Integrand size = 20, antiderivative size = 325 \[ \int \frac {1}{(a+b \cosh (c+d x) \sinh (c+d x))^{5/2}} \, dx=-\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 \left (4 a^2+b^2\right ) d (2 a+b \sinh (2 c+2 d x))^{3/2}}-\frac {32 \sqrt {2} a b \cosh (2 c+2 d x)}{3 \left (4 a^2+b^2\right )^2 d \sqrt {2 a+b \sinh (2 c+2 d x)}}-\frac {32 i \sqrt {2} a E\left (\frac {1}{2} \left (2 i c-\frac {\pi }{2}+2 i d x\right )|\frac {2 b}{2 i a+b}\right ) \sqrt {2 a+b \sinh (2 c+2 d x)}}{3 \left (4 a^2+b^2\right )^2 d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}+\frac {4 i \sqrt {2} \operatorname {EllipticF}\left (\frac {1}{2} \left (2 i c-\frac {\pi }{2}+2 i d x\right ),\frac {2 b}{2 i a+b}\right ) \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}{3 \left (4 a^2+b^2\right ) d \sqrt {2 a+b \sinh (2 c+2 d x)}} \] Output:
-4/3*2^(1/2)*b*cosh(2*d*x+2*c)/(4*a^2+b^2)/d/(2*a+b*sinh(2*d*x+2*c))^(3/2) -32/3*2^(1/2)*a*b*cosh(2*d*x+2*c)/(4*a^2+b^2)^2/d/(2*a+b*sinh(2*d*x+2*c))^ (1/2)+32/3*I*2^(1/2)*a*EllipticE(cos(I*c+1/4*Pi+I*d*x),2^(1/2)*(b/(2*I*a+b ))^(1/2))*(2*a+b*sinh(2*d*x+2*c))^(1/2)/(4*a^2+b^2)^2/d/((2*a+b*sinh(2*d*x +2*c))/(2*a-I*b))^(1/2)+4/3*I*2^(1/2)*InverseJacobiAM(I*c-1/4*Pi+I*d*x,2^( 1/2)*(b/(2*I*a+b))^(1/2))*((2*a+b*sinh(2*d*x+2*c))/(2*a-I*b))^(1/2)/(4*a^2 +b^2)/d/(2*a+b*sinh(2*d*x+2*c))^(1/2)
Time = 1.56 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(a+b \cosh (c+d x) \sinh (c+d x))^{5/2}} \, dx=\frac {4 \sqrt {2} \left (8 i a (2 a-i b)^2 E\left (\frac {1}{4} (-4 i c+\pi -4 i d x)|-\frac {2 i b}{2 a-i b}\right ) \left (\frac {2 a+b \sinh (2 (c+d x))}{2 a-i b}\right )^{3/2}+(2 a-i b)^2 (-2 i a+b) \operatorname {EllipticF}\left (\frac {1}{4} (-4 i c+\pi -4 i d x),-\frac {2 i b}{2 a-i b}\right ) \left (\frac {2 a+b \sinh (2 (c+d x))}{2 a-i b}\right )^{3/2}-b \cosh (2 (c+d x)) \left (20 a^2+b^2+8 a b \sinh (2 (c+d x))\right )\right )}{3 \left (4 a^2+b^2\right )^2 d (2 a+b \sinh (2 (c+d x)))^{3/2}} \] Input:
Integrate[(a + b*Cosh[c + d*x]*Sinh[c + d*x])^(-5/2),x]
Output:
(4*Sqrt[2]*((8*I)*a*(2*a - I*b)^2*EllipticE[((-4*I)*c + Pi - (4*I)*d*x)/4, ((-2*I)*b)/(2*a - I*b)]*((2*a + b*Sinh[2*(c + d*x)])/(2*a - I*b))^(3/2) + (2*a - I*b)^2*((-2*I)*a + b)*EllipticF[((-4*I)*c + Pi - (4*I)*d*x)/4, ((- 2*I)*b)/(2*a - I*b)]*((2*a + b*Sinh[2*(c + d*x)])/(2*a - I*b))^(3/2) - b*C osh[2*(c + d*x)]*(20*a^2 + b^2 + 8*a*b*Sinh[2*(c + d*x)])))/(3*(4*a^2 + b^ 2)^2*d*(2*a + b*Sinh[2*(c + d*x)])^(3/2))
Time = 1.37 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {3042, 3145, 3042, 3143, 25, 27, 3042, 3233, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
| \(\displaystyle \int \frac {1}{(a+b \sinh (c+d x) \cosh (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a-i b \sin (i c+i d x) \cos (i c+i d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3145 |
| \(\displaystyle \int \frac {1}{\left (a+\frac {1}{2} b \sinh (2 c+2 d x)\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a-\frac {1}{2} i b \sin (2 i c+2 i d x)\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3143 |
