Optimal. Leaf size=62 \[ -\frac {\cosh (a+b x)}{2 b}+\frac {\cosh (a-2 c+(b-2 d) x)}{4 (b-2 d)}+\frac {\cosh (a+2 c+(b+2 d) x)}{4 (b+2 d)} \]
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Rubi [A]
time = 0.05, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5732, 2718}
\begin {gather*} \frac {\cosh (a+x (b-2 d)-2 c)}{4 (b-2 d)}+\frac {\cosh (a+x (b+2 d)+2 c)}{4 (b+2 d)}-\frac {\cosh (a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 5732
Rubi steps
\begin {align*} \int \sinh (a+b x) \sinh ^2(c+d x) \, dx &=\int \left (-\frac {1}{2} \sinh (a+b x)+\frac {1}{4} \sinh (a-2 c+(b-2 d) x)+\frac {1}{4} \sinh (a+2 c+(b+2 d) x)\right ) \, dx\\ &=\frac {1}{4} \int \sinh (a-2 c+(b-2 d) x) \, dx+\frac {1}{4} \int \sinh (a+2 c+(b+2 d) x) \, dx-\frac {1}{2} \int \sinh (a+b x) \, dx\\ &=-\frac {\cosh (a+b x)}{2 b}+\frac {\cosh (a-2 c+(b-2 d) x)}{4 (b-2 d)}+\frac {\cosh (a+2 c+(b+2 d) x)}{4 (b+2 d)}\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 69, normalized size = 1.11 \begin {gather*} \frac {1}{4} \left (-\frac {2 \cosh (a) \cosh (b x)}{b}+\frac {\cosh (a-2 c+b x-2 d x)}{b-2 d}+\frac {\cosh (a+2 c+b x+2 d x)}{b+2 d}-\frac {2 \sinh (a) \sinh (b x)}{b}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.57, size = 57, normalized size = 0.92
method | result | size |
default | \(-\frac {\cosh \left (b x +a \right )}{2 b}+\frac {\cosh \left (a -2 c +\left (b -2 d \right ) x \right )}{4 b -8 d}+\frac {\cosh \left (a +2 c +\left (b +2 d \right ) x \right )}{4 b +8 d}\) | \(57\) |
risch | \(-\frac {{\mathrm e}^{b x +a}}{4 b}-\frac {{\mathrm e}^{-b x -a}}{4 b}+\frac {\left (b \,{\mathrm e}^{2 b x +2 a}-2 \,{\mathrm e}^{2 b x +2 a} d +b +2 d \right ) {\mathrm e}^{-b x +2 d x -a +2 c}}{8 \left (b +2 d \right ) \left (b -2 d \right )}+\frac {\left (b \,{\mathrm e}^{2 b x +2 a}+2 \,{\mathrm e}^{2 b x +2 a} d +b -2 d \right ) {\mathrm e}^{-b x -2 d x -a -2 c}}{8 \left (b +2 d \right ) \left (b -2 d \right )}\) | \(147\) |
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. 120 vs.
\(2 (56) = 112\).
time = 0.40, size = 120, normalized size = 1.94 \begin {gather*} \frac {b^{2} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} - 4 \, b d \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) \sinh \left (d x + c\right ) + b^{2} \cosh \left (b x + a\right ) \sinh \left (d x + c\right )^{2} - {\left (b^{2} - 4 \, d^{2}\right )} \cosh \left (b x + a\right )}{2 \, {\left ({\left (b^{3} - 4 \, b d^{2}\right )} \cosh \left (b x + a\right )^{2} - {\left (b^{3} - 4 \, b d^{2}\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. 408 vs.
\(2 (49) = 98\).
time = 0.85, size = 408, normalized size = 6.58 \begin {gather*} \begin {cases} x \sinh {\left (a \right )} \sinh ^{2}{\left (c \right )} & \text {for}\: b = 0 \wedge d = 0 \\\left (\frac {x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac {x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {\sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d}\right ) \sinh {\left (a \right )} & \text {for}\: b = 0 \\\frac {x \sinh {\left (a - 2 d x \right )} \sinh ^{2}{\left (c + d x \right )}}{4} + \frac {x \sinh {\left (a - 2 d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {x \sinh {\left (c + d x \right )} \cosh {\left (a - 2 d x \right )} \cosh {\left (c + d x \right )}}{2} + \frac {3 \sinh {\left (a - 2 d x \right )} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{4 d} + \frac {\cosh {\left (a - 2 d x \right )} \cosh ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: b = - 2 d \\\frac {x \sinh {\left (a + 2 d x \right )} \sinh ^{2}{\left (c + d x \right )}}{4} + \frac {x \sinh {\left (a + 2 d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} - \frac {x \sinh {\left (c + d x \right )} \cosh {\left (a + 2 d x \right )} \cosh {\left (c + d x \right )}}{2} + \frac {3 \sinh {\left (a + 2 d x \right )} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{4 d} - \frac {\cosh {\left (a + 2 d x \right )} \cosh ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: b = 2 d \\\frac {b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (a + b x \right )}}{b^{3} - 4 b d^{2}} - \frac {2 b d \sinh {\left (a + b x \right )} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{b^{3} - 4 b d^{2}} - \frac {2 d^{2} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (a + b x \right )}}{b^{3} - 4 b d^{2}} + \frac {2 d^{2} \cosh {\left (a + b x \right )} \cosh ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} & \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. 120 vs.
\(2 (56) = 112\).
time = 0.40, size = 120, normalized size = 1.94 \begin {gather*} \frac {e^{\left (b x + 2 \, d x + a + 2 \, c\right )}}{8 \, {\left (b + 2 \, d\right )}} + \frac {e^{\left (b x - 2 \, d x + a - 2 \, c\right )}}{8 \, {\left (b - 2 \, d\right )}} - \frac {e^{\left (b x + a\right )}}{4 \, b} + \frac {e^{\left (-b x + 2 \, d x - a + 2 \, c\right )}}{8 \, {\left (b - 2 \, d\right )}} + \frac {e^{\left (-b x - 2 \, d x - a - 2 \, c\right )}}{8 \, {\left (b + 2 \, d\right )}} - \frac {e^{\left (-b x - a\right )}}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 76, normalized size = 1.23 \begin {gather*} \frac {b^2\,\left (\mathrm {cosh}\left (a+b\,x\right )-\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {cosh}\left (c+d\,x\right )}^2\right )-2\,d^2\,\mathrm {cosh}\left (a+b\,x\right )+2\,b\,d\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )}{4\,b\,d^2-b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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