Optimal. Leaf size=88 \[ -\frac {2 d^2 e^{a+b x}}{b \left (b^2-4 d^2\right )}+\frac {b e^{a+b x} \cosh ^2(c+d x)}{b^2-4 d^2}-\frac {2 d e^{a+b x} \cosh (c+d x) \sinh (c+d x)}{b^2-4 d^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5585, 2225}
\begin {gather*} \frac {b e^{a+b x} \cosh ^2(c+d x)}{b^2-4 d^2}-\frac {2 d e^{a+b x} \sinh (c+d x) \cosh (c+d x)}{b^2-4 d^2}-\frac {2 d^2 e^{a+b x}}{b \left (b^2-4 d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 5585
Rubi steps
\begin {align*} \int e^{a+b x} \cosh ^2(c+d x) \, dx &=\frac {b e^{a+b x} \cosh ^2(c+d x)}{b^2-4 d^2}-\frac {2 d e^{a+b x} \cosh (c+d x) \sinh (c+d x)}{b^2-4 d^2}-\frac {\left (2 d^2\right ) \int e^{a+b x} \, dx}{b^2-4 d^2}\\ &=-\frac {2 d^2 e^{a+b x}}{b \left (b^2-4 d^2\right )}+\frac {b e^{a+b x} \cosh ^2(c+d x)}{b^2-4 d^2}-\frac {2 d e^{a+b x} \cosh (c+d x) \sinh (c+d x)}{b^2-4 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 56, normalized size = 0.64 \begin {gather*} \frac {e^{a+b x} \left (b^2-4 d^2+b^2 \cosh (2 (c+d x))-2 b d \sinh (2 (c+d x))\right )}{2 \left (b^3-4 b d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.48, size = 112, normalized size = 1.27
method | result | size |
risch | \(\frac {{\mathrm e}^{b x +a}}{2 b}+\frac {{\mathrm e}^{b x +2 d x +a +2 c}}{4 b +8 d}+\frac {{\mathrm e}^{b x -2 d x +a -2 c}}{4 b -8 d}\) | \(57\) |
default | \(\frac {\sinh \left (b x +a \right )}{2 b}+\frac {\sinh \left (a -2 c +\left (b -2 d \right ) x \right )}{4 b -8 d}+\frac {\sinh \left (a +2 c +\left (b +2 d \right ) x \right )}{4 b +8 d}+\frac {\cosh \left (b x +a \right )}{2 b}+\frac {\cosh \left (a -2 c +\left (b -2 d \right ) x \right )}{4 b -8 d}+\frac {\cosh \left (a +2 c +\left (b +2 d \right ) x \right )}{4 b +8 d}\) | \(112\) |
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 [A]
time = 0.36, size = 146, normalized size = 1.66 \begin {gather*} \frac {b^{2} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} + {\left (b^{2} \cosh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )^{2} + {\left (b^{2} - 4 \, d^{2}\right )} \cosh \left (b x + a\right ) + {\left (b^{2} \cosh \left (d x + c\right )^{2} + b^{2} - 4 \, d^{2}\right )} \sinh \left (b x + a\right ) - 4 \, {\left (b d \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) + b d \cosh \left (d x + c\right ) \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left (b^{3} - 4 \, b d^{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. 428 vs.
\(2 (78) = 156\).
time = 1.01, size = 428, normalized size = 4.86 \begin {gather*} \begin {cases} x e^{a} \cosh ^{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 ) e^{a} & \text {for}\: b = 0 \\\frac {x e^{a} e^{- 2 d x} \sinh ^{2}{\left (c + d x \right )}}{4} + \frac {x e^{a} e^{- 2 d x} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2} + \frac {x e^{a} e^{- 2 d x} \cosh ^{2}{\left (c + d x \right )}}{4} - \frac {e^{a} e^{- 2 d x} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{4 d} - \frac {e^{a} e^{- 2 d x} \cosh ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: b = - 2 d \\\frac {x e^{a} e^{2 d x} \sinh ^{2}{\left (c + d x \right )}}{4} - \frac {x e^{a} e^{2 d x} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2} + \frac {x e^{a} e^{2 d x} \cosh ^{2}{\left (c + d x \right )}}{4} - \frac {e^{a} e^{2 d x} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{4 d} + \frac {e^{a} e^{2 d x} \cosh ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: b = 2 d \\\frac {b^{2} e^{a} e^{b x} \cosh ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} - \frac {2 b d e^{a} e^{b x} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{b^{3} - 4 b d^{2}} + \frac {2 d^{2} e^{a} e^{b x} \sinh ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} - \frac {2 d^{2} e^{a} e^{b x} \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 [A]
time = 0.41, size = 56, normalized size = 0.64 \begin {gather*} \frac {e^{\left (b x + 2 \, d x + a + 2 \, c\right )}}{4 \, {\left (b + 2 \, d\right )}} + \frac {e^{\left (b x - 2 \, d x + a - 2 \, c\right )}}{4 \, {\left (b - 2 \, d\right )}} + \frac {e^{\left (b x + a\right )}}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.23, size = 68, normalized size = 0.77 \begin {gather*} \frac {2\,d^2\,{\mathrm {e}}^{a+b\,x}-b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {e}}^{a+b\,x}+2\,b\,d\,\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {e}}^{a+b\,x}\,\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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