Optimal. Leaf size=144 \[ \frac {\sinh (2 a-3 c+(2 b-3 d) x)}{16 (2 b-3 d)}+\frac {3 \sinh (2 a-c+(2 b-d) x)}{16 (2 b-d)}+\frac {3 \sinh (c+d x)}{8 d}+\frac {\sinh (3 c+3 d x)}{24 d}+\frac {3 \sinh (2 a+c+(2 b+d) x)}{16 (2 b+d)}+\frac {\sinh (2 a+3 c+(2 b+3 d) x)}{16 (2 b+3 d)} \]
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Rubi [A]
time = 0.08, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {5733, 2717}
\begin {gather*} \frac {\sinh (2 a+x (2 b-3 d)-3 c)}{16 (2 b-3 d)}+\frac {3 \sinh (2 a+x (2 b-d)-c)}{16 (2 b-d)}+\frac {3 \sinh (2 a+x (2 b+d)+c)}{16 (2 b+d)}+\frac {\sinh (2 a+x (2 b+3 d)+3 c)}{16 (2 b+3 d)}+\frac {3 \sinh (c+d x)}{8 d}+\frac {\sinh (3 c+3 d x)}{24 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 5733
Rubi steps
\begin {align*} \int \cosh ^2(a+b x) \cosh ^3(c+d x) \, dx &=\int \left (\frac {1}{16} \cosh (2 a-3 c+(2 b-3 d) x)+\frac {3}{16} \cosh (2 a-c+(2 b-d) x)+\frac {3}{8} \cosh (c+d x)+\frac {1}{8} \cosh (3 c+3 d x)+\frac {3}{16} \cosh (2 a+c+(2 b+d) x)+\frac {1}{16} \cosh (2 a+3 c+(2 b+3 d) x)\right ) \, dx\\ &=\frac {1}{16} \int \cosh (2 a-3 c+(2 b-3 d) x) \, dx+\frac {1}{16} \int \cosh (2 a+3 c+(2 b+3 d) x) \, dx+\frac {1}{8} \int \cosh (3 c+3 d x) \, dx+\frac {3}{16} \int \cosh (2 a-c+(2 b-d) x) \, dx+\frac {3}{16} \int \cosh (2 a+c+(2 b+d) x) \, dx+\frac {3}{8} \int \cosh (c+d x) \, dx\\ &=\frac {\sinh (2 a-3 c+(2 b-3 d) x)}{16 (2 b-3 d)}+\frac {3 \sinh (2 a-c+(2 b-d) x)}{16 (2 b-d)}+\frac {3 \sinh (c+d x)}{8 d}+\frac {\sinh (3 c+3 d x)}{24 d}+\frac {3 \sinh (2 a+c+(2 b+d) x)}{16 (2 b+d)}+\frac {\sinh (2 a+3 c+(2 b+3 d) x)}{16 (2 b+3 d)}\\ \end {align*}
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Mathematica [A]
time = 1.10, size = 158, normalized size = 1.10 \begin {gather*} \frac {1}{48} \left (\frac {18 \cosh (d x) \sinh (c)}{d}+\frac {2 \cosh (3 d x) \sinh (3 c)}{d}+\frac {18 \cosh (c) \sinh (d x)}{d}+\frac {2 \cosh (3 c) \sinh (3 d x)}{d}+\frac {3 \sinh (2 a-3 c+2 b x-3 d x)}{2 b-3 d}+\frac {9 \sinh (2 a-c+2 b x-d x)}{2 b-d}+\frac {9 \sinh (2 a+c+2 b x+d x)}{2 b+d}+\frac {3 \sinh (2 a+3 c+2 b x+3 d x)}{2 b+3 d}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.52, size = 133, normalized size = 0.92
method | result | size |
default | \(\frac {\sinh \left (2 a -3 c +\left (2 b -3 d \right ) x \right )}{32 b -48 d}+\frac {3 \sinh \left (2 a -c +\left (2 b -d \right ) x \right )}{16 \left (2 b -d \right )}+\frac {3 \sinh \left (d x +c \right )}{8 d}+\frac {\sinh \left (3 d x +3 c \right )}{24 d}+\frac {3 \sinh \left (2 a +c +\left (2 b +d \right ) x \right )}{16 \left (2 b +d \right )}+\frac {\sinh \left (2 a +3 c +\left (2 b +3 d \right ) x \right )}{32 b +48 d}\) | \(133\) |
risch | \(\frac {\left (6 d \,{\mathrm e}^{4 b x +4 a} b -9 d^{2} {\mathrm e}^{4 b x +4 a}+8 b^{2} {\mathrm e}^{2 b x +2 a}-18 d^{2} {\mathrm e}^{2 b x +2 a}-6 b d -9 d^{2}\right ) {\mathrm e}^{-2 b x +3 d x -2 a +3 c}}{96 \left (2 b +3 d \right ) \left (2 b -3 d \right ) d}+\frac {3 \left (2 d \,{\mathrm e}^{4 b x +4 a} b -d^{2} {\mathrm e}^{4 b x +4 a}+8 b^{2} {\mathrm e}^{2 b x +2 a}-2 d^{2} {\mathrm e}^{2 b x +2 a}-2 b d -d^{2}\right ) {\mathrm e}^{-2 b x +d x -2 a +c}}{32 \left (2 b +d \right ) \left (2 b -d \right ) d}-\frac {3 \left (-2 d \,{\mathrm e}^{4 b x +4 a} b -d^{2} {\mathrm e}^{4 b x +4 a}+8 b^{2} {\mathrm e}^{2 b x +2 a}-2 d^{2} {\mathrm e}^{2 b x +2 a}+2 b d -d^{2}\right ) {\mathrm e}^{-2 b x -d x -2 a -c}}{32 \left (2 b +d \right ) \left (2 b -d \right ) d}-\frac {\left (-6 d \,{\mathrm e}^{4 b x +4 a} b -9 d^{2} {\mathrm e}^{4 b x +4 a}+8 b^{2} {\mathrm e}^{2 b x +2 a}-18 d^{2} {\mathrm e}^{2 b x +2 a}+6 b d -9 d^{2}\right ) {\mathrm e}^{-2 b x -3 d x -2 a -3 c}}{96 \left (2 b +3 d \right ) \left (2 b -3 d \right ) d}\) | \(411\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 397 vs.
