3.1.87 \(\int (a+b \sinh ^2(c+d x))^{5/2} \, dx\) [87]

3.1.87.1 Optimal result

Integrand size = 16, antiderivative size = 232 \[ \int \left (a+b \sinh ^2(c+d x)\right )^{5/2} \, dx=\frac {4 (2 a-b) b \cosh (c+d x) \sinh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{15 d}+\frac {b \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}-\frac {i \left (23 a^2-23 a b+8 b^2\right ) E\left (i c+i d x\left |\frac {b}{a}\right .\right ) \sqrt {a+b \sinh ^2(c+d x)}}{15 d \sqrt {1+\frac {b \sinh ^2(c+d x)}{a}}}+\frac {4 i a (a-b) (2 a-b) \operatorname {EllipticF}\left (i c+i d x,\frac {b}{a}\right ) \sqrt {1+\frac {b \sinh ^2(c+d x)}{a}}}{15 d \sqrt {a+b \sinh ^2(c+d x)}} \]

output

1/5*b*cosh(d*x+c)*sinh(d*x+c)*(a+b*sinh(d*x+c)^2)^(3/2)/d+4/15*(2*a-b)*b*c 
osh(d*x+c)*sinh(d*x+c)*(a+b*sinh(d*x+c)^2)^(1/2)/d-1/15*I*(23*a^2-23*a*b+8 
*b^2)*(cos(I*c+I*d*x)^2)^(1/2)/cos(I*c+I*d*x)*EllipticE(sin(I*c+I*d*x),(b/ 
a)^(1/2))*(a+b*sinh(d*x+c)^2)^(1/2)/d/(1+b*sinh(d*x+c)^2/a)^(1/2)+4/15*I*a 
*(a-b)*(2*a-b)*(cos(I*c+I*d*x)^2)^(1/2)/cos(I*c+I*d*x)*EllipticF(sin(I*c+I 
*d*x),(b/a)^(1/2))*(1+b*sinh(d*x+c)^2/a)^(1/2)/d/(a+b*sinh(d*x+c)^2)^(1/2)
 

3.1.87.2 Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.90 \[ \int \left (a+b \sinh ^2(c+d x)\right )^{5/2} \, dx=\frac {-16 i a \left (23 a^2-23 a b+8 b^2\right ) \sqrt {\frac {2 a-b+b \cosh (2 (c+d x))}{a}} E\left (i (c+d x)\left |\frac {b}{a}\right .\right )+64 i a \left (2 a^2-3 a b+b^2\right ) \sqrt {\frac {2 a-b+b \cosh (2 (c+d x))}{a}} \operatorname {EllipticF}\left (i (c+d x),\frac {b}{a}\right )+\sqrt {2} b \left (88 a^2-88 a b+25 b^2+28 (2 a-b) b \cosh (2 (c+d x))+3 b^2 \cosh (4 (c+d x))\right ) \sinh (2 (c+d x))}{240 d \sqrt {2 a-b+b \cosh (2 (c+d x))}} \]

input

Integrate[(a + b*Sinh[c + d*x]^2)^(5/2),x]
 

output

((-16*I)*a*(23*a^2 - 23*a*b + 8*b^2)*Sqrt[(2*a - b + b*Cosh[2*(c + d*x)])/ 
a]*EllipticE[I*(c + d*x), b/a] + (64*I)*a*(2*a^2 - 3*a*b + b^2)*Sqrt[(2*a 
- b + b*Cosh[2*(c + d*x)])/a]*EllipticF[I*(c + d*x), b/a] + Sqrt[2]*b*(88* 
a^2 - 88*a*b + 25*b^2 + 28*(2*a - b)*b*Cosh[2*(c + d*x)] + 3*b^2*Cosh[4*(c 
 + d*x)])*Sinh[2*(c + d*x)])/(240*d*Sqrt[2*a - b + b*Cosh[2*(c + d*x)]])
 

3.1.87.3 Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {3042, 3659, 3042, 3649, 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.

