Optimal. Leaf size=155 \[ -\frac{b^5 \log (a+b \tanh (x))}{\left (a^2-b^2\right )^3}-\frac{\left (3 a^2+9 a b+8 b^2\right ) \log (1-\tanh (x))}{16 (a+b)^3}+\frac{\left (3 a^2-9 a b+8 b^2\right ) \log (\tanh (x)+1)}{16 (a-b)^3}-\frac{\cosh ^4(x) (b-a \tanh (x))}{4 \left (a^2-b^2\right )}+\frac{\cosh ^2(x) \left (4 b^3-a b^2 \left (7-\frac{3 a^2}{b^2}\right ) \tanh (x)\right )}{8 \left (a^2-b^2\right )^2} \]
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Rubi [A] time = 0.236122, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3506, 741, 823, 801} \[ -\frac{b^5 \log (a+b \tanh (x))}{\left (a^2-b^2\right )^3}-\frac{\left (3 a^2+9 a b+8 b^2\right ) \log (1-\tanh (x))}{16 (a+b)^3}+\frac{\left (3 a^2-9 a b+8 b^2\right ) \log (\tanh (x)+1)}{16 (a-b)^3}-\frac{\cosh ^4(x) (b-a \tanh (x))}{4 \left (a^2-b^2\right )}+\frac{\cosh ^2(x) \left (4 b^3-a b^2 \left (7-\frac{3 a^2}{b^2}\right ) \tanh (x)\right )}{8 \left (a^2-b^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 3506
Rule 741
Rule 823
Rule 801
Rubi steps
\begin{align*} \int \frac{\cosh ^4(x)}{a+b \tanh (x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x) \left (1-\frac{x^2}{b^2}\right )^3} \, dx,x,b \tanh (x)\right )}{b}\\ &=-\frac{\cosh ^4(x) (b-a \tanh (x))}{4 \left (a^2-b^2\right )}+\frac{b \operatorname{Subst}\left (\int \frac{-4+\frac{3 a^2}{b^2}+\frac{3 a x}{b^2}}{(a+x) \left (1-\frac{x^2}{b^2}\right )^2} \, dx,x,b \tanh (x)\right )}{4 \left (a^2-b^2\right )}\\ &=-\frac{\cosh ^4(x) (b-a \tanh (x))}{4 \left (a^2-b^2\right )}+\frac{\cosh ^2(x) \left (4 b^3-a \left (7-\frac{3 a^2}{b^2}\right ) b^2 \tanh (x)\right )}{8 \left (a^2-b^2\right )^2}-\frac{b^5 \operatorname{Subst}\left (\int \frac{-\frac{3 a^4-7 a^2 b^2+8 b^4}{b^6}+\frac{a \left (7-\frac{3 a^2}{b^2}\right ) x}{b^4}}{(a+x) \left (1-\frac{x^2}{b^2}\right )} \, dx,x,b \tanh (x)\right )}{8 \left (a^2-b^2\right )^2}\\ &=-\frac{\cosh ^4(x) (b-a \tanh (x))}{4 \left (a^2-b^2\right )}+\frac{\cosh ^2(x) \left (4 b^3-a \left (7-\frac{3 a^2}{b^2}\right ) b^2 \tanh (x)\right )}{8 \left (a^2-b^2\right )^2}-\frac{b^5 \operatorname{Subst}\left (\int \left (-\frac{(a-b)^2 \left (3 a^2+9 a b+8 b^2\right )}{2 b^5 (a+b) (b-x)}+\frac{8}{(a-b) (a+b) (a+x)}-\frac{(a+b)^2 \left (3 a^2-9 a b+8 b^2\right )}{2 (a-b) b^5 (b+x)}\right ) \, dx,x,b \tanh (x)\right )}{8 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (3 a^2+9 a b+8 b^2\right ) \log (1-\tanh (x))}{16 (a+b)^3}+\frac{\left (3 a^2-9 a b+8 b^2\right ) \log (1+\tanh (x))}{16 (a-b)^3}-\frac{b^5 \log (a+b \tanh (x))}{\left (a^2-b^2\right )^3}-\frac{\cosh ^4(x) (b-a \tanh (x))}{4 \left (a^2-b^2\right )}+\frac{\cosh ^2(x) \left (4 b^3-a \left (7-\frac{3 a^2}{b^2}\right ) b^2 \tanh (x)\right )}{8 \left (a^2-b^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.194281, size = 156, normalized size = 1.01 \[ \frac{-40 a^3 b^2 x-24 a^3 b^2 \sinh (2 x)-2 a^3 b^2 \sinh (4 x)-4 b \left (-4 a^2 b^2+a^4+3 b^4\right ) \cosh (2 x)-b \left (a^2-b^2\right )^2 \cosh (4 x)+12 a^5 x+8 a^5 \sinh (2 x)+a^5 \sinh (4 x)+60 a b^4 x+16 a b^4 \sinh (2 x)+a b^4 \sinh (4 x)-32 b^5 \log (a \cosh (x)+b \sinh (x))}{32 (a-b)^3 (a+b)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.049, size = 354, normalized size = 2.3 \begin{align*} -{\frac{1}{4\,a-4\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-4}}+2\,{\frac{1}{ \left ( 4\,a-4\,b \right ) \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{3}}}+{\frac{5\,a}{8\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{7\,b}{8\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{7\,a}{8\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}+{\frac{9\,b}{8\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}+{\frac{3\,{a}^{2}}{8\, \left ( a-b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{9\,ab}{8\, \left ( a-b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+{\frac{{b}^{2}}{ \left ( a-b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{{b}^{5}}{ \left ( a-b \right ) ^{3} \left ( a+b \right ) ^{3}}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+2\,\tanh \left ( x/2 \right ) b+a \right ) }+{\frac{1}{4\,a+4\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-4}}+2\,{\frac{1}{ \left ( 4\,a+4\,b \right ) \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{3}}}+{\frac{7\,a}{8\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{\frac{9\,b}{8\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{\frac{5\,a}{8\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{7\,b}{8\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}-{\frac{3\,{a}^{2}}{8\, \left ( a+b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }-{\frac{9\,ab}{8\, \left ( a+b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }-{\frac{{b}^{2}}{ \left ( a+b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16831, size = 223, normalized size = 1.44 \begin{align*} -\frac{b^{5} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac{{\left (3 \, a^{2} + 9 \, a b + 8 \, b^{2}\right )} x}{8 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac{{\left (4 \,{\left (2 \, a + 3 \, b\right )} e^{\left (-2 \, x\right )} + a + b\right )} e^{\left (4 \, x\right )}}{64 \,{\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac{4 \,{\left (2 \, a - 3 \, b\right )} e^{\left (-2 \, x\right )} +{\left (a - b\right )} e^{\left (-4 \, x\right )}}{64 \,{\left (a^{2} - 2 \, a b + b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.39283, size = 2882, normalized size = 18.59 \begin{align*} \text{result too large to display} \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 = 1.24017, size = 306, normalized size = 1.97 \begin{align*} -\frac{b^{5} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac{{\left (3 \, a^{2} - 9 \, a b + 8 \, b^{2}\right )} x}{8 \,{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} - \frac{{\left (18 \, a^{2} e^{\left (4 \, x\right )} - 54 \, a b e^{\left (4 \, x\right )} + 48 \, b^{2} e^{\left (4 \, x\right )} + 8 \, a^{2} e^{\left (2 \, x\right )} - 20 \, a b e^{\left (2 \, x\right )} + 12 \, b^{2} e^{\left (2 \, x\right )} + a^{2} - 2 \, a b + b^{2}\right )} e^{\left (-4 \, x\right )}}{64 \,{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac{a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 8 \, a e^{\left (2 \, x\right )} + 12 \, b e^{\left (2 \, x\right )}}{64 \,{\left (a^{2} + 2 \, a b + b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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