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]

-8/3*(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 = 0.08, antiderivative size = 96, normalized size of antiderivative = 1.00, 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 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]

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]

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]  time = 73.74, 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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fricas [B]  time = 0.44, size = 329, normalized size = 3.43 \[ \frac {\sqrt {\frac {1}{2}} {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x)^{4} + 4 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x) \sinh \relax (x)^{3} + {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \relax (x)^{4} - 18 \, {\left (b^{2} - c^{2}\right )} \cosh \relax (x)^{2} + 6 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x)^{2} - 3 \, b^{2} + 3 \, c^{2}\right )} \sinh \relax (x)^{2} + b^{2} - 2 \, b c + c^{2} + 4 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x)^{3} - 9 \, {\left (b^{2} - c^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x) - 8 \, {\left ({\left (b + c\right )} \cosh \relax (x)^{3} + 3 \, {\left (b + c\right )} \cosh \relax (x) \sinh \relax (x)^{2} + {\left (b + c\right )} \sinh \relax (x)^{3} + {\left (b - c\right )} \cosh \relax (x) + {\left (3 \, {\left (b + c\right )} \cosh \relax (x)^{2} + b - c\right )} \sinh \relax (x)\right )} \sqrt {b^{2} - c^{2}}\right )} \sqrt {\frac {{\left (b + c\right )} \cosh \relax (x)^{2} + 2 \, {\left (b + c\right )} \cosh \relax (x) \sinh \relax (x) + {\left (b + c\right )} \sinh \relax (x)^{2} - 2 \, \sqrt {b^{2} - c^{2}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + b - c}{\cosh \relax (x) + \sinh \relax (x)}}}{3 \, {\left ({\left (b + c\right )} \cosh \relax (x)^{3} + 3 \, {\left (b + c\right )} \cosh \relax (x) \sinh \relax (x)^{2} + {\left (b + c\right )} \sinh \relax (x)^{3} - {\left (b - c\right )} \cosh \relax (x) + {\left (3 \, {\left (b + c\right )} \cosh \relax (x)^{2} - b + c\right )} \sinh \relax (x)\right )}} \]

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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giac [B]  time = 0.20, size = 302, normalized size = 3.15 \[ -\frac {\sqrt {2} {\left ({\left (\sqrt {b^{2} - c^{2}} b \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right ) + \sqrt {b^{2} - c^{2}} c \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right )\right )} e^{\left (\frac {3}{2} \, x\right )} - 9 \, {\left (b^{2} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right ) - c^{2} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right )\right )} e^{\left (\frac {1}{2} \, x\right )} - 9 \, {\left (\sqrt {b^{2} - c^{2}} b \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right ) - \sqrt {b^{2} - c^{2}} c \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right )\right )} e^{\left (-\frac {1}{2} \, x\right )} + {\left (b^{2} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right ) - 2 \, b c \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right ) + c^{2} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right )\right )} e^{\left (-\frac {3}{2} \, x\right )}\right )}}{6 \, \sqrt {b - c}} \]

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)

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maple [B]  time = 0.81, size = 190, normalized size = 1.98 \[ \frac {\left (2 b^{2}-2 c^{2}\right ) \cosh \relax (x )}{\sqrt {-\frac {\sinh \relax (x ) b^{2}-\sinh \relax (x ) c^{2}+b^{2}-c^{2}}{\sqrt {b^{2}-c^{2}}}}}+\frac {\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )+1\right ) \left (\sinh ^{2}\relax (x )\right )}\, \arctan \left (\frac {\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )+1\right )}\, \cosh \relax (x )}{\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )+1\right ) \left (\sinh ^{2}\relax (x )\right )}}\right ) \left (b^{2}-c^{2}\right )}{\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )+1\right )}\, \sinh \relax (x ) \sqrt {-\frac {\sinh \relax (x ) b^{2}-\sinh \relax (x ) c^{2}+b^{2}-c^{2}}{\sqrt {b^{2}-c^{2}}}}} \]

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 = 0.98, size = 644, normalized size = 6.71 \[ \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 )}} \]

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

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

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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