\(2 (132) = 264\).
time = 0.37, size = 397, normalized size = 2.76 \begin {gather*} \frac {36 \, {\left (4 \, b^{3} d - b d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) \sinh \left (d x + c\right )^{2} + {\left (16 \, b^{4} - 40 \, b^{2} d^{2} + 9 \, d^{4} - 9 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (b x + a\right )^{2} - 9 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \sinh \left (b x + a\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 12 \, {\left ({\left (4 \, b^{3} d - b d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{3} + 3 \, {\left (4 \, b^{3} d - 9 \, b d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )\right )} \sinh \left (b x + a\right ) + 3 \, {\left (48 \, b^{4} - 120 \, b^{2} d^{2} + 27 \, d^{4} - 3 \, {\left (4 \, b^{2} d^{2} - 9 \, d^{4}\right )} \cosh \left (b x + a\right )^{2} + {\left (16 \, b^{4} - 40 \, b^{2} d^{2} + 9 \, d^{4} - 9 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right )^{2} - 3 \, {\left (4 \, b^{2} d^{2} - 9 \, d^{4} + 3 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (b x + a\right )^{2}\right )} \sinh \left (d x + c\right )}{24 \, {\left ({\left (16 \, b^{4} d - 40 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cosh \left (b x + a\right )^{2} - {\left (16 \, b^{4} d - 40 \, b^{2} d^{3} + 9 \, d^{5}\right )} \sinh \left (b x + a\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2001 vs.
\(2 (116) = 232\).
time = 6.87, size = 2001, normalized size = 13.90 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 260, normalized size = 1.81 \begin {gather*} \frac {e^{\left (2 \, b x + 3 \, d x + 2 \, a + 3 \, c\right )}}{32 \, {\left (2 \, b + 3 \, d\right )}} + \frac {3 \, e^{\left (2 \, b x + d x + 2 \, a + c\right )}}{32 \, {\left (2 \, b + d\right )}} + \frac {3 \, e^{\left (2 \, b x - d x + 2 \, a - c\right )}}{32 \, {\left (2 \, b - d\right )}} + \frac {e^{\left (2 \, b x - 3 \, d x + 2 \, a - 3 \, c\right )}}{32 \, {\left (2 \, b - 3 \, d\right )}} - \frac {e^{\left (-2 \, b x + 3 \, d x - 2 \, a + 3 \, c\right )}}{32 \, {\left (2 \, b - 3 \, d\right )}} - \frac {3 \, e^{\left (-2 \, b x + d x - 2 \, a + c\right )}}{32 \, {\left (2 \, b - d\right )}} - \frac {3 \, e^{\left (-2 \, b x - d x - 2 \, a - c\right )}}{32 \, {\left (2 \, b + d\right )}} - \frac {e^{\left (-2 \, b x - 3 \, d x - 2 \, a - 3 \, c\right )}}{32 \, {\left (2 \, b + 3 \, d\right )}} + \frac {e^{\left (3 \, d x + 3 \, c\right )}}{48 \, d} + \frac {3 \, e^{\left (d x + c\right )}}{16 \, d} - \frac {3 \, e^{\left (-d x - c\right )}}{16 \, d} - \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{48 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.95, size = 337, normalized size = 2.34 \begin {gather*} \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (8\,b^4-26\,b^2\,d^2+9\,d^4\right )}{d\,\left (16\,b^4-40\,b^2\,d^2+9\,d^4\right )}-{\mathrm {sinh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (\frac {3\,d^3}{16\,b^4-40\,b^2\,d^2+9\,d^4}-\frac {1}{3\,d}\right )-\frac {2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {cosh}\left (c+d\,x\right )}^3\,\mathrm {sinh}\left (a+b\,x\right )\,\left (7\,b\,d^2-4\,b^3\right )}{16\,b^4-40\,b^2\,d^2+9\,d^4}-{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (\frac {3\,d^3}{16\,b^4-40\,b^2\,d^2+9\,d^4}+\frac {1}{3\,d}\right )+\frac {12\,b\,d^2\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^2}{16\,b^4-40\,b^2\,d^2+9\,d^4}-\frac {2\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (4\,b^2-7\,d^2\right )}{d\,\left (16\,b^4-40\,b^2\,d^2+9\,d^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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