Optimal. Leaf size=144 \[ \frac {\sinh (2 a+x (2 b-3 d)-3 c)}{16 (2 b-3 d)}+\frac {3 \sinh (2 a+x (2 b-d)-c)}{16 (2 b-d)}+\frac {3 \sinh (2 a+x (2 b+d)+c)}{16 (2 b+d)}+\frac {\sinh (2 a+x (2 b+3 d)+3 c)}{16 (2 b+3 d)}-\frac {3 \sinh (c+d x)}{8 d}-\frac {\sinh (3 c+3 d x)}{24 d} \]
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Rubi [A] time = 0.09, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {5618, 2637} \[ \frac {\sinh (2 a+x (2 b-3 d)-3 c)}{16 (2 b-3 d)}+\frac {3 \sinh (2 a+x (2 b-d)-c)}{16 (2 b-d)}+\frac {3 \sinh (2 a+x (2 b+d)+c)}{16 (2 b+d)}+\frac {\sinh (2 a+x (2 b+3 d)+3 c)}{16 (2 b+3 d)}-\frac {3 \sinh (c+d x)}{8 d}-\frac {\sinh (3 c+3 d x)}{24 d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 5618
Rubi steps
\begin {align*} \int \cosh ^3(c+d x) \sinh ^2(a+b x) \, dx &=\int \left (\frac {1}{16} \cosh (2 a-3 c+(2 b-3 d) x)+\frac {3}{16} \cosh (2 a-c+(2 b-d) x)-\frac {3}{8} \cosh (c+d x)-\frac {1}{8} \cosh (3 c+3 d x)+\frac {3}{16} \cosh (2 a+c+(2 b+d) x)+\frac {1}{16} \cosh (2 a+3 c+(2 b+3 d) x)\right ) \, dx\\ &=\frac {1}{16} \int \cosh (2 a-3 c+(2 b-3 d) x) \, dx+\frac {1}{16} \int \cosh (2 a+3 c+(2 b+3 d) x) \, dx-\frac {1}{8} \int \cosh (3 c+3 d x) \, dx+\frac {3}{16} \int \cosh (2 a-c+(2 b-d) x) \, dx+\frac {3}{16} \int \cosh (2 a+c+(2 b+d) x) \, dx-\frac {3}{8} \int \cosh (c+d x) \, dx\\ &=\frac {\sinh (2 a-3 c+(2 b-3 d) x)}{16 (2 b-3 d)}+\frac {3 \sinh (2 a-c+(2 b-d) x)}{16 (2 b-d)}-\frac {3 \sinh (c+d x)}{8 d}-\frac {\sinh (3 c+3 d x)}{24 d}+\frac {3 \sinh (2 a+c+(2 b+d) x)}{16 (2 b+d)}+\frac {\sinh (2 a+3 c+(2 b+3 d) x)}{16 (2 b+3 d)}\\ \end {align*}
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Mathematica [A] time = 1.53, size = 158, normalized size = 1.10 \[ \frac {1}{48} \left (\frac {3 \sinh (2 a+2 b x-3 c-3 d x)}{2 b-3 d}+\frac {9 \sinh (2 a+2 b x-c-d x)}{2 b-d}+\frac {9 \sinh (2 a+2 b x+c+d x)}{2 b+d}+\frac {3 \sinh (2 a+2 b x+3 c+3 d x)}{2 b+3 d}-\frac {18 \sinh (c) \cosh (d x)}{d}-\frac {2 \sinh (3 c) \cosh (3 d x)}{d}-\frac {18 \cosh (c) \sinh (d x)}{d}-\frac {2 \cosh (3 c) \sinh (3 d x)}{d}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 398, normalized size = 2.76 \[ \frac {36 \, {\left (4 \, b^{3} d - b d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) \sinh \left (d x + c\right )^{2} - {\left (16 \, b^{4} - 40 \, b^{2} d^{2} + 9 \, d^{4} + 9 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (b x + a\right )^{2} + 9 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \sinh \left (b x + a\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 12 \, {\left ({\left (4 \, b^{3} d - b d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{3} + 3 \, {\left (4 \, b^{3} d - 9 \, b d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )\right )} \sinh \left (b x + a\right ) - 3 \, {\left (48 \, b^{4} - 120 \, b^{2} d^{2} + 27 \, d^{4} + 3 \, {\left (4 \, b^{2} d^{2} - 9 \, d^{4}\right )} \cosh \left (b x + a\right )^{2} + {\left (16 \, b^{4} - 40 \, b^{2} d^{2} + 9 \, d^{4} + 9 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (4 \, b^{2} d^{2} - 9 \, d^{4} + 3 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (b x + a\right )^{2}\right )} \sinh \left (d x + c\right )}{24 \, {\left ({\left (16 \, b^{4} d - 40 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cosh \left (b x + a\right )^{2} - {\left (16 \, b^{4} d - 40 \, b^{2} d^{3} + 9 \, d^{5}\right )} \sinh \left (b x + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 260, normalized size = 1.81 \[ \frac {e^{\left (2 \, b x + 3 \, d x + 2 \, a + 3 \, c\right )}}{32 \, {\left (2 \, b + 3 \, d\right )}} + \frac {3 \, e^{\left (2 \, b x + d x + 2 \, a + c\right )}}{32 \, {\left (2 \, b + d\right )}} + \frac {3 \, e^{\left (2 \, b x - d x + 2 \, a - c\right )}}{32 \, {\left (2 \, b - d\right )}} + \frac {e^{\left (2 \, b x - 3 \, d x + 2 \, a - 3 \, c\right )}}{32 \, {\left (2 \, b - 3 \, d\right )}} - \frac {e^{\left (-2 \, b x + 3 \, d x - 2 \, a + 3 \, c\right )}}{32 \, {\left (2 \, b - 3 \, d\right )}} - \frac {3 \, e^{\left (-2 \, b x + d x - 2 \, a + c\right )}}{32 \, {\left (2 \, b - d\right )}} - \frac {3 \, e^{\left (-2 \, b x - d x - 2 \, a - c\right )}}{32 \, {\left (2 \, b + d\right )}} - \frac {e^{\left (-2 \, b x - 3 \, d x - 2 \, a - 3 \, c\right )}}{32 \, {\left (2 \, b + 3 \, d\right )}} - \frac {e^{\left (3 \, d x + 3 \, c\right )}}{48 \, d} - \frac {3 \, e^{\left (d x + c\right )}}{16 \, d} + \frac {3 \, e^{\left (-d x - c\right )}}{16 \, d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 133, normalized size = 0.92 \[ \frac {\sinh \left (2 a -3 c +\left (2 b -3 d \right ) x \right )}{32 b -48 d}+\frac {3 \sinh \left (2 a -c +\left (2 b -d \right ) x \right )}{16 \left (2 b -d \right )}-\frac {3 \sinh \left (d x +c \right )}{8 d}-\frac {\sinh \left (3 d x +3 c \right )}{24 d}+\frac {3 \sinh \left (2 a +c +\left (2 b +d \right ) x \right )}{16 \left (2 b +d \right )}+\frac {\sinh \left (2 a +3 c +\left (2 b +3 d \right ) x \right )}{32 b +48 d} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.98, size = 337, normalized size = 2.34 \[ \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (8\,b^4-26\,b^2\,d^2+9\,d^4\right )}{d\,\left (16\,b^4-40\,b^2\,d^2+9\,d^4\right )}-{\mathrm {sinh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (\frac {3\,d^3}{16\,b^4-40\,b^2\,d^2+9\,d^4}+\frac {1}{3\,d}\right )-\frac {2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {cosh}\left (c+d\,x\right )}^3\,\mathrm {sinh}\left (a+b\,x\right )\,\left (7\,b\,d^2-4\,b^3\right )}{16\,b^4-40\,b^2\,d^2+9\,d^4}-\frac {2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (4\,b^4-7\,b^2\,d^2\right )}{d\,\left (16\,b^4-40\,b^2\,d^2+9\,d^4\right )}-{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (\frac {3\,d^3}{16\,b^4-40\,b^2\,d^2+9\,d^4}-\frac {1}{3\,d}\right )+\frac {12\,b\,d^2\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^2}{16\,b^4-40\,b^2\,d^2+9\,d^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 108.48, size = 2006, normalized size = 13.93 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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