3.8.69 \(\int (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x))^{3/2} \, dx\) [769]

Optimal. Leaf size=92 \[ \frac {8 \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x))}{3 \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}}+\frac {2}{3} (c \cosh (x)+b \sinh (x)) \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3192, 3191} \begin {gather*} \frac {2}{3} \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} (b \sinh (x)+c \cosh (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)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(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

Rule 3191

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] && E
qQ[a^2 - b^2 - c^2, 0]

Rule 3192

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] + Dist[a*((2*n - 1)/n), In
t[(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]

Rubi steps

\begin {align*} \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx &=\frac {2}{3} (c \cosh (x)+b \sinh (x)) \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}+\frac {1}{3} \left (4 \sqrt {b^2-c^2}\right ) \int \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, 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)}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 68.57, size = 4392, normalized size = 47.74 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(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*Co
sh[x])/3 + (2*b*Sinh[x])/3)*Sqrt[Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]] + (32*b*(-b + c)*(b + c)^2*(Elliptic
F[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, 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])*Sqrt[Sqrt[(b - c)*(b + c)] + b*Cosh[x] + c*Sinh[x]]*(-1 + Tanh[x/2])*Sqrt[-(((-b - c
+ Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]*(-c + (-b + Sqrt[b^2 - c^
2])*Tanh[x/2]))/(3*(b + c - Sqrt[b^2 - c^2])^2*(b + c + Sqrt[b^2 - c^2])*(1 + Cosh[x])*Sqrt[(Sqrt[(b - c)*(b +
c)] + b*Cosh[x] + c*Sinh[x])/(1 + Cosh[x])^2]*Sqrt[(-1 + Tanh[x/2]^2)*(-2*c*Tanh[x/2] + Sqrt[b^2 - c^2]*(-1 +
Tanh[x/2]^2) - b*(1 + Tanh[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*b^2*c^3*Tanh
[x/2]^2 - 2*b*c^4*Tanh[x/2]^2 + c^5*Tanh[x/2]^2 - 8*b^4*Sqrt[b^2 - c^2]*Tanh[x/2]^2 - 12*b^3*c*Sqrt[b^2 - c^2]
*Tanh[x/2]^2 + 5*b*c^3*Sqrt[b^2 - c^2]*Tanh[x/2]^2 + c^4*Sqrt[b^2 - c^2]*Tanh[x/2]^2 - 8*b^4*c*EllipticPi[-1,
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
]*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] + 8*b^2*
c^3*EllipticPi[-1, 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]*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tan
h[x/2])))] + 8*b^3*c*Sqrt[b^2 - c^2]*EllipticPi[-1, 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]*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b
+ c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] - 4*b*c^3*Sqrt[b^2 - c^2]*EllipticPi[-1, 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]*Sqrt[-(((-b - c + Sqrt[
b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] - 16*b^5*EllipticPi[-1, ArcSin[Sq
rt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x/
2]*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] + 8*b^4
*c*EllipticPi[-1, 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]*Tanh[x/2]*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*
(-1 + Tanh[x/2])))] + 20*b^3*c^2*EllipticPi[-1, 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]*Tanh[x/2]*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2])
)/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] - 8*b^2*c^3*EllipticPi[-1, 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]*Tanh[x/2]*Sqrt[-(((-b - c + Sqr
t[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] - 4*b*c^4*EllipticPi[-1, 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]*Tanh
[x/2]*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] + 16
*b^4*Sqrt[b^2 - c^2]*EllipticPi[-1, 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]*Tanh[x/2]*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c +
Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] - 8*b^3*c*Sqrt[b^2 - c^2]*EllipticPi[-1, 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]*Tanh[x/2]*Sqrt[-(((-b - c + S
qrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] - 12*b^2*c^2*Sqrt[b^2 - c^2]*
EllipticPi[-1, ArcSin[Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Ta
nh[x/2])))]], 1]*Tanh[x/2]*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1
+ Tanh[x/2])))] + 4*b*c^3*Sqrt[b^2 - c^2]*EllipticPi[-1, 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]*Tanh[x/2]*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 +
Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] + 16*b^5*EllipticPi[-1, ArcSin[Sqrt[-(((-b - c + S
qrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqr...

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(189\) vs. \(2(78)=156\).
time = 3.80, size = 190, normalized size = 2.07

