\(\int \frac {1}{(a+b \cosh (c+d x))^2} \, dx\) [68]

Optimal result

Integrand size = 12, antiderivative size = 86 \[ \int \frac {1}{(a+b \cosh (c+d x))^2} \, dx=\frac {2 a \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2} d}-\frac {b \sinh (c+d x)}{\left (a^2-b^2\right ) d (a+b \cosh (c+d x))} \]

[Out]

2*a*arctanh((a-b)^(1/2)*tanh(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(3/2)/(a+b)^(3/2)/d-b*sinh(d*x+c)/(a^2-b^2)/d/(
a+b*cosh(d*x+c))

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2743, 12, 2738, 211} \[ \int \frac {1}{(a+b \cosh (c+d x))^2} \, dx=\frac {2 a \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{3/2} (a+b)^{3/2}}-\frac {b \sinh (c+d x)}{d \left (a^2-b^2\right ) (a+b \cosh (c+d x))} \]

[In]

Int[(a + b*Cosh[c + d*x])^(-2),x]

[Out]

(2*a*ArcTanh[(Sqrt[a - b]*Tanh[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) - (b*Sinh[c + d*x])
/((a^2 - b^2)*d*(a + b*Cosh[c + d*x]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 2743

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((a + b*Sin[c + d*x])^(n
+ 1)/(d*(n + 1)*(a^2 - b^2))), x] + Dist[1/((n + 1)*(a^2 - b^2)), Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n +
1) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ
erQ[2*n]

Rubi steps \begin{align*} \text {integral}& = -\frac {b \sinh (c+d x)}{\left (a^2-b^2\right ) d (a+b \cosh (c+d x))}-\frac {\int \frac {a}{a+b \cosh (c+d x)} \, dx}{-a^2+b^2} \\ & = -\frac {b \sinh (c+d x)}{\left (a^2-b^2\right ) d (a+b \cosh (c+d x))}+\frac {a \int \frac {1}{a+b \cosh (c+d x)} \, dx}{a^2-b^2} \\ & = -\frac {b \sinh (c+d x)}{\left (a^2-b^2\right ) d (a+b \cosh (c+d x))}-\frac {(2 i a) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{\left (a^2-b^2\right ) d} \\ & = \frac {2 a \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2} d}-\frac {b \sinh (c+d x)}{\left (a^2-b^2\right ) d (a+b \cosh (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(a+b \cosh (c+d x))^2} \, dx=\frac {\frac {2 a \arctan \left (\frac {(a-b) \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{3/2}}-\frac {b \sinh (c+d x)}{(a-b) (a+b) (a+b \cosh (c+d x))}}{d} \]

[In]

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

[Out]

((2*a*ArcTan[((a - b)*Tanh[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - (b*Sinh[c + d*x])/((a - b)*(a
+ b)*(a + b*Cosh[c + d*x])))/d

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.37

method	result	size
derivativedivides	\(\frac {\frac {2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}+\frac {2 a \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{d}\)	\(118\)
default	\(\frac {\frac {2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}+\frac {2 a \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{d}\)	\(118\)
risch	\(\frac {2 a \,{\mathrm e}^{d x +c}+2 b}{d \left (a^{2}-b^{2}\right ) \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}+b \right )}+\frac {a \ln \left ({\mathrm e}^{d x +c}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}-\frac {a \ln \left ({\mathrm e}^{d x +c}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}\)	\(199\)

[In]

int(1/(a+b*cosh(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(2*b/(a^2-b^2)*tanh(1/2*d*x+1/2*c)/(tanh(1/2*d*x+1/2*c)^2*a-tanh(1/2*d*x+1/2*c)^2*b-a-b)+2*a/(a+b)/(a-b)/(
(a+b)*(a-b))^(1/2)*arctanh((a-b)*tanh(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (77) = 154\).

