3.8.62 \(\int (a+b \cosh (x)+c \sinh (x))^{3/2} \, dx\) [762]

3.8.62.1 Optimal result

Integrand size = 14, antiderivative size = 249 \[ \int (a+b \cosh (x)+c \sinh (x))^{3/2} \, dx=\frac {2}{3} (c \cosh (x)+b \sinh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)}-\frac {8 i a E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}{3 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}+\frac {2 i \left (a^2-b^2+c^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right ),\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}{3 \sqrt {a+b \cosh (x)+c \sinh (x)}} \]

output

2/3*(c*cosh(x)+b*sinh(x))*(a+b*cosh(x)+c*sinh(x))^(1/2)-8/3*I*a*(cos(1/2*I 
*x-1/2*arctan(b,-I*c))^2)^(1/2)/cos(1/2*I*x-1/2*arctan(b,-I*c))*EllipticE( 
sin(1/2*I*x-1/2*arctan(b,-I*c)),2^(1/2)*((b^2-c^2)^(1/2)/(a+(b^2-c^2)^(1/2 
)))^(1/2))*(a+b*cosh(x)+c*sinh(x))^(1/2)/((a+b*cosh(x)+c*sinh(x))/(a+(b^2- 
c^2)^(1/2)))^(1/2)+2/3*I*(a^2-b^2+c^2)*(cos(1/2*I*x-1/2*arctan(b,-I*c))^2) 
^(1/2)/cos(1/2*I*x-1/2*arctan(b,-I*c))*EllipticF(sin(1/2*I*x-1/2*arctan(b, 
-I*c)),2^(1/2)*((b^2-c^2)^(1/2)/(a+(b^2-c^2)^(1/2)))^(1/2))*((a+b*cosh(x)+ 
c*sinh(x))/(a+(b^2-c^2)^(1/2)))^(1/2)/(a+b*cosh(x)+c*sinh(x))^(1/2)
 

3.8.62.2 Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 6.12 (sec) , antiderivative size = 2292, normalized size of antiderivative = 9.20 \[ \int (a+b \cosh (x)+c \sinh (x))^{3/2} \, dx=\text {Result too large to show} \]

input

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

output

((8*a*b)/(3*c) + (2*c*Cosh[x])/3 + (2*b*Sinh[x])/3)*Sqrt[a + b*Cosh[x] + c 
*Sinh[x]] + (2*a^2*AppellF1[1/2, 1/2, 1/2, 3/2, ((-I)*(a + Sqrt[1 - b^2/c^ 
2]*c*Sinh[x + ArcTanh[b/c]]))/(Sqrt[1 - b^2/c^2]*(1 - (I*a)/(Sqrt[1 - b^2/ 
c^2]*c))*c), ((-I)*(a + Sqrt[1 - b^2/c^2]*c*Sinh[x + ArcTanh[b/c]]))/(Sqrt 
[1 - b^2/c^2]*(-1 - (I*a)/(Sqrt[1 - b^2/c^2]*c))*c)]*Sech[x + ArcTanh[b/c] 
]*Sqrt[-1 + I*Sinh[x + ArcTanh[b/c]]]*Sqrt[(c*Sqrt[(-b^2 + c^2)/c^2] - I*c 
*Sqrt[(-b^2 + c^2)/c^2]*Sinh[x + ArcTanh[b/c]])/(I*a + c*Sqrt[(-b^2 + c^2) 
/c^2])]*Sqrt[(c*Sqrt[(-b^2 + c^2)/c^2] + I*c*Sqrt[(-b^2 + c^2)/c^2]*Sinh[x 
 + ArcTanh[b/c]])/((-I)*a + c*Sqrt[(-b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(-b 
^2 + c^2)/c^2]*Sinh[x + ArcTanh[b/c]]])/(Sqrt[1 - b^2/c^2]*c*Sqrt[I*(I + S 
inh[x + ArcTanh[b/c]])]) + (2*b^2*AppellF1[1/2, 1/2, 1/2, 3/2, ((-I)*(a + 
Sqrt[1 - b^2/c^2]*c*Sinh[x + ArcTanh[b/c]]))/(Sqrt[1 - b^2/c^2]*(1 - (I*a) 
/(Sqrt[1 - b^2/c^2]*c))*c), ((-I)*(a + Sqrt[1 - b^2/c^2]*c*Sinh[x + ArcTan 
h[b/c]]))/(Sqrt[1 - b^2/c^2]*(-1 - (I*a)/(Sqrt[1 - b^2/c^2]*c))*c)]*Sech[x 
 + ArcTanh[b/c]]*Sqrt[-1 + I*Sinh[x + ArcTanh[b/c]]]*Sqrt[(c*Sqrt[(-b^2 + 
c^2)/c^2] - I*c*Sqrt[(-b^2 + c^2)/c^2]*Sinh[x + ArcTanh[b/c]])/(I*a + c*Sq 
rt[(-b^2 + c^2)/c^2])]*Sqrt[(c*Sqrt[(-b^2 + c^2)/c^2] + I*c*Sqrt[(-b^2 + c 
^2)/c^2]*Sinh[x + ArcTanh[b/c]])/((-I)*a + c*Sqrt[(-b^2 + c^2)/c^2])]*Sqrt 
[a + c*Sqrt[(-b^2 + c^2)/c^2]*Sinh[x + ArcTanh[b/c]]])/(3*Sqrt[1 - b^2/c^2 
]*c*Sqrt[I*(I + Sinh[x + ArcTanh[b/c]])]) - (2*c*AppellF1[1/2, 1/2, 1/2...
 

