\(\int \frac {1}{(a \cosh ^2(x))^{3/2}} \, dx\) [126]

Optimal result

Integrand size = 10, antiderivative size = 42 \[ \int \frac {1}{\left (a \cosh ^2(x)\right )^{3/2}} \, dx=\frac {\arctan (\sinh (x)) \cosh (x)}{2 a \sqrt {a \cosh ^2(x)}}+\frac {\tanh (x)}{2 a \sqrt {a \cosh ^2(x)}} \]

[Out]

1/2*arctan(sinh(x))*cosh(x)/a/(a*cosh(x)^2)^(1/2)+1/2*tanh(x)/a/(a*cosh(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3283, 3286, 3855} \[ \int \frac {1}{\left (a \cosh ^2(x)\right )^{3/2}} \, dx=\frac {\cosh (x) \arctan (\sinh (x))}{2 a \sqrt {a \cosh ^2(x)}}+\frac {\tanh (x)}{2 a \sqrt {a \cosh ^2(x)}} \]

[In]

Int[(a*Cosh[x]^2)^(-3/2),x]

[Out]

(ArcTan[Sinh[x]]*Cosh[x])/(2*a*Sqrt[a*Cosh[x]^2]) + Tanh[x]/(2*a*Sqrt[a*Cosh[x]^2])

Rule 3283

Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[Cot[e + f*x]*((b*Sin[e + f*x]^2)^(p + 1)/(b*f*(2
*p + 1))), x] + Dist[2*((p + 1)/(b*(2*p + 1))), Int[(b*Sin[e + f*x]^2)^(p + 1), x], x] /; FreeQ[{b, e, f}, x]
&&  !IntegerQ[p] && LtQ[p, -1]

Rule 3286

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\tanh (x)}{2 a \sqrt {a \cosh ^2(x)}}+\frac {\int \frac {1}{\sqrt {a \cosh ^2(x)}} \, dx}{2 a} \\ & = \frac {\tanh (x)}{2 a \sqrt {a \cosh ^2(x)}}+\frac {\cosh (x) \int \text {sech}(x) \, dx}{2 a \sqrt {a \cosh ^2(x)}} \\ & = \frac {\arctan (\sinh (x)) \cosh (x)}{2 a \sqrt {a \cosh ^2(x)}}+\frac {\tanh (x)}{2 a \sqrt {a \cosh ^2(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\left (a \cosh ^2(x)\right )^{3/2}} \, dx=\frac {\arctan (\sinh (x)) \cosh (x)+\tanh (x)}{2 a \sqrt {a \cosh ^2(x)}} \]

[In]

Integrate[(a*Cosh[x]^2)^(-3/2),x]

[Out]

(ArcTan[Sinh[x]]*Cosh[x] + Tanh[x])/(2*a*Sqrt[a*Cosh[x]^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(81\) vs. \(2(34)=68\).

Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.95

method	result	size
default	\(\frac {\sqrt {a \sinh \left (x \right )^{2}}\, \left (-\ln \left (\frac {2 \sqrt {-a}\, \sqrt {a \sinh \left (x \right )^{2}}-2 a}{\cosh \left (x \right )}\right ) a \cosh \left (x \right )^{2}+\sqrt {-a}\, \sqrt {a \sinh \left (x \right )^{2}}\right )}{2 a^{2} \cosh \left (x \right ) \sqrt {-a}\, \sinh \left (x \right ) \sqrt {a \cosh \left (x \right )^{2}}}\)	\(82\)
risch	\(\frac {{\mathrm e}^{2 x}-1}{a \left (1+{\mathrm e}^{2 x}\right ) \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}}+\frac {i \left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x} \ln \left ({\mathrm e}^{x}+i\right )}{2 a \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}}-\frac {i \left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x} \ln \left ({\mathrm e}^{x}-i\right )}{2 a \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}}\)	\(112\)

[In]

int(1/(a*cosh(x)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2/a^2/cosh(x)*(a*sinh(x)^2)^(1/2)*(-ln(2*((-a)^(1/2)*(a*sinh(x)^2)^(1/2)-a)/cosh(x))*a*cosh(x)^2+(-a)^(1/2)*
(a*sinh(x)^2)^(1/2))/(-a)^(1/2)/sinh(x)/(a*cosh(x)^2)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (34) = 68\).

