3.1.92 \(\int \sqrt {a+b \sinh ^2(x)} \, dx\) [92]

Optimal. Leaf size=42 \[ -\frac {i E\left (i x\left |\frac {b}{a}\right .\right ) \sqrt {a+b \sinh ^2(x)}}{\sqrt {1+\frac {b \sinh ^2(x)}{a}}} \]

[Out]

-I*(cosh(x)^2)^(1/2)/cosh(x)*EllipticE(I*sinh(x),(b/a)^(1/2))*(a+b*sinh(x)^2)^(1/2)/(1+b*sinh(x)^2/a)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3257, 3256} \begin {gather*} -\frac {i \sqrt {a+b \sinh ^2(x)} E\left (i x\left |\frac {b}{a}\right .\right )}{\sqrt {\frac {b \sinh ^2(x)}{a}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sinh[x]^2],x]

[Out]

((-I)*EllipticE[I*x, b/a]*Sqrt[a + b*Sinh[x]^2])/Sqrt[1 + (b*Sinh[x]^2)/a]

Rule 3256

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(Sqrt[a]/f)*EllipticE[e + f*x, -b/a], x] /
; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]

Rule 3257

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Dist[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[1 + b*(Sin
[e + f*x]^2/a)], Int[Sqrt[1 + (b*Sin[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int \sqrt {a+b \sinh ^2(x)} \, dx &=\frac {\sqrt {a+b \sinh ^2(x)} \int \sqrt {1+\frac {b \sinh ^2(x)}{a}} \, dx}{\sqrt {1+\frac {b \sinh ^2(x)}{a}}}\\ &=-\frac {i E\left (i x\left |\frac {b}{a}\right .\right ) \sqrt {a+b \sinh ^2(x)}}{\sqrt {1+\frac {b \sinh ^2(x)}{a}}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 54, normalized size = 1.29 \begin {gather*} -\frac {i a \sqrt {\frac {2 a-b+b \cosh (2 x)}{a}} E\left (i x\left |\frac {b}{a}\right .\right )}{\sqrt {2 a-b+b \cosh (2 x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*Sinh[x]^2],x]

[Out]

((-I)*a*Sqrt[(2*a - b + b*Cosh[2*x])/a]*EllipticE[I*x, b/a])/Sqrt[2*a - b + b*Cosh[2*x]]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(108\) vs. \(2(49)=98\).
time = 1.18, size = 109, normalized size = 2.60

method	result	size
default	\(\frac {\sqrt {\frac {a +b \left (\sinh ^{2}\left (x \right )\right )}{a}}\, \sqrt {\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \left (a \EllipticF \left (\sinh \left (x \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-b \EllipticF \left (\sinh \left (x \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+b \EllipticE \left (\sinh \left (x \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \cosh \left (x \right ) \sqrt {a +b \left (\sinh ^{2}\left (x \right )\right )}}\)	\(109\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a+b*sinh(x)^2)/a)^(1/2)*(cosh(x)^2)^(1/2)*(a*EllipticF(sinh(x)*(-1/a*b)^(1/2),(a/b)^(1/2))-b*EllipticF(sinh(
x)*(-1/a*b)^(1/2),(a/b)^(1/2))+b*EllipticE(sinh(x)*(-1/a*b)^(1/2),(a/b)^(1/2)))/(-1/a*b)^(1/2)/cosh(x)/(a+b*si
nh(x)^2)^(1/2)

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

[Out]

integrate(sqrt(b*sinh(x)^2 + a), x)

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Fricas [F]
time = 0.10, size = 12, normalized size = 0.29 \begin {gather*} {\rm integral}\left (\sqrt {b \sinh \left (x\right )^{2} + a}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*sinh(x)^2 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(sqrt(b*sinh(x)^2 + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \sqrt {b\,{\mathrm {sinh}\left (x\right )}^2+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sinh(x)^2)^(1/2),x)

[Out]

int((a + b*sinh(x)^2)^(1/2), x)

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