3.8.62.3 Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 3599, 27, 3042, 3628, 3042, 3598, 3042, 3132, 3606, 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.

\(\displaystyle \int (a+b \cosh (x)+c \sinh (x))^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+b \cos (i x)-i c \sin (i x))^{3/2}dx\)

\(\Big \downarrow \) 3599

\(\displaystyle \frac {2}{3} \int \frac {3 a^2+4 b \cosh (x) a+4 c \sinh (x) a+b^2-c^2}{2 \sqrt {a+b \cosh (x)+c \sinh (x)}}dx+\frac {2}{3} (b \sinh (x)+c \cosh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3628

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3598

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{3} (b \sinh (x)+c \cosh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)}+\frac {1}{3} \left (\frac {4 a \sqrt {a+b \cosh (x)+c \sinh (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2-c^2}}+\frac {\sqrt {b^2-c^2} \sin \left (i x-\tan ^{-1}(b,-i c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2-c^2}}}dx}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}-\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {a+b \cos (i x)-i c \sin (i x)}}dx\right )\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {2}{3} (b \sinh (x)+c \cosh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)}+\frac {1}{3} \left (-\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {a+b \cos (i x)-i c \sin (i x)}}dx-\frac {8 i a \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}\right )\)

\(\Big \downarrow \) 3606

\(\displaystyle \frac {2}{3} (b \sinh (x)+c \cosh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)}+\frac {1}{3} \left (-\frac {\left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2-c^2}}+\frac {\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}{a+\sqrt {b^2-c^2}}}}dx}{\sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {8 i a \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{3} (b \sinh (x)+c \cosh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)}+\frac {1}{3} \left (-\frac {\left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2-c^2}}+\frac {\sqrt {b^2-c^2} \sin \left (i x-\tan ^{-1}(b,-i c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2-c^2}}}}dx}{\sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {8 i a \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}\right )\)

\(\Big \downarrow \) 3140

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

input

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

output

(2*(c*Cosh[x] + b*Sinh[x])*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/3 + (((-8*I)*a 
*EllipticE[(I*x - ArcTan[b, (-I)*c])/2, (2*Sqrt[b^2 - c^2])/(a + Sqrt[b^2 
- c^2])]*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/Sqrt[(a + b*Cosh[x] + c*Sinh[x]) 
/(a + Sqrt[b^2 - c^2])] + ((2*I)*(a^2 - b^2 + c^2)*EllipticF[(I*x - ArcTan 
[b, (-I)*c])/2, (2*Sqrt[b^2 - c^2])/(a + Sqrt[b^2 - c^2])]*Sqrt[(a + b*Cos 
h[x] + c*Sinh[x])/(a + Sqrt[b^2 - c^2])])/Sqrt[a + b*Cosh[x] + c*Sinh[x]]) 
/3
 

