Optimal. Leaf size=317 \[ -\frac{2 b^{4/3} \tanh ^{-1}\left (\frac{\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^{2/3}+b^{2/3}}}\right )}{3 a^2 d \sqrt{a^{2/3}+b^{2/3}}}-\frac{2 b^{4/3} \tan ^{-1}\left (\frac{(-1)^{5/6} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{-(-1)^{2/3} a^{2/3}-b^{2/3}}}\right )}{3 a^2 d \sqrt{-(-1)^{2/3} a^{2/3}-b^{2/3}}}-\frac{2 b^{4/3} \tan ^{-1}\left (\frac{\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^2 d \sqrt{\sqrt [3]{-1} a^{2/3}-b^{2/3}}}+\frac{b \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac{\coth ^3(c+d x)}{3 a d}+\frac{\coth (c+d x)}{a d} \]
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Rubi [A] time = 0.432501, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {3220, 3770, 3767, 2660, 618, 206, 204} \[ -\frac{2 b^{4/3} \tanh ^{-1}\left (\frac{\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^{2/3}+b^{2/3}}}\right )}{3 a^2 d \sqrt{a^{2/3}+b^{2/3}}}-\frac{2 b^{4/3} \tan ^{-1}\left (\frac{(-1)^{5/6} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{-(-1)^{2/3} a^{2/3}-b^{2/3}}}\right )}{3 a^2 d \sqrt{-(-1)^{2/3} a^{2/3}-b^{2/3}}}-\frac{2 b^{4/3} \tan ^{-1}\left (\frac{\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^2 d \sqrt{\sqrt [3]{-1} a^{2/3}-b^{2/3}}}+\frac{b \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac{\coth ^3(c+d x)}{3 a d}+\frac{\coth (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 3220
Rule 3770
Rule 3767
Rule 2660
Rule 618
Rule 206
Rule 204
Rubi steps
\begin{align*} \int \frac{\text{csch}^4(c+d x)}{a+b \sinh ^3(c+d x)} \, dx &=\int \left (-\frac{b \text{csch}(c+d x)}{a^2}+\frac{\text{csch}^4(c+d x)}{a}-\frac{b^2 \sinh ^2(c+d x)}{a^2 \left (-a-b \sinh ^3(c+d x)\right )}\right ) \, dx\\ &=\frac{\int \text{csch}^4(c+d x) \, dx}{a}-\frac{b \int \text{csch}(c+d x) \, dx}{a^2}-\frac{b^2 \int \frac{\sinh ^2(c+d x)}{-a-b \sinh ^3(c+d x)} \, dx}{a^2}\\ &=\frac{b \tanh ^{-1}(\cosh (c+d x))}{a^2 d}+\frac{b^2 \int \left (-\frac{i}{3 b^{2/3} \left (-i \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)\right )}-\frac{i}{3 b^{2/3} \left (\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)\right )}-\frac{i}{3 b^{2/3} \left ((-1)^{5/6} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)\right )}\right ) \, dx}{a^2}+\frac{i \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth (c+d x)\right )}{a d}\\ &=\frac{b \tanh ^{-1}(\cosh (c+d x))}{a^2 d}+\frac{\coth (c+d x)}{a d}-\frac{\coth ^3(c+d x)}{3 a d}-\frac{\left (i b^{4/3}\right ) \int \frac{1}{-i \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a^2}-\frac{\left (i b^{4/3}\right ) \int \frac{1}{\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a^2}-\frac{\left (i b^{4/3}\right ) \int \frac{1}{(-1)^{5/6} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a^2}\\ &=\frac{b \tanh ^{-1}(\cosh (c+d x))}{a^2 d}+\frac{\coth (c+d x)}{a d}-\frac{\coth ^3(c+d x)}{3 a d}-\frac{\left (2 b^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-i \sqrt [3]{a}-2 \sqrt [3]{b} x-i \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{3 a^2 d}-\frac{\left (2 b^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [6]{-1} \sqrt [3]{a}-2 \sqrt [3]{b} x+\sqrt [6]{-1} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{3 a^2 d}-\frac{\left (2 b^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{(-1)^{5/6} \sqrt [3]{a}-2 \sqrt [3]{b} x+(-1)^{5/6} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{3 