\(\int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx\) [586]

Optimal result

Integrand size = 11, antiderivative size = 17 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {\sinh (x)}{a (a \cosh (x)+b \sinh (x))} \]

[Out]

sinh(x)/a/(a*cosh(x)+b*sinh(x))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3154} \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {\sinh (x)}{a (a \cosh (x)+b \sinh (x))} \]

[In]

Int[(a*Cosh[x] + b*Sinh[x])^(-2),x]

[Out]

Sinh[x]/(a*(a*Cosh[x] + b*Sinh[x]))

Rule 3154

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

Rubi steps \begin{align*} \text {integral}& = \frac {\sinh (x)}{a (a \cosh (x)+b \sinh (x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {\sinh (x)}{a (a \cosh (x)+b \sinh (x))} \]

[In]

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

[Out]

Sinh[x]/(a*(a*Cosh[x] + b*Sinh[x]))

Maple [A] (verified)

Time = 5.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.59

method	result	size
risch	\(-\frac {2}{\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{2 x}+a -b \right ) \left (a +b \right )}\)	\(27\)
default	\(\frac {2 \tanh \left (\frac {x}{2}\right )}{a \left (\tanh \left (\frac {x}{2}\right )^{2} a +2 b \tanh \left (\frac {x}{2}\right )+a \right )}\)	\(29\)

[In]

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

[Out]

-2/(a*exp(2*x)+b*exp(2*x)+a-b)/(a+b)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (17) = 34\).

Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 3.65 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx=-\frac {2}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} - b^{2}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 604 vs. \(2 (14) = 28\).

Time = 136.94 (sec) , antiderivative size = 604, normalized size of antiderivative = 35.53 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {\tanh {\left (\frac {x}{2} \right )}}{2} - \frac {1}{2 \tanh {\left (\frac {x}{2} \right )}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {\tanh {\left (\frac {x}{2} \right )}}{2} - \frac {1}{2 \tanh {\left (\frac {x}{2} \right )}}}{b^{2}} & \text {for}\: a = 0 \\\frac {x \tanh ^{4}{\left (\frac {x}{2} \right )}}{2 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{4}{\left (\frac {x}{2} \right )} + 4 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sinh ^{2}{\left (x \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh {\left (\frac {x}{2} \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}} - \frac {2 x \tanh ^{2}{\left (\frac {x}{2} \right )}}{2 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{4}{\left (\frac {x}{2} \right )} + 4 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sinh ^{2}{\left (x \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh {\left (\frac {x}{2} \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}} + \frac {x}{2 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{4}{\left (\frac {x}{2} \right )} + 4 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sinh ^{2}{\left (x \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh {\left (\frac {x}{2} \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}} + \frac {2 \tanh ^{3}{\left (\frac {x}{2} \right )}}{2 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{4}{\left (\frac {x}{2} \right )} + 4 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sinh ^{2}{\left (x \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh {\left (\frac {x}{2} \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}} + \frac {2 \tanh {\left (\frac {x}{2} \right )}}{2 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{4}{\left (\frac {x}{2} \right )} + 4 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sinh ^{2}{\left (x \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh {\left (\frac {x}{2} \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}} & \text {for}\: a = - \frac {2 b \tanh {\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} + 1} \\\frac {2 \tanh {\left (\frac {x}{2} \right )}}{a^{2} \tanh ^{2}{\left (\frac {x}{2} \right )} + a^{2} + 2 a b \tanh {\left (\frac {x}{2} \right )}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((zoo*(-tanh(x/2)/2 - 1/(2*tanh(x/2))), Eq(a, 0) & Eq(b, 0)), ((-tanh(x/2)/2 - 1/(2*tanh(x/2)))/b**2,
Eq(a, 0)), (x*tanh(x/2)**4/(2*b**2*sinh(x)**2*tanh(x/2)**4 + 4*b**2*sinh(x)**2*tanh(x/2)**2 + 2*b**2*sinh(x)*
*2 - 8*b**2*sinh(x)*cosh(x)*tanh(x/2)**3 - 8*b**2*sinh(x)*cosh(x)*tanh(x/2) + 8*b**2*cosh(x)**2*tanh(x/2)**2)
- 2*x*tanh(x/2)**2/(2*b**2*sinh(x)**2*tanh(x/2)**4 + 4*b**2*sinh(x)**2*tanh(x/2)**2 + 2*b**2*sinh(x)**2 - 8*b*
*2*sinh(x)*cosh(x)*tanh(x/2)**3 - 8*b**2*sinh(x)*cosh(x)*tanh(x/2) + 8*b**2*cosh(x)**2*tanh(x/2)**2) + x/(2*b*
*2*sinh(x)**2*tanh(x/2)**4 + 4*b**2*sinh(x)**2*tanh(x/2)**2 + 2*b**2*sinh(x)**2 - 8*b**2*sinh(x)*cosh(x)*tanh(
x/2)**3 - 8*b**2*sinh(x)*cosh(x)*tanh(x/2) + 8*b**2*cosh(x)**2*tanh(x/2)**2) + 2*tanh(x/2)**3/(2*b**2*sinh(x)*
*2*tanh(x/2)**4 + 4*b**2*sinh(x)**2*tanh(x/2)**2 + 2*b**2*sinh(x)**2 - 8*b**2*sinh(x)*cosh(x)*tanh(x/2)**3 - 8
*b**2*sinh(x)*cosh(x)*tanh(x/2) + 8*b**2*cosh(x)**2*tanh(x/2)**2) + 2*tanh(x/2)/(2*b**2*sinh(x)**2*tanh(x/2)**
4 + 4*b**2*sinh(x)**2*tanh(x/2)**2 + 2*b**2*sinh(x)**2 - 8*b**2*sinh(x)*cosh(x)*tanh(x/2)**3 - 8*b**2*sinh(x)*
cosh(x)*tanh(x/2) + 8*b**2*cosh(x)**2*tanh(x/2)**2), Eq(a, -2*b*tanh(x/2)/(tanh(x/2)**2 + 1))), (2*tanh(x/2)/(
a**2*tanh(x/2)**2 + a**2 + 2*a*b*tanh(x/2)), True))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.71 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {2}{a^{2} - b^{2} + {\left (a^{2} - 2 \, a b + b^{2}\right )} e^{\left (-2 \, x\right )}} \]

[In]

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

[Out]

2/(a^2 - b^2 + (a^2 - 2*a*b + b^2)*e^(-2*x))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx=-\frac {2}{{\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )} {\left (a + b\right )}} \]

[In]

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

[Out]

-2/((a*e^(2*x) + b*e^(2*x) + a - b)*(a + b))

Mupad [B] (verification not implemented)

Time = 2.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx=-\frac {2}{\left (a+b\right )\,\left (a-b+{\mathrm {e}}^{2\,x}\,\left (a+b\right )\right )} \]

[In]

int(1/(a*cosh(x) + b*sinh(x))^2,x)

[Out]

-2/((a + b)*(a - b + exp(2*x)*(a + b)))