3.768 \(\int (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x))^{5/2} \, dx\)
Optimal. Leaf size=140 \[ \frac {2}{5} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}+\frac {16}{15} \sqrt {b^2-c^2} (b \sinh (x)+c \cosh (x)) \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}+\frac {64 \left (b^2-c^2\right ) (b \sinh (x)+c \cosh (x))}{15 \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \]
[Out]
2/5*(c*cosh(x)+b*sinh(x))*(b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^(3/2)+64/15*(b^2-c^2)*(c*cosh(x)+b*sinh(x))/(b
*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^(1/2)+16/15*(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)
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Rubi [A] time = 0.12, antiderivative size = 140, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used =
{3113, 3112} \[ \frac {2}{5} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}+\frac {16}{15} \sqrt {b^2-c^2} (b \sinh (x)+c \cosh (x)) \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}+\frac {64 \left (b^2-c^2\right ) (b \sinh (x)+c \cosh (x))}{15 \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \]
Antiderivative was successfully verified.
[In]
Int[(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(5/2),x]
[Out]
(64*(b^2 - c^2)*(c*Cosh[x] + b*Sinh[x]))/(15*Sqrt[Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]]) + (16*Sqrt[b^2 - c
^2]*(c*Cosh[x] + b*Sinh[x])*Sqrt[Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]])/15 + (2*(c*Cosh[x] + b*Sinh[x])*(Sq
rt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2))/5
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 )^{5/2} \, dx &=\frac {2}{5} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}+\frac {1}{5} \left (8 \sqrt {b^2-c^2}\right ) \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx\\ &=\frac {16}{15} \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x)) \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}+\frac {2}{5} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}+\frac {1}{15} \left (32 \left (b^2-c^2\right )\right ) \int \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx\\ &=\frac {64 \left (b^2-c^2\right ) (c \cosh (x)+b \sinh (x))}{15 \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}}+\frac {16}{15} \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x)) \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}+\frac {2}{5} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}\\ \end {align*}
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Mathematica [C] time = 75.52, size = 10223, normalized size = 73.02 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(5/2),x]
[Out]
Result too large to show
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fricas [B] time = 0.44, size = 784, normalized size = 5.60 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^(5/2),x, algorithm="fricas")
[Out]
