Integrand size = 28, antiderivative size = 96 \[ \int \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx=-\frac {8 \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x))}{3 \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}}+\frac {2}{3} (c \cosh (x)+b \sinh (x)) \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \]
-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 = 31.28 (sec) , antiderivative size = 4260, normalized size of antiderivative = 44.38 \[ \int \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx=\text {Result too large to show} \]
(-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[-S qrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]] - (32*b*c*(-b + c)*(b + c)*(-b^2 + 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, Ar cSin[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])*(-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2 ]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2]))))^(3/2)*(c + (b + Sqrt[b^ 2 - c^2])*Tanh[x/2])*(-1 + Tanh[x/2]^2))/(3*(b + c + Sqrt[b^2 - c^2])^3*(- b^2 + c^2 + b*Sqrt[b^2 - c^2])*(1 + Cosh[x])*Sqrt[(-Sqrt[(b - c)*(b + c)] + b*Cosh[x] + c*Sinh[x])/(1 + Cosh[x])^2]*(1 + Tanh[x/2])^2*Sqrt[-((-1 + T anh[x/2]^2)*(2*c*Tanh[x/2] + Sqrt[b^2 - c^2]*(-1 + Tanh[x/2]^2) + b*(1 + T anh[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*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] - 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...
Time = 0.37 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, 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 |
| \(\displaystyle \frac {2}{3} (b \sinh (x)+c \cosh (x)) \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}-\frac {4}{3} \sqrt {b^2-c^2} \int \sqrt {b \cosh (x)+c \sinh (x)-\sqrt {b^2-c^2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} (b \sinh (x)+c \cosh (x)) \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}-\frac {4}{3} \sqrt {b^2-c^2} \int \sqrt {b \cos (i x)-i c \sin (i x)-\sqrt {b^2-c^2}}dx\) |
| \(\Big \downarrow \) 3591 |
| \(\displaystyle \frac {2}{3} (b \sinh (x)+c \cosh (x)) \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}-\frac {8 \sqrt {b^2-c^2} (b \sinh (x)+c \cosh (x))}{3 \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}}\) |
(-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
3.8.75.3.1 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. \(2(82)=164\).
Time = 0.22 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.97
| 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)*cos h(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-si nh(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. \(2 (82) = 164\).
Time = 0.29 (sec) , antiderivative size = 329, normalized size of antiderivative = 3.43 \[ \int \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx=\frac {\sqrt {\frac {1}{2}} {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \left (x\right )^{4} - 18 \, {\left (b^{2} - c^{2}\right )} \cosh \left (x\right )^{2} + 6 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{2} - 3 \, b^{2} + 3 \, c^{2}\right )} \sinh \left (x\right )^{2} + b^{2} - 2 \, b c + c^{2} + 4 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{3} - 9 \, {\left (b^{2} - c^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) - 8 \, {\left ({\left (b + c\right )} \cosh \left (x\right )^{3} + 3 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (b + c\right )} \sinh \left (x\right )^{3} + {\left (b - c\right )} \cosh \left (x\right ) + {\left (3 \, {\left (b + c\right )} \cosh \left (x\right )^{2} + b - c\right )} \sinh \left (x\right )\right )} \sqrt {b^{2} - c^{2}}\right )} \sqrt {\frac {{\left (b + c\right )} \cosh \left (x\right )^{2} + 2 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (b + c\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {b^{2} - c^{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + b - c}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{3 \, {\left ({\left (b + c\right )} \cosh \left (x\right )^{3} + 3 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (b + c\right )} \sinh \left (x\right )^{3} - {\left (b - c\right )} \cosh \left (x\right ) + {\left (3 \, {\left (b + c\right )} \cosh \left (x\right )^{2} - b + c\right )} \sinh \left (x\right )\right )}} \]
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. 644 vs. \(2 (82) = 164\).
Time = 0.47 (sec) , antiderivative size = 644, normalized size of antiderivative = 6.71 \[ \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*sq rt(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)*sq rt(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)*s qrt(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. 184 vs. \(2 (82) = 164\).
Time = 0.28 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.92 \[ \int \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx=-\frac {\sqrt {2} {\left ({\left (\sqrt {b^{2} - c^{2}} b + \sqrt {b^{2} - c^{2}} c\right )} e^{\left (\frac {3}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right ) - 9 \, {\left (b^{2} - c^{2}\right )} e^{\left (\frac {1}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right ) - {\left (9 \, \sqrt {b^{2} - c^{2}} {\left (b - c\right )} e^{x} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right ) - {\left (b^{2} - 2 \, b c + c^{2}\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right )\right )} e^{\left (-\frac {3}{2} \, x\right )}\right )}}{6 \, \sqrt {b - c}} \]
-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. \[ \int \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx=\int {\left (b\,\mathrm {cosh}\left (x\right )-\sqrt {b^2-c^2}+c\,\mathrm {sinh}\left (x\right )\right )}^{3/2} \,d x \]