3.1.53 \(\int \frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}} \, dx\) [53]

Optimal. Leaf size=54 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )}{b}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )}{b} \]

[Out]

-arctan(cosh(b*x+a)^(1/2)/sinh(b*x+a)^(1/2))/b+arctanh(cosh(b*x+a)^(1/2)/sinh(b*x+a)^(1/2))/b

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Rubi [A]
time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2655, 304, 209, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )}{b}-\frac {\text {ArcTan}\left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[Cosh[a + b*x]]/Sqrt[Sinh[a + b*x]],x]

[Out]

-(ArcTan[Sqrt[Cosh[a + b*x]]/Sqrt[Sinh[a + b*x]]]/b) + ArcTanh[Sqrt[Cosh[a + b*x]]/Sqrt[Sinh[a + b*x]]]/b

Rule 209

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

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 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 2655

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> With[{k = Denomina
tor[m]}, Dist[(-k)*a*(b/f), Subst[Int[x^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Cos[e + f*x])^(1/k)/(b*
Sin[e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.02, size = 57, normalized size = 1.06 \begin {gather*} \frac {2 \sqrt [4]{\cosh ^2(a+b x)} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};-\sinh ^2(a+b x)\right ) \sqrt {\sinh (a+b x)}}{b \sqrt {\cosh (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Cosh[a + b*x]]/Sqrt[Sinh[a + b*x]],x]

[Out]

(2*(Cosh[a + b*x]^2)^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, -Sinh[a + b*x]^2]*Sqrt[Sinh[a + b*x]])/(b*Sqrt[Cos
h[a + b*x]])

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Maple [F]
time = 0.68, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {\cosh }\left (b x +a \right )}{\sqrt {\sinh \left (b x +a \right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (46) = 92\).
time = 0.38, size = 144, normalized size = 2.67 \begin {gather*} \frac {2 \, \arctan \left (-\cosh \left (b x + a\right )^{2} + 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )} \sqrt {\sinh \left (b x + a\right )} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - \sinh \left (b x + a\right )^{2}\right ) - \log \left (-\cosh \left (b x + a\right )^{2} + 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )} \sqrt {\sinh \left (b x + a\right )} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - \sinh \left (b x + a\right )^{2}\right )}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*arctan(-cosh(b*x + a)^2 + 2*(cosh(b*x + a) + sinh(b*x + a))*sqrt(cosh(b*x + a))*sqrt(sinh(b*x + a)) - 2
*cosh(b*x + a)*sinh(b*x + a) - sinh(b*x + a)^2) - log(-cosh(b*x + a)^2 + 2*(cosh(b*x + a) + sinh(b*x + a))*sqr
t(cosh(b*x + a))*sqrt(sinh(b*x + a)) - 2*cosh(b*x + a)*sinh(b*x + a) - sinh(b*x + a)^2))/b

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\cosh {\left (a + b x \right )}}}{\sqrt {\sinh {\left (a + b x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(cosh(a + b*x))/sqrt(sinh(a + b*x)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {\mathrm {cosh}\left (a+b\,x\right )}}{\sqrt {\mathrm {sinh}\left (a+b\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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