Integrand size = 26, antiderivative size = 92 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx=\frac {8 \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x))}{3 \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}}+\frac {2}{3} (c \cosh (x)+b \sinh (x)) \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \]
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)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 35.93 (sec) , antiderivative size = 4392, normalized size of antiderivative = 47.74 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx=\text {Result too large to show} \]
(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*Cosh[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*(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] - 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])*Ta nh[x/2]))/(3*(b + c - Sqrt[b^2 - c^2])^2*(b + c + Sqrt[b^2 - c^2])*(1 + Co sh[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*Sq rt[b^2 - c^2] - 3*b*c^3*Sqrt[b^2 - c^2] - c^4*Sqrt[b^2 - c^2] + 8*b^4*c*Ta nh[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...
Time = 0.37 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {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.
| \(\displaystyle \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (\sqrt {b^2-c^2}+b \cos (i x)-i c \sin (i x)\right )^{3/2}dx\) |
| \(\Big \downarrow \) 3592 |
| \(\displaystyle \frac {4}{3} \sqrt {b^2-c^2} \int \sqrt {b \cosh (x)+c \sinh (x)+\sqrt {b^2-c^2}}dx+\frac {2}{3} \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} (b \sinh (x)+c \cosh (x))\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} (b \sinh (x)+c \cosh (x))+\frac {4}{3} \sqrt {b^2-c^2} \int \sqrt {b \cos (i x)-i c \sin (i x)+\sqrt {b^2-c^2}}dx\) |
| \(\Big \downarrow \) 3591 |
| \(\displaystyle \frac {2}{3} \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} (b \sinh (x)+c \cosh (x))+\frac {8 \sqrt {b^2-c^2} (b \sinh (x)+c \cosh (x))}{3 \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}}\) |
(8*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x]))/(3*Sqrt[Sqrt[b^2 - c^2] + b*Co sh[x] + c*Sinh[x]]) + (2*(c*Cosh[x] + b*Sinh[x])*Sqrt[Sqrt[b^2 - c^2] + b* Cosh[x] + c*Sinh[x]])/3
3.8.69.3.1 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. \(188\) vs. \(2(78)=156\).
Time = 0.21 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.05
| method | result | size |
| default | \(\frac {2 \left (-b^{2}+c^{2}\right ) \cosh \left (x \right )}{\sqrt {-\frac {\sinh \left (x \right ) b^{2}-\sinh \left (x \right ) c^{2}-b^{2}+c^{2}}{\sqrt {b^{2}-c^{2}}}}}+\frac {\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )-1\right ) \sinh \left (x \right )^{2}}\, \left (b^{2}-c^{2}\right ) \arctan \left (\frac {\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )-1\right )}\, \cosh \left (x \right )}{\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )-1\right ) \sinh \left (x \right )^{2}}}\right )}{\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )-1\right )}\, \sinh \left (x \right ) \sqrt {-\frac {\sinh \left (x \right ) b^{2}-\sinh \left (x \right ) c^{2}-b^{2}+c^{2}}{\sqrt {b^{2}-c^{2}}}}}\) | \(189\) |
2*(-b^2+c^2)/(-(sinh(x)*b^2-sinh(x)*c^2-b^2+c^2)/(b^2-c^2)^(1/2))^(1/2)*co sh(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)*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-s inh(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. 329 vs. \(2 (78) = 156\).
Time = 0.30 (sec) , antiderivative size = 329, normalized size of antiderivative = 3.58 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx=\frac {\sqrt {\frac {1}{2}} {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \left (x\right )^{4} - 18 \, {\left (b^{2} - c^{2}\right )} \cosh \left (x\right )^{2} + 6 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{2} - 3 \, b^{2} + 3 \, c^{2}\right )} \sinh \left (x\right )^{2} + b^{2} - 2 \, b c + c^{2} + 4 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{3} - 9 \, {\left (b^{2} - c^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 8 \, {\left ({\left (b + c\right )} \cosh \left (x\right )^{3} + 3 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (b + c\right )} \sinh \left (x\right )^{3} + {\left (b - c\right )} \cosh \left (x\right ) + {\left (3 \, {\left (b + c\right )} \cosh \left (x\right )^{2} + b - c\right )} \sinh \left (x\right )\right )} \sqrt {b^{2} - c^{2}}\right )} \sqrt {\frac {{\left (b + c\right )} \cosh \left (x\right )^{2} + 2 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (b + c\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {b^{2} - c^{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + b - c}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{3 \, {\left ({\left (b + c\right )} \cosh \left (x\right )^{3} + 3 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (b + c\right )} \sinh \left (x\right )^{3} - {\left (b - c\right )} \cosh \left (x\right ) + {\left (3 \, {\left (b + c\right )} \cosh \left (x\right )^{2} - b + c\right )} \sinh \left (x\right )\right )}} \]
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)*sinh(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) + sinh(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))
\[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx=\int \left (b \cosh {\left (x \right )} + c \sinh {\left (x \right )} + \sqrt {b^{2} - c^{2}}\right )^{\frac {3}{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 640 vs. \(2 (78) = 156\).
Time = 0.53 (sec) , antiderivative size = 640, normalized size of antiderivative = 6.96 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx=\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 )}} \]
1/6*sqrt(2)*(sqrt(b + c)*sqrt(b - c)*b + sqrt(b + c)*sqrt(b - c)*c)*(2*sqr t(b + c)*sqrt(b - c)*e^(-x) + (b - c)*e^(-2*x) + b + c)^(3/2)*e^(3/2*x)/(s qrt(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)*sq rt(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)*sqr t(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))
Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (78) = 156\).
Time = 0.28 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.99 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx=-\frac {\sqrt {2} {\left ({\left (\sqrt {b^{2} - c^{2}} b + \sqrt {b^{2} - c^{2}} c\right )} e^{\left (\frac {3}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + 9 \, {\left (b^{2} - c^{2}\right )} e^{\left (\frac {1}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) - {\left (9 \, \sqrt {b^{2} - c^{2}} {\left (b - c\right )} e^{x} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + {\left (b^{2} - 2 \, b c + c^{2}\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right )\right )} e^{\left (-\frac {3}{2} \, x\right )}\right )}}{6 \, \sqrt {b - c}} \]
-1/6*sqrt(2)*((sqrt(b^2 - c^2)*b + sqrt(b^2 - c^2)*c)*e^(3/2*x)*sgn(-sqrt( b^2 - c^2)*e^x - b + c) + 9*(b^2 - c^2)*e^(1/2*x)*sgn(-sqrt(b^2 - c^2)*e^x - b + c) - (9*sqrt(b^2 - c^2)*(b - c)*e^x*sgn(-sqrt(b^2 - c^2)*e^x - b + c) + (b^2 - 2*b*c + c^2)*sgn(-sqrt(b^2 - c^2)*e^x - b + c))*e^(-3/2*x))/sq rt(b - c)
Timed out. \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx=\int {\left (b\,\mathrm {cosh}\left (x\right )+\sqrt {b^2-c^2}+c\,\mathrm {sinh}\left (x\right )\right )}^{3/2} \,d x \]