Optimal. Leaf size=33 \[ i \text {Int}\left (-\frac {i \sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)},x\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx &=i \int -\frac {i \sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx\\ \end {align*}
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Mathematica [A] time = 0.53, size = 826, normalized size = 25.03 \[ \frac {\cosh (3 (c+d x)) a^3+27 b \sinh (c+d x) a^2-b \sinh (3 (c+d x)) a^2-9 \left (a^2+3 b^2\right ) \cosh (c+d x) a-b^2 \cosh (3 (c+d x)) a-2 b \text {RootSum}\left [a \text {$\#$1}^6+b \text {$\#$1}^6+3 a \text {$\#$1}^4-3 b \text {$\#$1}^4+3 a \text {$\#$1}^2+3 b \text {$\#$1}^2+a-b\& ,\frac {3 a^2 c \text {$\#$1}^4+3 b^2 c \text {$\#$1}^4-3 a b c \text {$\#$1}^4+3 a^2 d x \text {$\#$1}^4+3 b^2 d x \text {$\#$1}^4-3 a b d x \text {$\#$1}^4+6 a^2 \log \left (\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+6 b^2 \log \left (\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-6 a b \log \left (\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+2 a^2 c \text {$\#$1}^2-2 b^2 c \text {$\#$1}^2+2 a^2 d x \text {$\#$1}^2-2 b^2 d x \text {$\#$1}^2+4 a^2 \log \left (\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-4 b^2 \log \left (\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+3 a^2 c+3 b^2 c+3 a b c+3 a^2 d x+3 b^2 d x+3 a b d x+6 a^2 \log \left (\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+6 b^2 \log \left (\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+6 a b \log \left (\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )}{a \text {$\#$1}^5+b \text {$\#$1}^5+2 a \text {$\#$1}^3-2 b \text {$\#$1}^3+a \text {$\#$1}+b \text {$\#$1}}\& \right ] a+9 b^3 \sinh (c+d x)+b^3 \sinh (3 (c+d x))}{12 (a-b)^2 (a+b)^2 d} \]
Antiderivative was successfully verified.
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fricas [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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giac [A] time = 3.16, size = 344, normalized size = 10.42 \[ -\frac {\frac {{\left (9 \, a e^{\left (2 \, d x + 2 \, c\right )} + 9 \, b e^{\left (2 \, d x + 2 \, c\right )} - a + b\right )} e^{\left (-3 \, d x\right )}}{a^{2} e^{\left (3 \, c\right )} - 2 \, a b e^{\left (3 \, c\right )} + b^{2} e^{\left (3 \, c\right )}} - \frac {a^{2} e^{\left (3 \, d x + 30 \, c\right )} + 2 \, a b e^{\left (3 \, d x + 30 \, c\right )} + b^{2} e^{\left (3 \, d x + 30 \, c\right )} - 9 \, a^{2} e^{\left (d x + 28 \, c\right )} + 9 \, b^{2} e^{\left (d x + 28 \, c\right )}}{a^{3} e^{\left (27 \, c\right )} + 3 \, a^{2} b e^{\left (27 \, c\right )} + 3 \, a b^{2} e^{\left (27 \, c\right )} + b^{3} e^{\left (27 \, c\right )}}}{24 \, d} - \frac {\frac {6 \, {\left (a^{3} b e^{c} + a^{2} b^{2} e^{c} + a b^{3} e^{c}\right )} d x}{a - b} - \frac {{\left (a^{3} b e^{c} + a^{2} b^{2} e^{c} + a b^{3} e^{c}\right )} \log \left ({\left | a e^{\left (6 \, d x + 6 \, c\right )} + b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + a - b \right |}\right )}{a - b}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.47, size = 346, normalized size = 10.48 \[ -\frac {a b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (2 a^{2}+b^{2}\right ) \textit {\_R}^{4}-6 \textit {\_R}^{3} a b +2 \left (4 a^{2}+5 b^{2}\right ) \textit {\_R}^{2}-6 a b \textit {\_R} +2 a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 d \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {16}{3 d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (16 a +16 b \right )}-\frac {8}{d \left (16 a +16 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {a}{2 d \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {b}{d \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {8}{d \left (16 a -16 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {16}{3 d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3} \left (16 a -16 b \right )}-\frac {a}{2 d \left (a -b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {b}{d \left (a -b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (a^{3} + a^{2} b - a b^{2} - b^{3} + {\left (a^{3} e^{\left (6 \, c\right )} - a^{2} b e^{\left (6 \, c\right )} - a b^{2} e^{\left (6 \, c\right )} + b^{3} e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} - 9 \, {\left (a^{3} e^{\left (4 \, c\right )} - 3 \, a^{2} b e^{\left (4 \, c\right )} + 3 \, a b^{2} e^{\left (4 \, c\right )} - b^{3} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - 9 \, {\left (a^{3} e^{\left (2 \, c\right )} + 3 \, a^{2} b e^{\left (2 \, c\right )} + 3 \, a b^{2} e^{\left (2 \, c\right )} + b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )} e^{\left (-3 \, d x\right )}}{24 \, {\left (a^{4} d e^{\left (3 \, c\right )} - 2 \, a^{2} b^{2} d e^{\left (3 \, c\right )} + b^{4} d e^{\left (3 \, c\right )}\right )}} - \frac {1}{8} \, \int \frac {16 \, {\left (3 \, {\left (a^{3} b e^{\left (5 \, c\right )} - a^{2} b^{2} e^{\left (5 \, c\right )} + a b^{3} e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} + 2 \, {\left (a^{3} b e^{\left (3 \, c\right )} - a b^{3} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + 3 \, {\left (a^{3} b e^{c} + a^{2} b^{2} e^{c} + a b^{3} e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5} + {\left (a^{5} e^{\left (6 \, c\right )} + a^{4} b e^{\left (6 \, c\right )} - 2 \, a^{3} b^{2} e^{\left (6 \, c\right )} - 2 \, a^{2} b^{3} e^{\left (6 \, c\right )} + a b^{4} e^{\left (6 \, c\right )} + b^{5} e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 3 \, {\left (a^{5} e^{\left (4 \, c\right )} - a^{4} b e^{\left (4 \, c\right )} - 2 \, a^{3} b^{2} e^{\left (4 \, c\right )} + 2 \, a^{2} b^{3} e^{\left (4 \, c\right )} + a b^{4} e^{\left (4 \, c\right )} - b^{5} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 3 \, {\left (a^{5} e^{\left (2 \, c\right )} + a^{4} b e^{\left (2 \, c\right )} - 2 \, a^{3} b^{2} e^{\left (2 \, c\right )} - 2 \, a^{2} b^{3} e^{\left (2 \, c\right )} + a b^{4} e^{\left (2 \, c\right )} + b^{5} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.03 \[ \text {Hanged} \]
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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