\(\int (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x))^{3/2} \, dx\) [535]

Optimal result

Integrand size = 28, antiderivative size = 96 \[ \int \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx=-\frac {8 \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x))}{3 \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}}+\frac {2}{3} (c \cosh (x)+b \sinh (x)) \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \] Output:

-8/3*(b^2-c^2)^(1/2)*(c*cosh(x)+b*sinh(x))/(-(b^2-c^2)^(1/2)+b*cosh(x)+c*s 
inh(x))^(1/2)+2/3*(c*cosh(x)+b*sinh(x))*(-(b^2-c^2)^(1/2)+b*cosh(x)+c*sinh 
(x))^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 31.23 (sec) , antiderivative size = 4260, normalized size of antiderivative = 44.38 \[ \int \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx=\text {Result too large to show} \] Input:

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

Output:

(-2*b*Sqrt[b^2 - c^2]*Sqrt[-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]])/c + 
((-2*b*Sqrt[b^2 - c^2])/(3*c) + (2*c*Cosh[x])/3 + (2*b*Sinh[x])/3)*Sqrt[-S 
qrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]] - (32*b*c*(-b + c)*(b + c)*(-b^2 + 
 c^2)*(EllipticF[ArcSin[Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2])) 
/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1] - 2*EllipticPi[-1, Ar 
cSin[Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 
 - c^2])*(-1 + Tanh[x/2])))]], 1])*Sqrt[-Sqrt[(b - c)*(b + c)] + b*Cosh[x] 
 + c*Sinh[x]]*(-1 + Tanh[x/2])*(-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2 
]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2]))))^(3/2)*(c + (b + Sqrt[b^ 
2 - c^2])*Tanh[x/2])*(-1 + Tanh[x/2]^2))/(3*(b + c + Sqrt[b^2 - c^2])^3*(- 
b^2 + c^2 + b*Sqrt[b^2 - c^2])*(1 + Cosh[x])*Sqrt[(-Sqrt[(b - c)*(b + c)] 
+ b*Cosh[x] + c*Sinh[x])/(1 + Cosh[x])^2]*(1 + Tanh[x/2])^2*Sqrt[-((-1 + T 
anh[x/2]^2)*(2*c*Tanh[x/2] + Sqrt[b^2 - c^2]*(-1 + Tanh[x/2]^2) + b*(1 + T 
anh[x/2]^2)))]) + (16*(b - c)*(b + c)*Sqrt[-Sqrt[(b - c)*(b + c)] + b*Cosh 
[x] + c*Sinh[x]]*(2*b^3*c^2 + 3*b^2*c^3 - c^5 + 2*b^2*c^2*Sqrt[b^2 - c^2] 
+ 3*b*c^3*Sqrt[b^2 - c^2] + c^4*Sqrt[b^2 - c^2] + 8*b^4*c*Tanh[x/2] + 12*b 
^3*c^2*Tanh[x/2] - 2*b^2*c^3*Tanh[x/2] - 8*b*c^4*Tanh[x/2] - 2*c^5*Tanh[x/ 
2] + 8*b^3*c*Sqrt[b^2 - c^2]*Tanh[x/2] + 12*b^2*c^2*Sqrt[b^2 - c^2]*Tanh[x 
/2] + 2*b*c^3*Sqrt[b^2 - c^2]*Tanh[x/2] - 2*c^4*Sqrt[b^2 - c^2]*Tanh[x/2] 
+ 8*b^5*Tanh[x/2]^2 + 12*b^4*c*Tanh[x/2]^2 - 4*b^3*c^2*Tanh[x/2]^2 - 11...
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3592, 3042, 3591}

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.

