Integrand size = 25, antiderivative size = 305 \[ \int \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh ^4(e+f x) \, dx=-\frac {(3 a-8 b) \cosh (e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {8 (a-2 b) E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(3 a-8 b) \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {8 (a-2 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 f}+\frac {(a-2 b) \sinh ^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{f}-\frac {\left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh ^3(e+f x)}{3 f} \]
-1/3*(3*a-8*b)*cosh(f*x+e)*sinh(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/f-8/3*(a- 2*b)*(1/(1+sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*EllipticE(sinh(f* x+e)/(1+sinh(f*x+e)^2)^(1/2),(1-b/a)^(1/2))*sech(f*x+e)*(a+b*sinh(f*x+e)^2 )^(1/2)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+1/3*(3*a-8*b)*(1/(1+ sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)/(1+sin h(f*x+e)^2)^(1/2),(1-b/a)^(1/2))*sech(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/f/( sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+8/3*(a-2*b)*(a+b*sinh(f*x+e)^2) ^(1/2)*tanh(f*x+e)/f+(a-2*b)*sinh(f*x+e)^2*(a+b*sinh(f*x+e)^2)^(1/2)*tanh( f*x+e)/f-1/3*(a+b*sinh(f*x+e)^2)^(3/2)*tanh(f*x+e)^3/f
Result contains complex when optimal does not.
Time = 2.35 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.73 \[ \int \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh ^4(e+f x) \, dx=\frac {-32 i a (a-2 b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+4 i a (5 a-8 b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )-\frac {\left (32 a^2-108 a b+18 b^2+\left (64 a^2-160 a b+17 b^2\right ) \cosh (2 (e+f x))+2 (6 a-17 b) b \cosh (4 (e+f x))-b^2 \cosh (6 (e+f x))\right ) \text {sech}^2(e+f x) \tanh (e+f x)}{4 \sqrt {2}}}{12 f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \]
((-32*I)*a*(a - 2*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*( e + f*x), b/a] + (4*I)*a*(5*a - 8*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/ a]*EllipticF[I*(e + f*x), b/a] - ((32*a^2 - 108*a*b + 18*b^2 + (64*a^2 - 1 60*a*b + 17*b^2)*Cosh[2*(e + f*x)] + 2*(6*a - 17*b)*b*Cosh[4*(e + f*x)] - b^2*Cosh[6*(e + f*x)])*Sech[e + f*x]^2*Tanh[e + f*x])/(4*Sqrt[2]))/(12*f*S qrt[2*a - b + b*Cosh[2*(e + f*x)]])
Time = 0.57 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.24, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 3675, 369, 27, 439, 444, 27, 406, 320, 388, 313}
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.
| \(\displaystyle \int \tanh ^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (i e+i f x)^4 \left (a-b \sin (i e+i f x)^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 3675 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \frac {\sinh ^4(e+f x) \left (b \sinh ^2(e+f x)+a\right )^{3/2}}{\left (\sinh ^2(e+f x)+1\right )^{5/2}}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 369 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \int \frac {3 \sinh ^2(e+f x) \sqrt {b \sinh ^2(e+f x)+a} \left (2 b \sinh ^2(e+f x)+a\right )}{\left (\sinh ^2(e+f x)+1\right )^{3/2}}d\sinh (e+f x)-\frac {\sinh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{3 \left (\sinh ^2(e+f x)+1\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\int \frac {\sinh ^2(e+f x) \sqrt {b \sinh ^2(e+f x)+a} \left (2 b \sinh ^2(e+f x)+a\right )}{\left (\sinh ^2(e+f x)+1\right )^{3/2}}d\sinh (e+f x)-\frac {\sinh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{3 \left (\sinh ^2(e+f x)+1\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 439 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\int \frac {\sinh ^2(e+f x) \left ((3 a-8 b) b \sinh ^2(e+f x)+2 a (a-3 b)\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)-\frac {\sinh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{3 \left (\sinh ^2(e+f x)+1\right )^{3/2}}+\frac {(a-2 b) \sinh ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {\sinh ^2(e+f x)+1}}\right )}{f}\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\int \frac {b \left (8 (a-2 b) b \sinh ^2(e+f x)+a (3 a-8 b)\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{3 b}-\frac {1}{3} (3 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}-\frac {\sinh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{3 \left (\sinh ^2(e+f x)+1\right )^{3/2}}+\frac {(a-2 b) \sinh ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {\sinh ^2(e+f x)+1}}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \int \frac {8 (a-2 b) b \sinh ^2(e+f x)+a (3 a-8 b)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)-\frac {1}{3} (3 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}-\frac {\sinh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{3 \left (\sinh ^2(e+f x)+1\right )^{3/2}}+\frac {(a-2 b) \sinh ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {\sinh ^2(e+f x)+1}}\right )}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \left (a (3 a-8 b) \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+8 b (a-2 b) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)\right )-\frac {1}{3} (3 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}-\frac {\sinh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{3 \left (\sinh ^2(e+f x)+1\right )^{3/2}}+\frac {(a-2 b) \sinh ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {\sinh ^2(e+f x)+1}}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \left (8 b (a-2 b) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {(3 a-8 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )-\frac {1}{3} (3 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}-\frac {\sinh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{3 \left (\sinh ^2(e+f x)+1\right )^{3/2}}+\frac {(a-2 b) \sinh ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {\sinh ^2(e+f x)+1}}\right )}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \left (8 b (a-2 b) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\int \frac {\sqrt {b \sinh ^2(e+f x)+a}}{\left (\sinh ^2(e+f x)+1\right )^{3/2}}d\sinh (e+f x)}{b}\right )+\frac {(3 a-8 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )-\frac {1}{3} (3 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}-\frac {\sinh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{3 \left (\sinh ^2(e+f x)+1\right )^{3/2}}+\frac {(a-2 b) \sinh ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {\sinh ^2(e+f x)+1}}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \left (\frac {(3 a-8 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}+8 b (a-2 b) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\sqrt {a+b \sinh ^2(e+f x)} E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{b \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )\right )-\frac {1}{3} (3 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}-\frac {\sinh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{3 \left (\sinh ^2(e+f x)+1\right )^{3/2}}+\frac {(a-2 b) \sinh ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {\sinh ^2(e+f x)+1}}\right )}{f}\) |
(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*(((a - 2*b)*Sinh[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2])/Sqrt[1 + Sinh[e + f*x]^2] - ((3*a - 8*b)*Sinh[e + f*x] *Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[a + b*Sinh[e + f*x]^2])/3 - (Sinh[e + f*x] ^3*(a + b*Sinh[e + f*x]^2)^(3/2))/(3*(1 + Sinh[e + f*x]^2)^(3/2)) + (((3*a - 8*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x] ^2])/(Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[ e + f*x]^2))]) + 8*(a - 2*b)*b*((Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2] )/(b*Sqrt[1 + Sinh[e + f*x]^2]) - (EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/ a]*Sqrt[a + b*Sinh[e + f*x]^2])/(b*Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[(a + b*S inh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2))])))/3))/f
3.5.74.3.1 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[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2* b*(p + 1))), x] - Simp[e^2/(2*b*(p + 1)) Int[(e*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(m - 1) + d*(m + 2*q - 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 0 ] && GtQ[m, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*b*g*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*b*e*( p + 1) + (b*e - a*f)*(m + 1)) + d*(2*b*e*(p + 1) + (b*e - a*f)*(m + 2*q + 1 ))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && G tQ[q, 0] && !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*g*(g*x)^(m - 1)*(a + b*x^2)^ (p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q + 1) + 1))), x] - Simp[g^2/ (b*d*(m + 2*(p + q + 1) + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2) ^q*Simp[a*f*c*(m - 1) + (a*f*d*(m + 2*q + 1) + b*(f*c*(m + 2*p + 1) - e*d*( m + 2*(p + q + 1) + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && GtQ[m, 1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^ (m_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff^(m + 1 )*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])) Subst[Int[x^m*((a + b*ff^2*x^2) ^p/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b , e, f, p}, x] && IntegerQ[m/2] && !IntegerQ[p]
Time = 3.41 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.28
| method | result | size |
| default | \(-\frac {-\sqrt {-\frac {b}{a}}\, b^{2} \cosh \left (f x +e \right )^{6} \sinh \left (f x +e \right )+\left (3 \sqrt {-\frac {b}{a}}\, a b -7 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \cosh \left (f x +e \right )^{4} \sinh \left (f x +e \right )+\left (4 \sqrt {-\frac {b}{a}}\, a^{2}-13 \sqrt {-\frac {b}{a}}\, a b +9 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \cosh \left (f x +e \right )^{2} \sinh \left (f x +e \right )-\sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \left (3 \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2}-16 \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b +16 \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}+8 \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b -16 \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}\right ) \cosh \left (f x +e \right )^{2}+\left (-\sqrt {-\frac {b}{a}}\, a^{2}+2 \sqrt {-\frac {b}{a}}\, a b -\sqrt {-\frac {b}{a}}\, b^{2}\right ) \sinh \left (f x +e \right )}{3 \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right )^{3} \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(389\) |
-1/3*(-(-b/a)^(1/2)*b^2*cosh(f*x+e)^6*sinh(f*x+e)+(3*(-b/a)^(1/2)*a*b-7*(- b/a)^(1/2)*b^2)*cosh(f*x+e)^4*sinh(f*x+e)+(4*(-b/a)^(1/2)*a^2-13*(-b/a)^(1 /2)*a*b+9*(-b/a)^(1/2)*b^2)*cosh(f*x+e)^2*sinh(f*x+e)-(cosh(f*x+e)^2)^(1/2 )*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(3*EllipticF(sinh(f*x+e)*(-b/a)^(1/2), (a/b)^(1/2))*a^2-16*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*a*b+16 *EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*b^2+8*EllipticE(sinh(f*x+ e)*(-b/a)^(1/2),(a/b)^(1/2))*a*b-16*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/ b)^(1/2))*b^2)*cosh(f*x+e)^2+(-(-b/a)^(1/2)*a^2+2*(-b/a)^(1/2)*a*b-(-b/a)^ (1/2)*b^2)*sinh(f*x+e))/(-b/a)^(1/2)/cosh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(1/ 2)/f
\[ \int \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh ^4(e+f x) \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tanh \left (f x + e\right )^{4} \,d x } \]
Timed out. \[ \int \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh ^4(e+f x) \, dx=\text {Timed out} \]
\[ \int \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh ^4(e+f x) \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tanh \left (f x + e\right )^{4} \,d x } \]
Exception generated. \[ \int \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh ^4(e+f x) \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh ^4(e+f x) \, dx=\int {\mathrm {tanh}\left (e+f\,x\right )}^4\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]