Optimal. Leaf size=36 \[ -\frac {a \left (a+b \cosh ^2(x)\right )^4}{8 b^2}+\frac {\left (a+b \cosh ^2(x)\right )^5}{10 b^2} \]
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Rubi [A]
time = 0.06, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4420, 272, 45}
\begin {gather*} \frac {\left (a+b \cosh ^2(x)\right )^5}{10 b^2}-\frac {a \left (a+b \cosh ^2(x)\right )^4}{8 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 4420
Rubi steps
\begin {align*} \int \cosh ^3(x) \left (a+b \cosh ^2(x)\right )^3 \sinh (x) \, dx &=\text {Subst}\left (\int x^3 \left (a+b x^2\right )^3 \, dx,x,\cosh (x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int x (a+b x)^3 \, dx,x,\cosh ^2(x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {a (a+b x)^3}{b}+\frac {(a+b x)^4}{b}\right ) \, dx,x,\cosh ^2(x)\right )\\ &=-\frac {a \left (a+b \cosh ^2(x)\right )^4}{8 b^2}+\frac {\left (a+b \cosh ^2(x)\right )^5}{10 b^2}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(136\) vs. \(2(36)=72\).
time = 0.21, size = 136, normalized size = 3.78 \begin {gather*} \frac {1}{32} \left (12 a^2 b \cosh ^4(x)+8 a b^2 \cosh ^6(x)+2 b^3 \cosh ^8(x)+4 a^3 \cosh (2 x)+4 a^2 b \cosh ^3(x) \cosh (3 x)+a^3 \cosh (4 x)+\frac {1}{32} a b^2 (48 \cosh (2 x)+36 \cosh (4 x)+16 \cosh (6 x)+3 \cosh (8 x))+\frac {1}{320} b^3 (140 \cosh (2 x)+100 \cosh (4 x)+50 \cosh (6 x)+15 \cosh (8 x)+2 \cosh (10 x))\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.54, size = 32, normalized size = 0.89
method | result | size |
derivativedivides | \(-\frac {\frac {a \left (a +b \left (\cosh ^{2}\left (x \right )\right )\right )^{4}}{4}-\frac {\left (a +b \left (\cosh ^{2}\left (x \right )\right )\right )^{5}}{5}}{2 b^{2}}\) | \(32\) |
default | \(-\frac {\frac {a \left (a +b \left (\cosh ^{2}\left (x \right )\right )\right )^{4}}{4}-\frac {\left (a +b \left (\cosh ^{2}\left (x \right )\right )\right )^{5}}{5}}{2 b^{2}}\) | \(32\) |
risch | \(\frac {3 \,{\mathrm e}^{-4 x} a^{2} b}{64}+\frac {{\mathrm e}^{-6 x} a^{2} b}{128}+\frac {3 \,{\mathrm e}^{-6 x} a \,b^{2}}{256}+\frac {3 \,{\mathrm e}^{8 x} a \,b^{2}}{2048}+\frac {{\mathrm e}^{6 x} a^{2} b}{128}+\frac {{\mathrm e}^{4 x} a^{3}}{64}+\frac {3 \,{\mathrm e}^{4 x} b^{3}}{256}+\frac {{\mathrm e}^{2 x} a^{3}}{16}+\frac {21 \,{\mathrm e}^{2 x} b^{3}}{1024}+\frac {3 \,{\mathrm e}^{-8 x} a \,b^{2}}{2048}+\frac {3 \,{\mathrm e}^{6 x} a \,b^{2}}{256}+\frac {3 \,{\mathrm e}^{4 x} a^{2} b}{64}+\frac {21 \,{\mathrm e}^{4 x} a \,b^{2}}{512}+\frac {15 \,{\mathrm e}^{-2 x} a^{2} b}{128}+\frac {21 \,{\mathrm e}^{-2 x} a \,b^{2}}{256}+\frac {15 \,{\mathrm e}^{2 x} a^{2} b}{128}+\frac {21 \,{\mathrm e}^{2 x} a \,b^{2}}{256}+\frac {21 \,{\mathrm e}^{-4 x} a \,b^{2}}{512}+\frac {b^{3} {\mathrm e}^{-10 x}}{10240}+\frac {9 \,{\mathrm e}^{6 x} b^{3}}{2048}+\frac {9 \,{\mathrm e}^{-6 x} b^{3}}{2048}+\frac {{\mathrm e}^{8 x} b^{3}}{1024}+\frac {{\mathrm e}^{-2 x} a^{3}}{16}+\frac {{\mathrm e}^{-8 x} b^{3}}{1024}+\frac {21 \,{\mathrm e}^{-2 x} b^{3}}{1024}+\frac {{\mathrm e}^{-4 x} a^{3}}{64}+\frac {3 \,{\mathrm e}^{-4 x} b^{3}}{256}+\frac {b^{3} {\mathrm e}^{10 x}}{10240}\) | \(268\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 39, normalized size = 1.08 \begin {gather*} \frac {1}{10} \, b^{3} \cosh \left (x\right )^{10} + \frac {3}{8} \, a b^{2} \cosh \left (x\right )^{8} + \frac {1}{2} \, a^{2} b \cosh \left (x\right )^{6} + \frac {1}{4} \, a^{3} \cosh \left (x\right )^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 386 vs.
