Optimal. Leaf size=215 \[ \frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 a^{4/3} d}+\frac{b \log \left (a+b \tanh ^3(c+d x)\right )}{3 a^2 d}-\frac{b \log (\tanh (c+d x))}{a^2 d}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 a^{4/3} d}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} d}-\frac{\coth ^3(c+d x)}{3 a d}+\frac{\coth (c+d x)}{a d} \]
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Rubi [A] time = 0.238659, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {3663, 1834, 1871, 12, 292, 31, 634, 617, 204, 628, 260} \[ \frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 a^{4/3} d}+\frac{b \log \left (a+b \tanh ^3(c+d x)\right )}{3 a^2 d}-\frac{b \log (\tanh (c+d x))}{a^2 d}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 a^{4/3} d}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} d}-\frac{\coth ^3(c+d x)}{3 a d}+\frac{\coth (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 3663
Rule 1834
Rule 1871
Rule 12
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rule 260
Rubi steps
\begin{align*} \int \frac{\text{csch}^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1-x^2}{x^4 \left (a+b x^3\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a x^4}-\frac{1}{a x^2}-\frac{b}{a^2 x}+\frac{b x (a+b x)}{a^2 \left (a+b x^3\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\coth (c+d x)}{a d}-\frac{\coth ^3(c+d x)}{3 a d}-\frac{b \log (\tanh (c+d x))}{a^2 d}+\frac{b \operatorname{Subst}\left (\int \frac{x (a+b x)}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{a^2 d}\\ &=\frac{\coth (c+d x)}{a d}-\frac{\coth ^3(c+d x)}{3 a d}-\frac{b \log (\tanh (c+d x))}{a^2 d}+\frac{b \operatorname{Subst}\left (\int \frac{a x}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{a^2 d}+\frac{b^2 \operatorname{Subst}\left (\int \frac{x^2}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{a^2 d}\\ &=\frac{\coth (c+d x)}{a d}-\frac{\coth ^3(c+d x)}{3 a d}-\frac{b \log (\tanh (c+d x))}{a^2 d}+\frac{b \log \left (a+b \tanh ^3(c+d x)\right )}{3 a^2 d}+\frac{b \operatorname{Subst}\left (\int \frac{x}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{a d}\\ &=\frac{\coth (c+d x)}{a d}-\frac{\coth ^3(c+d x)}{3 a d}-\frac{b \log (\tanh (c+d x))}{a^2 d}+\frac{b \log \left (a+b \tanh ^3(c+d x)\right )}{3 a^2 d}-\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\tanh (c+d x)\right )}{3 a^{4/3} d}+\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tanh (c+d x)\right )}{3 a^{4/3} d}\\ &=\frac{\coth (c+d x)}{a d}-\frac{\coth ^3(c+d x)}{3 a d}-\frac{b \log (\tanh (c+d x))}{a^2 d}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 a^{4/3} d}+\frac{b \log \left (a+b \tanh ^3(c+d x)\right )}{3 a^2 d}+\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tanh (c+d x)\right )}{6 a^{4/3} d}+\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tanh (c+d x)\right )}{2 a d}\\ &=\frac{\coth (c+d x)}{a d}-\frac{\coth ^3(c+d x)}{3 a d}-\frac{b \log (\tanh (c+d x))}{a^2 d}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 a^{4/3} d}+\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 a^{4/3} d}+\frac{b \log \left (a+b \tanh ^3(c+d x)\right )}{3 a^2 d}+\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt [3]{a}}\right )}{a^{4/3} d}\\ &=-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} a^{4/3} d}+\frac{\coth (c+d x)}{a d}-\frac{\coth ^3(c+d x)}{3 a d}-\frac{b \log (\tanh (c+d x))}{a^2 d}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 a^{4/3} d}+\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 a^{4/3} d}+\frac{b \log \left (a+b \tanh ^3(c+d x)\right )}{3 a^2 d}\\ \end{align*}
