Optimal. Leaf size=75 \[ -\frac {2 (c+d x) \cosh (a+b x)}{3 b}+\frac {2 d \sinh (a+b x)}{3 b^2}+\frac {(c+d x) \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {d \sinh ^3(a+b x)}{9 b^2} \]
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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 = {3391, 3377,
2717} \begin {gather*} -\frac {d \sinh ^3(a+b x)}{9 b^2}+\frac {2 d \sinh (a+b x)}{3 b^2}-\frac {2 (c+d x) \cosh (a+b x)}{3 b}+\frac {(c+d x) \sinh ^2(a+b x) \cosh (a+b x)}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 3377
Rule 3391
Rubi steps
\begin {align*} \int (c+d x) \sinh ^3(a+b x) \, dx &=\frac {(c+d x) \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {d \sinh ^3(a+b x)}{9 b^2}-\frac {2}{3} \int (c+d x) \sinh (a+b x) \, dx\\ &=-\frac {2 (c+d x) \cosh (a+b x)}{3 b}+\frac {(c+d x) \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {d \sinh ^3(a+b x)}{9 b^2}+\frac {(2 d) \int \cosh (a+b x) \, dx}{3 b}\\ &=-\frac {2 (c+d x) \cosh (a+b x)}{3 b}+\frac {2 d \sinh (a+b x)}{3 b^2}+\frac {(c+d x) \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {d \sinh ^3(a+b x)}{9 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 59, normalized size = 0.79 \begin {gather*} \frac {-27 b (c+d x) \cosh (a+b x)+3 b (c+d x) \cosh (3 (a+b x))+d (27 \sinh (a+b x)-\sinh (3 (a+b x)))}{36 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 125, normalized size = 1.67
method | result | size |
risch | \(\frac {\left (3 b d x +3 b c -d \right ) {\mathrm e}^{3 b x +3 a}}{72 b^{2}}-\frac {3 \left (b d x +b c -d \right ) {\mathrm e}^{b x +a}}{8 b^{2}}-\frac {3 \left (b d x +b c +d \right ) {\mathrm e}^{-b x -a}}{8 b^{2}}+\frac {\left (3 b d x +3 b c +d \right ) {\mathrm e}^{-3 b x -3 a}}{72 b^{2}}\) | \(99\) |
default | \(\frac {-\frac {3 d \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{4 b}+\frac {3 a d \cosh \left (b x +a \right )}{4 b}-\frac {3 c \cosh \left (b x +a \right )}{4}}{b}+\frac {\frac {d \left (\left (3 b x +3 a \right ) \cosh \left (3 b x +3 a \right )-\sinh \left (3 b x +3 a \right )\right )}{b}-\frac {3 a d \cosh \left (3 b x +3 a \right )}{b}+3 c \cosh \left (3 b x +3 a \right )}{36 b}\) | \(125\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs.
\(2 (67) = 134\).
time = 0.28, size = 141, normalized size = 1.88 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 97, normalized size = 1.29 \begin {gather*} \frac {3 \, {\left (b d x + b c\right )} \cosh \left (b x + a\right )^{3} + 9 \, {\left (b d x + b c\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - d \sinh \left (b x + a\right )^{3} - 27 \, {\left (b d x + b c\right )} \cosh \left (b x + a\right ) - 3 \, {\left (d \cosh \left (b x + a\right )^{2} - 9 \, d\right )} \sinh \left (b x + a\right )}{36 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 126, normalized size = 1.68 \begin {gather*} \begin {cases} \frac {c \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {2 c \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac {d x \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {2 d x \cosh ^{3}{\left (a + b x \right )}}{3 b} - \frac {7 d \sinh ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {2 d \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \sinh ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 98, normalized size = 1.31 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 79, normalized size = 1.05 \begin {gather*} \frac {7\,d\,\mathrm {sinh}\left (a+b\,x\right )}{9\,b^2}-\frac {c\,\mathrm {cosh}\left (a+b\,x\right )-\frac {c\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3}+d\,x\,\mathrm {cosh}\left (a+b\,x\right )-\frac {d\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3}}{b}-\frac {d\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{9\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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