| \(\displaystyle -\frac {8 \int -\frac {6 a-b \sinh (2 c+2 d x)}{\sqrt {2} (2 a+b \sinh (2 c+2 d x))^{3/2}}dx}{3 \left (4 a^2+b^2\right )}-\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 d \left (4 a^2+b^2\right ) (2 a+b \sinh (2 c+2 d x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {8 \int \frac {6 a-b \sinh (2 c+2 d x)}{\sqrt {2} (2 a+b \sinh (2 c+2 d x))^{3/2}}dx}{3 \left (4 a^2+b^2\right )}-\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 d \left (4 a^2+b^2\right ) (2 a+b \sinh (2 c+2 d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 \sqrt {2} \int \frac {6 a-b \sinh (2 c+2 d x)}{(2 a+b \sinh (2 c+2 d x))^{3/2}}dx}{3 \left (4 a^2+b^2\right )}-\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 d \left (4 a^2+b^2\right ) (2 a+b \sinh (2 c+2 d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 d \left (4 a^2+b^2\right ) (2 a+b \sinh (2 c+2 d x))^{3/2}}+\frac {4 \sqrt {2} \int \frac {6 a+i b \sin (2 i c+2 i d x)}{(2 a-i b \sin (2 i c+2 i d x))^{3/2}}dx}{3 \left (4 a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {4 \sqrt {2} \left (-\frac {2 \int -\frac {12 a^2+8 b \sinh (2 c+2 d x) a-b^2}{2 \sqrt {2 a+b \sinh (2 c+2 d x)}}dx}{4 a^2+b^2}-\frac {8 a b \cosh (2 c+2 d x)}{d \left (4 a^2+b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)}}\right )}{3 \left (4 a^2+b^2\right )}-\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 d \left (4 a^2+b^2\right ) (2 a+b \sinh (2 c+2 d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 \sqrt {2} \left (\frac {\int \frac {12 a^2+8 b \sinh (2 c+2 d x) a-b^2}{\sqrt {2 a+b \sinh (2 c+2 d x)}}dx}{4 a^2+b^2}-\frac {8 a b \cosh (2 c+2 d x)}{d \left (4 a^2+b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)}}\right )}{3 \left (4 a^2+b^2\right )}-\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 d \left (4 a^2+b^2\right ) (2 a+b \sinh (2 c+2 d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 d \left (4 a^2+b^2\right ) (2 a+b \sinh (2 c+2 d x))^{3/2}}+\frac {4 \sqrt {2} \left (-\frac {8 a b \cosh (2 c+2 d x)}{d \left (4 a^2+b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)}}+\frac {\int \frac {12 a^2-8 i b \sin (2 i c+2 i d x) a-b^2}{\sqrt {2 a-i b \sin (2 i c+2 i d x)}}dx}{4 a^2+b^2}\right )}{3 \left (4 a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {4 \sqrt {2} \left (\frac {8 a \int \sqrt {2 a+b \sinh (2 c+2 d x)}dx-\left (4 a^2+b^2\right ) \int \frac {1}{\sqrt {2 a+b \sinh (2 c+2 d x)}}dx}{4 a^2+b^2}-\frac {8 a b \cosh (2 c+2 d x)}{d \left (4 a^2+b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)}}\right )}{3 \left (4 a^2+b^2\right )}-\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 d \left (4 a^2+b^2\right ) (2 a+b \sinh (2 c+2 d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 d \left (4 a^2+b^2\right ) (2 a+b \sinh (2 c+2 d x))^{3/2}}+\frac {4 \sqrt {2} \left (-\frac {8 a b \cosh (2 c+2 d x)}{d \left (4 a^2+b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)}}+\frac {8 a \int \sqrt {2 a-i b \sin (2 i c+2 i d x)}dx-\left (4 a^2+b^2\right ) \int \frac {1}{\sqrt {2 a-i b \sin (2 i c+2 i d x)}}dx}{4 a^2+b^2}\right )}{3 \left (4 a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 d \left (4 a^2+b^2\right ) (2 a+b \sinh (2 c+2 d x))^{3/2}}+\frac {4 \sqrt {2} \left (-\frac {8 a b \cosh (2 c+2 d x)}{d \left (4 a^2+b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)}}+\frac {\frac {8 a \sqrt {2 a+b \sinh (2 c+2 d x)} \int \sqrt {\frac {2 a}{2 a-i b}+\frac {b \sinh (2 c+2 d x)}{2 a-i b}}dx}{\sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}-\left (4 a^2+b^2\right ) \int \frac {1}{\sqrt {2 a-i b \sin (2 i c+2 i d x)}}dx}{4 a^2+b^2}\right )}{3 \left (4 a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 d \left (4 a^2+b^2\right ) (2 a+b \sinh (2 c+2 d x))^{3/2}}+\frac {4 \sqrt {2} \left (-\frac {8 a b \cosh (2 c+2 d x)}{d \left (4 a^2+b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)}}+\frac {\frac {8 a \sqrt {2 a+b \sinh (2 c+2 d x)} \int \sqrt {\frac {2 a}{2 a-i b}-\frac {i b \sin (2 i c+2 i d x)}{2 a-i