\(\int (a+b \sinh ^2(e+f x))^{3/2} \, dx\) [334]

Optimal result

Integrand size = 16, antiderivative size = 176 \[ \int \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {b \cosh (e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {2 i (2 a-b) E\left (i e+i f x\left |\frac {b}{a}\right .\right ) \sqrt {a+b \sinh ^2(e+f x)}}{3 f \sqrt {\frac {a+b \sinh ^2(e+f x)}{a}}}+\frac {i a (a-b) \operatorname {EllipticF}\left (i e+i f x,\frac {b}{a}\right ) \sqrt {\frac {a+b \sinh ^2(e+f x)}{a}}}{3 f \sqrt {a+b \sinh ^2(e+f x)}} \] Output:

1/3*b*cosh(f*x+e)*sinh(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/f-2/3*I*(2*a-b)*El 
lipticE(sin(I*e+I*f*x),(b/a)^(1/2))*(a+b*sinh(f*x+e)^2)^(1/2)/f/((a+b*sinh 
(f*x+e)^2)/a)^(1/2)+1/3*I*a*(a-b)*InverseJacobiAM(I*e+I*f*x,(b/a)^(1/2))*( 
(a+b*sinh(f*x+e)^2)/a)^(1/2)/f/(a+b*sinh(f*x+e)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.96 \[ \int \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {-4 i \sqrt {2} a (2 a-b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+2 i \sqrt {2} a (a-b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )+b (2 a-b+b \cosh (2 (e+f x))) \sinh (2 (e+f x))}{6 f \sqrt {4 a-2 b+2 b \cosh (2 (e+f x))}} \] Input:

Integrate[(a + b*Sinh[e + f*x]^2)^(3/2),x]
 

Output:

((-4*I)*Sqrt[2]*a*(2*a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*Ellipt 
icE[I*(e + f*x), b/a] + (2*I)*Sqrt[2]*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*( 
e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + b*(2*a - b + b*Cosh[2*(e + f*x 
)])*Sinh[2*(e + f*x)])/(6*f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {3042, 3659, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3659

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3651

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3657

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3656

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

\(\Big \downarrow \) 3662

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3661

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

Input:

Int[(a + b*Sinh[e + f*x]^2)^(3/2),x]
 

Output:

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

Defintions of rubi rules used

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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(427\) vs. \(2(159)=318\).

Time = 3.86 (sec) , antiderivative size = 428, normalized size of antiderivative = 2.43

method	result	size
default	\(\frac {\sqrt {-\frac {b}{a}}\, b^{2} \cosh \left (f x +e \right )^{4} \sinh \left (f x +e \right )+\sqrt {-\frac {b}{a}}\, a b \cosh \left (f x +e \right )^{2} \sinh \left (f x +e \right )-\sqrt {-\frac {b}{a}}\, b^{2} \cosh \left (f x +e \right )^{2} \sinh \left (f x +e \right )+3 a^{2} \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-5 a \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b +2 \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}+4 \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b -2 \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}}{3 \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\)	\(428\)

Input:

int((a+b*sinh(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/3*((-b/a)^(1/2)*b^2*cosh(f*x+e)^4*sinh(f*x+e)+(-b/a)^(1/2)*a*b*cosh(f*x+ 
e)^2*sinh(f*x+e)-(-b/a)^(1/2)*b^2*cosh(f*x+e)^2*sinh(f*x+e)+3*a^2*(b/a*cos 
h(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))-5*a*(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+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^2+4*(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*b-2*(b/a*cos 
h(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^2)/(-b/a)^(1/2)/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^ 
(1/2)/f
 

Fricas [F]

\[ \int \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \,d x } \] Input:

integrate((a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")
 

Output:

integral((b*sinh(f*x + e)^2 + a)^(3/2), x)
 

Sympy [F]

\[ \int \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int \left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \] Input:

integrate((a+b*sinh(f*x+e)**2)**(3/2),x)
 

Output:

Integral((a + b*sinh(e + f*x)**2)**(3/2), x)
 

Maxima [F]

\[ \int \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \,d x } \] Input:

integrate((a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")
 

Output:

integrate((b*sinh(f*x + e)^2 + a)^(3/2), x)
 

Giac [F]

\[ \int \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \,d x } \] Input:

integrate((a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")
 

Output:

integrate((b*sinh(f*x + e)^2 + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int {\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \] Input:

int((a + b*sinh(e + f*x)^2)^(3/2),x)
 

Output:

int((a + b*sinh(e + f*x)^2)^(3/2), x)
 

Reduce [F]

\[ \int \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\left (\int \sqrt {\sinh \left (f x +e \right )^{2} b +a}d x \right ) a +\left (\int \sqrt {\sinh \left (f x +e \right )^{2} b +a}\, \sinh \left (f x +e \right )^{2}d x \right ) b \] Input:

int((a+b*sinh(f*x+e)^2)^(3/2),x)
 

Output:

int(sqrt(sinh(e + f*x)**2*b + a),x)*a + int(sqrt(sinh(e + f*x)**2*b + a)*s 
inh(e + f*x)**2,x)*b