Integrand size = 25, antiderivative size = 355 \[ \int \cosh ^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=-\frac {\left (2 a^2-9 a b-b^2\right ) \cosh (e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{35 b f}+\frac {3 (a+b) \cosh (e+f x) \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{35 b f}+\frac {\cosh ^3(e+f x) \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{7 f}+\frac {2 (a+b) \left (a^2-6 a b+b^2\right ) E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{35 b^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {\left (a^2-18 a b+b^2\right ) \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{35 b f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {2 (a+b) \left (a^2-6 a b+b^2\right ) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{35 b^2 f} \] Output:
-1/35*(2*a^2-9*a*b-b^2)*cosh(f*x+e)*sinh(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/ b/f+3/35*(a+b)*cosh(f*x+e)*sinh(f*x+e)*(a+b*sinh(f*x+e)^2)^(3/2)/b/f+1/7*c osh(f*x+e)^3*sinh(f*x+e)*(a+b*sinh(f*x+e)^2)^(3/2)/f+2/35*(a+b)*(a^2-6*a*b +b^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)/b^2/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a) ^(1/2)-1/35*(a^2-18*a*b+b^2)*InverseJacobiAM(arctan(sinh(f*x+e)),(1-b/a)^( 1/2))*sech(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/b/f/(sech(f*x+e)^2*(a+b*sinh(f *x+e)^2)/a)^(1/2)-2/35*(a+b)*(a^2-6*a*b+b^2)*(a+b*sinh(f*x+e)^2)^(1/2)*tan h(f*x+e)/b^2/f
Result contains complex when optimal does not.
Time = 2.01 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.72 \[ \int \cosh ^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {128 i a \left (a^3-5 a^2 b-5 a b^2+b^3\right ) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-64 i a \left (2 a^3-11 a^2 b+8 a b^2+b^3\right ) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )+\sqrt {2} b \left (32 a^3+400 a^2 b-212 a b^2+30 b^3+b \left (144 a^2+192 a b-37 b^2\right ) \cosh (2 (e+f x))+2 b^2 (26 a+b) \cosh (4 (e+f x))+5 b^3 \cosh (6 (e+f x))\right ) \sinh (2 (e+f x))}{2240 b^2 f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \] Input:
Integrate[Cosh[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2),x]
Output:
((128*I)*a*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*Sqrt[(2*a - b + b*Cosh[2*(e + f *x)])/a]*EllipticE[I*(e + f*x), b/a] - (64*I)*a*(2*a^3 - 11*a^2*b + 8*a*b^ 2 + b^3)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/ a] + Sqrt[2]*b*(32*a^3 + 400*a^2*b - 212*a*b^2 + 30*b^3 + b*(144*a^2 + 192 *a*b - 37*b^2)*Cosh[2*(e + f*x)] + 2*b^2*(26*a + b)*Cosh[4*(e + f*x)] + 5* b^3*Cosh[6*(e + f*x)])*Sinh[2*(e + f*x)])/(2240*b^2*f*Sqrt[2*a - b + b*Cos h[2*(e + f*x)]])
Time = 0.59 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.17, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 3671, 318, 403, 27, 403, 25, 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.
| \(\displaystyle \int \cosh ^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (i e+i f x)^4 \left (a-b \sin (i e+i f x)^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 3671 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \left (\sinh ^2(e+f x)+1\right )^{3/2} \left (b \sinh ^2(e+f x)+a\right )^{3/2}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{7} \int \frac {\left (\sinh ^2(e+f x)+1\right )^{3/2} \left (2 (4 a-b) b \sinh ^2(e+f x)+a (7 a-b)\right )}{\sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {1}{7} b \sinh (e+f x) \left (\sinh ^2(e+f x)+1\right )^{5/2} \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{7} \left (\frac {\int \frac {3 b \sqrt {\sinh ^2(e+f x)+1} \left (\left (a^2+9 b a-2 b^2\right ) \sinh ^2(e+f x)+a (9 a-b)\right )}{\sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{5 b}+\frac {2}{5} (4 a-b) \sinh (e+f x) \left (\sinh ^2(e+f x)+1\right )^{3/2} \sqrt {a+b \sinh ^2(e+f x)}\right )+\frac {1}{7} b \sinh (e+f x) \left (\sinh ^2(e+f x)+1\right )^{5/2} \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{7} \left (\frac {3}{5} \int \frac {\sqrt {\sinh ^2(e+f x)+1} \left (\left (a^2+9 b a-2 b^2\right ) \sinh ^2(e+f x)+a (9 a-b)\right )}{\sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {2}{5} (4 a-b) \sinh (e+f x) \left (\sinh ^2(e+f x)+1\right )^{3/2} \sqrt {a+b \sinh ^2(e+f x)}\right )+\frac {1}{7} b \sinh (e+f x) \left (\sinh ^2(e+f x)+1\right )^{5/2} \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{7} \left (\frac {3}{5} \left (\frac {\int -\frac {2 (a+b) \left (a^2-6 b a+b^2\right ) \sinh ^2(e+f x)+a \left (a^2-18 b a+b^2\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{3 b}+\frac {\left (a^2+9 a b-2 b^2\right ) \sqrt {\sinh ^2(e+f x)+1} \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 