b}}dx}{\sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}-\left (4 a^2+b^2\right ) \int \frac {1}{\sqrt {2 a-i b \sin (2 i c+2 i d x)}}dx}{4 a^2+b^2}\right )}{3 \left (4 a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 d \left (4 a^2+b^2\right ) (2 a+b \sinh (2 c+2 d x))^{3/2}}+\frac {4 \sqrt {2} \left (-\frac {8 a b \cosh (2 c+2 d x)}{d \left (4 a^2+b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)}}+\frac {-\left (4 a^2+b^2\right ) \int \frac {1}{\sqrt {2 a-i b \sin (2 i c+2 i d x)}}dx-\frac {8 i a \sqrt {2 a+b \sinh (2 c+2 d x)} E\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}}{4 a^2+b^2}\right )}{3 \left (4 a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 d \left (4 a^2+b^2\right ) (2 a+b \sinh (2 c+2 d x))^{3/2}}+\frac {4 \sqrt {2} \left (-\frac {8 a b \cosh (2 c+2 d x)}{d \left (4 a^2+b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)}}+\frac {-\frac {\left (4 a^2+b^2\right ) \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}} \int \frac {1}{\sqrt {\frac {2 a}{2 a-i b}+\frac {b \sinh (2 c+2 d x)}{2 a-i b}}}dx}{\sqrt {2 a+b \sinh (2 c+2 d x)}}-\frac {8 i a \sqrt {2 a+b \sinh (2 c+2 d x)} E\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}}{4 a^2+b^2}\right )}{3 \left (4 a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 d \left (4 a^2+b^2\right ) (2 a+b \sinh (2 c+2 d x))^{3/2}}+\frac {4 \sqrt {2} \left (-\frac {8 a b \cosh (2 c+2 d x)}{d \left (4 a^2+b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)}}+\frac {-\frac {\left (4 a^2+b^2\right ) \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}} \int \frac {1}{\sqrt {\frac {2 a}{2 a-i b}-\frac {i b \sin (2 i c+2 i d x)}{2 a-i b}}}dx}{\sqrt {2 a+b \sinh (2 c+2 d x)}}-\frac {8 i a \sqrt {2 a+b \sinh (2 c+2 d x)} E\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}}{4 a^2+b^2}\right )}{3 \left (4 a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {4 \sqrt {2} b \cosh (2 c+2 d x)}{3 d \left (4 a^2+b^2\right ) (2 a+b \sinh (2 c+2 d x))^{3/2}}+\frac {4 \sqrt {2} \left (-\frac {8 a b \cosh (2 c+2 d x)}{d \left (4 a^2+b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)}}+\frac {\frac {i \left (4 a^2+b^2\right ) \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right ),\frac {2 b}{2 i a+b}\right )}{d \sqrt {2 a+b \sinh (2 c+2 d x)}}-\frac {8 i a \sqrt {2 a+b \sinh (2 c+2 d x)} E\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}}{4 a^2+b^2}\right )}{3 \left (4 a^2+b^2\right )}\) |
Input:
Int[(a + b*Cosh[c + d*x]*Sinh[c + d*x])^(-5/2),x]
Output:
(-4*Sqrt[2]*b*Cosh[2*c + 2*d*x])/(3*(4*a^2 + b^2)*d*(2*a + b*Sinh[2*c + 2* d*x])^(3/2)) + (4*Sqrt[2]*((-8*a*b*Cosh[2*c + 2*d*x])/((4*a^2 + b^2)*d*Sqr t[2*a + b*Sinh[2*c + 2*d*x]]) + (((-8*I)*a*EllipticE[((2*I)*c - Pi/2 + (2* I)*d*x)/2, (2*b)/((2*I)*a + b)]*Sqrt[2*a + b*Sinh[2*c + 2*d*x]])/(d*Sqrt[( 2*a + b*Sinh[2*c + 2*d*x])/(2*a - I*b)]) + (I*(4*a^2 + b^2)*EllipticF[((2* I)*c - Pi/2 + (2*I)*d*x)/2, (2*b)/((2*I)*a + b)]*Sqrt[(2*a + b*Sinh[2*c + 2*d*x])/(2*a - I*b)])/(d*Sqrt[2*a + b*Sinh[2*c + 2*d*x]]))/(4*a^2 + b^2))) /(3*(4*a^2 + b^2))
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_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 - b^2))), x] + Simp [1/((n + 1)*(a^2 - b^2)) Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Int[((a_) + cos[(c_.) + (d_.)*(x_)]*(b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_ Symbol] :> Int[(a + b*(Sin[2*c + 2*d*x]/2))^n, x] /; FreeQ[{a, b, c, d, n}, x]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( m + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 640 vs. \(2 (286 ) = 572\).
