Integrand size = 26, antiderivative size = 140 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2} \, dx=\frac {64 \left (b^2-c^2\right ) (c \cosh (x)+b \sinh (x))}{15 \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}}+\frac {16}{15} \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x)) \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}+\frac {2}{5} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \]
2/5*(c*cosh(x)+b*sinh(x))*(b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^(3/2)+64/1 5*(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)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 56.31 (sec) , antiderivative size = 4500, normalized size of antiderivative = 32.14 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2} \, dx=\text {Result too large to show} \]
Sqrt[b^2 - c^2]*((4*b*Sqrt[b^2 - c^2])/(3*c) + (4*c*Cosh[x])/3 + (4*b*Sinh [x])/3)*Sqrt[Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]] + Sqrt[Sqrt[b^2 - c^ 2] + b*Cosh[x] + c*Sinh[x]]*((44*b*(b^2 - c^2))/(15*c) + (2*c*Sqrt[b^2 - c ^2]*Cosh[x])/15 + (2*b*c*Cosh[2*x])/5 + (2*b*Sqrt[b^2 - c^2]*Sinh[x])/15 + ((b^2 + c^2)*Sinh[2*x])/5) + (256*b*(-b + c)*(b + c)^2*Sqrt[b^2 - c^2]*(E llipticF[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[S qrt[-(((-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]))/(15*(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))]) + (128*(b - c)^2*(b + c)^2*Sqrt[S qrt[(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] + ...
Time = 0.52 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3042, 3592, 3042, 3592, 3042, 3591}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (\sqrt {b^2-c^2}+b \cos (i x)-i c \sin (i x)\right )^{5/2}dx\) |
\(\Big \downarrow \) 3592 |
\(\displaystyle \frac {8}{5} \sqrt {b^2-c^2} \int \left (b \cosh (x)+c \sinh (x)+\sqrt {b^2-c^2}\right )^{3/2}dx+\frac {2}{5} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{5} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}+\frac {8}{5} \sqrt {b^2-c^2} \int \left (b \cos (i x)-i c \sin (i x)+\sqrt {b^2-c^2}\right )^{3/2}dx\) |
\(\Big \downarrow \) 3592 |
\(\displaystyle \frac {8}{5} \sqrt {b^2-c^2} \left (\frac {4}{3} \sqrt {b^2-c^2} \int \sqrt {b \cosh (x)+c \sinh (x)+\sqrt {b^2-c^2}}dx+\frac {2}{3} \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} (b \sinh (x)+c \cosh (x))\right )+\frac {2}{5} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{5} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}+\frac {8}{5} \sqrt {b^2-c^2} \left (\frac {2}{3} \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} (b \sinh (x)+c \cosh (x))+\frac {4}{3} \sqrt {b^2-c^2} \int \sqrt {b \cos (i x)-i c \sin (i x)+\sqrt {b^2-c^2}}dx\right )\) |
\(\Big \downarrow \) 3591 |
\(\displaystyle \frac {2}{5} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}+\frac {8}{5} \sqrt {b^2-c^2} \left (\frac {2}{3} \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} (b \sinh (x)+c \cosh (x))+\frac {8 \sqrt {b^2-c^2} (b \sinh (x)+c \cosh (x))}{3 \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}}\right )\) |
(2*(c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2) )/5 + (8*Sqrt[b^2 - c^2]*((8*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x]))/(3*S qrt[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))/5
3.8.68.3.1 Defintions of rubi rules used
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] && EqQ[a^ 2 - b^2 - c^2, 0]
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] + Simp[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]
Leaf count of result is larger than twice the leaf count of optimal. \(288\) vs. \(2(120)=240\).
