Integrand size = 12, antiderivative size = 45 \[ \int \left (1-\sinh ^2(x)\right )^{3/2} \, dx=-2 i E(i x|-1)+\frac {2}{3} i \operatorname {EllipticF}(i x,-1)-\frac {1}{3} \cosh (x) \sinh (x) \sqrt {1-\sinh ^2(x)} \]
-2*I*(cosh(x)^2)^(1/2)/cosh(x)*EllipticE(I*sinh(x),I)+2/3*I*(cosh(x)^2)^(1 /2)/cosh(x)*EllipticF(I*sinh(x),I)-1/3*cosh(x)*sinh(x)*(1-sinh(x)^2)^(1/2)
Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \left (1-\sinh ^2(x)\right )^{3/2} \, dx=\frac {1}{12} \left (-24 i E(i x|-1)+8 i \operatorname {EllipticF}(i x,-1)-\sqrt {6-2 \cosh (2 x)} \sinh (2 x)\right ) \]
((-24*I)*EllipticE[I*x, -1] + (8*I)*EllipticF[I*x, -1] - Sqrt[6 - 2*Cosh[2 *x]]*Sinh[2*x])/12
Time = 0.38 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3659, 27, 3042, 3651, 3042, 3656, 3661}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (1-\sinh ^2(x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (1+\sin (i x)^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 3659 |
\(\displaystyle \frac {1}{3} \int \frac {2 \left (2-3 \sinh ^2(x)\right )}{\sqrt {1-\sinh ^2(x)}}dx-\frac {1}{3} \sinh (x) \sqrt {1-\sinh ^2(x)} \cosh (x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2}{3} \int \frac {2-3 \sinh ^2(x)}{\sqrt {1-\sinh ^2(x)}}dx-\frac {1}{3} \sinh (x) \sqrt {1-\sinh ^2(x)} \cosh (x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{3} \sinh (x) \sqrt {1-\sinh ^2(x)} \cosh (x)+\frac {2}{3} \int \frac {3 \sin (i x)^2+2}{\sqrt {\sin (i x)^2+1}}dx\) |
\(\Big \downarrow \) 3651 |
\(\displaystyle \frac {2}{3} \left (3 \int \sqrt {1-\sinh ^2(x)}dx-\int \frac {1}{\sqrt {1-\sinh ^2(x)}}dx\right )-\frac {1}{3} \sinh (x) \sqrt {1-\sinh ^2(x)} \cosh (x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{3} \sinh (x) \sqrt {1-\sinh ^2(x)} \cosh (x)+\frac {2}{3} \left (3 \int \sqrt {\sin (i x)^2+1}dx-\int \frac {1}{\sqrt {\sin (i x)^2+1}}dx\right )\) |
\(\Big \downarrow \) 3656 |
\(\displaystyle -\frac {1}{3} \sinh (x) \sqrt {1-\sinh ^2(x)} \cosh (x)+\frac {2}{3} \left (-\int \frac {1}{\sqrt {\sin (i x)^2+1}}dx-3 i E(i x|-1)\right )\) |
\(\Big \downarrow \) 3661 |
\(\displaystyle -\frac {1}{3} \sinh (x) \sqrt {1-\sinh ^2(x)} \cosh (x)+\frac {2}{3} (i \operatorname {EllipticF}(i x,-1)-3 i E(i x|-1))\) |
(2*((-3*I)*EllipticE[I*x, -1] + I*EllipticF[I*x, -1]))/3 - (Cosh[x]*Sinh[x ]*Sqrt[1 - Sinh[x]^2])/3
3.1.95.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[B/b Int[Sqrt[a + b*Sin[e + f*x]^2], x] , x] + Simp[(A*b - a*B)/b Int[1/Sqrt[a + b*Sin[e + f*x]^2], x], x] /; Fre eQ[{a, b, e, f, A, B}, x]
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]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*C os[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x]^2)^(p - 1)/(2*f*p)), x] + Sim p[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]
Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(1/(S qrt[a]*f))*EllipticF[e + f*x, -b/a], x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]
Time = 0.42 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.29
method | result | size |
default | \(\frac {\sqrt {-\left (-1+\sinh \left (x \right )^{2}\right ) \cosh \left (x \right )^{2}}\, \left (\cosh \left (x \right )^{4} \sinh \left (x \right )+10 \sqrt {-\cosh \left (x \right )^{2}+2}\, \sqrt {\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \operatorname {EllipticF}\left (\sinh \left (x \right ), i\right )-6 \sqrt {-\cosh \left (x \right )^{2}+2}\, \sqrt {\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \operatorname {EllipticE}\left (\sinh \left (x \right ), i\right )-2 \sinh \left (x \right ) \cosh \left (x \right )^{2}\right )}{3 \sqrt {1-\sinh \left (x \right )^{4}}\, \cosh \left (x \right ) \sqrt {1-\sinh \left (x \right )^{2}}}\) | \(103\) |
1/3*(-(-1+sinh(x)^2)*cosh(x)^2)^(1/2)*(cosh(x)^4*sinh(x)+10*(-cosh(x)^2+2) ^(1/2)*(cosh(x)^2)^(1/2)*EllipticF(sinh(x),I)-6*(-cosh(x)^2+2)^(1/2)*(cosh (x)^2)^(1/2)*EllipticE(sinh(x),I)-2*sinh(x)*cosh(x)^2)/(1-sinh(x)^4)^(1/2) /cosh(x)/(1-sinh(x)^2)^(1/2)
\[ \int \left (1-\sinh ^2(x)\right )^{3/2} \, dx=\int { {\left (-\sinh \left (x\right )^{2} + 1\right )}^{\frac {3}{2}} \,d x } \]
\[ \int \left (1-\sinh ^2(x)\right )^{3/2} \, dx=\int \left (1 - \sinh ^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int \left (1-\sinh ^2(x)\right )^{3/2} \, dx=\int { {\left (-\sinh \left (x\right )^{2} + 1\right )}^{\frac {3}{2}} \,d x } \]
\[ \int \left (1-\sinh ^2(x)\right )^{3/2} \, dx=\int { {\left (-\sinh \left (x\right )^{2} + 1\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \left (1-\sinh ^2(x)\right )^{3/2} \, dx=\int {\left (1-{\mathrm {sinh}\left (x\right )}^2\right )}^{3/2} \,d x \]