Time = 0.66 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.97
| method | result | size |
| default | \(\frac {4 \sqrt {\cosh \left (2 d x +2 c \right )^{2} \left (2 a +b \sinh \left (2 d x +2 c \right )\right )}\, \left (-\frac {2 \sqrt {\cosh \left (2 d x +2 c \right )^{2} \left (2 a +b \sinh \left (2 d x +2 c \right )\right )}}{3 b \left (4 a^{2}+b^{2}\right ) \left (\sinh \left (2 d x +2 c \right )+\frac {2 a}{b}\right )^{2}}-\frac {16 b \cosh \left (2 d x +2 c \right )^{2} a}{3 \left (4 a^{2}+b^{2}\right )^{2} \sqrt {\cosh \left (2 d x +2 c \right )^{2} \left (2 a +b \sinh \left (2 d x +2 c \right )\right )}}+\frac {2 \left (12 a^{2}-b^{2}\right ) \left (\frac {2 a}{b}-i\right ) \sqrt {\frac {-2 a -b \sinh \left (2 d x +2 c \right )}{i b -2 a}}\, \sqrt {\frac {\left (-\sinh \left (2 d x +2 c \right )+i\right ) b}{i b +2 a}}\, \sqrt {\frac {b \left (\sinh \left (2 d x +2 c \right )+i\right )}{i b -2 a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {-2 a -b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {\frac {-i b +2 a}{i b +2 a}}\right )}{\left (48 a^{4}+24 b^{2} a^{2}+3 b^{4}\right ) \sqrt {\cosh \left (2 d x +2 c \right )^{2} \left (2 a +b \sinh \left (2 d x +2 c \right )\right )}}+\frac {16 a b \left (\frac {2 a}{b}-i\right ) \sqrt {\frac {-2 a -b \sinh \left (2 d x +2 c \right )}{i b -2 a}}\, \sqrt {\frac {\left (-\sinh \left (2 d x +2 c \right )+i\right ) b}{i b +2 a}}\, \sqrt {\frac {b \left (\sinh \left (2 d x +2 c \right )+i\right )}{i b -2 a}}\, \left (\left (-\frac {2 a}{b}-i\right ) \operatorname {EllipticE}\left (\sqrt {\frac {-2 a -b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {\frac {-i b +2 a}{i b +2 a}}\right )+i \operatorname {EllipticF}\left (\sqrt {\frac {-2 a -b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {\frac {-i b +2 a}{i b +2 a}}\right )\right )}{3 \left (4 a^{2}+b^{2}\right )^{2} \sqrt {\cosh \left (2 d x +2 c \right )^{2} \left (2 a +b \sinh \left (2 d x +2 c \right )\right )}}\right )}{\cosh \left (2 d x +2 c \right ) \sqrt {4 a +2 b \sinh \left (2 d x +2 c \right )}\, d}\) | \(641\) |
| risch | \(\text {Expression too large to display}\) | \(6261\) |
Input:
int(1/(a+b*cosh(d*x+c)*sinh(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
Output:
