Integrand size = 21, antiderivative size = 29 \[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=\frac {2}{\sqrt {1-\sinh ^2(x)}}+2 \sqrt {1-\sinh ^2(x)} \] Output:
2/(1-sinh(x)^2)^(1/2)+2*(1-sinh(x)^2)^(1/2)
Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=\frac {5-\cosh (2 x)}{\sqrt {1-\sinh ^2(x)}} \] Input:
Integrate[(Sinh[x]^2*Sinh[2*x])/(1 - Sinh[x]^2)^(3/2),x]
Output:
(5 - Cosh[2*x])/Sqrt[1 - Sinh[x]^2]
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 26, 4878, 27, 243, 53, 2009}
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.
\(\displaystyle \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \sin (i x)^2 \sin (2 i x)}{\left (1+\sin (i x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\sin (i x)^2 \sin (2 i x)}{\left (\sin (i x)^2+1\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4878 |
\(\displaystyle i \int -\frac {2 i \sinh ^3(x)}{\left (1-\sinh ^2(x)\right )^{3/2}}d\sinh (x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {\sinh ^3(x)}{\left (1-\sinh ^2(x)\right )^{3/2}}d\sinh (x)\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \int \frac {\sinh ^2(x)}{\left (1-\sinh ^2(x)\right )^{3/2}}d\sinh ^2(x)\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \int \left (\frac {1}{\left (1-\sinh ^2(x)\right )^{3/2}}-\frac {1}{\sqrt {1-\sinh ^2(x)}}\right )d\sinh ^2(x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \sqrt {1-\sinh ^2(x)}+\frac {2}{\sqrt {1-\sinh ^2(x)}}\) |
Input:
Int[(Sinh[x]^2*Sinh[2*x])/(1 - Sinh[x]^2)^(3/2),x]
Output:
2/Sqrt[1 - Sinh[x]^2] + 2*Sqrt[1 - Sinh[x]^2]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFa ctors[Sin[v], x]}, d/Coefficient[v, x, 1] Subst[Int[SubstFor[1, Sin[v]/d, u/Cos[v], x], x], x, Sin[v]/d]], x] /; !FalseQ[v] && FunctionOfQ[NonfreeF actors[Sin[v], x], u/Cos[v], x]]
Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(-\frac {2 \sinh \left (x \right )^{2}}{\sqrt {1-\sinh \left (x \right )^{2}}}+\frac {4}{\sqrt {1-\sinh \left (x \right )^{2}}}\) | \(30\) |
default | \(-\frac {2 \sinh \left (x \right )^{2}}{\sqrt {1-\sinh \left (x \right )^{2}}}+\frac {4}{\sqrt {1-\sinh \left (x \right )^{2}}}\) | \(30\) |
Input:
int(sinh(x)^2*sinh(2*x)/(1-sinh(x)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2*sinh(x)^2/(1-sinh(x)^2)^(1/2)+4/(1-sinh(x)^2)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (25) = 50\).
Time = 0.07 (sec) , antiderivative size = 161, normalized size of antiderivative = 5.55 \[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=\frac {\sqrt {2} {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 5\right )} \sinh \left (x\right )^{2} - 10 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \sqrt {-\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} + 2 \, {\left (5 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{3} - 6 \, \cosh \left (x\right )^{3} + 2 \, {\left (5 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (5 \, \cosh \left (x\right )^{4} - 18 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) + \cosh \left (x\right )} \] Input:
integrate(sinh(x)^2*sinh(2*x)/(1-sinh(x)^2)^(3/2),x, algorithm="fricas")
Output:
sqrt(2)*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 5) *sinh(x)^2 - 10*cosh(x)^2 + 4*(cosh(x)^3 - 5*cosh(x))*sinh(x) + 1)*sqrt(-( cosh(x)^2 + sinh(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(c osh(x)^5 + 5*cosh(x)*sinh(x)^4 + sinh(x)^5 + 2*(5*cosh(x)^2 - 3)*sinh(x)^3 - 6*cosh(x)^3 + 2*(5*cosh(x)^3 - 9*cosh(x))*sinh(x)^2 + (5*cosh(x)^4 - 18 *cosh(x)^2 + 1)*sinh(x) + cosh(x))
\[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=\int \frac {\sinh ^{2}{\left (x \right )} \sinh {\left (2 x \right )}}{\left (- \left (\sinh {\left (x \right )} - 1\right ) \left (\sinh {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(sinh(x)**2*sinh(2*x)/(1-sinh(x)**2)**(3/2),x)
Output:
Integral(sinh(x)**2*sinh(2*x)/(-(sinh(x) - 1)*(sinh(x) + 1))**(3/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (25) = 50\).
Time = 0.15 (sec) , antiderivative size = 177, normalized size of antiderivative = 6.10 \[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=-\frac {16 \, e^{\left (-x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {62 \, e^{\left (-3 \, x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {16 \, e^{\left (-5 \, x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {e^{\left (-7 \, x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {e^{x}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} \] Input:
integrate(sinh(x)^2*sinh(2*x)/(1-sinh(x)^2)^(3/2),x, algorithm="maxima")
Output:
-16*e^(-x)/((2*e^(-x) + e^(-2*x) - 1)^(3/2)*(2*e^(-x) - e^(-2*x) + 1)^(3/2 )) + 62*e^(-3*x)/((2*e^(-x) + e^(-2*x) - 1)^(3/2)*(2*e^(-x) - e^(-2*x) + 1 )^(3/2)) - 16*e^(-5*x)/((2*e^(-x) + e^(-2*x) - 1)^(3/2)*(2*e^(-x) - e^(-2* x) + 1)^(3/2)) + e^(-7*x)/((2*e^(-x) + e^(-2*x) - 1)^(3/2)*(2*e^(-x) - e^( -2*x) + 1)^(3/2)) + e^x/((2*e^(-x) + e^(-2*x) - 1)^(3/2)*(2*e^(-x) - e^(-2 *x) + 1)^(3/2))
\[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=\int { \frac {\sinh \left (2 \, x\right ) \sinh \left (x\right )^{2}}{{\left (-\sinh \left (x\right )^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(sinh(x)^2*sinh(2*x)/(1-sinh(x)^2)^(3/2),x, algorithm="giac")
Output:
integrate(sinh(2*x)*sinh(x)^2/(-sinh(x)^2 + 1)^(3/2), x)
Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=\frac {2\,\sqrt {1-{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^2}\,\left ({\mathrm {e}}^{4\,x}-10\,{\mathrm {e}}^{2\,x}+1\right )}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1} \] Input:
int((sinh(2*x)*sinh(x)^2)/(1 - sinh(x)^2)^(3/2),x)
Output:
(2*(1 - (exp(-x)/2 - exp(x)/2)^2)^(1/2)*(exp(4*x) - 10*exp(2*x) + 1))/(exp (4*x) - 6*exp(2*x) + 1)
\[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=\int \frac {\sqrt {-\sinh \left (x \right )^{2}+1}\, \sinh \left (2 x \right ) \sinh \left (x \right )^{2}}{\sinh \left (x \right )^{4}-2 \sinh \left (x \right )^{2}+1}d x \] Input:
int(sinh(x)^2*sinh(2*x)/(1-sinh(x)^2)^(3/2),x)
Output:
int((sqrt( - sinh(x)**2 + 1)*sinh(2*x)*sinh(x)**2)/(sinh(x)**4 - 2*sinh(x) **2 + 1),x)