a^2 d}\\ &=\frac{b \tanh ^{-1}(\cosh (c+d x))}{a^2 d}+\frac{\coth (c+d x)}{a d}-\frac{\coth ^3(c+d x)}{3 a d}+\frac{\left (4 b^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (\sqrt [3]{-1} a^{2/3}-b^{2/3}\right )-x^2} \, dx,x,-2 \sqrt [3]{b}+2 \sqrt [6]{-1} \sqrt [3]{a} \tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{3 a^2 d}+\frac{\left (4 b^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (a^{2/3}+b^{2/3}\right )-x^2} \, dx,x,-2 \sqrt [3]{b}-2 i \sqrt [3]{a} \tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{3 a^2 d}+\frac{\left (4 b^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left ((-1)^{2/3} a^{2/3}+b^{2/3}\right )-x^2} \, dx,x,-2 \sqrt [3]{b}+2 (-1)^{5/6} \sqrt [3]{a} \tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{3 a^2 d}\\ &=\frac{2 b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{-(-1)^{2/3} a^{2/3}-b^{2/3}}}\right )}{3 a^2 \sqrt{-(-1)^{2/3} a^{2/3}-b^{2/3}} d}+\frac{2 b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^2 \sqrt{\sqrt [3]{-1} a^{2/3}-b^{2/3}} d}+\frac{b \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac{2 b^{4/3} \tanh ^{-1}\left (\frac{\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^{2/3}+b^{2/3}}}\right )}{3 a^2 \sqrt{a^{2/3}+b^{2/3}} d}+\frac{\coth (c+d x)}{a d}-\frac{\coth ^3(c+d x)}{3 a d}\\ \end{align*}
Mathematica [C] time = 6.06156, size = 370, normalized size = 1.17 \[ \frac{4 b^2 \text{RootSum}\left [8 \text{$\#$1}^3 a+\text{$\#$1}^6 b-3 \text{$\#$1}^4 b+3 \text{$\#$1}^2 b-b\& ,\frac{2 \text{$\#$1}^4 \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )-4 \text{$\#$1}^2 \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+\text{$\#$1}^4 c-2 \text{$\#$1}^2 c+\text{$\#$1}^4 d x-2 \text{$\#$1}^2 d x+2 \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+c+d x}{4 \text{$\#$1}^2 a+\text{$\#$1}^5 b-2 \text{$\#$1}^3 b+\text{$\#$1} b}\& \right ]+8 a \tanh \left (\frac{1}{2} (c+d x)\right )+8 a \coth \left (\frac{1}{2} (c+d x)\right )+8 a \sinh ^4\left (\frac{1}{2} (c+d x)\right ) \text{csch}^3(c+d x)-\frac{1}{2} a \sinh (c+d x) \text{csch}^4\left (\frac{1}{2} (c+d x)\right )-24 b \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{24 a^2 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.087, size = 178, normalized size = 0.6 \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{4\,{b}^{2}}{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}}^{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{4 \,{\left (3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}}{3 \,{\left (a d e^{\left (6 \, d x + 6 \, c\right )} - 3 \, a d e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a d e^{\left (2 \, d x + 2 \, c\right )} - a 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} + 16 \, \int \frac{b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 2 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )} + b^{2} e^{\left (d x + c\right )}}{8 \,{\left (a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 3 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{3} e^{\left (3 \, d x + 3 \, c\right )} + 3 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - a^{2} b\right )}}\,{d x} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}\left (d x + c\right )^{4}}{b \sinh \left (d x + c\right )^{3} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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