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

Optimal. Leaf size=87 \[ \frac{2 i \sqrt{1-\sinh ^2(x)} \text{EllipticF}(i x,-1)}{3 \sqrt{\sinh ^2(x)-1}}+\frac{1}{3} \sinh (x) \sqrt{\sinh ^2(x)-1} \cosh (x)+\frac{2 i \sqrt{\sinh ^2(x)-1} E(i x|-1)}{\sqrt{1-\sinh ^2(x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0855633, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {3180, 3172, 3178, 3177, 3183, 3182} \[ \frac{1}{3} \sinh (x) \sqrt{\sinh ^2(x)-1} \cosh (x)+\frac{2 i \sqrt{1-\sinh ^2(x)} F(i x|-1)}{3 \sqrt{\sinh ^2(x)-1}}+\frac{2 i \sqrt{\sinh ^2(x)-1} E(i x|-1)}{\sqrt{1-\sinh ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[(-1 + Sinh[x]^2)^(3/2),x]

[Out]

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

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 3178

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]

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 3183

Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Dist[Sqrt[1 + (b*Sin[e + f*x]^2)/a]/Sqrt[a +
b*Sin[e + f*x]^2], Int[1/Sqrt[1 + (b*Sin[e + f*x]^2)/a], x], 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\\ &=\frac{1}{3} \cosh (x) \sinh (x) \sqrt{-1+\sinh ^2(x)}-\frac{\left (2 \sqrt{1-\sinh ^2(x)}\right ) \int \frac{1}{\sqrt{1-\sinh ^2(x)}} \, dx}{3 \sqrt{-1+\sinh ^2(x)}}-\frac{\left (2 \sqrt{-1+\sinh ^2(x)}\right ) \int \sqrt{1-\sinh ^2(x)} \, dx}{\sqrt{1-\sinh ^2(x)}}\\ &=\frac{2 i F(i x|-1) \sqrt{1-\sinh ^2(x)}}{3 \sqrt{-1+\sinh ^2(x)}}+\frac{1}{3} \cosh (x) \sinh (x) \sqrt{-1+\sinh ^2(x)}+\frac{2 i E(i x|-1) \sqrt{-1+\sinh ^2(x)}}{\sqrt{1-\sinh ^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.12193, size = 78, normalized size = 0.9 \[ \frac{8 i \sqrt{3-\cosh (2 x)} \text{EllipticF}(i x,-1)+\frac{\sinh (4 x)-6 \sinh (2 x)}{\sqrt{2}}-24 i \sqrt{3-\cosh (2 x)} E(i x|-1)}{12 \sqrt{\cosh (2 x)-3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + Sinh[x]^2)^(3/2),x]

[Out]

((-24*I)*Sqrt[3 - Cosh[2*x]]*EllipticE[I*x, -1] + (8*I)*Sqrt[3 - Cosh[2*x]]*EllipticF[I*x, -1] + (-6*Sinh[2*x]
+ Sinh[4*x])/Sqrt[2])/(12*Sqrt[-3 + Cosh[2*x]])

________________________________________________________________________________________

Maple [A]  time = 0.079, size = 106, normalized size = 1.2 \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}+2\,i\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{2}}\sqrt{- \left ( \cosh \left ( x \right ) \right ) ^{2}+2}{\it EllipticF} \left ( i\sinh \left ( x \right ) ,i \right ) -6\,i\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{2}}\sqrt{- \left ( \cosh \left ( x \right ) \right ) ^{2}+2}{\it EllipticE} \left ( i\sinh \left ( x \right ) ,i \right ) -2\, \left ( \cosh \left ( x \right ) \right ) ^{2}\sinh \left ( x \right ) \right ){\frac{1}{\sqrt{ \left ( \sinh \left ( x \right ) \right ) ^{4}-1}}}{\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+2*I*(cosh(x)^2)^(1/2)*(-cosh(x)^2+2)^(1/2)*EllipticF(I
*sinh(x),I)-6*I*(cosh(x)^2)^(1/2)*(-cosh(x)^2+2)^(1/2)*EllipticE(I*sinh(x),I)-2*cosh(x)^2*sinh(x))/(sinh(x)^4-
1)^(1/2)/cosh(x)/(-1+sinh(x)^2)^(1/2)

________________________________________________________________________________________

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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

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

________________________________________________________________________________________

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)