\(\displaystyle \int \left (a+b \sinh ^2(c+d x)\right )^{5/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a-b \sin (i c+i d x)^2\right )^{5/2}dx\)

\(\Big \downarrow \) 3659

\(\displaystyle \frac {1}{5} \int \sqrt {b \sinh ^2(c+d x)+a} \left (4 (2 a-b) b \sinh ^2(c+d x)+a (5 a-b)\right )dx+\frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}+\frac {1}{5} \int \sqrt {a-b \sin (i c+i d x)^2} \left (a (5 a-b)-4 (2 a-b) b \sin (i c+i d x)^2\right )dx\)

\(\Big \downarrow \) 3649

\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {b \left (23 a^2-23 b a+8 b^2\right ) \sinh ^2(c+d x)+a \left (15 a^2-11 b a+4 b^2\right )}{\sqrt {b \sinh ^2(c+d x)+a}}dx+\frac {4 b (2 a-b) \sinh (c+d x) \cosh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{3 d}\right )+\frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}+\frac {1}{5} \left (\frac {4 b (2 a-b) \sinh (c+d x) \cosh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{3 d}+\frac {1}{3} \int \frac {a \left (15 a^2-11 b a+4 b^2\right )-b \left (23 a^2-23 b a+8 b^2\right ) \sin (i c+i d x)^2}{\sqrt {a-b \sin (i c+i d x)^2}}dx\right )\)

\(\Big \downarrow \) 3651

\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\left (23 a^2-23 a b+8 b^2\right ) \int \sqrt {b \sinh ^2(c+d x)+a}dx-4 a (a-b) (2 a-b) \int \frac {1}{\sqrt {b \sinh ^2(c+d x)+a}}dx\right )+\frac {4 b (2 a-b) \sinh (c+d x) \cosh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{3 d}\right )+\frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}+\frac {1}{5} \left (\frac {4 b (2 a-b) \sinh (c+d x) \cosh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{3 d}+\frac {1}{3} \left (\left (23 a^2-23 a b+8 b^2\right ) \int \sqrt {a-b \sin (i c+i d x)^2}dx-4 a (a-b) (2 a-b) \int \frac {1}{\sqrt {a-b \sin (i c+i d x)^2}}dx\right )\right )\)

\(\Big \downarrow \) 3657

\(\displaystyle \frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}+\frac {1}{5} \left (\frac {4 b (2 a-b) \sinh (c+d x) \cosh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{3 d}+\frac {1}{3} \left (\frac {\left (23 a^2-23 a b+8 b^2\right ) \sqrt {a+b \sinh ^2(c+d x)} \int \sqrt {\frac {b \sinh ^2(c+d x)}{a}+1}dx}{\sqrt {\frac {b \sinh ^2(c+d x)}{a}+1}}-4 a (a-b) (2 a-b) \int \frac {1}{\sqrt {a-b \sin (i c+i d x)^2}}dx\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}+\frac {1}{5} \left (\frac {4 b (2 a-b) \sinh (c+d x) \cosh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{3 d}+\frac {1}{3} \left (\frac {\left (23 a^2-23 a b+8 b^2\right ) \sqrt {a+b \sinh ^2(c+d x)} \int \sqrt {1-\frac {b \sin (i c+i d x)^2}{a}}dx}{\sqrt {\frac {b \sinh ^2(c+d x)}{a}+1}}-4 a (a-b) (2 a-b) \int \frac {1}{\sqrt {a-b \sin (i c+i d x)^2}}dx\right )\right )\)

\(\Big \downarrow \) 3656

\(\displaystyle \frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}+\frac {1}{5} \left (\frac {4 b (2 a-b) \sinh (c+d x) \cosh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{3 d}+\frac {1}{3} \left (-4 a (a-b) (2 a-b) \int \frac {1}{\sqrt {a-b \sin (i c+i d x)^2}}dx-\frac {i \left (23 a^2-23 a b+8 b^2\right ) \sqrt {a+b \sinh ^2(c+d x)} E\left (i c+i d x\left |\frac {b}{a}\right .\right )}{d \sqrt {\frac {b \sinh ^2(c+d x)}{a}+1}}\right )\right )\)

