3.2.23 \(\int \frac {1}{(a+b \sinh ^2(e+f x))^{5/2}} \, dx\) [123]

3.2.23.1 Optimal result

Integrand size = 16, antiderivative size = 251 \[ \int \frac {1}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=-\frac {b \cosh (e+f x) \sinh (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 (2 a-b) b \cosh (e+f x) \sinh (e+f x)}{3 a^2 (a-b)^2 f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {2 i (2 a-b) E\left (i e+i f x\left |\frac {b}{a}\right .\right ) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 (a-b)^2 f \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}}+\frac {i \operatorname {EllipticF}\left (i e+i f x,\frac {b}{a}\right ) \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}}{3 a (a-b) f \sqrt {a+b \sinh ^2(e+f x)}} \]

output

-1/3*b*cosh(f*x+e)*sinh(f*x+e)/a/(a-b)/f/(a+b*sinh(f*x+e)^2)^(3/2)-2/3*(2* 
a-b)*b*cosh(f*x+e)*sinh(f*x+e)/a^2/(a-b)^2/f/(a+b*sinh(f*x+e)^2)^(1/2)-2/3 
*I*(2*a-b)*(cos(I*e+I*f*x)^2)^(1/2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f*x 
),(b/a)^(1/2))*(a+b*sinh(f*x+e)^2)^(1/2)/a^2/(a-b)^2/f/(1+b*sinh(f*x+e)^2/ 
a)^(1/2)+1/3*I*(cos(I*e+I*f*x)^2)^(1/2)/cos(I*e+I*f*x)*EllipticF(sin(I*e+I 
*f*x),(b/a)^(1/2))*(1+b*sinh(f*x+e)^2/a)^(1/2)/a/(a-b)/f/(a+b*sinh(f*x+e)^ 
2)^(1/2)
 

3.2.23.2 Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\frac {-2 i a^2 (2 a-b) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+i a^2 (a-b) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )+\sqrt {2} b \left (-5 a^2+5 a b-b^2+b (-2 a+b) \cosh (2 (e+f x))\right ) \sinh (2 (e+f x))}{3 a^2 (a-b)^2 f (2 a-b+b \cosh (2 (e+f x)))^{3/2}} \]

input

Integrate[(a + b*Sinh[e + f*x]^2)^(-5/2),x]
 

output

((-2*I)*a^2*(2*a - b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[ 
I*(e + f*x), b/a] + I*a^2*(a - b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2 
)*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*b*(-5*a^2 + 5*a*b - b^2 + b*(-2*a 
+ b)*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(3*a^2*(a - b)^2*f*(2*a - b + b 
*Cosh[2*(e + f*x)])^(3/2))
 

3.2.23.3 Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {3042, 3663, 25, 3042, 3652, 3042, 3651, 3042, 3657, 3042, 3656, 3662, 3042, 3661}

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.

\(\displaystyle \int \frac {1}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\left (a-b \sin (i e+i f x)^2\right )^{5/2}}dx\)

\(\Big \downarrow \) 3663

\(\displaystyle -\frac {\int -\frac {-b \sinh ^2(e+f x)+3 a-2 b}{\left (b \sinh ^2(e+f x)+a\right )^{3/2}}dx}{3 a (a-b)}-\frac {b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {-b \sinh ^2(e+f x)+3 a-2 b}{\left (b \sinh ^2(e+f x)+a\right )^{3/2}}dx}{3 a (a-b)}-\frac {b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\int \frac {b \sin (i e+i f x)^2+3 a-2 b}{\left (a-b \sin (i e+i f x)^2\right )^{3/2}}dx}{3 a (a-b)}\)

\(\Big \downarrow \) 3652

\(\displaystyle \frac {\frac {\int \frac {2 (2 a-b) b \sinh ^2(e+f x)+a (3 a-b)}{\sqrt {b \sinh ^2(e+f x)+a}}dx}{a (a-b)}-\frac {2 b (2 a-b) \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}}}{3 a (a-b)}-\frac {b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {-\frac {2 b (2 a-b) \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\int \frac {a (3 a-b)-2 (2 a-b) b \sin (i e+i f x)^2}{\sqrt {a-b \sin (i e+i f x)^2}}dx}{a (a-b)}}{3 a (a-b)}\)