4*(cosh(2*d*x+2*c)^2*(2*a+b*sinh(2*d*x+2*c)))^(1/2)*(-2/3/b/(4*a^2+b^2)*(c osh(2*d*x+2*c)^2*(2*a+b*sinh(2*d*x+2*c)))^(1/2)/(sinh(2*d*x+2*c)+2/b*a)^2- 16/3*b*cosh(2*d*x+2*c)^2/(4*a^2+b^2)^2*a/(cosh(2*d*x+2*c)^2*(2*a+b*sinh(2* d*x+2*c)))^(1/2)+2*(12*a^2-b^2)/(48*a^4+24*a^2*b^2+3*b^4)*(2/b*a-I)*((-2*a -b*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2)*((-sinh(2*d*x+2*c)+I)*b/(I*b+2*a))^(1 /2)*(b*(sinh(2*d*x+2*c)+I)/(I*b-2*a))^(1/2)/(cosh(2*d*x+2*c)^2*(2*a+b*sinh (2*d*x+2*c)))^(1/2)*EllipticF(((-2*a-b*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2),( (2*a-I*b)/(I*b+2*a))^(1/2))+16/3*a*b/(4*a^2+b^2)^2*(2/b*a-I)*((-2*a-b*sinh (2*d*x+2*c))/(I*b-2*a))^(1/2)*((-sinh(2*d*x+2*c)+I)*b/(I*b+2*a))^(1/2)*(b* (sinh(2*d*x+2*c)+I)/(I*b-2*a))^(1/2)/(cosh(2*d*x+2*c)^2*(2*a+b*sinh(2*d*x+ 2*c)))^(1/2)*((-2/b*a-I)*EllipticE(((-2*a-b*sinh(2*d*x+2*c))/(I*b-2*a))^(1 /2),((2*a-I*b)/(I*b+2*a))^(1/2))+I*EllipticF(((-2*a-b*sinh(2*d*x+2*c))/(I* b-2*a))^(1/2),((2*a-I*b)/(I*b+2*a))^(1/2))))/cosh(2*d*x+2*c)/(4*a+2*b*sinh (2*d*x+2*c))^(1/2)/d
Leaf count of result is larger than twice the leaf count of optimal. 4231 vs. \(2 (276) = 552\).
Time = 0.19 (sec) , antiderivative size = 4231, normalized size of antiderivative = 13.02 \[ \int \frac {1}{(a+b \cosh (c+d x) \sinh (c+d x))^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b*cosh(d*x+c)*sinh(d*x+c))^(5/2),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{(a+b \cosh (c+d x) \sinh (c+d x))^{5/2}} \, dx=\int \frac {1}{\left (a + b \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(a+b*cosh(d*x+c)*sinh(d*x+c))**(5/2),x)
Output:
Integral((a + b*sinh(c + d*x)*cosh(c + d*x))**(-5/2), x)
\[ \int \frac {1}{(a+b \cosh (c+d x) \sinh (c+d x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(a+b*cosh(d*x+c)*sinh(d*x+c))^(5/2),x, algorithm="maxima")
Output:
integrate((b*cosh(d*x + c)*sinh(d*x + c) + a)^(-5/2), x)
\[ \int \frac {1}{(a+b \cosh (c+d x) \sinh (c+d x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(a+b*cosh(d*x+c)*sinh(d*x+c))^(5/2),x, algorithm="giac")
Output:
integrate((b*cosh(d*x + c)*sinh(d*x + c) + a)^(-5/2), x)
Timed out. \[ \int \frac {1}{(a+b \cosh (c+d x) \sinh (c+d x))^{5/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )\right )}^{5/2}} \,d x \] Input:
int(1/(a + b*cosh(c + d*x)*sinh(c + d*x))^(5/2),x)
Output:
int(1/(a + b*cosh(c + d*x)*sinh(c + d*x))^(5/2), x)
\[ \int \frac {1}{(a+b \cosh (c+d x) \sinh (c+d x))^{5/2}} \, dx=\int \frac {\sqrt {\cosh \left (d x +c \right ) \sinh \left (d x +c \right ) b +a}}{\cosh \left (d x +c \right )^{3} \sinh \left (d x +c \right )^{3} b^{3}+3 \cosh \left (d x +c \right )^{2} \sinh \left (d x +c \right )^{2} a \,b^{2}+3 \cosh \left (d x +c \right ) \sinh \left (d x +c \right ) a^{2} b +a^{3}}d x \] Input:
int(1/(a+b*cosh(d*x+c)*sinh(d*x+c))^(5/2),x)
Output:
int(sqrt(cosh(c + d*x)*sinh(c + d*x)*b + a)/(cosh(c + d*x)**3*sinh(c + d*x )**3*b**3 + 3*cosh(c + d*x)**2*sinh(c + d*x)**2*a*b**2 + 3*cosh(c + d*x)*s inh(c + d*x)*a**2*b + a**3),x)