\(\displaystyle \int \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3592

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3591

\(\displaystyle \frac {2}{3} (b \sinh (x)+c \cosh (x)) \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}-\frac {8 \sqrt {b^2-c^2} (b \sinh (x)+c \cosh (x))}{3 \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}}\)

Input:

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

Output:

(-8*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x]))/(3*Sqrt[-Sqrt[b^2 - c^2] + b* 
Cosh[x] + c*Sinh[x]]) + (2*(c*Cosh[x] + b*Sinh[x])*Sqrt[-Sqrt[b^2 - c^2] + 
 b*Cosh[x] + c*Sinh[x]])/3
 

Defintions of rubi rules used

rule 3042

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

rule 3591

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

rule 3592

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[a*((2*n - 1)/n)   Int[(a 
 + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[a^2 - b^2 - c^2, 0] && GtQ[n, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(188\) vs. \(2(82)=164\).

Time = 0.20 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.97

method	result	size
default	\(\frac {2 \left (b^{2}-c^{2}\right ) \cosh \left (x \right )}{\sqrt {-\frac {b^{2} \sinh \left (x \right )-\sinh \left (x \right ) c^{2}+b^{2}-c^{2}}{\sqrt {b^{2}-c^{2}}}}}+\frac {\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right ) \sinh \left (x \right )^{2}}\, \left (b^{2}-c^{2}\right ) \arctan \left (\frac {\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right )}\, \cosh \left (x \right )}{\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right ) \sinh \left (x \right )^{2}}}\right )}{\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right )}\, \sinh \left (x \right ) \sqrt {-\frac {b^{2} \sinh \left (x \right )-\sinh \left (x \right ) c^{2}+b^{2}-c^{2}}{\sqrt {b^{2}-c^{2}}}}}\)	\(189\)

Input:

int((-(b^2-c^2)^(1/2)+b*cosh(x)+c*sinh(x))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

2*(b^2-c^2)/(-(b^2*sinh(x)-sinh(x)*c^2+b^2-c^2)/(b^2-c^2)^(1/2))^(1/2)*cos 
h(x)+(-(b^2-c^2)^(1/2)*(sinh(x)+1)*sinh(x)^2)^(1/2)*(b^2-c^2)/((b^2-c^2)^( 
1/2)*(sinh(x)+1))^(1/2)*arctan(((b^2-c^2)^(1/2)*(sinh(x)+1))^(1/2)*cosh(x) 
/(-(b^2-c^2)^(1/2)*(sinh(x)+1)*sinh(x)^2)^(1/2))/sinh(x)/(-(b^2*sinh(x)-si 
nh(x)*c^2+b^2-c^2)/(b^2-c^2)^(1/2))^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (82) = 164\).

Time = 0.12 (sec) , antiderivative size = 329, normalized size of antiderivative = 3.43 \[ \int \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx=\frac {\sqrt {\frac {1}{2}} {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \left (x\right )^{4} - 18 \, {\left (b^{2} - c^{2}\right )} \cosh \left (x\right )^{2} + 6 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{2} - 3 \, b^{2} + 3 \, c^{2}\right )} \sinh \left (x\right )^{2} + b^{2} - 2 \, b c + c^{2} + 4 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{3} - 9 \, {\left (b^{2} - c^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) - 8 \, {\left ({\left (b + c\right )} \cosh \left (x\right )^{3} + 3 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (b + c\right )} \sinh \left (x\right )^{3} + {\left (b - c\right )} \cosh \left (x\right ) + {\left (3 \, {\left (b + c\right )} \cosh \left (x\right )^{2} + b - c\right )} \sinh \left (x\right )\right )} \sqrt {b^{2} - c^{2}}\right )} \sqrt {\frac {{\left (b + c\right )} \cosh \left (x\right )^{2} + 2 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (b + c\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {b^{2} - c^{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + b - c}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{3 \, {\left ({\left (b + c\right )} \cosh \left (x\right )^{3} + 3 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (b + c\right )} \sinh \left (x\right )^{3} - {\left (b - c\right )} \cosh \left (x\right ) + {\left (3 \, {\left (b + c\right )} \cosh \left (x\right )^{2} - b + c\right )} \sinh \left (x\right )\right )}} \] Input:

integrate((-(b^2-c^2)^(1/2)+b*cosh(x)+c*sinh(x))^(3/2),x, algorithm="frica 
s")
 

Output:

