Optimal. Leaf size=157 \[ \frac {\sqrt [3]{b} \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} d}-\frac {\coth (c+d x)}{a d}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 a^{4/3} d}-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 a^{4/3} d} \]
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Rubi [A]
time = 0.09, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3744, 331, 298,
31, 648, 631, 210, 642} \begin {gather*} \frac {\sqrt [3]{b} \text {ArcTan}\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 {\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 {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 a^{4/3} d}-\frac {\coth (c+d x)}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 298
Rule 331
Rule 631
Rule 642
Rule 648
Rule 3744
Rubi steps
\begin {align*} \int \frac {\text {csch}^2(c+d x)}{a+b \tanh ^3(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^3\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {\coth (c+d x)}{a d}-\frac {b \text {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 {b^{2/3} \text {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} \text {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 {\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} \text {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} \text {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 {\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 {\sqrt [3]{b} \text {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 {\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}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.13, size = 190, normalized size = 1.21 \begin {gather*} -\frac {3 \coth (c+d x)+2 b \text {RootSum}\left [a-b+3 a \text {$\#$1}+3 b \text {$\#$1}+3 a \text {$\#$1}^2-3 b \text {$\#$1}^2+a \text {$\#$1}^3+b \text {$\#$1}^3\&,\frac {-c-d x-\log (-\cosh (c+d x)-\sinh (c+d x)+\cosh (c+d x) \text {$\#$1}-\sinh (c+d x) \text {$\#$1})+c \text {$\#$1}+d x \text {$\#$1}+\log (-\cosh (c+d x)-\sinh (c+d x)+\cosh (c+d x) \text {$\#$1}-\sinh (c+d x) \text {$\#$1}) \text {$\#$1}}{a+b+2 a \text {$\#$1}-2 b \text {$\#$1}+a \text {$\#$1}^2+b \text {$\#$1}^2}\&\right ]}{3 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 4.23, size = 116, normalized size = 0.74
method | result | size |
derivativedivides | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+\frac {2 b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R} \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(116\) |
default | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+\frac {2 b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R} \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(116\) |
risch | \(-\frac {2}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )}+4 \left (\munderset {\textit {\_R} =\RootOf \left (1728 a^{4} d^{3} \textit {\_Z}^{3}-b \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {288 a^{3} d^{2} \textit {\_R}^{2}}{\left (a +b \right ) \left (\frac {b}{a +b}+\frac {a}{a +b}\right )}-\frac {24 a^{2} d \textit {\_R}}{\left (a +b \right ) \left (\frac {b}{a +b}+\frac {a}{a +b}\right )}+\frac {-\frac {b}{a +b}+\frac {a}{a +b}}{\frac {b}{a +b}+\frac {a}{a +b}}\right )\right )\) | \(153\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 640 vs.
\(2 (123) = 246\).
time = 0.40, size = 640, normalized size = 4.08 \begin {gather*} -\frac {2 \, {\left (\sqrt {3} \cosh \left (d x + c\right )^{2} + 2 \, \sqrt {3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sqrt {3} \sinh \left (d x + c\right )^{2} - \sqrt {3}\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} b \cosh \left (d x + c\right )^{2} + 2 \, \sqrt {3} b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sqrt {3} b \sinh \left (d x + c\right )^{2} - {\left (\sqrt {3} a \cosh \left (d x + c\right )^{2} + 2 \, \sqrt {3} a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sqrt {3} a \sinh \left (d x + c\right )^{2} + \sqrt {3} a\right )} \left (\frac {b}{a}\right )^{\frac {2}{3}} - {\left (\sqrt {3} b \cosh \left (d x + c\right )^{2} + 2 \, \sqrt {3} b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sqrt {3} b \sinh \left (d x + c\right )^{2} - \sqrt {3} b\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}}}{3 \, b}\right ) + {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left ({\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + {\left (a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 2 \, {\left (a \cosh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a \sinh \left (d x + c\right )^{2} - a\right )} \left (\frac {b}{a}\right )^{\frac {2}{3}} + 2 \, {\left (a \cosh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a \sinh \left (d x + c\right )^{2} + a\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} + a + b\right ) - 2 \, {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left ({\left (a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a + b\right )} \sinh \left (d x + c\right )^{2} + 2 \, a \left (\frac {b}{a}\right )^{\frac {2}{3}} - 2 \, a \left (\frac {b}{a}\right )^{\frac {1}{3}} + a - b\right ) + 12}{6 \, {\left (a d \cosh \left (d x + c\right )^{2} + 2 \, a d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a d \sinh \left (d x + c\right )^{2} - a d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {csch}^{2}{\left (c + d x \right )}}{a + b \tanh ^{3}{\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 21, normalized size = 0.13 \begin {gather*} -\frac {2}{a d {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.71, size = 669, normalized size = 4.26 \begin {gather*} \frac {b^{1/3}\,\ln \left (a^{1/3}-b^{1/3}+a^{1/3}\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^{1/3}\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{3\,a^{4/3}\,d}-\frac {2}{a\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}+\frac {b^{1/3}\,\ln \left (\frac {256\,b^3\,\left (19\,a^2\,b-24\,a\,b^2+6\,a^3-b^3+8\,a^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+70\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+113\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^4\,{\left (a+b\right )}^6}+\frac {b^{1/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1536\,b^3\,d\,\left (8\,a^2-8\,b^2+15\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+66\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^2\,{\left (a+b\right )}^6}+\frac {768\,b^{7/3}\,d\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (24\,a^2\,b-19\,a\,b^2+a^3-6\,b^3+a^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+8\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+113\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+70\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^{7/3}\,{\left (a+b\right )}^6}\right )}{3\,a^{4/3}\,d}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{4/3}\,d}-\frac {b^{1/3}\,\ln \left (\frac {256\,b^3\,\left (19\,a^2\,b-24\,a\,b^2+6\,a^3-b^3+8\,a^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+70\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+113\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^4\,{\left (a+b\right )}^6}-\frac {b^{1/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1536\,b^3\,d\,\left (8\,a^2-8\,b^2+15\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+66\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^2\,{\left (a+b\right )}^6}-\frac {768\,b^{7/3}\,d\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (24\,a^2\,b-19\,a\,b^2+a^3-6\,b^3+a^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+8\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+113\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+70\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^{7/3}\,{\left (a+b\right )}^6}\right )}{3\,a^{4/3}\,d}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{4/3}\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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