Mathematica [C] time = 3.15715, size = 322, normalized size = 1.5 \[ \frac{b \text{RootSum}\left [\text{$\#$1}^3 a+3 \text{$\#$1}^2 a+\text{$\#$1}^3 b-3 \text{$\#$1}^2 b+3 \text{$\#$1} a+3 \text{$\#$1} b+a-b\& ,\frac{-\text{$\#$1}^2 a \log \left (e^{2 (c+d x)}-\text{$\#$1}\right )+2 \text{$\#$1}^2 a c+2 \text{$\#$1}^2 a d x-\text{$\#$1}^2 b \log \left (e^{2 (c+d x)}-\text{$\#$1}\right )+2 \text{$\#$1}^2 b c+2 \text{$\#$1}^2 b d x+4 \text{$\#$1} a \log \left (e^{2 (c+d x)}-\text{$\#$1}\right )+a \log \left (e^{2 (c+d x)}-\text{$\#$1}\right )-8 \text{$\#$1} a c-8 \text{$\#$1} a d x+2 \text{$\#$1} b \log \left (e^{2 (c+d x)}-\text{$\#$1}\right )-b \log \left (e^{2 (c+d x)}-\text{$\#$1}\right )-4 \text{$\#$1} b c-4 \text{$\#$1} b d x-2 a c-2 a d x+2 b c+2 b d x}{\text{$\#$1}^2 a-\text{$\#$1}^2 b+2 \text{$\#$1} a+2 \text{$\#$1} b+a-b}\& \right ]-a \coth (c+d x) \left (\text{csch}^2(c+d x)-2\right )+3 b (-\log (\sinh (c+d x))+c+d x)}{3 a^2 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.131, size = 187, normalized size = 0.9 \begin{align*} -{\frac{1}{24\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{3}{8\,da}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{1}{24\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}+{\frac{3}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{b}{d{a}^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+{\frac{b}{3\,d{a}^{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{{{\it \_R}}^{5}a+4\,{{\it \_R}}^{2}b+3\,{\it \_R}\,a}{{{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} 2 \, a b{\left (\frac{-{\left (a - b\right )} \int \frac{1}{{\left (a e^{\left (6 \, c\right )} + b e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 3 \,{\left (a e^{\left (4 \, c\right )} - b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 3 \,{\left (a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} + a - b}\,{d x} + x}{a^{3} - a^{2} b} - \frac{d x + c}{{\left (a^{3} - a^{2} b\right )} d}\right )} - 2 \, b^{2}{\left (\frac{-{\left (a - b\right )} \int \frac{1}{{\left (a e^{\left (6 \, c\right )} + b e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 3 \,{\left (a e^{\left (4 \, c\right )} - b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 3 \,{\left (a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} + a - b}\,{d x} + x}{a^{3} - a^{2} b} - \frac{d x + c}{{\left (a^{3} - a^{2} b\right )} d}\right )} + \frac{0 \, }{a} + \frac{0 \, }{a^{2}} - \frac{0 \, }{a} - \frac{0 \, }{a^{2}} + \frac{2 \,{\left (3 \, b d x e^{\left (6 \, d x + 6 \, c\right )} - 9 \, b d x e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b d x + 3 \,{\left (3 \, b d x e^{\left (2 \, c\right )} - 2 \, a e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} + 2 \, a\right )}}{3 \,{\left (a^{2} d e^{\left (6 \, d x + 6 \, c\right )} - 3 \, a^{2} d e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a^{2} d e^{\left (2 \, d x + 2 \, c\right )} - a^{2} d\right )}} - \frac{b \log \left ({\left (e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{a^{2} d} - \frac{b \log \left ({\left (e^{\left (d x + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 17.1023, size = 4415, normalized size = 20.53 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}^{4}{\left (c + d x \right )}}{a + b \tanh ^{3}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.42274, size = 243, normalized size = 1.13 \begin{align*} \frac{\frac{2 \, b \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^{2}} - \frac{6 \, b \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{a^{2}} + \frac{11 \, b e^{\left (6 \, d x + 6 \, c\right )} - 33 \, b e^{\left (4 \, d x + 4 \, c\right )} - 24 \, a e^{\left (2 \, d x + 2 \, c\right )} + 33 \, b e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a - 11 \, b}{a^{2}{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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