\(\Big \downarrow \) 3662

\(\displaystyle \frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}+\frac {1}{5} \left (\frac {4 b (2 a-b) \sinh (c+d x) \cosh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{3 d}+\frac {1}{3} \left (-\frac {4 a (a-b) (2 a-b) \sqrt {\frac {b \sinh ^2(c+d x)}{a}+1} \int \frac {1}{\sqrt {\frac {b \sinh ^2(c+d x)}{a}+1}}dx}{\sqrt {a+b \sinh ^2(c+d x)}}-\frac {i \left (23 a^2-23 a b+8 b^2\right ) \sqrt {a+b \sinh ^2(c+d x)} E\left (i c+i d x\left |\frac {b}{a}\right .\right )}{d \sqrt {\frac {b \sinh ^2(c+d x)}{a}+1}}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}+\frac {1}{5} \left (\frac {4 b (2 a-b) \sinh (c+d x) \cosh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{3 d}+\frac {1}{3} \left (-\frac {4 a (a-b) (2 a-b) \sqrt {\frac {b \sinh ^2(c+d x)}{a}+1} \int \frac {1}{\sqrt {1-\frac {b \sin (i c+i d x)^2}{a}}}dx}{\sqrt {a+b \sinh ^2(c+d x)}}-\frac {i \left (23 a^2-23 a b+8 b^2\right ) \sqrt {a+b \sinh ^2(c+d x)} E\left (i c+i d x\left |\frac {b}{a}\right .\right )}{d \sqrt {\frac {b \sinh ^2(c+d x)}{a}+1}}\right )\right )\)

\(\Big \downarrow \) 3661

\(\displaystyle \frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^{3/2}}{5 d}+\frac {1}{5} \left (\frac {4 b (2 a-b) \sinh (c+d x) \cosh (c+d x) \sqrt {a+b \sinh ^2(c+d x)}}{3 d}+\frac {1}{3} \left (\frac {4 i a (a-b) (2 a-b) \sqrt {\frac {b \sinh ^2(c+d x)}{a}+1} \operatorname {EllipticF}\left (i c+i d x,\frac {b}{a}\right )}{d \sqrt {a+b \sinh ^2(c+d x)}}-\frac {i \left (23 a^2-23 a b+8 b^2\right ) \sqrt {a+b \sinh ^2(c+d x)} E\left (i c+i d x\left |\frac {b}{a}\right .\right )}{d \sqrt {\frac {b \sinh ^2(c+d x)}{a}+1}}\right )\right )\)

input

Int[(a + b*Sinh[c + d*x]^2)^(5/2),x]
 

output

(b*Cosh[c + d*x]*Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2)^(3/2))/(5*d) + ((4* 
(2*a - b)*b*Cosh[c + d*x]*Sinh[c + d*x]*Sqrt[a + b*Sinh[c + d*x]^2])/(3*d) 
 + (((-I)*(23*a^2 - 23*a*b + 8*b^2)*EllipticE[I*c + I*d*x, b/a]*Sqrt[a + b 
*Sinh[c + d*x]^2])/(d*Sqrt[1 + (b*Sinh[c + d*x]^2)/a]) + ((4*I)*a*(a - b)* 
(2*a - b)*EllipticF[I*c + I*d*x, b/a]*Sqrt[1 + (b*Sinh[c + d*x]^2)/a])/(d* 
Sqrt[a + b*Sinh[c + d*x]^2]))/3)/5
 

3.1.87.3.1 Defintions of rubi rules used

rule 3042

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

rule 3649

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

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 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 3659

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*f*p)), x] + Sim 
p[1/(2*p)   Int[(a + b*Sin[e + f*x]^2)^(p - 2)*Simp[a*(b + 2*a*p) + b*(2*a 
+ b)*(2*p - 1)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[ 
a + b, 0] && GtQ[p, 1]
 

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]
 

3.1.87.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(608\) vs. \(2(268)=536\).

Time = 1.99 (sec) , antiderivative size = 609, normalized size of antiderivative = 2.62

method	result	size
default	\(\frac {3 \sqrt {-\frac {b}{a}}\, b^{3} \cosh \left (d x +c \right )^{6} \sinh \left (d x +c \right )+\left (14 \sqrt {-\frac {b}{a}}\, a \,b^{2}-10 \sqrt {-\frac {b}{a}}\, b^{3}\right ) \cosh \left (d x +c \right )^{4} \sinh \left (d x +c \right )+\left (11 \sqrt {-\frac {b}{a}}\, a^{2} b -18 \sqrt {-\frac {b}{a}}\, a \,b^{2}+7 \sqrt {-\frac {b}{a}}\, b^{3}\right ) \cosh \left (d x +c \right )^{2} \sinh \left (d x +c \right )+15 a^{3} \sqrt {\frac {b \cosh \left (d x +c \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-34 a^{2} b \sqrt {\frac {b \cosh \left (d x +c \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+27 \sqrt {\frac {b \cosh \left (d x +c \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2}-8 \sqrt {\frac {b \cosh \left (d x +c \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{3}+23 a^{2} b \sqrt {\frac {b \cosh \left (d x +c \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-23 \sqrt {\frac {b \cosh \left (d x +c \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2}+8 \sqrt {\frac {b \cosh \left (d x +c \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 d x +2 c \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (d x +c \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{3}}{15 \sqrt {-\frac {b}{a}}\, \cosh \left (d x +c \right ) \sqrt {a +b \sinh \left (d x +c \right )^{2}}\, d}\)	\(609\)