Time = 0.26 (sec) , antiderivative size = 299, normalized size of antiderivative = 7.12 \[ \int \frac {1}{\left (a \cosh ^2(x)\right )^{3/2}} \, dx=\frac {{\left (3 \, \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{2} + e^{x} \sinh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} e^{x} \sinh \left (x\right ) + {\left (4 \, \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{3} + e^{x} \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} e^{x} \sinh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} e^{x} \sinh \left (x\right ) + {\left (\cosh \left (x\right )^{4} + 2 \, \cosh \left (x\right )^{2} + 1\right )} e^{x}\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} e^{x}\right )} \sqrt {a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a} e^{\left (-x\right )}}{a^{2} \cosh \left (x\right )^{4} + {\left (a^{2} e^{\left (2 \, x\right )} + a^{2}\right )} \sinh \left (x\right )^{4} + 2 \, a^{2} \cosh \left (x\right )^{2} + 4 \, {\left (a^{2} \cosh \left (x\right ) e^{\left (2 \, x\right )} + a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \, {\left (3 \, a^{2} \cosh \left (x\right )^{2} + a^{2} + {\left (3 \, a^{2} \cosh \left (x\right )^{2} + a^{2}\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right )^{2} + a^{2} + {\left (a^{2} \cosh \left (x\right )^{4} + 2 \, a^{2} \cosh \left (x\right )^{2} + a^{2}\right )} e^{\left (2 \, x\right )} + 4 \, {\left (a^{2} \cosh \left (x\right )^{3} + a^{2} \cosh \left (x\right ) + {\left (a^{2} \cosh \left (x\right )^{3} + a^{2} \cosh \left (x\right )\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right )} \]

[In]

integrate(1/(a*cosh(x)^2)^(3/2),x, algorithm="fricas")

[Out]

(3*cosh(x)*e^x*sinh(x)^2 + e^x*sinh(x)^3 + (3*cosh(x)^2 - 1)*e^x*sinh(x) + (4*cosh(x)*e^x*sinh(x)^3 + e^x*sinh
(x)^4 + 2*(3*cosh(x)^2 + 1)*e^x*sinh(x)^2 + 4*(cosh(x)^3 + cosh(x))*e^x*sinh(x) + (cosh(x)^4 + 2*cosh(x)^2 + 1
)*e^x)*arctan(cosh(x) + sinh(x)) + (cosh(x)^3 - cosh(x))*e^x)*sqrt(a*e^(4*x) + 2*a*e^(2*x) + a)*e^(-x)/(a^2*co
sh(x)^4 + (a^2*e^(2*x) + a^2)*sinh(x)^4 + 2*a^2*cosh(x)^2 + 4*(a^2*cosh(x)*e^(2*x) + a^2*cosh(x))*sinh(x)^3 +
2*(3*a^2*cosh(x)^2 + a^2 + (3*a^2*cosh(x)^2 + a^2)*e^(2*x))*sinh(x)^2 + a^2 + (a^2*cosh(x)^4 + 2*a^2*cosh(x)^2
+ a^2)*e^(2*x) + 4*(a^2*cosh(x)^3 + a^2*cosh(x) + (a^2*cosh(x)^3 + a^2*cosh(x))*e^(2*x))*sinh(x))

Sympy [F]

\[ \int \frac {1}{\left (a \cosh ^2(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a \cosh ^{2}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/(a*cosh(x)**2)**(3/2),x)

[Out]

Integral((a*cosh(x)**2)**(-3/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (a \cosh ^2(x)\right )^{3/2}} \, dx=\frac {e^{\left (3 \, x\right )} - e^{x}}{a^{\frac {3}{2}} e^{\left (4 \, x\right )} + 2 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + a^{\frac {3}{2}}} + \frac {\arctan \left (e^{x}\right )}{a^{\frac {3}{2}}} \]

[In]

integrate(1/(a*cosh(x)^2)^(3/2),x, algorithm="maxima")

[Out]

(e^(3*x) - e^x)/(a^(3/2)*e^(4*x) + 2*a^(3/2)*e^(2*x) + a^(3/2)) + arctan(e^x)/a^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.33 \[ \int \frac {1}{\left (a \cosh ^2(x)\right )^{3/2}} \, dx=\frac {\frac {\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )}{\sqrt {a}} - \frac {4 \, {\left (e^{\left (-x\right )} - e^{x}\right )}}{{\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )} \sqrt {a}}}{4 \, a} \]

[In]

integrate(1/(a*cosh(x)^2)^(3/2),x, algorithm="giac")

[Out]

1/4*((pi + 2*arctan(1/2*(e^(2*x) - 1)*e^(-x)))/sqrt(a) - 4*(e^(-x) - e^x)/(((e^(-x) - e^x)^2 + 4)*sqrt(a)))/a

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a \cosh ^2(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (a\,{\mathrm {cosh}\left (x\right )}^2\right )}^{3/2}} \,d x \]

[In]

int(1/(a*cosh(x)^2)^(3/2),x)

[Out]

int(1/(a*cosh(x)^2)^(3/2), x)