Optimal. Leaf size=195 \[ -\frac {3 \cosh (a+x (b-3 d)-3 c)}{32 (b-3 d)}-\frac {9 \cosh (a+x (b-d)-c)}{32 (b-d)}+\frac {\cosh (3 (a-c)+3 x (b-d))}{96 (b-d)}+\frac {3 \cosh (3 a+x (3 b-d)-c)}{32 (3 b-d)}-\frac {9 \cosh (a+x (b+d)+c)}{32 (b+d)}+\frac {\cosh (3 (a+c)+3 x (b+d))}{96 (b+d)}+\frac {3 \cosh (3 a+x (3 b+d)+c)}{32 (3 b+d)}-\frac {3 \cosh (a+x (b+3 d)+3 c)}{32 (b+3 d)} \]
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Rubi [A] time = 0.15, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {5618, 2638} \[ -\frac {3 \cosh (a+x (b-3 d)-3 c)}{32 (b-3 d)}-\frac {9 \cosh (a+x (b-d)-c)}{32 (b-d)}+\frac {\cosh (3 (a-c)+3 x (b-d))}{96 (b-d)}+\frac {3 \cosh (3 a+x (3 b-d)-c)}{32 (3 b-d)}-\frac {9 \cosh (a+x (b+d)+c)}{32 (b+d)}+\frac {\cosh (3 (a+c)+3 x (b+d))}{96 (b+d)}+\frac {3 \cosh (3 a+x (3 b+d)+c)}{32 (3 b+d)}-\frac {3 \cosh (a+x (b+3 d)+3 c)}{32 (b+3 d)} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 5618
Rubi steps
\begin {align*} \int \cosh ^3(c+d x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac {3}{32} \sinh (a-3 c+(b-3 d) x)-\frac {9}{32} \sinh (a-c+(b-d) x)+\frac {1}{32} \sinh (3 (a-c)+3 (b-d) x)+\frac {3}{32} \sinh (3 a-c+(3 b-d) x)-\frac {9}{32} \sinh (a+c+(b+d) x)+\frac {1}{32} \sinh (3 (a+c)+3 (b+d) x)+\frac {3}{32} \sinh (3 a+c+(3 b+d) x)-\frac {3}{32} \sinh (a+3 c+(b+3 d) x)\right ) \, dx\\ &=\frac {1}{32} \int \sinh (3 (a-c)+3 (b-d) x) \, dx+\frac {1}{32} \int \sinh (3 (a+c)+3 (b+d) x) \, dx-\frac {3}{32} \int \sinh (a-3 c+(b-3 d) x) \, dx+\frac {3}{32} \int \sinh (3 a-c+(3 b-d) x) \, dx+\frac {3}{32} \int \sinh (3 a+c+(3 b+d) x) \, dx-\frac {3}{32} \int \sinh (a+3 c+(b+3 d) x) \, dx-\frac {9}{32} \int \sinh (a-c+(b-d) x) \, dx-\frac {9}{32} \int \sinh (a+c+(b+d) x) \, dx\\ &=-\frac {3 \cosh (a-3 c+(b-3 d) x)}{32 (b-3 d)}-\frac {9 \cosh (a-c+(b-d) x)}{32 (b-d)}+\frac {\cosh (3 (a-c)+3 (b-d) x)}{96 (b-d)}+\frac {3 \cosh (3 a-c+(3 b-d) x)}{32 (3 b-d)}-\frac {9 \cosh (a+c+(b+d) x)}{32 (b+d)}+\frac {\cosh (3 (a+c)+3 (b+d) x)}{96 (b+d)}+\frac {3 \cosh (3 a+c+(3 b+d) x)}{32 (3 b+d)}-\frac {3 \cosh (a+3 c+(b+3 d) x)}{32 (b+3 d)}\\ \end {align*}
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Mathematica [A] time = 1.69, size = 176, normalized size = 0.90 \[ \frac {1}{96} \left (-\frac {9 \cosh (a+b x-3 c-3 d x)}{b-3 d}-\frac {27 \cosh (a+b x-c-d x)}{b-d}+\frac {\cosh (3 (a+b x-c-d x))}{b-d}+\frac {9 \cosh (3 a+3 b x-c-d x)}{3 b-d}+\frac {9 \cosh (3 a+3 b x+c+d x)}{3 b+d}-\frac {9 \cosh (a+b x+3 c+3 d x)}{b+3 d}-\frac {27 \cosh (a+x (b+d)+c)}{b+d}+\frac {\cosh (3 (a+x (b+d)+c))}{b+d}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 729, normalized size = 3.74 \[ \frac {{\left ({\left (9 \, b^{5} - 82 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cosh \left (b x + a\right )^{3} - 9 \, {\left (9 \, b^{5} - 10 \, b^{3} d^{2} + b d^{4}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right )^{3} - {\left ({\left (9 \, b^{4} d - 82 \, b^{2} d^{3} + 9 \, d^{5}\right )} \sinh \left (b x + a\right )^{3} - 3 \, {\left (81 \, b^{4} d - 90 \, b^{2} d^{3} + 9 \, d^{5} - {\left (9 \, b^{4} d - 82 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left ({\left (9 \, b^{5} - 82 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{3} + 27 \, {\left (b^{5} - 10 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )\right )} \sinh \left (b x + a\right )^{2} + 3 \, {\left (3 \, {\left (9 \, b^{5} - 82 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} + {\left ({\left (9 \, b^{5} - 82 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cosh \left (b x + a\right )^{3} - 9 \, {\left (9 \, b^{5} - 10 \, b^{3} d^{2} + b d^{4}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 27 \, {\left ({\left (b^{5} - 10 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cosh \left (b x + a\right )^{3} - {\left (9 \, b^{5} - 82 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) - 3 \, {\left ({\left (3 \, b^{4} d - 30 \, b^{2} d^{3} + 27 \, d^{5} + {\left (9 \, b^{4} d - 82 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (b x + a\right )^{3} - 3 \, {\left (27 \, b^{4} d - 246 \, b^{2} d^{3} + 27 \, d^{5} - 3 \, {\left (b^{4} d - 10 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cosh \left (b x + a\right )^{2} + {\left (81 \, b^{4} d - 90 \, b^{2} d^{3} + 9 \, d^{5} - {\left (9 \, b^{4} d - 82 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{48 \, {\left ({\left (9 \, b^{6} - 91 \, b^{4} d^{2} + 91 \, b^{2} d^{4} - 9 \, d^{6}\right )} \cosh \left (b x + a\right )^{4} - 2 \, {\left (9 \, b^{6} - 91 \, b^{4} d^{2} + 91 \, b^{2} d^{4} - 9 \, d^{6}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + {\left (9 \, b^{6} - 91 \, b^{4} d^{2} + 91 \, b^{2} d^{4} - 9 \, d^{6}\right )} \sinh \left (b x + a\right )^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 373, normalized size = 1.91 \[ \frac {e^{\left (3 \, b x + 3 \, d x + 3 \, a + 3 \, c\right )}}{192 \, {\left (b + d\right )}} + \frac {3 \, e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{64 \, {\left (3 \, b + d\right )}} + \frac {3 \, e^{\left (3 \, b x - d x + 3 \, a - c\right )}}{64 \, {\left (3 \, b - d\right )}} + \frac {e^{\left (3 \, b x - 3 \, d x + 3 \, a - 3 \, c\right )}}{192 \, {\left (b - d\right )}} - \frac {3 \, e^{\left (b x + 3 \, d x + a + 3 \, c\right )}}{64 \, {\left (b + 3 \, d\right )}} - \frac {9 \, e^{\left (b x + d x + a + c\right )}}{64 \, {\left (b + d\right )}} - \frac {9 \, e^{\left (b x - d x + a - c\right )}}{64 \, {\left (b - d\right )}} - \frac {3 \, e^{\left (b x - 3 \, d x + a - 3 \, c\right )}}{64 \, {\left (b - 3 \, d\right )}} - \frac {3 \, e^{\left (-b x + 3 \, d x - a + 3 \, c\right )}}{64 \, {\left (b - 3 \, d\right )}} - \frac {9 \, e^{\left (-b x + d x - a + c\right )}}{64 \, {\left (b - d\right )}} - \frac {9 \, e^{\left (-b x - d x - a - c\right )}}{64 \, {\left (b + d\right )}} - \frac {3 \, e^{\left (-b x - 3 \, d x - a - 3 \, c\right )}}{64 \, {\left (b + 3 \, d\right )}} + \frac {e^{\left (-3 \, b x + 3 \, d x - 3 \, a + 3 \, c\right )}}{192 \, {\left (b - d\right )}} + \frac {3 \, e^{\left (-3 \, b x + d x - 3 \, a + c\right )}}{64 \, {\left (3 \, b - d\right )}} + \frac {3 \, e^{\left (-3 \, b x - d x - 3 \, a - c\right )}}{64 \, {\left (3 \, b + d\right )}} + \frac {e^{\left (-3 \, b x - 3 \, d x - 3 \, a - 3 \, c\right )}}{192 \, {\left (b + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 184, normalized size = 0.94 \[ -\frac {3 \cosh \left (a -3 c +\left (b -3 d \right ) x \right )}{32 \left (b -3 d \right )}-\frac {9 \cosh \left (a -c +\left (b -d \right ) x \right )}{32 \left (b -d \right )}-\frac {9 \cosh \left (a +c +\left (b +d \right ) x \right )}{32 \left (b +d \right )}-\frac {3 \cosh \left (a +3 c +\left (b +3 d \right ) x \right )}{32 \left (b +3 d \right )}+\frac {\cosh \left (\left (3 b -3 d \right ) x +3 a -3 c \right )}{96 b -96 d}+\frac {3 \cosh \left (3 a -c +\left (3 b -d \right ) x \right )}{32 \left (3 b -d \right )}+\frac {3 \cosh \left (3 a +c +\left (3 b +d \right ) x \right )}{32 \left (3 b +d \right )}+\frac {\cosh \left (\left (3 b +3 d \right ) x +3 a +3 c \right )}{96 b +96 d} \]
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 = 2.09, size = 908, normalized size = 4.66 \[ -{\mathrm {e}}^{3\,a+c+3\,b\,x+d\,x}\,\left (\frac {-9\,b^3+3\,b^2\,d+9\,b\,d^2-3\,d^3}{576\,b^4-640\,b^2\,d^2+64\,d^4}+\frac {{\mathrm {e}}^{-6\,a-6\,b\,x}\,\left (-9\,b^3-3\,b^2\,d+9\,b\,d^2+3\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}-\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}\,\left (-81\,b^3+81\,b^2\,d+9\,b\,d^2-9\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}-\frac {{\mathrm {e}}^{-4\,a-4\,b\,x}\,\left (-81\,b^3-81\,b^2\,d+9\,b\,d^2+9\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}\right )-{\mathrm {e}}^{3\,a-c+3\,b\,x-d\,x}\,\left (\frac {-9\,b^3-3\,b^2\,d+9\,b\,d^2+3\,d^3}{576\,b^4-640\,b^2\,d^2+64\,d^4}+\frac {{\mathrm {e}}^{-6\,a-6\,b\,x}\,\left (-9\,b^3+3\,b^2\,d+9\,b\,d^2-3\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}-\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}\,\left (-81\,b^3-81\,b^2\,d+9\,b\,d^2+9\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}-\frac {{\mathrm {e}}^{-4\,a-4\,b\,x}\,\left (-81\,b^3+81\,b^2\,d+9\,b\,d^2-9\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}\right )-{\mathrm {e}}^{3\,a-3\,c+3\,b\,x-3\,d\,x}\,\left (\frac {-b^3-b^2\,d+9\,b\,d^2+9\,d^3}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}+\frac {{\mathrm {e}}^{-6\,a-6\,b\,x}\,\left (-b^3+b^2\,d+9\,b\,d^2-9\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}-\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}\,\left (-9\,b^3-27\,b^2\,d+9\,b\,d^2+27\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}-\frac {{\mathrm {e}}^{-4\,a-4\,b\,x}\,\left (-9\,b^3+27\,b^2\,d+9\,b\,d^2-27\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}\right )-{\mathrm {e}}^{3\,a+3\,c+3\,b\,x+3\,d\,x}\,\left (\frac {-b^3+b^2\,d+9\,b\,d^2-9\,d^3}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}+\frac {{\mathrm {e}}^{-6\,a-6\,b\,x}\,\left (-b^3-b^2\,d+9\,b\,d^2+9\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}-\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}\,\left (-9\,b^3+27\,b^2\,d+9\,b\,d^2-27\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}-\frac {{\mathrm {e}}^{-4\,a-4\,b\,x}\,\left (-9\,b^3-27\,b^2\,d+9\,b\,d^2+27\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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