3.8.62.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

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

rule 3132

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]
 

rule 3140

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]
 

rule 3598

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_ 
)]], x_Symbol] :> Simp[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + 
b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]   Int[Sqrt[a/(a + S 
qrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - Arc 
Tan[b, c]]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] 
 && NeQ[b^2 + c^2, 0] &&  !GtQ[a + Sqrt[b^2 + c^2], 0]
 

rule 3599

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ 
(n_), x_Symbol] :> Simp[(-(c*Cos[d + e*x] - b*Sin[d + e*x]))*((a + b*Cos[d 
+ e*x] + c*Sin[d + e*x])^(n - 1)/(e*n)), x] + Simp[1/n   Int[Simp[n*a^2 + ( 
n - 1)*(b^2 + c^2) + a*b*(2*n - 1)*Cos[d + e*x] + a*c*(2*n - 1)*Sin[d + e*x 
], x]*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 2), x], x] /; FreeQ[{a, b, 
 c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && GtQ[n, 1]
 

rule 3606

Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*( 
x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sq 
rt[b^2 + c^2])]/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]   Int[1/Sqrt[a/(a 
 + Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - 
 ArcTan[b, c]]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2 
, 0] && NeQ[b^2 + c^2, 0] &&  !GtQ[a + Sqrt[b^2 + c^2], 0]
 

rule 3628

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]) 
/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]] 
, x_Symbol] :> Simp[B/b   Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x] 
, x] + Simp[(A*b - a*B)/b   Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]] 
, x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[B*c - b*C, 0] && NeQ[ 
A*b - a*B, 0]
 

3.8.62.4 Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.29

method	result	size
default	\(\frac {2 a \left (-b^{2}+c^{2}\right ) \cosh \left (x \right )}{\sqrt {b^{2}-c^{2}}\, \sqrt {\frac {-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}+\frac {\sqrt {\frac {\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \left (x \right )^{2}}{\sqrt {b^{2}-c^{2}}}}\, a^{2} \ln \left (\frac {-\sinh \left (x \right ) \cosh \left (x \right ) b^{2}+\sinh \left (x \right ) \cosh \left (x \right ) c^{2}+\cosh \left (x \right ) \sqrt {b^{2}-c^{2}}\, a +\sqrt {\frac {\left (-b^{2}+c^{2}\right ) \sinh \left (x \right )^{3}}{\sqrt {b^{2}-c^{2}}}+a \sinh \left (x \right )^{2}}\, \sqrt {b^{2}-c^{2}}\, \sqrt {\frac {\left (-b^{2}+c^{2}\right ) \sinh \left (x \right )}{\sqrt {b^{2}-c^{2}}}+a}}{\sqrt {b^{2}-c^{2}}\, \sqrt {\frac {-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}\right ) \sqrt {b^{2}-c^{2}}}{\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \left (x \right )}\)	\(321\)

input

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

output

2*a/(b^2-c^2)^(1/2)*(-b^2+c^2)/((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2 
))/(b^2-c^2)^(1/2))^(1/2)*cosh(x)+((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^( 
1/2))/(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2)*a^2*ln((-sinh(x)*cosh(x)*b^2+sinh(x 
)*cosh(x)*c^2+cosh(x)*(b^2-c^2)^(1/2)*a+((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x 
)^3+a*sinh(x)^2)^(1/2)*(b^2-c^2)^(1/2)*((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x) 
+a)^(1/2))/(b^2-c^2)^(1/2)/((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/( 
b^2-c^2)^(1/2))^(1/2))/(-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2))*(b^2-c 
^2)^(1/2)/sinh(x)
 