method	result	size
default	\(\frac {\cosh \left (x \right ) \left (-2 b^{2}+2 c^{2}\right )}{\sqrt {-\frac {\sinh \left (x \right ) b^{2}-\sinh \left (x \right ) c^{2}-b^{2}+c^{2}}{\sqrt {b^{2}-c^{2}}}}}+\frac {\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )-1\right ) \left (\sinh ^{2}\left (x \right )\right )}\, \arctan \left (\frac {\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )-1\right )}\, \cosh \left (x \right )}{\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )-1\right ) \left (\sinh ^{2}\left (x \right )\right )}}\right ) \left (b^{2}-c^{2}\right )}{\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )-1\right )}\, \sinh \left (x \right ) \sqrt {-\frac {\sinh \left (x \right ) b^{2}-\sinh \left (x \right ) c^{2}-b^{2}+c^{2}}{\sqrt {b^{2}-c^{2}}}}}\)	\(190\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 640 vs. \(2 (78) = 156\).
time = 0.65, size = 640, normalized size = 6.96 \begin {gather*} \frac {\sqrt {2} {\left (\sqrt {b + c} \sqrt {b - c} b + \sqrt {b + c} \sqrt {b - c} c\right )} {\left (2 \, \sqrt {b + c} \sqrt {b - c} e^{\left (-x\right )} + {\left (b - c\right )} e^{\left (-2 \, x\right )} + b + c\right )}^{\frac {3}{2}} e^{\left (\frac {3}{2} \, x\right )}}{6 \, {\left (\sqrt {b + c} \sqrt {b - c} b + \sqrt {b + c} \sqrt {b - c} c + 3 \, {\left (b^{2} - c^{2}\right )} e^{\left (-x\right )} + 3 \, {\left (\sqrt {b + c} \sqrt {b - c} b - \sqrt {b + c} \sqrt {b - c} c\right )} e^{\left (-2 \, x\right )} + {\left (b^{2} - 2 \, b c + c^{2}\right )} e^{\left (-3 \, x\right )}\right )}} + \frac {3 \, \sqrt {2} {\left (b^{2} - c^{2}\right )} {\left (2 \, \sqrt {b + c} \sqrt {b - c} e^{\left (-x\right )} + {\left (b - c\right )} e^{\left (-2 \, x\right )} + b + c\right )}^{\frac {3}{2}} e^{\left (\frac {1}{2} \, x\right )}}{2 \, {\left (\sqrt {b + c} \sqrt {b - c} b + \sqrt {b + c} \sqrt {b - c} c + 3 \, {\left (b^{2} - c^{2}\right )} e^{\left (-x\right )} + 3 \, {\left (\sqrt {b + c} \sqrt {b - c} b - \sqrt {b + c} \sqrt {b - c} c\right )} e^{\left (-2 \, x\right )} + {\left (b^{2} - 2 \, b c + c^{2}\right )} e^{\left (-3 \, x\right )}\right )}} - \frac {3 \, \sqrt {2} {\left (\sqrt {b + c} \sqrt {b - c} b - \sqrt {b + c} \sqrt {b - c} c\right )} {\left (2 \, \sqrt {b + c} \sqrt {b - c} e^{\left (-x\right )} + {\left (b - c\right )} e^{\left (-2 \, x\right )} + b + c\right )}^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, x\right )}}{2 \, {\left (\sqrt {b + c} \sqrt {b - c} b + \sqrt {b + c} \sqrt {b - c} c + 3 \, {\left (b^{2} - c^{2}\right )} e^{\left (-x\right )} + 3 \, {\left (\sqrt {b + c} \sqrt {b - c} b - \sqrt {b + c} \sqrt {b - c} c\right )} e^{\left (-2 \, x\right )} + {\left (b^{2} - 2 \, b c + c^{2}\right )} e^{\left (-3 \, x\right )}\right )}} - \frac {\sqrt {2} {\left (b^{2} - 2 \, b c + c^{2}\right )} {\left (2 \, \sqrt {b + c} \sqrt {b - c} e^{\left (-x\right )} + {\left (b - c\right )} e^{\left (-2 \, x\right )} + b + c\right )}^{\frac {3}{2}} e^{\left (-\frac {3}{2} \, x\right )}}{6 \, {\left (\sqrt {b + c} \sqrt {b - c} b + \sqrt {b + c} \sqrt {b - c} c + 3 \, {\left (b^{2} - c^{2}\right )} e^{\left (-x\right )} + 3 \, {\left (\sqrt {b + c} \sqrt {b - c} b - \sqrt {b + c} \sqrt {b - c} c\right )} e^{\left (-2 \, x\right )} + {\left (b^{2} - 2 \, b c + c^{2}\right )} e^{\left (-3 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (78) = 156\).
time = 0.45, size = 329, normalized size = 3.58 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)*s
inh(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) + si
nh(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))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \cosh {\left (x \right )} + c \sinh {\left (x \right )} + \sqrt {b^{2} - c^{2}}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (78) = 156\).
time = 0.43, size = 183, normalized size = 1.99 \begin {gather*} -\frac {\sqrt {2} {\left ({\left (\sqrt {b^{2} - c^{2}} b + \sqrt {b^{2} - c^{2}} c\right )} e^{\left (\frac {3}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + 9 \, {\left (b^{2} - c^{2}\right )} e^{\left (\frac {1}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) - {\left (9 \, \sqrt {b^{2} - c^{2}} {\left (b - c\right )} e^{x} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + {\left (b^{2} - 2 \, b c + c^{2}\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right )\right )} e^{\left (-\frac {3}{2} \, x\right )}\right )}}{6 \, \sqrt {b - c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(2)*((sqrt(b^2 - c^2)*b + sqrt(b^2 - c^2)*c)*e^(3/2*x)*sgn(-sqrt(b^2 - c^2)*e^x - b + c) + 9*(b^2 - c
^2)*e^(1/2*x)*sgn(-sqrt(b^2 - c^2)*e^x - b + c) - (9*sqrt(b^2 - c^2)*(b - c)*e^x*sgn(-sqrt(b^2 - c^2)*e^x - b
+ c) + (b^2 - 2*b*c + c^2)*sgn(-sqrt(b^2 - c^2)*e^x - b + c))*e^(-3/2*x))/sqrt(b - c)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (b\,\mathrm {cosh}\left (x\right )+\sqrt {b^2-c^2}+c\,\mathrm {sinh}\left (x\right )\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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