3.858 \(\int \frac{1}{a+b \cosh (c+d x) \sinh (c+d x)} \, dx\)

Optimal. Leaf size=44 \[ -\frac{2 \tanh ^{-1}\left (\frac{b-2 a \tanh (c+d x)}{\sqrt{4 a^2+b^2}}\right )}{d \sqrt{4 a^2+b^2}} \]

[Out]

(-2*ArcTanh[(b - 2*a*Tanh[c + d*x])/Sqrt[4*a^2 + b^2]])/(Sqrt[4*a^2 + b^2]*d)

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Rubi [A]  time = 0.123194, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2666, 2660, 618, 204} \[ -\frac{2 \tanh ^{-1}\left (\frac{b-2 a \tanh (c+d x)}{\sqrt{4 a^2+b^2}}\right )}{d \sqrt{4 a^2+b^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[c + d*x]*Sinh[c + d*x])^(-1),x]

[Out]

(-2*ArcTanh[(b - 2*a*Tanh[c + d*x])/Sqrt[4*a^2 + b^2]])/(Sqrt[4*a^2 + b^2]*d)

Rule 2666

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

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*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 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{a+b \cosh (c+d x) \sinh (c+d x)} \, dx &=\int \frac{1}{a+\frac{1}{2} b \sinh (2 c+2 d x)} \, dx\\ &=-\frac{i \operatorname{Subst}\left (\int \frac{1}{a-i b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (2 i c+2 i d x)\right )\right )}{d}\\ &=\frac{(2 i) \operatorname{Subst}\left (\int \frac{1}{-4 a^2-b^2-x^2} \, dx,x,-i b+2 a \tan \left (\frac{1}{2} (2 i c+2 i d x)\right )\right )}{d}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{b-2 a \tanh (c+d x)}{\sqrt{4 a^2+b^2}}\right )}{\sqrt{4 a^2+b^2} d}\\ \end{align*}

Mathematica [A]  time = 0.0795756, size = 48, normalized size = 1.09 \[ \frac{2 \tan ^{-1}\left (\frac{b-2 a \tanh (c+d x)}{\sqrt{-4 a^2-b^2}}\right )}{d \sqrt{-4 a^2-b^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cosh[c + d*x]*Sinh[c + d*x])^(-1),x]

[Out]

(2*ArcTan[(b - 2*a*Tanh[c + d*x])/Sqrt[-4*a^2 - b^2]])/(Sqrt[-4*a^2 - b^2]*d)

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Maple [B]  time = 0.087, size = 207, normalized size = 4.7 \begin{align*}{\frac{1}{d}\ln \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}a+\sqrt{4\,{a}^{2}+{b}^{2}}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +b\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +a \right ){\frac{1}{\sqrt{4\,{a}^{2}+{b}^{2}}}}}-4\,{\frac{{a}^{2}\ln \left ( - \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+\sqrt{4\,{a}^{2}+{b}^{2}}\tanh \left ( 1/2\,dx+c/2 \right ) -b\tanh \left ( 1/2\,dx+c/2 \right ) -a \right ) }{d \left ( 4\,{a}^{2}+{b}^{2} \right ) ^{3/2}}}-{\frac{{b}^{2}}{d}\ln \left ( - \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}a+\sqrt{4\,{a}^{2}+{b}^{2}}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -b\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -a \right ) \left ( 4\,{a}^{2}+{b}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*cosh(d*x+c)*sinh(d*x+c)),x)

[Out]

1/d/(4*a^2+b^2)^(1/2)*ln(tanh(1/2*d*x+1/2*c)^2*a+(4*a^2+b^2)^(1/2)*tanh(1/2*d*x+1/2*c)+b*tanh(1/2*d*x+1/2*c)+a
)-4/d*a^2/(4*a^2+b^2)^(3/2)*ln(-tanh(1/2*d*x+1/2*c)^2*a+(4*a^2+b^2)^(1/2)*tanh(1/2*d*x+1/2*c)-b*tanh(1/2*d*x+1
/2*c)-a)-1/d/(4*a^2+b^2)^(3/2)*ln(-tanh(1/2*d*x+1/2*c)^2*a+(4*a^2+b^2)^(1/2)*tanh(1/2*d*x+1/2*c)-b*tanh(1/2*d*
x+1/2*c)-a)*b^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.3383, size = 772, normalized size = 17.55 \begin{align*} \frac{\log \left (\frac{b^{2} \cosh \left (d x + c\right )^{4} + 4 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{2} \sinh \left (d x + c\right )^{4} + 4 \, a b \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, b^{2} \cosh \left (d x + c\right )^{2} + 2 \, a b\right )} \sinh \left (d x + c\right )^{2} + 8 \, a^{2} + b^{2} + 4 \,{\left (b^{2} \cosh \left (d x + c\right )^{3} + 2 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 2 \,{\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + 2 \, a\right )} \sqrt{4 \, a^{2} + b^{2}}}{b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, b \cosh \left (d x + c\right )^{2} + 2 \, a\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left (b \cosh \left (d x + c\right )^{3} + 2 \, a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - b}\right )}{\sqrt{4 \, a^{2} + b^{2}} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*sinh(d*x + c)^4 + 4*a*b*cosh(d*x + c)^2 +
2*(3*b^2*cosh(d*x + c)^2 + 2*a*b)*sinh(d*x + c)^2 + 8*a^2 + b^2 + 4*(b^2*cosh(d*x + c)^3 + 2*a*b*cosh(d*x + c
))*sinh(d*x + c) - 2*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + 2*a)*sqrt(4*a^
2 + b^2))/(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 4*a*cosh(d*x + c)^2 + 2
*(3*b*cosh(d*x + c)^2 + 2*a)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + 2*a*cosh(d*x + c))*sinh(d*x + c) - b))/(
sqrt(4*a^2 + b^2)*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.18213, size = 107, normalized size = 2.43 \begin{align*} \frac{\log \left (\frac{{\left | 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + 4 \, a - 2 \, \sqrt{4 \, a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + 4 \, a + 2 \, \sqrt{4 \, a^{2} + b^{2}} \right |}}\right )}{\sqrt{4 \, a^{2} + b^{2}} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(2*b*e^(2*d*x + 2*c) + 4*a - 2*sqrt(4*a^2 + b^2))/abs(2*b*e^(2*d*x + 2*c) + 4*a + 2*sqrt(4*a^2 + b^2)))
/(sqrt(4*a^2 + b^2)*d)