\(\Big \downarrow \) 3651

\(\displaystyle \frac {\frac {2 (2 a-b) \int \sqrt {b \sinh ^2(e+f x)+a}dx-a (a-b) \int \frac {1}{\sqrt {b \sinh ^2(e+f x)+a}}dx}{a (a-b)}-\frac {2 b (2 a-b) \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}}}{3 a (a-b)}-\frac {b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {-\frac {2 b (2 a-b) \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}}+\frac {2 (2 a-b) \int \sqrt {a-b \sin (i e+i f x)^2}dx-a (a-b) \int \frac {1}{\sqrt {a-b \sin (i e+i f x)^2}}dx}{a (a-b)}}{3 a (a-b)}\)

\(\Big \downarrow \) 3657

\(\displaystyle -\frac {b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {-\frac {2 b (2 a-b) \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\frac {2 (2 a-b) \sqrt {a+b \sinh ^2(e+f x)} \int \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}dx}{\sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}}-a (a-b) \int \frac {1}{\sqrt {a-b \sin (i e+i f x)^2}}dx}{a (a-b)}}{3 a (a-b)}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {-\frac {2 b (2 a-b) \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\frac {2 (2 a-b) \sqrt {a+b \sinh ^2(e+f x)} \int \sqrt {1-\frac {b \sin (i e+i f x)^2}{a}}dx}{\sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}}-a (a-b) \int \frac {1}{\sqrt {a-b \sin (i e+i f x)^2}}dx}{a (a-b)}}{3 a (a-b)}\)

\(\Big \downarrow \) 3656

\(\displaystyle -\frac {b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {-\frac {2 b (2 a-b) \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}}+\frac {-a (a-b) \int \frac {1}{\sqrt {a-b \sin (i e+i f x)^2}}dx-\frac {2 i (2 a-b) \sqrt {a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac {b}{a}\right .\right )}{f \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}}}{a (a-b)}}{3 a (a-b)}\)

\(\Big \downarrow \) 3662

\(\displaystyle -\frac {b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {-\frac {2 b (2 a-b) \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}}+\frac {-\frac {a (a-b) \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}}dx}{\sqrt {a+b \sinh ^2(e+f x)}}-\frac {2 i (2 a-b) \sqrt {a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac {b}{a}\right .\right )}{f \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}}}{a (a-b)}}{3 a (a-b)}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {-\frac {2 b (2 a-b) \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}}+\frac {-\frac {a (a-b) \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {1-\frac {b \sin (i e+i f x)^2}{a}}}dx}{\sqrt {a+b \sinh ^2(e+f x)}}-\frac {2 i (2 a-b) \sqrt {a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac {b}{a}\right .\right )}{f \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}}}{a (a-b)}}{3 a (a-b)}\)

\(\Big \downarrow \) 3661

\(\displaystyle -\frac {b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {-\frac {2 b (2 a-b) \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\frac {i a (a-b) \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (i e+i f x,\frac {b}{a}\right )}{f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {2 i (2 a-b) \sqrt {a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac {b}{a}\right .\right )}{f \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}}}{a (a-b)}}{3 a (a-b)}\)

input

Int[(a + b*Sinh[e + f*x]^2)^(-5/2),x]
 

output

-1/3*(b*Cosh[e + f*x]*Sinh[e + f*x])/(a*(a - b)*f*(a + b*Sinh[e + f*x]^2)^ 
(3/2)) + ((-2*(2*a - b)*b*Cosh[e + f*x]*Sinh[e + f*x])/(a*(a - b)*f*Sqrt[a 
 + b*Sinh[e + f*x]^2]) + (((-2*I)*(2*a - b)*EllipticE[I*e + I*f*x, b/a]*Sq 
rt[a + b*Sinh[e + f*x]^2])/(f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a]) + (I*a*(a - 
 b)*EllipticF[I*e + I*f*x, b/a]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])/(f*Sqrt[a 
 + b*Sinh[e + f*x]^2]))/(a*(a - b)))/(3*a*(a - b))
 

