Optimal. Leaf size=97 \[ -\frac {3 \cosh (a-c+(b-d) x)}{8 (b-d)}+\frac {\cosh (3 a-c+(3 b-d) x)}{8 (3 b-d)}-\frac {3 \cosh (a+c+(b+d) x)}{8 (b+d)}+\frac {\cosh (3 a+c+(3 b+d) x)}{8 (3 b+d)} \]
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Rubi [A]
time = 0.06, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5737, 2718}
\begin {gather*} -\frac {3 \cosh (a+x (b-d)-c)}{8 (b-d)}+\frac {\cosh (3 a+x (3 b-d)-c)}{8 (3 b-d)}-\frac {3 \cosh (a+x (b+d)+c)}{8 (b+d)}+\frac {\cosh (3 a+x (3 b+d)+c)}{8 (3 b+d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 5737
Rubi steps
\begin {align*} \int \cosh (c+d x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac {3}{8} \sinh (a-c+(b-d) x)+\frac {1}{8} \sinh (3 a-c+(3 b-d) x)-\frac {3}{8} \sinh (a+c+(b+d) x)+\frac {1}{8} \sinh (3 a+c+(3 b+d) x)\right ) \, dx\\ &=\frac {1}{8} \int \sinh (3 a-c+(3 b-d) x) \, dx+\frac {1}{8} \int \sinh (3 a+c+(3 b+d) x) \, dx-\frac {3}{8} \int \sinh (a-c+(b-d) x) \, dx-\frac {3}{8} \int \sinh (a+c+(b+d) x) \, dx\\ &=-\frac {3 \cosh (a-c+(b-d) x)}{8 (b-d)}+\frac {\cosh (3 a-c+(3 b-d) x)}{8 (3 b-d)}-\frac {3 \cosh (a+c+(b+d) x)}{8 (b+d)}+\frac {\cosh (3 a+c+(3 b+d) x)}{8 (3 b+d)}\\ \end {align*}
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Mathematica [A]
time = 0.36, size = 90, normalized size = 0.93 \begin {gather*} \frac {1}{8} \left (-\frac {3 \cosh (a-c+b x-d x)}{b-d}+\frac {\cosh (3 a-c+3 b x-d x)}{3 b-d}+\frac {\cosh (3 a+c+3 b x+d x)}{3 b+d}-\frac {3 \cosh (a+c+(b+d) x)}{b+d}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.75, size = 90, normalized size = 0.93
method | result | size |
default | \(-\frac {3 \cosh \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}+\frac {\cosh \left (3 a -c +\left (3 b -d \right ) x \right )}{24 b -8 d}-\frac {3 \cosh \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}+\frac {\cosh \left (3 a +c +\left (3 b +d \right ) x \right )}{24 b +8 d}\) | \(90\) |
risch | \(\frac {\left (3 b^{3} {\mathrm e}^{6 b x +6 a}-b^{2} d \,{\mathrm e}^{6 b x +6 a}-3 b \,d^{2} {\mathrm e}^{6 b x +6 a}+d^{3} {\mathrm e}^{6 b x +6 a}-27 b^{3} {\mathrm e}^{4 b x +4 a}+27 b^{2} d \,{\mathrm e}^{4 b x +4 a}+3 b \,d^{2} {\mathrm e}^{4 b x +4 a}-3 d^{3} {\mathrm e}^{4 b x +4 a}-27 b^{3} {\mathrm e}^{2 b x +2 a}-27 b^{2} d \,{\mathrm e}^{2 b x +2 a}+3 b \,d^{2} {\mathrm e}^{2 b x +2 a}+3 d^{3} {\mathrm e}^{2 b x +2 a}+3 b^{3}+b^{2} d -3 d^{2} b -d^{3}\right ) {\mathrm e}^{-3 b x +d x -3 a +c}}{16 \left (3 b +d \right ) \left (b +d \right ) \left (3 b -d \right ) \left (b -d \right )}+\frac {\left (3 b^{3} {\mathrm e}^{6 b x +6 a}+b^{2} d \,{\mathrm e}^{6 b x +6 a}-3 b \,d^{2} {\mathrm e}^{6 b x +6 a}-d^{3} {\mathrm e}^{6 b x +6 a}-27 b^{3} {\mathrm e}^{4 b x +4 a}-27 b^{2} d \,{\mathrm e}^{4 b x +4 a}+3 b \,d^{2} {\mathrm e}^{4 b x +4 a}+3 d^{3} {\mathrm e}^{4 b x +4 a}-27 b^{3} {\mathrm e}^{2 b x +2 a}+27 b^{2} d \,{\mathrm e}^{2 b x +2 a}+3 b \,d^{2} {\mathrm e}^{2 b x +2 a}-3 d^{3} {\mathrm e}^{2 b x +2 a}+3 b^{3}-b^{2} d -3 d^{2} b +d^{3}\right ) {\mathrm e}^{-3 b x -d x -3 a -c}}{16 \left (3 b +d \right ) \left (b +d \right ) \left (3 b -d \right ) \left (b -d \right )}\) | \(480\) |
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. 243 vs.