1/30*sqrt(1/2)*(3*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^6 + 18*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)*sinh(
x)^5 + 3*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*sinh(x)^6 + 125*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^4 + 5*(25*b^3 + 2
5*b^2*c - 25*b*c^2 - 25*c^3 + 9*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^2)*sinh(x)^4 + 20*(3*(b^3 + 3*b^2*c +
3*b*c^2 + c^3)*cosh(x)^3 + 25*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x))*sinh(x)^3 + 3*b^3 - 9*b^2*c + 9*b*c^2 - 3*c
^3 + 125*(b^3 - b^2*c - b*c^2 + c^3)*cosh(x)^2 + 5*(9*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^4 + 25*b^3 - 25*
b^2*c - 25*b*c^2 + 25*c^3 + 150*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^2)*sinh(x)^2 + 2*(9*(b^3 + 3*b^2*c + 3*b*c
^2 + c^3)*cosh(x)^5 + 250*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^3 + 125*(b^3 - b^2*c - b*c^2 + c^3)*cosh(x))*sin
h(x) + 2*(11*(b^2 + 2*b*c + c^2)*cosh(x)^5 + 55*(b^2 + 2*b*c + c^2)*cosh(x)*sinh(x)^4 + 11*(b^2 + 2*b*c + c^2)
*sinh(x)^5 - 150*(b^2 - c^2)*cosh(x)^3 + 10*(11*(b^2 + 2*b*c + c^2)*cosh(x)^2 - 15*b^2 + 15*c^2)*sinh(x)^3 + 1
0*(11*(b^2 + 2*b*c + c^2)*cosh(x)^3 - 45*(b^2 - c^2)*cosh(x))*sinh(x)^2 + 11*(b^2 - 2*b*c + c^2)*cosh(x) + (55
*(b^2 + 2*b*c + c^2)*cosh(x)^4 - 450*(b^2 - c^2)*cosh(x)^2 + 11*b^2 - 22*b*c + 11*c^2)*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)^4 + 4*(b + c)*cosh(x)*sinh(x)^3 + (b + c)*sinh(x)^4 - (b -
c)*cosh(x)^2 + (6*(b + c)*cosh(x)^2 - b + c)*sinh(x)^2 + 2*(2*(b + c)*cosh(x)^3 - (b - c)*cosh(x))*sinh(x))
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giac [B] time = 0.33, size = 657, normalized size = 4.69 \[ -\frac {\sqrt {2} {\left (3 \, {\left (\sqrt {b^{2} - c^{2}} b^{2} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + 2 \, \sqrt {b^{2} - c^{2}} b c \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + \sqrt {b^{2} - c^{2}} c^{2} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right )\right )} e^{\left (\frac {5}{2} \, x\right )} + 25 \, {\left (b^{3} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + b^{2} c \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) - b c^{2} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) - c^{3} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right )\right )} e^{\left (\frac {3}{2} \, x\right )} + 150 \, {\left (\sqrt {b^{2} - c^{2}} b^{2} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) - \sqrt {b^{2} - c^{2}} c^{2} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right )\right )} e^{\left (\frac {1}{2} \, x\right )} - 150 \, {\left (b^{3} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) - b^{2} c \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) - b c^{2} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + c^{3} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right )\right )} e^{\left (-\frac {1}{2} \, x\right )} - 25 \, {\left (\sqrt {b^{2} - c^{2}} b^{2} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) - 