input

int((a+b*sinh(d*x+c)^2)^(5/2),x,method=_RETURNVERBOSE)
 

output

1/15*(3*(-b/a)^(1/2)*b^3*cosh(d*x+c)^6*sinh(d*x+c)+(14*(-b/a)^(1/2)*a*b^2- 
10*(-b/a)^(1/2)*b^3)*cosh(d*x+c)^4*sinh(d*x+c)+(11*(-b/a)^(1/2)*a^2*b-18*( 
-b/a)^(1/2)*a*b^2+7*(-b/a)^(1/2)*b^3)*cosh(d*x+c)^2*sinh(d*x+c)+15*a^3*(b/ 
a*cosh(d*x+c)^2+(a-b)/a)^(1/2)*(cosh(d*x+c)^2)^(1/2)*EllipticF(sinh(d*x+c) 
*(-b/a)^(1/2),(a/b)^(1/2))-34*a^2*b*(b/a*cosh(d*x+c)^2+(a-b)/a)^(1/2)*(cos 
h(d*x+c)^2)^(1/2)*EllipticF(sinh(d*x+c)*(-b/a)^(1/2),(a/b)^(1/2))+27*(b/a* 
cosh(d*x+c)^2+(a-b)/a)^(1/2)*(cosh(d*x+c)^2)^(1/2)*EllipticF(sinh(d*x+c)*( 
-b/a)^(1/2),(a/b)^(1/2))*a*b^2-8*(b/a*cosh(d*x+c)^2+(a-b)/a)^(1/2)*(cosh(d 
*x+c)^2)^(1/2)*EllipticF(sinh(d*x+c)*(-b/a)^(1/2),(a/b)^(1/2))*b^3+23*a^2* 
b*(b/a*cosh(d*x+c)^2+(a-b)/a)^(1/2)*(cosh(d*x+c)^2)^(1/2)*EllipticE(sinh(d 
*x+c)*(-b/a)^(1/2),(a/b)^(1/2))-23*(b/a*cosh(d*x+c)^2+(a-b)/a)^(1/2)*(cosh 
(d*x+c)^2)^(1/2)*EllipticE(sinh(d*x+c)*(-b/a)^(1/2),(a/b)^(1/2))*a*b^2+8*( 
b/a*cosh(d*x+c)^2+(a-b)/a)^(1/2)*(cosh(d*x+c)^2)^(1/2)*EllipticE(sinh(d*x+ 
c)*(-b/a)^(1/2),(a/b)^(1/2))*b^3)/(-b/a)^(1/2)/cosh(d*x+c)/(a+b*sinh(d*x+c 
)^2)^(1/2)/d
 

3.1.87.5 Fricas [F]

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

input

integrate((a+b*sinh(d*x+c)^2)^(5/2),x, algorithm="fricas")
 

output

integral((b^2*sinh(d*x + c)^4 + 2*a*b*sinh(d*x + c)^2 + a^2)*sqrt(b*sinh(d 
*x + c)^2 + a), x)
 

3.1.87.6 Sympy [F(-1)]

Timed out. \[ \int \left (a+b \sinh ^2(c+d x)\right )^{5/2} \, dx=\text {Timed out} \]

input

integrate((a+b*sinh(d*x+c)**2)**(5/2),x)
 

output

Timed out
 

3.1.87.7 Maxima [F]

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

input

integrate((a+b*sinh(d*x+c)^2)^(5/2),x, algorithm="maxima")
 

output

integrate((b*sinh(d*x + c)^2 + a)^(5/2), x)
 

3.1.87.8 Giac [F(-2)]

Exception generated. \[ \int \left (a+b \sinh ^2(c+d x)\right )^{5/2} \, dx=\text {Exception raised: TypeError} \]

input

integrate((a+b*sinh(d*x+c)^2)^(5/2),x, algorithm="giac")
 

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

3.1.87.9 Mupad [F(-1)]

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

input

int((a + b*sinh(c + d*x)^2)^(5/2),x)
 

output

int((a + b*sinh(c + d*x)^2)^(5/2), x)