1/3*sqrt(1/2)*((b^2 + 2*b*c + c^2)*cosh(x)^4 + 4*(b^2 + 2*b*c + c^2)*cosh( 
x)*sinh(x)^3 + (b^2 + 2*b*c + c^2)*sinh(x)^4 - 18*(b^2 - c^2)*cosh(x)^2 + 
6*((b^2 + 2*b*c + c^2)*cosh(x)^2 - 3*b^2 + 3*c^2)*sinh(x)^2 + b^2 - 2*b*c 
+ c^2 + 4*((b^2 + 2*b*c + c^2)*cosh(x)^3 - 9*(b^2 - c^2)*cosh(x))*sinh(x) 
- 8*((b + c)*cosh(x)^3 + 3*(b + c)*cosh(x)*sinh(x)^2 + (b + c)*sinh(x)^3 + 
 (b - c)*cosh(x) + (3*(b + c)*cosh(x)^2 + b - c)*sinh(x))*sqrt(b^2 - c^2)) 
*sqrt(((b + c)*cosh(x)^2 + 2*(b + c)*cosh(x)*sinh(x) + (b + c)*sinh(x)^2 - 
 2*sqrt(b^2 - c^2)*(cosh(x) + sinh(x)) + b - c)/(cosh(x) + sinh(x)))/((b + 
 c)*cosh(x)^3 + 3*(b + c)*cosh(x)*sinh(x)^2 + (b + c)*sinh(x)^3 - (b - c)* 
cosh(x) + (3*(b + c)*cosh(x)^2 - b + c)*sinh(x))
 

Sympy [F]

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

integrate((-(b**2-c**2)**(1/2)+b*cosh(x)+c*sinh(x))**(3/2),x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 644 vs. \(2 (82) = 164\).

Time = 0.32 (sec) , antiderivative size = 644, normalized size of antiderivative = 6.71 \[ \int \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx =\text {Too large to display} \] Input:

integrate((-(b^2-c^2)^(1/2)+b*cosh(x)+c*sinh(x))^(3/2),x, algorithm="maxim 
a")
 

Output:

1/6*sqrt(2)*(sqrt(b + c)*sqrt(b - c)*b + sqrt(b + c)*sqrt(b - c)*c)*(-2*sq 
rt(b + c)*sqrt(b - c)*e^(-x) + (b - c)*e^(-2*x) + b + c)^(3/2)*e^(3/2*x)/( 
sqrt(b + c)*sqrt(b - c)*b + sqrt(b + c)*sqrt(b - c)*c - 3*(b^2 - c^2)*e^(- 
x) + 3*(sqrt(b + c)*sqrt(b - c)*b - sqrt(b + c)*sqrt(b - c)*c)*e^(-2*x) - 
(b^2 - 2*b*c + c^2)*e^(-3*x)) - 3/2*sqrt(2)*(b^2 - c^2)*(-2*sqrt(b + c)*sq 
rt(b - c)*e^(-x) + (b - c)*e^(-2*x) + b + c)^(3/2)*e^(1/2*x)/(sqrt(b + c)* 
sqrt(b - c)*b + sqrt(b + c)*sqrt(b - c)*c - 3*(b^2 - c^2)*e^(-x) + 3*(sqrt 
(b + c)*sqrt(b - c)*b - sqrt(b + c)*sqrt(b - c)*c)*e^(-2*x) - (b^2 - 2*b*c 
 + c^2)*e^(-3*x)) - 3/2*sqrt(2)*(sqrt(b + c)*sqrt(b - c)*b - sqrt(b + c)*s 
qrt(b - c)*c)*(-2*sqrt(b + c)*sqrt(b - c)*e^(-x) + (b - c)*e^(-2*x) + b + 
c)^(3/2)*e^(-1/2*x)/(sqrt(b + c)*sqrt(b - c)*b + sqrt(b + c)*sqrt(b - c)*c 
 - 3*(b^2 - c^2)*e^(-x) + 3*(sqrt(b + c)*sqrt(b - c)*b - sqrt(b + c)*sqrt( 
b - c)*c)*e^(-2*x) - (b^2 - 2*b*c + c^2)*e^(-3*x)) + 1/6*sqrt(2)*(b^2 - 2* 
b*c + c^2)*(-2*sqrt(b + c)*sqrt(b - c)*e^(-x) + (b - c)*e^(-2*x) + b + c)^ 
(3/2)*e^(-3/2*x)/(sqrt(b + c)*sqrt(b - c)*b + sqrt(b + c)*sqrt(b - c)*c - 
3*(b^2 - c^2)*e^(-x) + 3*(sqrt(b + c)*sqrt(b - c)*b - sqrt(b + c)*sqrt(b - 
 c)*c)*e^(-2*x) - (b^2 - 2*b*c + c^2)*e^(-3*x))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (82) = 164\).

