Integrand size = 25, antiderivative size = 385 \[ \int \frac {\coth ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=-\frac {(a-b) \coth (e+f x) \text {csch}^2(e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 (a-3 b) \coth (e+f x) \text {csch}^2(e+f x)}{3 a^2 b f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {8 (a-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^4 f}+\frac {(3 a-8 b) \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 b f}-\frac {8 (a-2 b) E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^4 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(3 a-8 b) \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^4 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {8 (a-2 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^4 f} \]
-1/3*(a-b)*coth(f*x+e)*csch(f*x+e)^2/a/b/f/(a+b*sinh(f*x+e)^2)^(3/2)-2/3*( a-3*b)*coth(f*x+e)*csch(f*x+e)^2/a^2/b/f/(a+b*sinh(f*x+e)^2)^(1/2)-8/3*(a- 2*b)*coth(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/a^4/f+1/3*(3*a-8*b)*coth(f*x+e) *csch(f*x+e)^2*(a+b*sinh(f*x+e)^2)^(1/2)/a^3/b/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)/a^4/f/(sec h(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+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)/a^4/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)*tan h(f*x+e)/a^4/f
Result contains complex when optimal does not.
Time = 2.36 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.64 \[ \int \frac {\coth ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=-\frac {i \left (\frac {i b \left (8 a^3-63 a^2 b+92 a b^2-40 b^3-2 \left (8 a^3-38 a^2 b+63 a b^2-30 b^3\right ) \cosh (2 (e+f x))-b \left (13 a^2-36 a b+24 b^2\right ) \cosh (4 (e+f x))-2 a b^2 \cosh (6 (e+f x))+4 b^3 \cosh (6 (e+f x))\right ) \coth (e+f x) \text {csch}^2(e+f x)}{\sqrt {2}}+2 a^2 b \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} \left (8 (a-2 b) E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+(-5 a+8 b) \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )\right )\right )}{6 a^4 b f (2 a-b+b \cosh (2 (e+f x)))^{3/2}} \]
((-1/6*I)*((I*b*(8*a^3 - 63*a^2*b + 92*a*b^2 - 40*b^3 - 2*(8*a^3 - 38*a^2* b + 63*a*b^2 - 30*b^3)*Cosh[2*(e + f*x)] - b*(13*a^2 - 36*a*b + 24*b^2)*Co sh[4*(e + f*x)] - 2*a*b^2*Cosh[6*(e + f*x)] + 4*b^3*Cosh[6*(e + f*x)])*Cot h[e + f*x]*Csch[e + f*x]^2)/Sqrt[2] + 2*a^2*b*((2*a - b + b*Cosh[2*(e + f* x)])/a)^(3/2)*(8*(a - 2*b)*EllipticE[I*(e + f*x), b/a] + (-5*a + 8*b)*Elli pticF[I*(e + f*x), b/a])))/(a^4*b*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))
Time = 0.69 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.22, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 3675, 370, 441, 27, 445, 27, 445, 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.
| \(\displaystyle \int \frac {\coth ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (i e+i f x)^4 \left (a-b \sin (i e+i f x)^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3675 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \frac {\text {csch}^4(e+f x) \left (\sinh ^2(e+f x)+1\right )^{3/2}}{\left (b \sinh ^2(e+f x)+a\right )^{5/2}}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 370 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {\int \frac {\text {csch}^4(e+f x) \left ((2 a-5 b) \sinh ^2(e+f x)+3 (a-2 b)\right )}{\sqrt {\sinh ^2(e+f x)+1} \left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{3 a b}-\frac {(a-b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{3 a b \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 441 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {\frac {\int \frac {3 (a-b) \text {csch}^4(e+f x) \left (2 (a-3 b) \sinh ^2(e+f x)+3 a-8 b\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a (a-b)}+\frac {2 (a-3 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{a \sqrt {a+b \sinh ^2(e+f x)}}}{3 a b}-\frac {(a-b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{3 a b \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {\frac {3 \int \frac {\text {csch}^4(e+f x) \left (2 (a-3 b) \sinh ^2(e+f x)+3 a-8 b\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}+\frac {2 (a-3 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{a \sqrt {a+b \sinh ^2(e+f x)}}}{3 a b}-\frac {(a-b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{3 a b \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {\frac {3 \left (-\frac {\int \frac {b \text {csch}^2(e+f x) \left ((3 a-8 b) \sinh ^2(e+f x)+8 (a-2 b)\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{3 a}-\frac {(3 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{a}+\frac {2 (a-3 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{a \sqrt {a+b \sinh ^2(e+f x)}}}{3 a b}-\frac {(a-b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{3 a b \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {\frac {3 \left (-\frac {b \int \frac {\text {csch}^2(e+f x) \left ((3 a-8 b) \sinh ^2(e+f x)+8 (a-2 b)\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{3 a}-\frac {(3 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{a}+\frac {2 (a-3 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{a \sqrt {a+b \sinh ^2(e+f x)}}}{3 a b}-\frac {(a-b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{3 a b \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {\frac {3 \left (-\frac {b \left (-\frac {\int -\frac {8 (a-2 b) b \sinh ^2(e+f x)+a (3 a-8 b)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}-\frac {8 (a-2 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}\right )}{3 a}-\frac {(3 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{a}+\frac {2 (a-3 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{a \sqrt {a+b \sinh ^2(e+f x)}}}{3 a b}-\frac {(a-b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{3 a b \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {\frac {3 \left (-\frac {b \left (\frac {\int \frac {8 (a-2 b) b \sinh ^2(e+f x)+a (3 a-8 b)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}-\frac {8 (a-2 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}\right )}{3 a}-\frac {(3 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{a}+\frac {2 (a-3 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{a \sqrt {a+b \sinh ^2(e+f x)}}}{3 a b}-\frac {(a-b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{3 a b \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {\frac {3 \left (-\frac {b \left (\frac {a (3 a-8 b) \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+8 b (a-2 b) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}-\frac {8 (a-2 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}\right )}{3 a}-\frac {(3 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{a}+\frac {2 (a-3 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{a \sqrt {a+b \sinh ^2(e+f x)}}}{3 a b}-\frac {(a-b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{3 a b \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {\frac {3 \left (-\frac {b \left (\frac {8 b (a-2 b) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {(3 a-8 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}}{a}-\frac {8 (a-2 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}\right )}{3 a}-\frac {(3 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{a}+\frac {2 (a-3 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{a \sqrt {a+b \sinh ^2(e+f x)}}}{3 a b}-\frac {(a-b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{3 a b \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {\frac {3 \left (-\frac {b \left (\frac {8 b (a-2 b) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\int \frac {\sqrt {b \sinh ^2(e+f x)+a}}{\left (\sinh ^2(e+f x)+1\right )^{3/2}}d\sinh (e+f x)}{b}\right )+\frac {(3 a-8 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}}{a}-\frac {8 (a-2 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}\right )}{3 a}-\frac {(3 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{a}+\frac {2 (a-3 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{a \sqrt {a+b \sinh ^2(e+f x)}}}{3 a b}-\frac {(a-b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{3 a b \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {\frac {3 \left (-\frac {b \left (\frac {\frac {(3 a-8 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}+8 b (a-2 b) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\sqrt {a+b \sinh ^2(e+f x)} E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{b \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )}{a}-\frac {8 (a-2 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}\right )}{3 a}-\frac {(3 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{a}+\frac {2 (a-3 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{a \sqrt {a+b \sinh ^2(e+f x)}}}{3 a b}-\frac {(a-b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x)}{3 a b \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*(-1/3*((a - b)*Csch[e + f*x]^3*Sqrt[1 + Sinh[e + f*x]^2])/(a*b*(a + b*Sinh[e + f*x]^2)^(3/2)) - ((2*(a - 3*b)*C sch[e + f*x]^3*Sqrt[1 + Sinh[e + f*x]^2])/(a*Sqrt[a + b*Sinh[e + f*x]^2]) + (3*(-1/3*((3*a - 8*b)*Csch[e + f*x]^3*Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[a + b*Sinh[e + f*x]^2])/a - (b*((-8*(a - 2*b)*Csch[e + f*x]*Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[a + b*Sinh[e + f*x]^2])/a + (((3*a - 8*b)*EllipticF[ArcTan[S inh[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*Sinh[e + f*x]^2)/(a*(1 + Si nh[e + f*x]^2))])))/a))/(3*a)))/a)/(3*a*b)))/f
3.6.11.3.1 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[(-(b*c - a*d))*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(a*b*e*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(e*x) ^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*Simp[c*(b*c*2*(p + 1) + (b*c - a *d)*(m + 1)) + d*(b*c*2*(p + 1) + (b*c - a*d)*(m + 2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 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 + 1)/(a*g*2*(b*c - a*d)*(p + 1))), x] + Si mp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2 )^q*Simp[c*(b*e - a*f)*(m + 1) + e*2*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, x] && LtQ[p, -1]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[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]
Leaf count of result is larger than twice the leaf count of optimal. \(922\) vs. \(2(433)=866\).
Time = 4.11 (sec) , antiderivative size = 923, normalized size of antiderivative = 2.40
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(923\) |
| risch | \(\text {Expression too large to display}\) | \(1122960\) |
-1/3*(8*(-b/a)^(1/2)*a*b^2*sinh(f*x+e)^8-16*(-b/a)^(1/2)*b^3*sinh(f*x+e)^8 -3*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+ e)*(-b/a)^(1/2),(a/b)^(1/2))*a^2*b*sinh(f*x+e)^5+16*((a+b*sinh(f*x+e)^2)/a )^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/ 2))*a*b^2*sinh(f*x+e)^5-16*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^( 1/2)*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*b^3*sinh(f*x+e)^5-8*( (a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*( -b/a)^(1/2),(a/b)^(1/2))*a*b^2*sinh(f*x+e)^5+16*((a+b*sinh(f*x+e)^2)/a)^(1 /2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))* b^3*sinh(f*x+e)^5+13*(-b/a)^(1/2)*a^2*b*sinh(f*x+e)^6-16*(-b/a)^(1/2)*a*b^ 2*sinh(f*x+e)^6-16*(-b/a)^(1/2)*b^3*sinh(f*x+e)^6-3*((a+b*sinh(f*x+e)^2)/a )^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/ 2))*a^3*sinh(f*x+e)^3+16*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/ 2)*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*a^2*b*sinh(f*x+e)^3-16* ((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)* (-b/a)^(1/2),(a/b)^(1/2))*a*b^2*sinh(f*x+e)^3-8*((a+b*sinh(f*x+e)^2)/a)^(1 /2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))* a^2*b*sinh(f*x+e)^3+16*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2) *EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*a*b^2*sinh(f*x+e)^3+4*(-b /a)^(1/2)*a^3*sinh(f*x+e)^4+7*(-b/a)^(1/2)*a^2*b*sinh(f*x+e)^4-24*(-b/a...
Leaf count of result is larger than twice the leaf count of optimal. 13823 vs. \(2 (381) = 762\).
Time = 0.60 (sec) , antiderivative size = 13823, normalized size of antiderivative = 35.90 \[ \int \frac {\coth ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
\[ \int \frac {\coth ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\coth ^{4}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\coth ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\coth \left (f x + e\right )^{4}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Exception generated. \[ \int \frac {\coth ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\coth ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {coth}\left (e+f\,x\right )}^4}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]