Optimal. Leaf size=322 \[ \frac {2 (-1)^{2/3} b \text {ArcTan}\left (\frac {\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{5/3} \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}} d}+\frac {2 (-1)^{2/3} b \text {ArcTan}\left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^{5/3} \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} d}+\frac {\tanh ^{-1}(\cosh (c+d x))}{2 a d}+\frac {2 b \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 a^{5/3} \sqrt {a^{2/3}+b^{2/3}} d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \]
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Rubi [A]
time = 0.43, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3299, 3853,
3855, 3292, 2739, 632, 210} \begin {gather*} \frac {2 (-1)^{2/3} b \text {ArcTan}\left (\frac {\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{5/3} d \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac {2 (-1)^{2/3} b \text {ArcTan}\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^{5/3} d \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}+\frac {2 b \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^{5/3} d \sqrt {a^{2/3}+b^{2/3}}}+\frac {\tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 3292
Rule 3299
Rule 3853
Rule 3855
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(c+d x)}{a+b \sinh ^3(c+d x)} \, dx &=-\left (i \int \left (\frac {i \text {csch}^3(c+d x)}{a}-\frac {i b}{a \left (a+b \sinh ^3(c+d x)\right )}\right ) \, dx\right )\\ &=\frac {\int \text {csch}^3(c+d x) \, dx}{a}-\frac {b \int \frac {1}{a+b \sinh ^3(c+d x)} \, dx}{a}\\ &=-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {\int \text {csch}(c+d x) \, dx}{2 a}-\frac {b \int \left (\frac {\sqrt [6]{-1}}{3 a^{2/3} \left (\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)\right )}+\frac {\sqrt [6]{-1}}{3 a^{2/3} \left (\sqrt [6]{-1} \sqrt [3]{a}+\sqrt [6]{-1} \sqrt [3]{b} \sinh (c+d x)\right )}+\frac {\sqrt [6]{-1}}{3 a^{2/3} \left (\sqrt [6]{-1} \sqrt [3]{a}+(-1)^{5/6} \sqrt [3]{b} \sinh (c+d x)\right )}\right ) \, dx}{a}\\ &=\frac {\tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {\left (\sqrt [6]{-1} b\right ) \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a^{5/3}}-\frac {\left (\sqrt [6]{-1} b\right ) \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}+\sqrt [6]{-1} \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a^{5/3}}-\frac {\left (\sqrt [6]{-1} b\right ) \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}+(-1)^{5/6} \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a^{5/3}}\\ &=\frac {\tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}+\frac {\left (2 (-1)^{2/3} b\right ) \text {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^{5/3} d}+\frac {\left (2 (-1)^{2/3} b\right ) \text {Subst}\left (\int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}+2 \sqrt [3]{-1} \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^{5/3} d}+\frac {\left (2 (-1)^{2/3} b\right ) \text {Subst}\left (\int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}-2 (-1)^{2/3} \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^{5/3} d}\\ &=\frac {\tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {\left (4 (-1)^{2/3} b\right ) \text {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^{5/3} d}-\frac {\left (4 (-1)^{2/3} b\right ) \text {Subst}\left (\int \frac {1}{-4 \sqrt [3]{-1} \left (a^{2/3}+b^{2/3}\right )-x^2} \, dx,x,-2 (-1)^{2/3} \sqrt [3]{b}+2 \sqrt [6]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a^{5/3} d}-\frac {\left (4 (-1)^{2/3} b\right ) \text {Subst}\left (\int \frac {1}{-4 \left (\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{-1} \sqrt [3]{b}+2 \sqrt [6]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a^{5/3} d}\\ &=-\frac {2 (-1)^{2/3} b \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^{5/3} \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} d}+\frac {2 (-1)^{2/3} b \tan ^{-1}\left (\frac {\sqrt [3]{-1} \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}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{5/3} \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}} d}+\frac {\tanh ^{-1}(\cosh (c+d x))}{2 a d}+\frac {2 b \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^{5/3} \sqrt {a^{2/3}+b^{2/3}} d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a 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 = 0.36, size = 178, normalized size = 0.55 \begin {gather*} -\frac {16 b \text {RootSum}\left [-b+3 b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {c \text {$\#$1}+d x \text {$\#$1}+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}}{b+4 a \text {$\#$1}-2 b \text {$\#$1}^2+b \text {$\#$1}^4}\&\right ]+3 \left (\text {csch}^2\left (\frac {1}{2} (c+d x)\right )+4 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+\text {sech}^2\left (\frac {1}{2} (c+d x)\right )\right )}{24 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 = 2.87, size = 138, normalized size = 0.43
method | result | size |
derivativedivides | \(\frac {\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}+\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{d}\) | \(138\) |
default | \(\frac {\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}+\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{d}\) | \(138\) |
risch | \(-\frac {{\mathrm e}^{d x +c} \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+8 \left (\munderset {\textit {\_R} =\RootOf \left (\left (191102976 a^{12} d^{6}+191102976 a^{10} b^{2} d^{6}\right ) \textit {\_Z}^{6}-995328 a^{8} b^{2} d^{4} \textit {\_Z}^{4}+1728 a^{4} d^{2} \textit {\_Z}^{2} b^{4}-b^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\left (\frac {15925248 d^{5} a^{11}}{b^{6}}+\frac {15925248 d^{5} a^{9}}{b^{4}}\right ) \textit {\_R}^{5}+\left (\frac {331776 d^{4} a^{9}}{b^{5}}+\frac {331776 d^{4} a^{7}}{b^{3}}\right ) \textit {\_R}^{4}+\left (-\frac {69120 d^{3} a^{7}}{b^{4}}+\frac {13824 d^{3} a^{5}}{b^{2}}\right ) \textit {\_R}^{3}-\frac {1728 d^{2} a^{5} \textit {\_R}^{2}}{b^{3}}+\frac {72 d \,a^{3} \textit {\_R}}{b^{2}}+\frac {2 a}{b}\right )\right )+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d a}-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d a}\) | \(259\) |
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] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 29179 vs. \(2 (229) = 458\).
time = 8.96, size = 29179, normalized size = 90.62 \begin {gather*} \text {too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 89.25, size = 2500, normalized size = 7.76 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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