Optimal. Leaf size=136 \[ \frac {5}{2} \left (b^2-c^2\right )^{3/2} x+\frac {5}{2} c \left (b^2-c^2\right ) \cosh (x)+\frac {5}{2} b \left (b^2-c^2\right ) \sinh (x)+\frac {5}{6} \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )+\frac {1}{3} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2 \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3192, 2717,
2718} \begin {gather*} \frac {5}{2} x \left (b^2-c^2\right )^{3/2}+\frac {5}{2} b \left (b^2-c^2\right ) \sinh (x)+\frac {5}{2} c \left (b^2-c^2\right ) \cosh (x)+\frac {1}{3} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac {5}{6} \sqrt {b^2-c^2} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2717
Rule 2718
Rule 3192
Rubi steps
\begin {align*} \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3 \, dx &=\frac {1}{3} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac {1}{3} \left (5 \sqrt {b^2-c^2}\right ) \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2 \, dx\\ &=\frac {5}{6} \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )+\frac {1}{3} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac {1}{2} \left (5 \left (b^2-c^2\right )\right ) \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right ) \, dx\\ &=\frac {5}{2} \left (b^2-c^2\right )^{3/2} x+\frac {5}{6} \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )+\frac {1}{3} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2+\frac {1}{2} \left (5 b \left (b^2-c^2\right )\right ) \int \cosh (x) \, dx+\frac {1}{2} \left (5 c \left (b^2-c^2\right )\right ) \int \sinh (x) \, dx\\ &=\frac {5}{2} \left (b^2-c^2\right )^{3/2} x+\frac {5}{2} c \left (b^2-c^2\right ) \cosh (x)+\frac {5}{2} b \left (b^2-c^2\right ) \sinh (x)+\frac {5}{6} \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )+\frac {1}{3} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.19, size = 134, normalized size = 0.99 \begin {gather*} \frac {1}{12} \left (30 (b-c) (b+c) \sqrt {b^2-c^2} x+45 c \left (b^2-c^2\right ) \cosh (x)+18 b c \sqrt {b^2-c^2} \cosh (2 x)+c \left (3 b^2+c^2\right ) \cosh (3 x)+45 b \left (b^2-c^2\right ) \sinh (x)+9 \sqrt {b^2-c^2} \left (b^2+c^2\right ) \sinh (2 x)+b \left (b^2+3 c^2\right ) \sinh (3 x)\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(277\) vs.
\(2(118)=236\).
time = 1.80, size = 278, normalized size = 2.04
method | result | size |
risch | \(\frac {5 \left (b^{2}-c^{2}\right )^{\frac {3}{2}} x}{2}+\frac {{\mathrm e}^{3 x} b^{3}}{24}+\frac {{\mathrm e}^{3 x} b^{2} c}{8}+\frac {{\mathrm e}^{3 x} b \,c^{2}}{8}+\frac {{\mathrm e}^{3 x} c^{3}}{24}+\frac {3 \,{\mathrm e}^{2 x} \sqrt {b^{2}-c^{2}}\, b^{2}}{8}+\frac {3 \,{\mathrm e}^{2 x} \sqrt {b^{2}-c^{2}}\, b c}{4}+\frac {3 \,{\mathrm e}^{2 x} \sqrt {b^{2}-c^{2}}\, c^{2}}{8}+\frac {15 b^{3} {\mathrm e}^{x}}{8}+\frac {15 \,{\mathrm e}^{x} b^{2} c}{8}-\frac {15 \,{\mathrm e}^{x} b \,c^{2}}{8}-\frac {15 \,{\mathrm e}^{x} c^{3}}{8}-\frac {15 \,{\mathrm e}^{-x} b^{3}}{8}+\frac {15 \,{\mathrm e}^{-x} b^{2} c}{8}+\frac {15 \,{\mathrm e}^{-x} b \,c^{2}}{8}-\frac {15 \,{\mathrm e}^{-x} c^{3}}{8}-\frac {3 \,{\mathrm e}^{-2 x} \sqrt {b^{2}-c^{2}}\, b^{2}}{8}+\frac {3 \,{\mathrm e}^{-2 x} \sqrt {b^{2}-c^{2}}\, b c}{4}-\frac {3 \,{\mathrm e}^{-2 x} \sqrt {b^{2}-c^{2}}\, c^{2}}{8}-\frac {{\mathrm e}^{-3 x} b^{3}}{24}+\frac {{\mathrm e}^{-3 x} b^{2} c}{8}-\frac {{\mathrm e}^{-3 x} b \,c^{2}}{8}+\frac {{\mathrm e}^{-3 x} c^{3}}{24}\) | \(278\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 161, normalized size = 1.18 \begin {gather*} b^{2} c \cosh \left (x\right )^{3} + b c^{2} \sinh \left (x\right )^{3} + \frac {1}{24} \, c^{3} {\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} + e^{\left (-3 \, x\right )} - 9 \, e^{x}\right )} + \frac {1}{24} \, b^{3} {\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} - e^{\left (-3 \, x\right )} + 9 \, e^{x}\right )} + {\left (b^{2} - c^{2}\right )}^{\frac {3}{2}} x + 3 \, {\left (b^{2} - c^{2}\right )} {\left (c \cosh \left (x\right ) + b \sinh \left (x\right )\right )} + \frac {3}{8} \, {\left (8 \, b c \cosh \left (x\right )^{2} + b^{2} {\left (4 \, x + e^{\left (2 \, x\right )} - e^{\left (-2 \, x\right )}\right )} - c^{2} {\left (4 \, x - e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}\right )} \sqrt {b^{2} - c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 664 vs.