2 \, \sqrt {b^{2} - c^{2}} b c \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + \sqrt {b^{2} - c^{2}} c^{2} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right )\right )} e^{\left (-\frac {3}{2} \, x\right )} - 3 \, {\left (b^{3} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) - 3 \, b^{2} c \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + 3 \, b c^{2} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) - c^{3} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right )\right )} e^{\left (-\frac {5}{2} \, x\right )}\right )}}{60 \, \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))^(5/2),x, algorithm="giac")
[Out]
-1/60*sqrt(2)*(3*(sqrt(b^2 - c^2)*b^2*sgn(-sqrt(b^2 - c^2)*e^x - b + c) + 2*sqrt(b^2 - c^2)*b*c*sgn(-sqrt(b^2
- c^2)*e^x - b + c) + sqrt(b^2 - c^2)*c^2*sgn(-sqrt(b^2 - c^2)*e^x - b + c))*e^(5/2*x) + 25*(b^3*sgn(-sqrt(b^2
- c^2)*e^x - b + c) + b^2*c*sgn(-sqrt(b^2 - c^2)*e^x - b + c) - b*c^2*sgn(-sqrt(b^2 - c^2)*e^x - b + c) - c^3
*sgn(-sqrt(b^2 - c^2)*e^x - b + c))*e^(3/2*x) + 150*(sqrt(b^2 - c^2)*b^2*sgn(-sqrt(b^2 - c^2)*e^x - b + c) - s
qrt(b^2 - c^2)*c^2*sgn(-sqrt(b^2 - c^2)*e^x - b + c))*e^(1/2*x) - 150*(b^3*sgn(-sqrt(b^2 - c^2)*e^x - b + c) -
b^2*c*sgn(-sqrt(b^2 - c^2)*e^x - b + c) - b*c^2*sgn(-sqrt(b^2 - c^2)*e^x - b + c) + c^3*sgn(-sqrt(b^2 - c^2)*
e^x - b + c))*e^(-1/2*x) - 25*(sqrt(b^2 - c^2)*b^2*sgn(-sqrt(b^2 - c^2)*e^x - b + c) - 2*sqrt(b^2 - c^2)*b*c*s
gn(-sqrt(b^2 - c^2)*e^x - b + c) + sqrt(b^2 - c^2)*c^2*sgn(-sqrt(b^2 - c^2)*e^x - b + c))*e^(-3/2*x) - 3*(b^3*
sgn(-sqrt(b^2 - c^2)*e^x - b + c) - 3*b^2*c*sgn(-sqrt(b^2 - c^2)*e^x - b + c) + 3*b*c^2*sgn(-sqrt(b^2 - c^2)*e
^x - b + c) - c^3*sgn(-sqrt(b^2 - c^2)*e^x - b + c))*e^(-5/2*x))/sqrt(b - c)
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maple [B] time = 1.13, size = 518, normalized size = 3.70 \[ \frac {-\frac {\left (b^{2}-c^{2}\right )^{\frac {3}{2}} \left (\cosh ^{3}\relax (x )\right )}{3}-\frac {\left (-2 b^{2}+2 c^{2}\right ) \left (-b^{2}+c^{2}\right ) \cosh \relax (x )}{\sqrt {b^{2}-c^{2}}}}{\sqrt {-\frac {\sinh \relax (x ) b^{2}-\sinh \relax (x ) c^{2}-b^{2}+c^{2}}{\sqrt {b^{2}-c^{2}}}}}-\frac {\left (\cosh \relax (x ) \sqrt {\sinh \relax (x ) \sqrt {b^{2}-c^{2}}-\sqrt {b^{2}-c^{2}}}\, \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{3}\relax (x )\right )+\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{2}\relax (x )\right )}\, \left (b^{2}-c^{2}\right )-\sinh \relax (x ) \left (b^{2}-c^{2}\right )^{\frac {3}{2}} \arctan \left (\frac {\sqrt {\sinh \relax (x ) \sqrt {b^{2}-c^{2}}-\sqrt {b^{2}-c^{2}}}\, \cosh \relax (x )}{\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{3}\relax (x )\right )+\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{2}\relax (x )\right )}}\right )+\sqrt {b^{2}-c^{2}}\, \arctan \left (\frac {\sqrt {\sinh \relax (x ) \sqrt {b^{2}-c^{2}}-\sqrt {b^{2}-c^{2}}}\, \cosh \relax (x )}{\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{3}\relax (x )\right )+\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{2}\relax (x )\right )}}\right ) b^{2}-\sqrt {b^{2}-c^{2}}\, \arctan \left (\frac {\sqrt {\sinh \relax (x ) \sqrt {b^{2}-c^{2}}-\sqrt {b^{2}-c^{2}}}\, \cosh \relax (x )}{\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{3}\relax (x )\right )+\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{2}\relax (x )\right )}}\right ) c^{2}\right ) \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )-1\right ) \left (\sinh ^{2}\relax (x )\right )}}{2 \sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )-1\right )}\, \left (\sinh \relax (x )-1\right ) \sinh \relax (x ) \sqrt {-\frac {\sinh \relax (x ) b^{2}-\sinh \relax (x ) c^{2}-b^{2}+c^{2}}{\sqrt {b^{2}-c^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^(5/2),x)
[Out]
1/(-(sinh(x)*b^2-sinh(x)*c^2-b^2+c^2)/(b^2-c^2)^(1/2))^(1/2)*(-1/3*(b^2-c^2)^(3/2)*cosh(x)^3-(-2*b^2+2*c^2)*(-
b^2+c^2)/(b^2-c^2)^(1/2)*cosh(x))-1/2*(cosh(x)*(sinh(x)*(b^2-c^2)^(1/2)-(b^2-c^2)^(1/2))^(1/2)*(-(b^2-c^2)^(1/
2)*sinh(x)^3+(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2)*(b^2-c^2)-sinh(x)*(b^2-c^2)^(3/2)*arctan((sinh(x)*(b^2-c^2)^(1/2
)-(b^2-c^2)^(1/2))^(1/2)*cosh(x)/(-(b^2-c^2)^(1/2)*sinh(x)^3+(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2))+(b^2-c^2)^(1/2)
*arctan((sinh(x)*(b^2-c^2)^(1/2)-(b^2-c^2)^(1/2))^(1/2)*cosh(x)/(-(b^2-c^2)^(1/2)*sinh(x)^3+(b^2-c^2)^(1/2)*si
nh(x)^2)^(1/2))*b^2-(b^2-c^2)^(1/2)*arctan((sinh(x)*(b^2-c^2)^(1/2)-(b^2-c^2)^(1/2))^(1/2)*cosh(x)/(-(b^2-c^2)
^(1/2)*sinh(x)^3+(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2))*c^2)*(-(b^2-c^2)^(1/2)*(sinh(x)-1)*sinh(x)^2)^(1/2)/((b^2-c
^2)^(1/2)*(sinh(x)-1))^(1/2)/(sinh(x)-1)/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 = 4.42, size = 1783, normalized size = 12.74 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^(5/2),x, algorithm="maxima")
[Out]
1/20*sqrt(2)*(sqrt(b + c)*sqrt(b - c)*b^2 + 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b + c)*sqrt(b - c)*c^2)*(2*sq
rt(b + c)*sqrt(b - c)*e^(-x) + (b - c)*e^(-2*x) + b + c)^(5/2)*e^(5/2*x)/(sqrt(b + c)*sqrt(b - c)*b^2 + 2*sqrt
(b + c)*sqrt(b - c)*b*c + sqrt(b + c)*sqrt(b - c)*c^2 + 5*(b^3 + b^2*c - b*c^2 - c^3)*e^(-x) + 10*(sqrt(b + c)
*sqrt(b - c)*b^2 - sqrt(b + c)*sqrt(b - c)*c^2)*e^(-2*x) + 10*(b^3 - b^2*c - b*c^2 + c^3)*e^(-3*x) + 5*(sqrt(b
+ c)*sqrt(b - c)*b^2 - 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b + c)*sqrt(b - c)*c^2)*e^(-4*x) + (b^3 - 3*b^2*c
+ 3*b*c^2 - c^3)*e^(-5*x)) + 5/12*sqrt(2)*(b^3 + b^2*c - b*c^2 - c^3)*(2*sqrt(b + c)*sqrt(b - c)*e^(-x) + (b
- c)*e^(-2*x) + b + c)^(5/2)*e^(3/2*x)/(sqrt(b + c)*sqrt(b - c)*b^2 + 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b +
c)*sqrt(b - c)*c^2 + 5*(b^3 + b^2*c - b*c^2 - c^3)*e^(-x) + 10*(sqrt(b + c)*sqrt(b - c)*b^2 - sqrt(b + c)*sqr