\(2 (89) = 178\).
time = 0.45, size = 243, normalized size = 2.51 \begin {gather*} \frac {9 \, {\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} + 3 \, {\left ({\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{3} - {\left (9 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) - {\left ({\left (b^{2} d - d^{3}\right )} \sinh \left (b x + a\right )^{3} - 3 \, {\left (9 \, b^{2} d - d^{3} - {\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{4 \, {\left ({\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{4} - 2 \, {\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + {\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \sinh \left (b x + a\right )^{4}\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. 937 vs.
\(2 (76) = 152\).
time = 2.56, size = 937, normalized size = 9.66 \begin {gather*} \begin {cases} x \sinh ^{3}{\left (a \right )} \cosh {\left (c \right )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {3 x \sinh ^{3}{\left (a - d x \right )} \cosh {\left (c + d x \right )}}{8} + \frac {3 x \sinh ^{2}{\left (a - d x \right )} \sinh {\left (c + d x \right )} \cosh {\left (a - d x \right )}}{8} - \frac {3 x \sinh {\left (a - d x \right )} \cosh ^{2}{\left (a - d x \right )} \cosh {\left (c + d x \right )}}{8} - \frac {3 x \sinh {\left (c + d x \right )} \cosh ^{3}{\left (a - d x \right )}}{8} - \frac {\sinh ^{3}{\left (a - d x \right )} \sinh {\left (c + d x \right )}}{8 d} - \frac {3 \sinh ^{2}{\left (a - d x \right )} \cosh {\left (a - d x \right )} \cosh {\left (c + d x \right )}}{4 d} + \frac {3 \cosh ^{3}{\left (a - d x \right )} \cosh {\left (c + d x \right )}}{8 d} & \text {for}\: b = - d \\\frac {x \sinh ^{3}{\left (a - \frac {d x}{3} \right )} \cosh {\left (c + d x \right )}}{8} + \frac {3 x \sinh ^{2}{\left (a - \frac {d x}{3} \right )} \sinh {\left (c + d x \right )} \cosh {\left (a - \frac {d x}{3} \right )}}{8} + \frac {3 x \sinh {\left (a - \frac {d x}{3} \right )} \cosh ^{2}{\left (a - \frac {d x}{3} \right )} \cosh {\left (c + d x \right )}}{8} + \frac {x \sinh {\left (c + d x \right )} \cosh ^{3}{\left (a - \frac {d x}{3} \right )}}{8} + \frac {7 \sinh ^{3}{\left (a - \frac {d x}{3} \right )} \sinh {\left (c + d x \right )}}{8 d} - \frac {3 \sinh {\left (a - \frac {d x}{3} \right )} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (a - \frac {d x}{3} \right )}}{4 d} - \frac {3 \cosh ^{3}{\left (a - \frac {d x}{3} \right )} \cosh {\left (c + d x \right )}}{8 d} & \text {for}\: b = - \frac {d}{3} \\\frac {x \sinh ^{3}{\left (a + \frac {d x}{3} \right )} \cosh {\left (c + d x \right )}}{8} - \frac {3 x \sinh ^{2}{\left (a + \frac {d x}{3} \right )} \sinh {\left (c + d x \right )} \cosh {\left (a + \frac {d x}{3} \right )}}{8} + \frac {3 x \sinh {\left (a + \frac {d x}{3} \right )} \cosh ^{2}{\left (a + \frac {d x}{3} \right )} \cosh {\left (c + d x \right )}}{8} - \frac {x \sinh {\left (c + d x \right )} \cosh ^{3}{\left (a + \frac {d x}{3} \right )}}{8} + \frac {7 \sinh ^{3}{\left (a + \frac {d x}{3} \right )} \sinh {\left (c + d x \right )}}{8 d} - \frac {3 \sinh {\left (a + \frac {d x}{3} \right )} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (a + \frac {d x}{3} \right )}}{4 d} + \frac {3 \cosh ^{3}{\left (a + \frac {d x}{3} \right )} \cosh {\left (c + d x \right )}}{8 d} & \text {for}\: b = \frac {d}{3} \\\frac {3 x \sinh ^{3}{\left (a + d x \right )} \cosh {\left (c + d x \right )}}{8} - \frac {3 x \sinh ^{2}{\left (a + d x \right )} \sinh {\left (c + d x \right )} \cosh {\left (a + d x \right )}}{8} - \frac {3 x \sinh {\left (a + d x \right )} \cosh ^{2}{\left (a + d x \right )} \cosh {\left (c + d x \right )}}{8} + \frac {3 x \sinh {\left (c + d x \right )} \cosh ^{3}{\left (a + d x \right )}}{8} + \frac {5 \sinh ^{3}{\left (a + d x \right )} \sinh {\left (c + d x \right )}}{8 d} - \frac {3 \sinh {\left (a + d x \right )} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (a + d x \right )}}{4 d} + \frac {3 \cosh ^{3}{\left (a + d x \right )} \cosh {\left (c + d x \right )}}{8 d} & \text {for}\: b = d \\\frac {9 b^{3} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )} \cosh {\left (c + d x \right )}}{9 b^{4} - 10 b^{2} d^{2} + d^{4}} - \frac {6 b^{3} \cosh ^{3}{\left (a + b x \right )} \cosh {\left (c + d x \right )}}{9 b^{4} - 10 b^{2} d^{2} + d^{4}} - \frac {7 b^{2} d \sinh ^{3}{\left (a + b x \right )} \sinh {\left (c + d x \right )}}{9 b^{4} - 10 b^{2} d^{2} + d^{4}} + \frac {6 b^{2} d \sinh {\left (a + b x \right )} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (a + b x \right )}}{9 b^{4} - 10 b^{2} d^{2} + d^{4}} - \frac {3 b d^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )} \cosh {\left (c + d x \right )}}{9 b^{4} - 10 b^{2} d^{2} + d^{4}} + \frac {d^{3} \sinh ^{3}{\left (a + b x \right )} \sinh {\left (c + d x \right )}}{9 b^{4} - 10 b^{2} d^{2} + d^{4}} & \text {otherwise} \end {cases} \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. 183 vs.
\(2 (89) = 178\).
time = 0.41, size = 183, normalized size = 1.89 \begin {gather*} \frac {e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{16 \, {\left (3 \, b + d\right )}} + \frac {e^{\left (3 \, b x - d x + 3 \, a - c\right )}}{16 \, {\left (3 \, b - d\right )}} - \frac {3 \, e^{\left (b x + d x + a + c\right )}}{16 \, {\left (b + d\right )}} - \frac {3 \, e^{\left (b x - d x + a - c\right )}}{16 \, {\left (b - d\right )}} - \frac {3 \, e^{\left (-b x + d x - a + c\right )}}{16 \, {\left (b - d\right )}} - \frac {3 \, e^{\left (-b x - d x - a - c\right )}}{16 \, {\left (b + d\right )}} + \frac {e^{\left (-3 \, b x + d x - 3 \, a + c\right )}}{16 \, {\left (3 \, b - d\right )}} + \frac {e^{\left (-3 \, b x - d x - 3 \, a - c\right )}}{16 \, {\left (3 \, b + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.53, size = 183, normalized size = 1.89 \begin {gather*} \frac {6\,b^2\,d\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )}{9\,b^4-10\,b^2\,d^2+d^4}-\frac {d\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\mathrm {sinh}\left (c+d\,x\right )\,\left (7\,b^2-d^2\right )}{9\,b^4-10\,b^2\,d^2+d^4}-\frac {3\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (b\,d^2-3\,b^3\right )}{9\,b^4-10\,b^2\,d^2+d^4}-\frac {6\,b^3\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,\mathrm {cosh}\left (c+d\,x\right )}{9\,b^4-10\,b^2\,d^2+d^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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