3.95 \(\int (1-\sinh ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=45 \[ \frac{2}{3} i \text{EllipticF}(i x,-1)-2 i E(i x|-1)-\frac{1}{3} \sinh (x) \sqrt{1-\sinh ^2(x)} \cosh (x) \]

[Out]

(-2*I)*EllipticE[I*x, -1] + ((2*I)/3)*EllipticF[I*x, -1] - (Cosh[x]*Sinh[x]*Sqrt[1 - Sinh[x]^2])/3

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Rubi [A]  time = 0.0635546, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3180, 3172, 3177, 3182} \[ \frac{2}{3} i F(i x|-1)-2 i E(i x|-1)-\frac{1}{3} \sinh (x) \sqrt{1-\sinh ^2(x)} \cosh (x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - Sinh[x]^2)^(3/2),x]

[Out]

(-2*I)*EllipticE[I*x, -1] + ((2*I)/3)*EllipticF[I*x, -1] - (Cosh[x]*Sinh[x]*Sqrt[1 - Sinh[x]^2])/3

Rule 3180

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

Rule 3172

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Dist[
B/b, Int[Sqrt[a + b*Sin[e + f*x]^2], x], x] + Dist[(A*b - a*B)/b, Int[1/Sqrt[a + b*Sin[e + f*x]^2], x], x] /;
FreeQ[{a, b, e, f, A, B}, x]

Rule 3177

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

Rule 3182

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

Rubi steps

\begin{align*} \int \left (1-\sinh ^2(x)\right )^{3/2} \, dx &=-\frac{1}{3} \cosh (x) \sinh (x) \sqrt{1-\sinh ^2(x)}+\frac{1}{3} \int \frac{4-6 \sinh ^2(x)}{\sqrt{1-\sinh ^2(x)}} \, dx\\ &=-\frac{1}{3} \cosh (x) \sinh (x) \sqrt{1-\sinh ^2(x)}-\frac{2}{3} \int \frac{1}{\sqrt{1-\sinh ^2(x)}} \, dx+2 \int \sqrt{1-\sinh ^2(x)} \, dx\\ &=-2 i E(i x|-1)+\frac{2}{3} i F(i x|-1)-\frac{1}{3} \cosh (x) \sinh (x) \sqrt{1-\sinh ^2(x)}\\ \end{align*}

Mathematica [A]  time = 0.0688293, size = 45, normalized size = 1. \[ \frac{1}{12} \left (8 i \text{EllipticF}(i x,-1)-24 i E(i x|-1)-\sinh (2 x) \sqrt{6-2 \cosh (2 x)}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Sinh[x]^2)^(3/2),x]

[Out]

((-24*I)*EllipticE[I*x, -1] + (8*I)*EllipticF[I*x, -1] - Sqrt[6 - 2*Cosh[2*x]]*Sinh[2*x])/12

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Maple [A]  time = 0.077, size = 103, normalized size = 2.3 \begin{align*}{\frac{1}{3\,\cosh \left ( x \right ) }\sqrt{- \left ( -1+ \left ( \sinh \left ( x \right ) \right ) ^{2} \right ) \left ( \cosh \left ( x \right ) \right ) ^{2}} \left ( \sinh \left ( x \right ) \left ( \cosh \left ( x \right ) \right ) ^{4}+10\,\sqrt{- \left ( \cosh \left ( x \right ) \right ) ^{2}+2}\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\it EllipticF} \left ( \sinh \left ( x \right ) ,i \right ) -6\,\sqrt{- \left ( \cosh \left ( x \right ) \right ) ^{2}+2}\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\it EllipticE} \left ( \sinh \left ( x \right ) ,i \right ) -2\, \left ( \cosh \left ( x \right ) \right ) ^{2}\sinh \left ( x \right ) \right ){\frac{1}{\sqrt{1- \left ( \sinh \left ( x \right ) \right ) ^{4}}}}{\frac{1}{\sqrt{1- \left ( \sinh \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-sinh(x)^2)^(3/2),x)

[Out]

1/3*(-(-1+sinh(x)^2)*cosh(x)^2)^(1/2)*(sinh(x)*cosh(x)^4+10*(-cosh(x)^2+2)^(1/2)*(cosh(x)^2)^(1/2)*EllipticF(s
inh(x),I)-6*(-cosh(x)^2+2)^(1/2)*(cosh(x)^2)^(1/2)*EllipticE(sinh(x),I)-2*cosh(x)^2*sinh(x))/(1-sinh(x)^4)^(1/
2)/cosh(x)/(1-sinh(x)^2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-\sinh \left (x\right )^{2} + 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-sinh(x)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((-sinh(x)^2 + 1)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-\sinh \left (x\right )^{2} + 1\right )}^{\frac{3}{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-sinh(x)^2)^(3/2),x, algorithm="fricas")

[Out]

integral((-sinh(x)^2 + 1)^(3/2), x)

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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-sinh(x)**2)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-\sinh \left (x\right )^{2} + 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-sinh(x)^2)^(3/2),x, algorithm="giac")

[Out]

integrate((-sinh(x)^2 + 1)^(3/2), x)