Time = 0.26 (sec) , antiderivative size = 743, normalized size of antiderivative = 8.64 \[ \int \frac {1}{(a+b \cosh (c+d x))^2} \, dx=\left [\frac {2 \, a^{2} b - 2 \, b^{3} - {\left (a b \cosh \left (d x + c\right )^{2} + a b \sinh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) + a b + 2 \, {\left (a b \cosh \left (d x + c\right ) + a^{2}\right )} \sinh \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) + 2 \, {\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) + b}\right ) + 2 \, {\left (a^{3} - a b^{2}\right )} \cosh \left (d x + c\right ) + 2 \, {\left (a^{3} - a b^{2}\right )} \sinh \left (d x + c\right )}{{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \cosh \left (d x + c\right )^{2} + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \sinh \left (d x + c\right )^{2} + 2 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \cosh \left (d x + c\right ) + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d + 2 \, {\left ({\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \cosh \left (d x + c\right ) + {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d\right )} \sinh \left (d x + c\right )}, \frac {2 \, {\left (a^{2} b - b^{3} - {\left (a b \cosh \left (d x + c\right )^{2} + a b \sinh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) + a b + 2 \, {\left (a b \cosh \left (d x + c\right ) + a^{2}\right )} \sinh \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{a^{2} - b^{2}}\right ) + {\left (a^{3} - a b^{2}\right )} \cosh \left (d x + c\right ) + {\left (a^{3} - a b^{2}\right )} \sinh \left (d x + c\right )\right )}}{{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \cosh \left (d x + c\right )^{2} + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \sinh \left (d x + c\right )^{2} + 2 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \cosh \left (d x + c\right ) + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d + 2 \, {\left ({\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \cosh \left (d x + c\right ) + {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d\right )} \sinh \left (d x + c\right )}\right ] \]

[In]

integrate(1/(a+b*cosh(d*x+c))^2,x, algorithm="fricas")

[Out]

[(2*a^2*b - 2*b^3 - (a*b*cosh(d*x + c)^2 + a*b*sinh(d*x + c)^2 + 2*a^2*cosh(d*x + c) + a*b + 2*(a*b*cosh(d*x +
c) + a^2)*sinh(d*x + c))*sqrt(a^2 - b^2)*log((b^2*cosh(d*x + c)^2 + b^2*sinh(d*x + c)^2 + 2*a*b*cosh(d*x + c)
+ 2*a^2 - b^2 + 2*(b^2*cosh(d*x + c) + a*b)*sinh(d*x + c) + 2*sqrt(a^2 - b^2)*(b*cosh(d*x + c) + b*sinh(d*x +
c) + a))/(b*cosh(d*x + c)^2 + b*sinh(d*x + c)^2 + 2*a*cosh(d*x + c) + 2*(b*cosh(d*x + c) + a)*sinh(d*x + c) +
b)) + 2*(a^3 - a*b^2)*cosh(d*x + c) + 2*(a^3 - a*b^2)*sinh(d*x + c))/((a^4*b - 2*a^2*b^3 + b^5)*d*cosh(d*x +
c)^2 + (a^4*b - 2*a^2*b^3 + b^5)*d*sinh(d*x + c)^2 + 2*(a^5 - 2*a^3*b^2 + a*b^4)*d*cosh(d*x + c) + (a^4*b - 2*
a^2*b^3 + b^5)*d + 2*((a^4*b - 2*a^2*b^3 + b^5)*d*cosh(d*x + c) + (a^5 - 2*a^3*b^2 + a*b^4)*d)*sinh(d*x + c)),
2*(a^2*b - b^3 - (a*b*cosh(d*x + c)^2 + a*b*sinh(d*x + c)^2 + 2*a^2*cosh(d*x + c) + a*b + 2*(a*b*cosh(d*x + c
) + a^2)*sinh(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cosh(d*x + c) + b*sinh(d*x + c) + a)/(a^2
- b^2)) + (a^3 - a*b^2)*cosh(d*x + c) + (a^3 - a*b^2)*sinh(d*x + c))/((a^4*b - 2*a^2*b^3 + b^5)*d*cosh(d*x +
c)^2 + (a^4*b - 2*a^2*b^3 + b^5)*d*sinh(d*x + c)^2 + 2*(a^5 - 2*a^3*b^2 + a*b^4)*d*cosh(d*x + c) + (a^4*b - 2*
a^2*b^3 + b^5)*d + 2*((a^4*b - 2*a^2*b^3 + b^5)*d*cosh(d*x + c) + (a^5 - 2*a^3*b^2 + a*b^4)*d)*sinh(d*x + c))]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2332 vs. \(2 (70) = 140\).