3.2.23.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3651

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2)/Sqrt[(a_) + (b_.)*sin[(e_.) + 
 (f_.)*(x_)]^2], x_Symbol] :> Simp[B/b   Int[Sqrt[a + b*Sin[e + f*x]^2], x] 
, x] + Simp[(A*b - a*B)/b   Int[1/Sqrt[a + b*Sin[e + f*x]^2], x], x] /; Fre 
eQ[{a, b, e, f, A, B}, x]
 

rule 3652

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b - a*B))*Cos[e + f*x]*Sin[e + f*x 
]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(a + b)*(p + 1))), x] - Simp[1/(2* 
a*(a + b)*(p + 1))   Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[a*B - A*(2*a*( 
p + 1) + b*(2*p + 3)) + 2*(A*b - a*B)*(p + 2)*Sin[e + f*x]^2, x], x], x] /; 
 FreeQ[{a, b, e, f, A, B}, x] && LtQ[p, -1] && NeQ[a + b, 0]
 

rule 3656

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(Sqrt[a 
]/f)*EllipticE[e + f*x, -b/a], x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]
 

rule 3657

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[Sqrt[a 
+ b*Sin[e + f*x]^2]/Sqrt[1 + b*(Sin[e + f*x]^2/a)]   Int[Sqrt[1 + (b*Sin[e 
+ f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] &&  !GtQ[a, 0]
 

rule 3661

Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(1/(S 
qrt[a]*f))*EllipticF[e + f*x, -b/a], x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 
 0]
 

rule 3662

Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[Sqrt[ 
1 + b*(Sin[e + f*x]^2/a)]/Sqrt[a + b*Sin[e + f*x]^2]   Int[1/Sqrt[1 + (b*Si 
n[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] &&  !GtQ[a, 0]
 

rule 3663

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*C 
os[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(p + 1)*(a 
+ b))), x] + Simp[1/(2*a*(p + 1)*(a + b))   Int[(a + b*Sin[e + f*x]^2)^(p + 
 1)*Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x], x] 
 /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && LtQ[p, -1]
 

3.2.23.4 Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.62

method	result	size
default	\(\frac {\sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}\, \left (-\frac {\sinh \left (f x +e \right ) \sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}}{3 b a \left (a -b \right ) \left (\sinh \left (f x +e \right )^{2}+\frac {a}{b}\right )^{2}}-\frac {2 b \cosh \left (f x +e \right )^{2} \sinh \left (f x +e \right ) \left (2 a -b \right )}{3 a^{2} \left (a -b \right )^{2} \sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}}+\frac {\left (3 a -b \right ) \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )}{\left (3 a^{3}-6 a^{2} b +3 a \,b^{2}\right ) \sqrt {-\frac {b}{a}}\, \sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}}-\frac {2 b \left (2 a -b \right ) \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \left (\operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-\operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )\right )}{3 \left (a -b \right )^{2} a^{2} \sqrt {-\frac {b}{a}}\, \sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}}\right )}{\cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\)	\(406\)
risch	\(\text {Expression too large to display}\)	\(16744\)

input

int(1/(a+b*sinh(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)
 

output

((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(-1/3/b/a/(a-b)*sinh(f*x+e)*((a+ 
b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)/(sinh(f*x+e)^2+a/b)^2-2/3*b*cosh(f*x 
+e)^2/a^2/(a-b)^2*sinh(f*x+e)*(2*a-b)/((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^ 
(1/2)+(3*a-b)/(3*a^3-6*a^2*b+3*a*b^2)/(-b/a)^(1/2)*((a+b*sinh(f*x+e)^2)/a) 
^(1/2)*(cosh(f*x+e)^2)^(1/2)/((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*Ell 
ipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))-2/3*b*(2*a-b)/(a-b)^2/a^2/(-b 
/a)^(1/2)*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)/((a+b*sinh(f 
*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1 
/2))-EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))))/cosh(f*x+e)/(a+b*si 
nh(f*x+e)^2)^(1/2)/f
 

3.2.23.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5442 vs. \(2 (259) = 518\).

Time = 0.22 (sec) , antiderivative size = 5442, normalized size of antiderivative = 21.68 \[ \int \frac {1}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]

input

integrate(1/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="fricas")
 

output

Too large to include
 

3.2.23.6 Sympy [F]

\[ \int \frac {1}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]

input

integrate(1/(a+b*sinh(f*x+e)**2)**(5/2),x)
 

output

Integral((a + b*sinh(e + f*x)**2)**(-5/2), x)
 

3.2.23.7 Maxima [F]

\[ \int \frac {1}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate(1/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="maxima")
 

output

integrate((b*sinh(f*x + e)^2 + a)^(-5/2), x)
 

3.2.23.8 Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]

input

integrate(1/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="giac")
 

output

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:Error: Bad Argument Type
 

3.2.23.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]

input

int(1/(a + b*sinh(e + f*x)^2)^(5/2),x)
 

output

int(1/(a + b*sinh(e + f*x)^2)^(5/2), x)