Time = 0.25 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.68 \[ \int \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx=\frac {3 \, {\left (\sqrt {2} b^{2} + 2 \, \sqrt {2} b c + \sqrt {2} c^{2}\right )} \sqrt {b^{2} - c^{2}} \sqrt {b e^{x} + c e^{x}} {\left | b + c \right |} - {\left (\sqrt {2} b^{3} + 3 \, \sqrt {2} b^{2} c + 3 \, \sqrt {2} b c^{2} + \sqrt {2} c^{3}\right )} {\left (b e^{x} + c e^{x}\right )}^{\frac {3}{2}} + 6 \, {\left (\sqrt {2} b^{3} + 3 \, \sqrt {2} b^{2} c + 3 \, \sqrt {2} b c^{2} + \sqrt {2} c^{3}\right )} \sqrt {b^{2} - c^{2}} \sqrt {b e^{x} + c e^{x}}}{6 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )}} + \frac {\sqrt {2} {\left (6 \, {\left (b e^{x} + c e^{x}\right )} {\left (b - c\right )} {\left | b + c \right |} - \sqrt {b^{2} - c^{2}} {\left (b - c\right )} {\left | b + c \right |} + 3 \, {\left (b^{2} - c^{2}\right )} {\left (b e^{x} + c e^{x}\right )}\right )}}{6 \, {\left (b e^{x} + c e^{x}\right )}^{\frac {3}{2}}} \] Input:

integrate((-(b^2-c^2)^(1/2)+b*cosh(x)+c*sinh(x))^(3/2),x, algorithm="giac" 
)
 

Output:

1/6*(3*(sqrt(2)*b^2 + 2*sqrt(2)*b*c + sqrt(2)*c^2)*sqrt(b^2 - c^2)*sqrt(b* 
e^x + c*e^x)*abs(b + c) - (sqrt(2)*b^3 + 3*sqrt(2)*b^2*c + 3*sqrt(2)*b*c^2 
 + sqrt(2)*c^3)*(b*e^x + c*e^x)^(3/2) + 6*(sqrt(2)*b^3 + 3*sqrt(2)*b^2*c + 
 3*sqrt(2)*b*c^2 + sqrt(2)*c^3)*sqrt(b^2 - c^2)*sqrt(b*e^x + c*e^x))/(b^3 
+ 3*b^2*c + 3*b*c^2 + c^3) + 1/6*sqrt(2)*(6*(b*e^x + c*e^x)*(b - c)*abs(b 
+ c) - sqrt(b^2 - c^2)*(b - c)*abs(b + c) + 3*(b^2 - c^2)*(b*e^x + c*e^x)) 
/(b*e^x + c*e^x)^(3/2)
 

Mupad [F(-1)]

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

int((b*cosh(x) - (b^2 - c^2)^(1/2) + c*sinh(x))^(3/2),x)
 

Output:

int((b*cosh(x) - (b^2 - c^2)^(1/2) + c*sinh(x))^(3/2), x)
 

Reduce [F]

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

int((-(b^2-c^2)^(1/2)+b*cosh(x)+c*sinh(x))^(3/2),x)
 

Output:

int((-(b^2-c^2)^(1/2)+b*cosh(x)+c*sinh(x))^(3/2),x)