Time = 40.23 (sec) , antiderivative size = 2332, normalized size of antiderivative = 27.12 \[ \int \frac {1}{(a+b \cosh (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a+b*cosh(d*x+c))**2,x)

[Out]

Piecewise((zoo*x/cosh(c)**2, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (-tanh(c/2 + d*x/2)**3/(6*b**2*d) + tanh(c/2 + d
*x/2)/(2*b**2*d), Eq(a, b)), (1/(2*b**2*d*tanh(c/2 + d*x/2)) - 1/(6*b**2*d*tanh(c/2 + d*x/2)**3), Eq(a, -b)),
(x/(a + b*cosh(c))**2, Eq(d, 0)), (-a**2*log(-sqrt(a/(a - b) + b/(a - b)) + tanh(c/2 + d*x/2))*tanh(c/2 + d*x/
2)**2/(a**4*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - a**4*d*sqrt(a/(a - b) + b/(a - b)) - 2*a**3*b
*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 + 2*a**2*b**2*d*sqrt(a/(a - b) + b/(a - b)) + 2*a*b**3*d*s
qrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - b**4*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - b*
*4*d*sqrt(a/(a - b) + b/(a - b))) + a**2*log(-sqrt(a/(a - b) + b/(a - b)) + tanh(c/2 + d*x/2))/(a**4*d*sqrt(a/
(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - a**4*d*sqrt(a/(a - b) + b/(a - b)) - 2*a**3*b*d*sqrt(a/(a - b) + b
/(a - b))*tanh(c/2 + d*x/2)**2 + 2*a**2*b**2*d*sqrt(a/(a - b) + b/(a - b)) + 2*a*b**3*d*sqrt(a/(a - b) + b/(a
- b))*tanh(c/2 + d*x/2)**2 - b**4*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - b**4*d*sqrt(a/(a - b) +
b/(a - b))) + a**2*log(sqrt(a/(a - b) + b/(a - b)) + tanh(c/2 + d*x/2))*tanh(c/2 + d*x/2)**2/(a**4*d*sqrt(a/(
a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - a**4*d*sqrt(a/(a - b) + b/(a - b)) - 2*a**3*b*d*sqrt(a/(a - b) + b/
(a - b))*tanh(c/2 + d*x/2)**2 + 2*a**2*b**2*d*sqrt(a/(a - b) + b/(a - b)) + 2*a*b**3*d*sqrt(a/(a - b) + b/(a -
b))*tanh(c/2 + d*x/2)**2 - b**4*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - b**4*d*sqrt(a/(a - b) +
b/(a - b))) - a**2*log(sqrt(a/(a - b) + b/(a - b)) + tanh(c/2 + d*x/2))/(a**4*d*sqrt(a/(a - b) + b/(a - b))*ta
nh(c/2 + d*x/2)**2 - a**4*d*sqrt(a/(a - b) + b/(a - b)) - 2*a**3*b*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*
x/2)**2 + 2*a**2*b**2*d*sqrt(a/(a - b) + b/(a - b)) + 2*a*b**3*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)
**2 - b**4*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - b**4*d*sqrt(a/(a - b) + b/(a - b))) + 2*a*b*sq
rt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)/(a**4*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - a**4*d*
sqrt(a/(a - b) + b/(a - b)) - 2*a**3*b*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 + 2*a**2*b**2*d*sqrt
(a/(a - b) + b/(a - b)) + 2*a*b**3*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - b**4*d*sqrt(a/(a - b)
+ b/(a - b))*tanh(c/2 + d*x/2)**2 - b**4*d*sqrt(a/(a - b) + b/(a - b))) + a*b*log(-sqrt(a/(a - b) + b/(a - b))
+ tanh(c/2 + d*x/2))*tanh(c/2 + d*x/2)**2/(a**4*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - a**4*d*s
qrt(a/(a - b) + b/(a - b)) - 2*a**3*b*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 + 2*a**2*b**2*d*sqrt(
a/(a - b) + b/(a - b)) + 2*a*b**3*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - b**4*d*sqrt(a/(a - b) +
b/(a - b))*tanh(c/2 + d*x/2)**2 - b**4*d*sqrt(a/(a - b) + b/(a - b))) + a*b*log(-sqrt(a/(a - b) + b/(a - b))
+ tanh(c/2 + d*x/2))/(a**4*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - a**4*d*sqrt(a/(a - b) + b/(a -
b)) - 2*a**3*b*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 + 2*a**2*b**2*d*sqrt(a/(a - b) + b/(a - b))
+ 2*a*b**3*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - b**4*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 +
d*x/2)**2 - b**4*d*sqrt(a/(a - b) + b/(a - b))) - a*b*log(sqrt(a/(a - b) + b/(a - b)) + tanh(c/2 + d*x/2))*ta
nh(c/2 + d*x/2)**2/(a**4*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - a**4*d*sqrt(a/(a - b) + b/(a - b
)) - 2*a**3*b*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 + 2*a**2*b**2*d*sqrt(a/(a - b) + b/(a - b)) +
2*a*b**3*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - b**4*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d
*x/2)**2 - b**4*d*sqrt(a/(a - b) + b/(a - b))) - a*b*log(sqrt(a/(a - b) + b/(a - b)) + tanh(c/2 + d*x/2))/(a**
4*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - a**4*d*sqrt(a/(a - b) + b/(a - b)) - 2*a**3*b*d*sqrt(a/
(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 + 2*a**2*b**2*d*sqrt(a/(a - b) + b/(a - b)) + 2*a*b**3*d*sqrt(a/(a -
b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - b**4*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - b**4*d*sqrt(
a/(a - b) + b/(a - b))) - 2*b**2*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)/(a**4*d*sqrt(a/(a - b) + b/(a -
b))*tanh(c/2 + d*x/2)**2 - a**4*d*sqrt(a/(a - b) + b/(a - b)) - 2*a**3*b*d*sqrt(a/(a - b) + b/(a - b))*tanh(c
/2 + d*x/2)**2 + 2*a**2*b**2*d*sqrt(a/(a - b) + b/(a - b)) + 2*a*b**3*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 +
d*x/2)**2 - b**4*d*sqrt(a/(a - b) + b/(a - b))*tanh(c/2 + d*x/2)**2 - b**4*d*sqrt(a/(a - b) + b/(a - b))), Tr
ue))

