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

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

[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

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Rubi [A]  time = 0.0774329, antiderivative size = 92, normalized size of antiderivative = 1., 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 = {3113, 3112} \[ \frac{2}{3} \sqrt{\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)} (b \sinh (x)+c \cosh (x))+\frac{8 \sqrt{b^2-c^2} (b \sinh (x)+c \cosh (x))}{3 \sqrt{\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)}} \]

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 3113

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, 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]

Rule 3112

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]

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*}

Mathematica [C]  time = 70.9472, size = 4392, normalized size = 47.74 \[ \text{Result too large to show} \]

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 + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x/2]^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*EllipticP
i[-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]^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^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]^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]^2*Sqrt[-(((-b - c +
Sqrt[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 +
Tanh[x/2])))]], 1]*Tanh[x/2]^2*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])
*(-1 + Tanh[x/2])))] + 2*c*EllipticE[ArcSin[Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqr
t[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*(-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^4*Tanh[x/2] + c^3*(Sqrt[b^2 - c^2] + c*Tanh[x/2]) - b^2*c*(
2*Sqrt[b^2 - c^2] + 5*c*Tanh[x/2]) + b^3*(2*c - 4*Sqrt[b^2 - c^2]*Tanh[x/2]) + b*c^2*(-2*c + 3*Sqrt[b^2 - c^2]
*Tanh[x/2])) + 2*b*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]*(-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^4*Tanh[x/2] - c^3*(Sqrt[b^2 - c^2] + c*Tanh[x/2]) + b^2*c*(2*Sqrt[
b^2 - c^2] + 5*c*Tanh[x/2]) + b*c^2*(2*c - 3*Sqrt[b^2 - c^2]*Tanh[x/2]) + b^3*(-2*c + 4*Sqrt[b^2 - c^2]*Tanh[x
/2]))))/(3*c*(b + c - Sqrt[b^2 - c^2])^2*(-b + Sqrt[b^2 - c^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] + Sqr
t[b^2 - c^2]*(-1 + Tanh[x/2]^2) - b*(1 + Tanh[x/2]^2))])

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Maple [B]  time = 0.513, size = 190, normalized size = 2.1 \begin{align*}{ \left ( -2\,{b}^{2}+2\,{c}^{2} \right ) \cosh \left ( x \right ){\frac{1}{\sqrt{-{(\sinh \left ( x \right ){b}^{2}-\sinh \left ( x \right ){c}^{2}-{b}^{2}+{c}^{2}){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}}}}}}+{\frac{{b}^{2}-{c}^{2}}{\sinh \left ( x \right ) }\sqrt{-\sqrt{{b}^{2}-{c}^{2}} \left ( \sinh \left ( x \right ) -1 \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}}\arctan \left ({\cosh \left ( x \right ) \sqrt{\sqrt{{b}^{2}-{c}^{2}} \left ( \sinh \left ( x \right ) -1 \right ) }{\frac{1}{\sqrt{-\sqrt{{b}^{2}-{c}^{2}} \left ( \sinh \left ( x \right ) -1 \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}}}}} \right ){\frac{1}{\sqrt{\sqrt{{b}^{2}-{c}^{2}} \left ( \sinh \left ( x \right ) -1 \right ) }}}{\frac{1}{\sqrt{-{(\sinh \left ( x \right ){b}^{2}-\sinh \left ( x \right ){c}^{2}-{b}^{2}+{c}^{2}){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}}}}}} \end{align*}

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)

[Out]

(-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)*cosh(x)+(-(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]  time = 3.21733, size = 864, normalized size = 9.39 \begin{align*} \text{result too large to display} \end{align*}

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]  time = 2.42817, size = 967, normalized size = 10.51 \begin{align*} \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{align*}

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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [B]  time = 1.26444, size = 409, normalized size = 4.45 \begin{align*} -\frac{\sqrt{2}{\left ({\left (\sqrt{b^{2} - c^{2}} b \mathrm{sgn}\left (-\sqrt{b^{2} - c^{2}} e^{x} - b + c\right ) + \sqrt{b^{2} - c^{2}} c \mathrm{sgn}\left (-\sqrt{b^{2} - c^{2}} e^{x} - b + c\right )\right )} e^{\left (\frac{3}{2} \, x\right )} + 9 \,{\left (b^{2} \mathrm{sgn}\left (-\sqrt{b^{2} - c^{2}} e^{x} - b + c\right ) - c^{2} \mathrm{sgn}\left (-\sqrt{b^{2} - c^{2}} e^{x} - b + c\right )\right )} e^{\left (\frac{1}{2} \, x\right )} - 9 \,{\left (\sqrt{b^{2} - c^{2}} b \mathrm{sgn}\left (-\sqrt{b^{2} - c^{2}} e^{x} - b + c\right ) - \sqrt{b^{2} - c^{2}} c \mathrm{sgn}\left (-\sqrt{b^{2} - c^{2}} e^{x} - b + c\right )\right )} e^{\left (-\frac{1}{2} \, x\right )} -{\left (b^{2} \mathrm{sgn}\left (-\sqrt{b^{2} - c^{2}} e^{x} - b + c\right ) - 2 \, b c \mathrm{sgn}\left (-\sqrt{b^{2} - c^{2}} e^{x} - b + c\right ) + c^{2} \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{align*}

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