Time = 0.60 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.06
method | result | size |
default | \(-\frac {\left (b -c \right )^{2} \left (b +c \right )^{2} \left (\frac {\cosh \left (x \right )^{3}}{3}+2 \cosh \left (x \right )\right )}{\sqrt {-\frac {\sinh \left (x \right ) b^{2}-\sinh \left (x \right ) c^{2}-b^{2}+c^{2}}{\sqrt {b^{2}-c^{2}}}}\, \sqrt {\left (b -c \right ) \left (b +c \right )}}-\frac {\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )-1\right ) \sinh \left (x \right )^{2}}\, \sqrt {b^{2}-c^{2}}\, \left (-b^{2}+c^{2}\right ) \left (-\frac {\cosh \left (x \right ) \sqrt {b^{2}-c^{2}}\, \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )-1\right ) \sinh \left (x \right )^{2}}}{2 \left (\sinh \left (x \right ) b^{2}-\sinh \left (x \right ) c^{2}-b^{2}+c^{2}\right )}+\frac {\arctan \left (\frac {\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )-1\right )}\, \cosh \left (x \right )}{\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )-1\right ) \sinh \left (x \right )^{2}}}\right )}{2 \sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )-1\right )}}\right )}{\sinh \left (x \right ) \sqrt {-\frac {\sinh \left (x \right ) b^{2}-\sinh \left (x \right ) c^{2}-b^{2}+c^{2}}{\sqrt {b^{2}-c^{2}}}}}\) | \(289\) |
-1/(-(sinh(x)*b^2-sinh(x)*c^2-b^2+c^2)/(b^2-c^2)^(1/2))^(1/2)*(b-c)^2*(b+c )^2/((b-c)*(b+c))^(1/2)*(1/3*cosh(x)^3+2*cosh(x))-(-(b^2-c^2)^(1/2)*(sinh( x)-1)*sinh(x)^2)^(1/2)*(b^2-c^2)^(1/2)*(-b^2+c^2)*(-1/2*cosh(x)/(sinh(x)*b ^2-sinh(x)*c^2-b^2+c^2)*(b^2-c^2)^(1/2)*(-(b^2-c^2)^(1/2)*(sinh(x)-1)*sinh (x)^2)^(1/2)+1/2/((b^2-c^2)^(1/2)*(sinh(x)-1))^(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)))/sinh(x)/(-(sinh(x)*b^2-sinh(x)*c^2-b^2+c^2)/(b^2-c^2)^(1/2))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 784 vs. \(2 (120) = 240\).
Time = 0.32 (sec) , antiderivative size = 784, normalized size of antiderivative = 5.60 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2} \, dx=\text {Too large to display} \]
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 + 25* 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)*s inh(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))*sinh(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 + 10*(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)*s inh(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)*s...
Timed out. \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1783 vs. \(2 (120) = 240\).
Time = 2.15 (sec) , antiderivative size = 1783, normalized size of antiderivative = 12.74 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2} \, dx=\text {Too large to display} \]
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*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)*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)*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*sqrt(2)*(s qrt(b + c)*sqrt(b - c)*b^2 - sqrt(b + c)*sqrt(b - c)*c^2)*(2*sqrt(b + c)*s qrt(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 +...
Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (120) = 240\).
Time = 0.30 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.25 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2} \, dx=-\frac {\sqrt {2} {\left (150 \, {\left (b^{2} - c^{2}\right )}^{\frac {3}{2}} e^{\left (\frac {1}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + 3 \, {\left (\sqrt {b^{2} - c^{2}} b^{2} + 2 \, \sqrt {b^{2} - c^{2}} b c + \sqrt {b^{2} - c^{2}} c^{2}\right )} e^{\left (\frac {5}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + 25 \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} e^{\left (\frac {3}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) - {\left (25 \, {\left (b^{2} - 2 \, b c + c^{2}\right )} \sqrt {b^{2} - c^{2}} e^{x} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + 150 \, {\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + 3 \, {\left (b^{3} - 3 \, b^{2} c + 3 \, b c^{2} - c^{3}\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right )\right )} e^{\left (-\frac {5}{2} \, x\right )}\right )}}{60 \, \sqrt {b - c}} \]
-1/60*sqrt(2)*(150*(b^2 - c^2)^(3/2)*e^(1/2*x)*sgn(-sqrt(b^2 - c^2)*e^x - b + c) + 3*(sqrt(b^2 - c^2)*b^2 + 2*sqrt(b^2 - c^2)*b*c + sqrt(b^2 - c^2)* c^2)*e^(5/2*x)*sgn(-sqrt(b^2 - c^2)*e^x - b + c) + 25*(b^3 + b^2*c - b*c^2 - c^3)*e^(3/2*x)*sgn(-sqrt(b^2 - c^2)*e^x - b + c) - (25*(b^2 - 2*b*c + c ^2)*sqrt(b^2 - c^2)*e^x*sgn(-sqrt(b^2 - c^2)*e^x - b + c) + 150*(b^3 - b^2 *c - b*c^2 + c^3)*e^(2*x)*sgn(-sqrt(b^2 - c^2)*e^x - b + c) + 3*(b^3 - 3*b ^2*c + 3*b*c^2 - c^3)*sgn(-sqrt(b^2 - c^2)*e^x - b + c))*e^(-5/2*x))/sqrt( b - c)
Timed out. \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2} \, dx=\int {\left (b\,\mathrm {cosh}\left (x\right )+\sqrt {b^2-c^2}+c\,\mathrm {sinh}\left (x\right )\right )}^{5/2} \,d x \]