3.88 \(\int \sqrt{1+\sinh ^2(x)} \, dx\)

Optimal. Leaf size=11 \[ \sqrt{\cosh ^2(x)} \tanh (x) \]

[Out]

Sqrt[Cosh[x]^2]*Tanh[x]

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Rubi [A]  time = 0.0239073, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3176, 3207, 2637} \[ \sqrt{\cosh ^2(x)} \tanh (x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Sinh[x]^2],x]

[Out]

Sqrt[Cosh[x]^2]*Tanh[x]

Rule 3176

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*cos[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]

Rule 3207

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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \sqrt{1+\sinh ^2(x)} \, dx &=\int \sqrt{\cosh ^2(x)} \, dx\\ &=\left (\sqrt{\cosh ^2(x)} \text{sech}(x)\right ) \int \cosh (x) \, dx\\ &=\sqrt{\cosh ^2(x)} \tanh (x)\\ \end{align*}

Mathematica [A]  time = 0.0065058, size = 11, normalized size = 1. \[ \sqrt{\cosh ^2(x)} \tanh (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Sinh[x]^2],x]

[Out]

Sqrt[Cosh[x]^2]*Tanh[x]

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Maple [A]  time = 0.049, size = 14, normalized size = 1.3 \begin{align*}{\frac{\sinh \left ( x \right ) }{\cosh \left ( x \right ) }\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+sinh(x)^2)^(1/2),x)

[Out]

(cosh(x)^2)^(1/2)*sinh(x)/cosh(x)

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Maxima [A]  time = 1.55792, size = 15, normalized size = 1.36 \begin{align*} -\frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+sinh(x)^2)^(1/2),x, algorithm="maxima")

[Out]

-1/2*e^(-x) + 1/2*e^x

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Fricas [A]  time = 1.87428, size = 12, normalized size = 1.09 \begin{align*} \sinh \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+sinh(x)^2)^(1/2),x, algorithm="fricas")

[Out]

sinh(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sinh ^{2}{\left (x \right )} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+sinh(x)**2)**(1/2),x)

[Out]

Integral(sqrt(sinh(x)**2 + 1), x)

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Giac [A]  time = 1.25695, size = 15, normalized size = 1.36 \begin{align*} -\frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+sinh(x)^2)^(1/2),x, algorithm="giac")

[Out]

-1/2*e^(-x) + 1/2*e^x