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(a+b \cosh (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(a+b*cosh(d*x+c))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.15 \[ \int \frac {1}{(a+b \cosh (c+d x))^2} \, dx=\frac {2 \, {\left (\frac {a \arctan \left (\frac {b e^{\left (d x + c\right )} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{{\left (a^{2} - b^{2}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {a e^{\left (d x + c\right )} + b}{{\left (a^{2} - b^{2}\right )} {\left (b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} + b\right )}}\right )}}{d} \]

[In]

integrate(1/(a+b*cosh(d*x+c))^2,x, algorithm="giac")

[Out]

2*(a*arctan((b*e^(d*x + c) + a)/sqrt(-a^2 + b^2))/((a^2 - b^2)*sqrt(-a^2 + b^2)) + (a*e^(d*x + c) + b)/((a^2 -
b^2)*(b*e^(2*d*x + 2*c) + 2*a*e^(d*x + c) + b)))/d

Mupad [B] (verification not implemented)

Time = 2.12 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.50 \[ \int \frac {1}{(a+b \cosh (c+d x))^2} \, dx=\frac {\frac {2\,b^2}{d\,\left (a^2\,b-b^3\right )}+\frac {2\,a\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left (a^2\,b-b^3\right )}}{b+2\,a\,{\mathrm {e}}^{c+d\,x}+b\,{\mathrm {e}}^{2\,c+2\,d\,x}}+\frac {a\,\ln \left (-\frac {2\,a\,{\mathrm {e}}^{c+d\,x}}{b\,\left (a^2-b^2\right )}-\frac {2\,a\,\left (b+a\,{\mathrm {e}}^{c+d\,x}\right )}{b\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}\right )}{d\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}-\frac {a\,\ln \left (\frac {2\,a\,\left (b+a\,{\mathrm {e}}^{c+d\,x}\right )}{b\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}-\frac {2\,a\,{\mathrm {e}}^{c+d\,x}}{b\,\left (a^2-b^2\right )}\right )}{d\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}} \]

[In]

int(1/(a + b*cosh(c + d*x))^2,x)

[Out]

((2*b^2)/(d*(a^2*b - b^3)) + (2*a*b*exp(c + d*x))/(d*(a^2*b - b^3)))/(b + 2*a*exp(c + d*x) + b*exp(2*c + 2*d*x
)) + (a*log(- (2*a*exp(c + d*x))/(b*(a^2 - b^2)) - (2*a*(b + a*exp(c + d*x)))/(b*(a + b)^(3/2)*(a - b)^(3/2)))
)/(d*(a + b)^(3/2)*(a - b)^(3/2)) - (a*log((2*a*(b + a*exp(c + d*x)))/(b*(a + b)^(3/2)*(a - b)^(3/2)) - (2*a*e
xp(c + d*x))/(b*(a^2 - b^2))))/(d*(a + b)^(3/2)*(a - b)^(3/2))