 2*(3*a^8 + 10*a^7*b + 13*a^6*b^2 + 12*a^5*b^3 + 13*a^4*b^4 +
10*a^3*b^5 + 3*a^2*b^6)*e^(4*d*x + 4*c) + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 - 5*a^4*b^4 - 4*a^3*b^5 - a^2*b^6)*e^(
2*d*x + 2*c))*d) - 1/16*(9*a^4*b + 4*a^3*b^2 - 22*a^2*b^3 - 20*a*b^4 - 3*b^5 + (27*a^4*b - 86*a^3*b^2 - 84*a^2
*b^3 + 38*a*b^4 + 9*b^5)*e^(-2*d*x - 2*c) + (27*a^4*b - 156*a^3*b^2 + 110*a^2*b^3 - 36*a*b^4 - 9*b^5)*e^(-4*d*
x - 4*c) + 3*(3*a^4*b - 22*a^3*b^2 - 20*a^2*b^3 + 6*a*b^4 + b^5)*e^(-6*d*x - 6*c))/((a^8 + 6*a^7*b + 15*a^6*b^
2 + 20*a^5*b^3 + 15*a^4*b^4 + 6*a^3*b^5 + a^2*b^6 + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 - 5*a^4*b^4 - 4*a^3*b^5 - a^2
*b^6)*e^(-2*d*x - 2*c) + 2*(3*a^8 + 10*a^7*b + 13*a^6*b^2 + 12*a^5*b^3 + 13*a^4*b^4 + 10*a^3*b^5 + 3*a^2*b^6)*
e^(-4*d*x - 4*c) + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 - 5*a^4*b^4 - 4*a^3*b^5 - a^2*b^6)*e^(-6*d*x - 6*c) + (a^8 + 6
*a^7*b + 15*a^6*b^2 + 20*a^5*b^3 + 15*a^4*b^4 + 6*a^3*b^5 + a^2*b^6)*e^(-8*d*x - 8*c))*d) + 1/8*(9*a^3*b + 21*
a^2*b^2 + 15*a*b^3 + 3*b^4 + (27*a^3*b + 13*a^2*b^2 - 23*a*b^3 - 9*b^4)*e^(-2*d*x - 2*c) + 3*(9*a^3*b - 3*a^2*
b^2 + 7*a*b^3 + 3*b^4)*e^(-4*d*x - 4*c) + (9*a^3*b - a^2*b^2 - 13*a*b^3 - 3*b^4)*e^(-6*d*x - 6*c))/((a^7 + 5*a
^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5 + 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 -
a^2*b^5)*e^(-2*d*x - 2*c) + 2*(3*a^7 + 7*a^6*b + 6*a^5*b^2 + 6*a^4*b^3 + 7*a^3*b^4 + 3*a^2*b^5)*e^(-4*d*x - 4*
c) + 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 - a^2*b^5)*e^(-6*d*x - 6*c) + (a^7 + 5*a^6*b + 10*a^
5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*e^(-8*d*x - 8*c))*d) + 1/2*(d*x + c)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3
)*d) + 1/8*e^(2*d*x + 2*c)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) - 1/8*e^(-2*d*x - 2*c)/((a^3 + 3*a^2*b + 3*a*b^
2 + b^3)*d)

Giac [F]

\[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int { \frac {\cosh \left (d x + c\right )^{2}}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \]

[In]

int(cosh(c + d*x)^2/(a + b*tanh(c + d*x)^2)^3,x)

[Out]

int(cosh(c + d*x)^2/(a + b*tanh(c + d*x)^2)^3, x)