b}\right )+\frac {2}{5} (4 a-b) \sinh (e+f x) \left (\sinh ^2(e+f x)+1\right )^{3/2} \sqrt {a+b \sinh ^2(e+f x)}\right )+\frac {1}{7} b \sinh (e+f x) \left (\sinh ^2(e+f x)+1\right )^{5/2} \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{7} \left (\frac {3}{5} \left (\frac {\left (a^2+9 a b-2 b^2\right ) \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1} \sqrt {a+b \sinh ^2(e+f x)}}{3 b}-\frac {\int \frac {2 (a+b) \left (a^2-6 b a+b^2\right ) \sinh ^2(e+f x)+a \left (a^2-18 b a+b^2\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{3 b}\right )+\frac {2}{5} (4 a-b) \sinh (e+f x) \left (\sinh ^2(e+f x)+1\right )^{3/2} \sqrt {a+b \sinh ^2(e+f x)}\right )+\frac {1}{7} b \sinh (e+f x) \left (\sinh ^2(e+f x)+1\right )^{5/2} \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{7} \left (\frac {3}{5} \left (\frac {\left (a^2+9 a b-2 b^2\right ) \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1} \sqrt {a+b \sinh ^2(e+f x)}}{3 b}-\frac {a \left (a^2-18 a b+b^2\right ) \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+2 (a+b) \left (a^2-6 a b+b^2\right ) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{3 b}\right )+\frac {2}{5} (4 a-b) \sinh (e+f x) \left (\sinh ^2(e+f x)+1\right )^{3/2} \sqrt {a+b \sinh ^2(e+f x)}\right )+\frac {1}{7} b \sinh (e+f x) \left (\sinh ^2(e+f x)+1\right )^{5/2} \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{7} \left (\frac {3}{5} \left (\frac {\left (a^2+9 a b-2 b^2\right ) \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1} \sqrt {a+b \sinh ^2(e+f x)}}{3 b}-\frac {2 (a+b) \left (a^2-6 a b+b^2\right ) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {\left (a^2-18 a b+b^2\right ) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}}{3 b}\right )+\frac {2}{5} (4 a-b) \sinh (e+f x) \left (\sinh ^2(e+f x)+1\right )^{3/2} \sqrt {a+b \sinh ^2(e+f x)}\right )+\frac {1}{7} b \sinh (e+f x) \left (\sinh ^2(e+f x)+1\right )^{5/2} \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{7} \left (\frac {3}{5} \left (\frac {\left (a^2+9 a b-2 b^2\right ) \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1} \sqrt {a+b \sinh ^2(e+f x)}}{3 b}-\frac {2 (a+b) \left (a^2-6 a b+b^2\right ) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\int \frac {\sqrt {b \sinh ^2(e+f x)+a}}{\left (\sinh ^2(e+f x)+1\right )^{3/2}}d\sinh (e+f x)}{b}\right )+\frac {\left (a^2-18 a b+b^2\right ) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}}{3 b}\right )+\frac {2}{5} (4 a-b) \sinh (e+f x) \left (\sinh ^2(e+f x)+1\right )^{3/2} \sqrt {a+b \sinh ^2(e+f x)}\right )+\frac {1}{7} b \sinh (e+f x) \left (\sinh ^2(e+f x)+1\right )^{5/2} \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{7} \left (\frac {3}{5} \left (\frac {\left (a^2+9 a b-2 b^2\right ) \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1} \sqrt {a+b \sinh ^2(e+f x)}}{3 b}-\frac {\frac {\left (a^2-18 a b+b^2\right ) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}+2 (a+b) \left (a^2-6 a b+b^2\right ) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\sqrt {a+b \sinh ^2(e+f x)} E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{b \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )}{3 b}\right )+\frac {2}{5} (4 a-b) \sinh (e+f x) \left (\sinh ^2(e+f x)+1\right )^{3/2} \sqrt {a+b \sinh ^2(e+f x)}\right )+\frac {1}{7} b \sinh (e+f x) \left (\sinh ^2(e+f x)+1\right )^{5/2} \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
Input:
Int[Cosh[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2),x]
Output:
(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*((b*Sinh[e + f*x]*(1 + Sinh[e + f*x]^ 2)^(5/2)*Sqrt[a + b*Sinh[e + f*x]^2])/7 + ((2*(4*a - b)*Sinh[e + f*x]*(1 + Sinh[e + f*x]^2)^(3/2)*Sqrt[a + b*Sinh[e + f*x]^2])/5 + (3*(((a^2 + 9*a*b - 2*b^2)*Sinh[e + f*x]*Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[a + b*Sinh[e + f*x] ^2])/(3*b) - (((a^2 - 18*a*b + b^2)*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*Si nh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2))]) + 2*(a + b)*(a^2 - 6*a*b + b^2) *((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*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2))])))/(3*b)))/5)/7))/f
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
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[(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[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
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[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff*(Sqrt[ Cos[e + f*x]^2]/(f*Cos[e + f*x])) Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && !IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(729\) vs. \(2(343)=686\).