t(b - c)*c^2)*e^(-2*x) + 10*(b^3 - b^2*c - b*c^2 + c^3)*e^(-3*x) + 5*(sqrt(b + c)*sqrt(b - c)*b^2 - 2*sqrt(b +
c)*sqrt(b - c)*b*c + sqrt(b + c)*sqrt(b - c)*c^2)*e^(-4*x) + (b^3 - 3*b^2*c + 3*b*c^2 - c^3)*e^(-5*x)) + 5/2*
sqrt(2)*(sqrt(b + c)*sqrt(b - c)*b^2 - sqrt(b + c)*sqrt(b - c)*c^2)*(2*sqrt(b + c)*sqrt(b - c)*e^(-x) + (b - c
)*e^(-2*x) + b + c)^(5/2)*e^(1/2*x)/(sqrt(b + c)*sqrt(b - c)*b^2 + 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b + c)
*sqrt(b - c)*c^2 + 5*(b^3 + b^2*c - b*c^2 - c^3)*e^(-x) + 10*(sqrt(b + c)*sqrt(b - c)*b^2 - sqrt(b + c)*sqrt(b
- c)*c^2)*e^(-2*x) + 10*(b^3 - b^2*c - b*c^2 + c^3)*e^(-3*x) + 5*(sqrt(b + c)*sqrt(b - c)*b^2 - 2*sqrt(b + c)
*sqrt(b - c)*b*c + sqrt(b + c)*sqrt(b - c)*c^2)*e^(-4*x) + (b^3 - 3*b^2*c + 3*b*c^2 - c^3)*e^(-5*x)) - 5/2*sqr
t(2)*(b^3 - b^2*c - b*c^2 + c^3)*(2*sqrt(b + c)*sqrt(b - c)*e^(-x) + (b - c)*e^(-2*x) + b + c)^(5/2)*e^(-1/2*x
)/(sqrt(b + c)*sqrt(b - c)*b^2 + 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b + c)*sqrt(b - c)*c^2 + 5*(b^3 + b^2*c
- b*c^2 - c^3)*e^(-x) + 10*(sqrt(b + c)*sqrt(b - c)*b^2 - sqrt(b + c)*sqrt(b - c)*c^2)*e^(-2*x) + 10*(b^3 - b^
2*c - b*c^2 + c^3)*e^(-3*x) + 5*(sqrt(b + c)*sqrt(b - c)*b^2 - 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b + c)*sqr
t(b - c)*c^2)*e^(-4*x) + (b^3 - 3*b^2*c + 3*b*c^2 - c^3)*e^(-5*x)) - 5/12*sqrt(2)*(sqrt(b + c)*sqrt(b - c)*b^2
- 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b + c)*sqrt(b - c)*c^2)*(2*sqrt(b + c)*sqrt(b - c)*e^(-x) + (b - c)*e^
(-2*x) + b + c)^(5/2)*e^(-3/2*x)/(sqrt(b + c)*sqrt(b - c)*b^2 + 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b + c)*sq
rt(b - c)*c^2 + 5*(b^3 + b^2*c - b*c^2 - c^3)*e^(-x) + 10*(sqrt(b + c)*sqrt(b - c)*b^2 - sqrt(b + c)*sqrt(b -
c)*c^2)*e^(-2*x) + 10*(b^3 - b^2*c - b*c^2 + c^3)*e^(-3*x) + 5*(sqrt(b + c)*sqrt(b - c)*b^2 - 2*sqrt(b + c)*sq
rt(b - c)*b*c + sqrt(b + c)*sqrt(b - c)*c^2)*e^(-4*x) + (b^3 - 3*b^2*c + 3*b*c^2 - c^3)*e^(-5*x)) - 1/20*sqrt(
2)*(b^3 - 3*b^2*c + 3*b*c^2 - c^3)*(2*sqrt(b + c)*sqrt(b - c)*e^(-x) + (b - c)*e^(-2*x) + b + c)^(5/2)*e^(-5/2
*x)/(sqrt(b + c)*sqrt(b - c)*b^2 + 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b + c)*sqrt(b - c)*c^2 + 5*(b^3 + b^2*
c - b*c^2 - c^3)*e^(-x) + 10*(sqrt(b + c)*sqrt(b - c)*b^2 - sqrt(b + c)*sqrt(b - c)*c^2)*e^(-2*x) + 10*(b^3 -
b^2*c - b*c^2 + c^3)*e^(-3*x) + 5*(sqrt(b + c)*sqrt(b - c)*b^2 - 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b + c)*s
qrt(b - c)*c^2)*e^(-4*x) + (b^3 - 3*b^2*c + 3*b*c^2 - c^3)*e^(-5*x))
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (b\,\mathrm {cosh}\relax (x)+\sqrt {b^2-c^2}+c\,\mathrm {sinh}\relax (x)\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*cosh(x) + (b^2 - c^2)^(1/2) + c*sinh(x))^(5/2),x)
[Out]
int((b*cosh(x) + (b^2 - c^2)^(1/2) + c*sinh(x))^(5/2), x)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*cosh(x)+c*sinh(x)+(b**2-c**2)**(1/2))**(5/2),x)
[Out]
Timed out
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