Optimal. Leaf size=32 \[ i \text{Unintegrable}\left (-\frac{i \sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)},x\right ) \]
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Rubi [A] time = 0.0461342, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}
Mathematica [A] time = 0.483965, size = 826, normalized size = 25.81 \[ \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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Maple [A] time = 0.11, size = 346, normalized size = 10.8 \begin{align*} -8\,{\frac{1}{d \left ( 16\,a-16\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}+{\frac{16}{3\,d \left ( 16\,a-16\,b \right ) } \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}-{\frac{a}{2\,d \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{b}{d \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{16}{3\,d \left ( 16\,a+16\,b \right ) } \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}-8\,{\frac{1}{d \left ( 16\,a+16\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{2}}}+{\frac{a}{2\,d \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{b}{d \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{ab}{3\,d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2}}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}+3\,a{{\it \_Z}}^{4}+8\,b{{\it \_Z}}^{3}+3\,a{{\it \_Z}}^{2}+a \right ) }{\frac{ \left ( 2\,{a}^{2}+{b}^{2} \right ){{\it \_R}}^{4}-6\,{{\it \_R}}^{3}ab+2\, \left ( 4\,{a}^{2}+5\,{b}^{2} \right ){{\it \_R}}^{2}-6\,ab{\it \_R}+2\,{a}^{2}+{b}^{2}}{{{\it \_R}}^{5}a+2\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{2}b+{\it \_R}\,a}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.05765, size = 473, normalized size = 14.78 \begin{align*} -\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 d - b d} - \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 d - b d}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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