3.1.88 \(\int \sqrt {1+\sinh ^2(x)} \, dx\) [88]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3255, 3286, 2717} \begin {gather*} \sqrt {\cosh ^2(x)} \tanh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Sinh[x]^2],x]

[Out]

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

Rule 2717

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

Rule 3255

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 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
]])

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 = 0.01, size = 11, normalized size = 1.00 \begin {gather*} \sqrt {\cosh ^2(x)} \tanh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Sinh[x]^2],x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.81, size = 14, normalized size = 1.27

method	result	size
default	\(\frac {\sinh \left (x \right ) \sqrt {\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}}{\cosh \left (x \right )}\)	\(14\)
risch	\(\frac {\sqrt {\left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}\, {\mathrm e}^{2 x}}{2+2 \,{\mathrm e}^{2 x}}-\frac {\sqrt {\left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}}{2 \left (1+{\mathrm e}^{2 x}\right )}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+sinh(x)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.48, size = 11, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \end {gather*}

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

________________________________________________________________________________________

Fricas [A]
time = 0.41, size = 2, normalized size = 0.18 \begin {gather*} \sinh \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sinh(x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\sinh ^{2}{\left (x \right )} + 1}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 11, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \end {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 0.07, size = 2, normalized size = 0.18 \begin {gather*} \mathrm {sinh}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sinh(x)^2 + 1)^(1/2),x)

[Out]

sinh(x)

________________________________________________________________________________________