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

Optimal. Leaf size=96 \[ \frac{2}{3} (b \sinh (x)+c \cosh (x)) \sqrt{-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)}-\frac{8 \sqrt{b^2-c^2} (b \sinh (x)+c \cosh (x))}{3 \sqrt{-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)}} \]

[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 = 0.0787381, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3113, 3112} \[ \frac{2}{3} (b \sinh (x)+c \cosh (x)) \sqrt{-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)}-\frac{8 \sqrt{b^2-c^2} (b \sinh (x)+c \cosh (x))}{3 \sqrt{-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)}} \]

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 = 73.3534, size = 9861, normalized size = 102.72 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 0.436, size = 190, normalized size = 2. \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 = 3.21121, size = 869, normalized size = 9.05 \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)*s
qrt(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 = 2.53898, size = 967, normalized size = 10.07 \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 = 1.28982, size = 408, normalized size = 4.25 \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)