Optimal. Leaf size=109 \[ -\frac{b^2 \left (17 a^2-2 b^2\right ) \coth (c+d x)}{3 d}-\frac{2 a b \left (2 a^2-b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{d}+a^4 x-\frac{4 a b^3 \coth (c+d x) \text{csch}(c+d x)}{3 d}-\frac{b^2 \coth (c+d x) (a+b \text{csch}(c+d x))^2}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.127954, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {3782, 4048, 3770, 3767, 8} \[ -\frac{b^2 \left (17 a^2-2 b^2\right ) \coth (c+d x)}{3 d}-\frac{2 a b \left (2 a^2-b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{d}+a^4 x-\frac{4 a b^3 \coth (c+d x) \text{csch}(c+d x)}{3 d}-\frac{b^2 \coth (c+d x) (a+b \text{csch}(c+d x))^2}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3782
Rule 4048
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int (a+b \text{csch}(c+d x))^4 \, dx &=-\frac{b^2 \coth (c+d x) (a+b \text{csch}(c+d x))^2}{3 d}+\frac{1}{3} \int (a+b \text{csch}(c+d x)) \left (3 a^3+b \left (9 a^2-2 b^2\right ) \text{csch}(c+d x)+8 a b^2 \text{csch}^2(c+d x)\right ) \, dx\\ &=-\frac{4 a b^3 \coth (c+d x) \text{csch}(c+d x)}{3 d}-\frac{b^2 \coth (c+d x) (a+b \text{csch}(c+d x))^2}{3 d}+\frac{1}{6} \int \left (6 a^4+12 a b \left (2 a^2-b^2\right ) \text{csch}(c+d x)+2 b^2 \left (17 a^2-2 b^2\right ) \text{csch}^2(c+d x)\right ) \, dx\\ &=a^4 x-\frac{4 a b^3 \coth (c+d x) \text{csch}(c+d x)}{3 d}-\frac{b^2 \coth (c+d x) (a+b \text{csch}(c+d x))^2}{3 d}+\frac{1}{3} \left (b^2 \left (17 a^2-2 b^2\right )\right ) \int \text{csch}^2(c+d x) \, dx+\left (2 a b \left (2 a^2-b^2\right )\right ) \int \text{csch}(c+d x) \, dx\\ &=a^4 x-\frac{2 a b \left (2 a^2-b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{4 a b^3 \coth (c+d x) \text{csch}(c+d x)}{3 d}-\frac{b^2 \coth (c+d x) (a+b \text{csch}(c+d x))^2}{3 d}-\frac{\left (i b^2 \left (17 a^2-2 b^2\right )\right ) \operatorname{Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{3 d}\\ &=a^4 x-\frac{2 a b \left (2 a^2-b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{b^2 \left (17 a^2-2 b^2\right ) \coth (c+d x)}{3 d}-\frac{4 a b^3 \coth (c+d x) \text{csch}(c+d x)}{3 d}-\frac{b^2 \coth (c+d x) (a+b \text{csch}(c+d x))^2}{3 d}\\ \end{align*}
Mathematica [B] time = 6.22972, size = 508, normalized size = 4.66 \[ \frac{\sinh ^4(c+d x) \text{csch}\left (\frac{1}{2} (c+d x)\right ) \left (b^4 \cosh \left (\frac{1}{2} (c+d x)\right )-9 a^2 b^2 \cosh \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \text{csch}(c+d x))^4}{3 d (a \sinh (c+d x)+b)^4}+\frac{\sinh ^4(c+d x) \text{sech}\left (\frac{1}{2} (c+d x)\right ) \left (b^4 \sinh \left (\frac{1}{2} (c+d x)\right )-9 a^2 b^2 \sinh \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \text{csch}(c+d x))^4}{3 d (a \sinh (c+d x)+b)^4}+\frac{2 a b \left (2 a^2-b^2\right ) \sinh ^4(c+d x) \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \text{csch}(c+d x))^4}{d (a \sinh (c+d x)+b)^4}+\frac{a^4 (c+d x) \sinh ^4(c+d x) (a+b \text{csch}(c+d x))^4}{d (a \sinh (c+d x)+b)^4}-\frac{a b^3 \sinh ^4(c+d x) \text{csch}^2\left (\frac{1}{2} (c+d x)\right ) (a+b \text{csch}(c+d x))^4}{2 d (a \sinh (c+d x)+b)^4}-\frac{b^4 \sinh ^4(c+d x) \coth \left (\frac{1}{2} (c+d x)\right ) \text{csch}^2\left (\frac{1}{2} (c+d x)\right ) (a+b \text{csch}(c+d x))^4}{24 d (a \sinh (c+d x)+b)^4}-\frac{a b^3 \sinh ^4(c+d x) \text{sech}^2\left (\frac{1}{2} (c+d x)\right ) (a+b \text{csch}(c+d x))^4}{2 d (a \sinh (c+d x)+b)^4}+\frac{b^4 \sinh ^4(c+d x) \tanh \left (\frac{1}{2} (c+d x)\right ) \text{sech}^2\left (\frac{1}{2} (c+d x)\right ) (a+b \text{csch}(c+d x))^4}{24 d (a \sinh (c+d x)+b)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.039, size = 92, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({a}^{4} \left ( dx+c \right ) -8\,{a}^{3}b{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) -6\,{a}^{2}{b}^{2}{\rm coth} \left (dx+c\right )+4\,a{b}^{3} \left ( -1/2\,{\rm csch} \left (dx+c\right ){\rm coth} \left (dx+c\right )+{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) +{b}^{4} \left ({\frac{2}{3}}-{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{3}} \right ){\rm coth} \left (dx+c\right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.02551, size = 316, normalized size = 2.9 \begin{align*} a^{4} x + 2 \, a b^{3}{\left (\frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac{2 \,{\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} + \frac{4}{3} \, b^{4}{\left (\frac{3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac{1}{d{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac{4 \, a^{3} b \log \left (\tanh \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}{d} + \frac{12 \, a^{2} b^{2}}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.73342, size = 3453, normalized size = 31.68 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{csch}{\left (c + d x \right )}\right )^{4}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13597, size = 236, normalized size = 2.17 \begin{align*} \frac{{\left (d x + c\right )} a^{4}}{d} - \frac{2 \,{\left (2 \, a^{3} b - a b^{3}\right )} \log \left (e^{\left (d x + c\right )} + 1\right )}{d} + \frac{2 \,{\left (2 \, a^{3} b - a b^{3}\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{d} - \frac{4 \,{\left (3 \, a b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 9 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 18 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} - 3 \, a b^{3} e^{\left (d x + c\right )} + 9 \, a^{2} b^{2} - b^{4}\right )}}{3 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]