\(2 (118) = 236\).
time = 0.40, size = 664, normalized size = 4.88 \begin {gather*} \frac {{\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{6} + 6 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{5} + {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \sinh \left (x\right )^{6} + 45 \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} \cosh \left (x\right )^{4} + 15 \, {\left (3 \, b^{3} + 3 \, b^{2} c - 3 \, b c^{2} - 3 \, c^{3} + {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{4} + 20 \, {\left ({\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{3} + 9 \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - b^{3} + 3 \, b^{2} c - 3 \, b c^{2} + c^{3} - 45 \, {\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} \cosh \left (x\right )^{2} + 15 \, {\left ({\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{4} - 3 \, b^{3} + 3 \, b^{2} c + 3 \, b c^{2} - 3 \, c^{3} + 18 \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 6 \, {\left ({\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{5} + 30 \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} \cosh \left (x\right )^{3} - 15 \, {\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 3 \, {\left (3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{5} + 15 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{4} + 3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \left (x\right )^{5} + 20 \, {\left (b^{2} - c^{2}\right )} x \cosh \left (x\right )^{3} + 10 \, {\left (3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (b^{2} - c^{2}\right )} x\right )} \sinh \left (x\right )^{3} + 30 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{3} + 2 \, {\left (b^{2} - c^{2}\right )} x \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - 3 \, {\left (b^{2} - 2 \, b c + c^{2}\right )} \cosh \left (x\right ) + 3 \, {\left (5 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{4} + 20 \, {\left (b^{2} - c^{2}\right )} x \cosh \left (x\right )^{2} - b^{2} + 2 \, b c - c^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b^{2} - c^{2}}}{24 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 298 vs.
\(2 (128) = 256\).
time = 0.19, size = 298, normalized size = 2.19 \begin {gather*} - \frac {2 b^{3} \sinh ^{3}{\left (x \right )}}{3} + b^{3} \sinh {\left (x \right )} \cosh ^{2}{\left (x \right )} + 3 b^{3} \sinh {\left (x \right )} + b^{2} c \cosh ^{3}{\left (x \right )} + 3 b^{2} c \cosh {\left (x \right )} - \frac {3 b^{2} x \sqrt {b^{2} - c^{2}} \sinh ^{2}{\left (x \right )}}{2} + \frac {3 b^{2} x \sqrt {b^{2} - c^{2}} \cosh ^{2}{\left (x \right )}}{2} + b^{2} x \sqrt {b^{2} - c^{2}} + \frac {3 b^{2} \sqrt {b^{2} - c^{2}} \sinh {\left (x \right )} \cosh {\left (x \right )}}{2} + b c^{2} \sinh ^{3}{\left (x \right )} - 3 b c^{2} \sinh {\left (x \right )} + 3 b c \sqrt {b^{2} - c^{2}} \cosh ^{2}{\left (x \right )} + c^{3} \sinh ^{2}{\left (x \right )} \cosh {\left (x \right )} - \frac {2 c^{3} \cosh ^{3}{\left (x \right )}}{3} - 3 c^{3} \cosh {\left (x \right )} + \frac {3 c^{2} x \sqrt {b^{2} - c^{2}} \sinh ^{2}{\left (x \right )}}{2} - \frac {3 c^{2} x \sqrt {b^{2} - c^{2}} \cosh ^{2}{\left (x \right )}}{2} - c^{2} x \sqrt {b^{2} - c^{2}} + \frac {3 c^{2} \sqrt {b^{2} - c^{2}} \sinh {\left (x \right )} \cosh {\left (x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.40, size = 194, normalized size = 1.43 \begin {gather*} \frac {5}{2} \, {\left (b^{2} - c^{2}\right )}^{\frac {3}{2}} x + \frac {3}{8} \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \sqrt {b^{2} - c^{2}} e^{\left (2 \, x\right )} + \frac {1}{24} \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} e^{\left (3 \, x\right )} - \frac {1}{24} \, {\left (b^{3} - 3 \, b^{2} c + 3 \, b c^{2} - c^{3} + 45 \, {\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} e^{\left (2 \, x\right )} + 9 \, {\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^{x}\right )} e^{\left (-3 \, x\right )} + \frac {15}{8} \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.66, size = 144, normalized size = 1.06 \begin {gather*} \frac {11\,b^3\,\mathrm {sinh}\left (x\right )}{3}+\frac {c^3\,{\mathrm {cosh}\left (x\right )}^3}{3}+\frac {5\,x\,{\left (b^2-c^2\right )}^{3/2}}{2}-4\,c^3\,\mathrm {cosh}\left (x\right )+\frac {b^3\,{\mathrm {cosh}\left (x\right )}^2\,\mathrm {sinh}\left (x\right )}{3}+3\,b^2\,c\,\mathrm {cosh}\left (x\right )-4\,b\,c^2\,\mathrm {sinh}\left (x\right )+b^2\,c\,{\mathrm {cosh}\left (x\right )}^3+3\,b\,c\,{\mathrm {cosh}\left (x\right )}^2\,\sqrt {b^2-c^2}+\frac {3\,b^2\,\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )\,\sqrt {b^2-c^2}}{2}+\frac {3\,c^2\,\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )\,\sqrt {b^2-c^2}}{2}+b\,c^2\,{\mathrm {cosh}\left (x\right )}^2\,\mathrm {sinh}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________