\(2 (32) = 64\).
time = 0.38, size = 386, normalized size = 10.72 \begin {gather*} \frac {1}{5120} \, b^{3} \cosh \left (x\right )^{10} + \frac {1}{5120} \, b^{3} \sinh \left (x\right )^{10} + \frac {1}{1024} \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (x\right )^{8} + \frac {1}{1024} \, {\left (9 \, b^{3} \cosh \left (x\right )^{2} + 3 \, a b^{2} + 2 \, b^{3}\right )} \sinh \left (x\right )^{8} + \frac {1}{1024} \, {\left (16 \, a^{2} b + 24 \, a b^{2} + 9 \, b^{3}\right )} \cosh \left (x\right )^{6} + \frac {1}{1024} \, {\left (42 \, b^{3} \cosh \left (x\right )^{4} + 16 \, a^{2} b + 24 \, a b^{2} + 9 \, b^{3} + 28 \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{6} + \frac {1}{256} \, {\left (8 \, a^{3} + 24 \, a^{2} b + 21 \, a b^{2} + 6 \, b^{3}\right )} \cosh \left (x\right )^{4} + \frac {1}{1024} \, {\left (42 \, b^{3} \cosh \left (x\right )^{6} + 70 \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (x\right )^{4} + 32 \, a^{3} + 96 \, a^{2} b + 84 \, a b^{2} + 24 \, b^{3} + 15 \, {\left (16 \, a^{2} b + 24 \, a b^{2} + 9 \, b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{4} + \frac {1}{512} \, {\left (64 \, a^{3} + 120 \, a^{2} b + 84 \, a b^{2} + 21 \, b^{3}\right )} \cosh \left (x\right )^{2} + \frac {1}{1024} \, {\left (9 \, b^{3} \cosh \left (x\right )^{8} + 28 \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (x\right )^{6} + 15 \, {\left (16 \, a^{2} b + 24 \, a b^{2} + 9 \, b^{3}\right )} \cosh \left (x\right )^{4} + 128 \, a^{3} + 240 \, a^{2} b + 168 \, a b^{2} + 42 \, b^{3} + 24 \, {\left (8 \, a^{3} + 24 \, a^{2} b + 21 \, a b^{2} + 6 \, b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.71, size = 44, normalized size = 1.22 \begin {gather*} \frac {a^{3} \cosh ^{4}{\left (x \right )}}{4} + \frac {a^{2} b \cosh ^{6}{\left (x \right )}}{2} + \frac {3 a b^{2} \cosh ^{8}{\left (x \right )}}{8} + \frac {b^{3} \cosh ^{10}{\left (x \right )}}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 224 vs.
\(2 (32) = 64\).
time = 0.40, size = 224, normalized size = 6.22 \begin {gather*} \frac {1}{10240} \, b^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{5} + \frac {3}{2048} \, a b^{2} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{4} + \frac {1}{1024} \, b^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{4} + \frac {1}{128} \, a^{2} b {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{3} + \frac {3}{256} \, a b^{2} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{3} + \frac {1}{256} \, b^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{3} + \frac {1}{64} \, a^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{2} + \frac {3}{64} \, a^{2} b {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{2} + \frac {9}{256} \, a b^{2} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{2} + \frac {1}{128} \, b^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{2} + \frac {1}{16} \, a^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} + \frac {3}{32} \, a^{2} b {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} + \frac {3}{64} \, a b^{2} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} + \frac {1}{128} \, b^{3} {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.89, size = 39, normalized size = 1.08 \begin {gather*} \frac {a^3\,{\mathrm {cosh}\left (x\right )}^4}{4}+\frac {a^2\,b\,{\mathrm {cosh}\left (x\right )}^6}{2}+\frac {3\,a\,b^2\,{\mathrm {cosh}\left (x\right )}^8}{8}+\frac {b^3\,{\mathrm {cosh}\left (x\right )}^{10}}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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