Time = 7.96 (sec) , antiderivative size = 730, normalized size of antiderivative = 2.06
| method | result | size |
| default | \(\frac {5 \sqrt {-\frac {b}{a}}\, b^{3} \cosh \left (f x +e \right )^{8} \sinh \left (f x +e \right )+\left (13 \sqrt {-\frac {b}{a}}\, a \,b^{2}-7 \sqrt {-\frac {b}{a}}\, b^{3}\right ) \cosh \left (f x +e \right )^{6} \sinh \left (f x +e \right )+\left (9 \sqrt {-\frac {b}{a}}\, a^{2} b -\sqrt {-\frac {b}{a}}\, a \,b^{2}\right ) \cosh \left (f x +e \right )^{4} \sinh \left (f x +e \right )+\left (\sqrt {-\frac {b}{a}}\, a^{3}+8 \sqrt {-\frac {b}{a}}\, a^{2} b -11 \sqrt {-\frac {b}{a}}\, a \,b^{2}+2 \sqrt {-\frac {b}{a}}\, b^{3}\right ) \cosh \left (f x +e \right )^{2} \sinh \left (f x +e \right )+\sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{3}+8 \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2} b -11 \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2}+2 \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{3}-2 \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{3}+10 \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2} b +10 \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2}-2 \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{3}}{35 b \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(730\) |
Input:
int(cosh(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/35*(5*(-b/a)^(1/2)*b^3*cosh(f*x+e)^8*sinh(f*x+e)+(13*(-b/a)^(1/2)*a*b^2- 7*(-b/a)^(1/2)*b^3)*cosh(f*x+e)^6*sinh(f*x+e)+(9*(-b/a)^(1/2)*a^2*b-(-b/a) ^(1/2)*a*b^2)*cosh(f*x+e)^4*sinh(f*x+e)+((-b/a)^(1/2)*a^3+8*(-b/a)^(1/2)*a ^2*b-11*(-b/a)^(1/2)*a*b^2+2*(-b/a)^(1/2)*b^3)*cosh(f*x+e)^2*sinh(f*x+e)+( b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+ e)*(-b/a)^(1/2),(1/b*a)^(1/2))*a^3+8*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(co sh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(1/b*a)^(1/2))*a^2*b -11*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh (f*x+e)*(-b/a)^(1/2),(1/b*a)^(1/2))*a*b^2+2*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1 /2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(1/b*a)^(1/2) )*b^3-2*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE( sinh(f*x+e)*(-b/a)^(1/2),(1/b*a)^(1/2))*a^3+10*(b/a*cosh(f*x+e)^2+(a-b)/a) ^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(1/b*a)^(1 /2))*a^2*b+10*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*Elli pticE(sinh(f*x+e)*(-b/a)^(1/2),(1/b*a)^(1/2))*a*b^2-2*(b/a*cosh(f*x+e)^2+( a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(1/ b*a)^(1/2))*b^3)/b/(-b/a)^(1/2)/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/f
\[ \int \cosh ^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \cosh \left (f x + e\right )^{4} \,d x } \] Input:
integrate(cosh(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")
Output:
integral((b*cosh(f*x + e)^4*sinh(f*x + e)^2 + a*cosh(f*x + e)^4)*sqrt(b*si nh(f*x + e)^2 + a), x)
Timed out. \[ \int \cosh ^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \] Input:
integrate(cosh(f*x+e)**4*(a+b*sinh(f*x+e)**2)**(3/2),x)
Output:
Timed out
\[ \int \cosh ^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \cosh \left (f x + e\right )^{4} \,d x } \] Input:
integrate(cosh(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")
Output:
integrate((b*sinh(f*x + e)^2 + a)^(3/2)*cosh(f*x + e)^4, x)
\[ \int \cosh ^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \cosh \left (f x + e\right )^{4} \,d x } \] Input:
integrate(cosh(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")
Output:
integrate((b*sinh(f*x + e)^2 + a)^(3/2)*cosh(f*x + e)^4, x)
Timed out. \[ \int \cosh ^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int {\mathrm {cosh}\left (e+f\,x\right )}^4\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \] Input:
int(cosh(e + f*x)^4*(a + b*sinh(e + f*x)^2)^(3/2),x)
Output:
int(cosh(e + f*x)^4*(a + b*sinh(e + f*x)^2)^(3/2), x)
\[ \int \cosh ^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\left (\int \sqrt {\sinh \left (f x +e \right )^{2} b +a}\, \cosh \left (f x +e \right )^{4} \sinh \left (f x +e \right )^{2}d x \right ) b +\left (\int \sqrt {\sinh \left (f x +e \right )^{2} b +a}\, \cosh \left (f x +e \right )^{4}d x \right ) a \] Input:
int(cosh(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(3/2),x)
Output:
int(sqrt(sinh(e + f*x)**2*b + a)*cosh(e + f*x)**4*sinh(e + f*x)**2,x)*b + int(sqrt(sinh(e + f*x)**2*b + a)*cosh(e + f*x)**4,x)*a