3.8.62.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.86 \[ \int (a+b \cosh (x)+c \sinh (x))^{3/2} \, dx=\frac {2 \, {\left (\sqrt {2} {\left (a^{2} + 3 \, b^{2} - 3 \, c^{2}\right )} \cosh \left (x\right ) + \sqrt {2} {\left (a^{2} + 3 \, b^{2} - 3 \, c^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b + c} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2} + 3 \, c^{2}\right )}}{3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2} + 9 \, a c^{2}\right )}}{27 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )}}, \frac {3 \, {\left (b + c\right )} \cosh \left (x\right ) + 3 \, {\left (b + c\right )} \sinh \left (x\right ) + 2 \, a}{3 \, {\left (b + c\right )}}\right ) - 24 \, {\left (\sqrt {2} {\left (a b + a c\right )} \cosh \left (x\right ) + \sqrt {2} {\left (a b + a c\right )} \sinh \left (x\right )\right )} \sqrt {b + c} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2} + 3 \, c^{2}\right )}}{3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2} + 9 \, a c^{2}\right )}}{27 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2} + 3 \, c^{2}\right )}}{3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2} + 9 \, a c^{2}\right )}}{27 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )}}, \frac {3 \, {\left (b + c\right )} \cosh \left (x\right ) + 3 \, {\left (b + c\right )} \sinh \left (x\right ) + 2 \, a}{3 \, {\left (b + c\right )}}\right )\right ) + 3 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{2} + {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \left (x\right )^{2} - b^{2} + c^{2} - 8 \, {\left (a b + a c\right )} \cosh \left (x\right ) - 2 \, {\left (4 \, a b + 4 \, a c - {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + c \sinh \left (x\right ) + a}}{9 \, {\left ({\left (b + c\right )} \cosh \left (x\right ) + {\left (b + c\right )} \sinh \left (x\right )\right )}} \]

input

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

output

1/9*(2*(sqrt(2)*(a^2 + 3*b^2 - 3*c^2)*cosh(x) + sqrt(2)*(a^2 + 3*b^2 - 3*c 
^2)*sinh(x))*sqrt(b + c)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2 + 3*c^2)/( 
b^2 + 2*b*c + c^2), -8/27*(8*a^3 - 9*a*b^2 + 9*a*c^2)/(b^3 + 3*b^2*c + 3*b 
*c^2 + c^3), 1/3*(3*(b + c)*cosh(x) + 3*(b + c)*sinh(x) + 2*a)/(b + c)) - 
24*(sqrt(2)*(a*b + a*c)*cosh(x) + sqrt(2)*(a*b + a*c)*sinh(x))*sqrt(b + c) 
*weierstrassZeta(4/3*(4*a^2 - 3*b^2 + 3*c^2)/(b^2 + 2*b*c + c^2), -8/27*(8 
*a^3 - 9*a*b^2 + 9*a*c^2)/(b^3 + 3*b^2*c + 3*b*c^2 + c^3), weierstrassPInv 
erse(4/3*(4*a^2 - 3*b^2 + 3*c^2)/(b^2 + 2*b*c + c^2), -8/27*(8*a^3 - 9*a*b 
^2 + 9*a*c^2)/(b^3 + 3*b^2*c + 3*b*c^2 + c^3), 1/3*(3*(b + c)*cosh(x) + 3* 
(b + c)*sinh(x) + 2*a)/(b + c))) + 3*((b^2 + 2*b*c + c^2)*cosh(x)^2 + (b^2 
 + 2*b*c + c^2)*sinh(x)^2 - b^2 + c^2 - 8*(a*b + a*c)*cosh(x) - 2*(4*a*b + 
 4*a*c - (b^2 + 2*b*c + c^2)*cosh(x))*sinh(x))*sqrt(b*cosh(x) + c*sinh(x) 
+ a))/((b + c)*cosh(x) + (b + c)*sinh(x))
 

3.8.62.6 Sympy [F]

\[ \int (a+b \cosh (x)+c \sinh (x))^{3/2} \, dx=\int \left (a + b \cosh {\left (x \right )} + c \sinh {\left (x \right )}\right )^{\frac {3}{2}}\, dx \]

input

integrate((a+b*cosh(x)+c*sinh(x))**(3/2),x)
 

output

Integral((a + b*cosh(x) + c*sinh(x))**(3/2), x)
 

3.8.62.7 Maxima [F]

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

input

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

output

integrate((b*cosh(x) + c*sinh(x) + a)^(3/2), x)
 

3.8.62.8 Giac [F]

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

input

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

output

integrate((b*cosh(x) + c*sinh(x) + a)^(3/2), x)
 

3.8.62.9 Mupad [F(-1)]

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

input

int((a + b*cosh(x) + c*sinh(x))^(3/2),x)
 

output

int((a + b*cosh(x) + c*sinh(x))^(3/2), x)