Optimal. Leaf size=75 \[ -\frac {d \cosh ^3(a+b x)}{9 b^2}-\frac {2 d \cosh (a+b x)}{3 b^2}+\frac {2 (c+d x) \sinh (a+b x)}{3 b}+\frac {(c+d x) \sinh (a+b x) \cosh ^2(a+b x)}{3 b} \]
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Rubi [A] time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3310, 3296, 2638} \[ -\frac {d \cosh ^3(a+b x)}{9 b^2}-\frac {2 d \cosh (a+b x)}{3 b^2}+\frac {2 (c+d x) \sinh (a+b x)}{3 b}+\frac {(c+d x) \sinh (a+b x) \cosh ^2(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3310
Rubi steps
\begin {align*} \int (c+d x) \cosh ^3(a+b x) \, dx &=-\frac {d \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x) \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {2}{3} \int (c+d x) \cosh (a+b x) \, dx\\ &=-\frac {d \cosh ^3(a+b x)}{9 b^2}+\frac {2 (c+d x) \sinh (a+b x)}{3 b}+\frac {(c+d x) \cosh ^2(a+b x) \sinh (a+b x)}{3 b}-\frac {(2 d) \int \sinh (a+b x) \, dx}{3 b}\\ &=-\frac {2 d \cosh (a+b x)}{3 b^2}-\frac {d \cosh ^3(a+b x)}{9 b^2}+\frac {2 (c+d x) \sinh (a+b x)}{3 b}+\frac {(c+d x) \cosh ^2(a+b x) \sinh (a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 52, normalized size = 0.69 \[ -\frac {-3 b (c+d x) (9 \sinh (a+b x)+\sinh (3 (a+b x)))+27 d \cosh (a+b x)+d \cosh (3 (a+b x))}{36 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 95, normalized size = 1.27 \[ -\frac {d \cosh \left (b x + a\right )^{3} + 3 \, d \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 3 \, {\left (b d x + b c\right )} \sinh \left (b x + a\right )^{3} + 27 \, d \cosh \left (b x + a\right ) - 9 \, {\left (3 \, b d x + {\left (b d x + b c\right )} \cosh \left (b x + a\right )^{2} + 3 \, b c\right )} \sinh \left (b x + a\right )}{36 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 98, normalized size = 1.31 \[ \frac {{\left (3 \, b d x + 3 \, b c - d\right )} e^{\left (3 \, b x + 3 \, a\right )}}{72 \, b^{2}} + \frac {3 \, {\left (b d x + b c - d\right )} e^{\left (b x + a\right )}}{8 \, b^{2}} - \frac {3 \, {\left (b d x + b c + d\right )} e^{\left (-b x - a\right )}}{8 \, b^{2}} - \frac {{\left (3 \, b d x + 3 \, b c + d\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{72 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 109, normalized size = 1.45 \[ \frac {\frac {d \left (\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{3}-\frac {2 \cosh \left (b x +a \right )}{3}-\frac {\left (\cosh ^{3}\left (b x +a \right )\right )}{9}\right )}{b}-\frac {d a \left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{3}\right ) \sinh \left (b x +a \right )}{b}+c \left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{3}\right ) \sinh \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 143, normalized size = 1.91 \[ \frac {1}{72} \, d {\left (\frac {{\left (3 \, b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{2}} + \frac {27 \, {\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{b^{2}} - \frac {27 \, {\left (b x + 1\right )} e^{\left (-b x - a\right )}}{b^{2}} - \frac {{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{2}}\right )} + \frac {1}{24} \, c {\left (\frac {e^{\left (3 \, b x + 3 \, a\right )}}{b} + \frac {9 \, e^{\left (b x + a\right )}}{b} - \frac {9 \, e^{\left (-b x - a\right )}}{b} - \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 77, normalized size = 1.03 \[ \frac {\frac {3\,c\,\mathrm {sinh}\left (a+b\,x\right )}{4}+\frac {c\,\mathrm {sinh}\left (3\,a+3\,b\,x\right )}{12}+\frac {d\,x\,\mathrm {sinh}\left (3\,a+3\,b\,x\right )}{12}+\frac {3\,d\,x\,\mathrm {sinh}\left (a+b\,x\right )}{4}}{b}-\frac {d\,\mathrm {cosh}\left (3\,a+3\,b\,x\right )}{36\,b^2}-\frac {3\,d\,\mathrm {cosh}\left (a+b\,x\right )}{4\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.94, size = 126, normalized size = 1.68 \[ \begin {cases} - \frac {2 c \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac {c \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} - \frac {2 d x \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac {d x \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} + \frac {2 d \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{3 b^{2}} - \frac {7 d \cosh ^{3}{\left (a + b x \right )}}{9 b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \cosh ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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