Optimal. Leaf size=317 \[ -\frac {2 b^{4/3} \text {ArcTan}\left (\frac {(-1)^{5/6} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}}}\right )}{3 a^2 \sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}} d}-\frac {2 b^{4/3} \text {ArcTan}\left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^2 \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} d}+\frac {b \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac {2 b^{4/3} \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 a^2 \sqrt {a^{2/3}+b^{2/3}} d}+\frac {\coth (c+d x)}{a d}-\frac {\coth ^3(c+d x)}{3 a d} \]
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Rubi [A]
time = 0.36, antiderivative size = 317, normalized size of antiderivative = 1.00, 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 = {3299, 3855,
3852, 2739, 632, 212, 210} \begin {gather*} \frac {b \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac {2 b^{4/3} \text {ArcTan}\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} \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^2 d \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}-\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 {\coth ^3(c+d x)}{3 a d}+\frac {\coth (c+d x)}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 632
Rule 2739
Rule 3299
Rule 3852
Rule 3855
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 \text {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 ) \text {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 ) \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^2 d}-\frac {\left (2 b^{4/3}\right ) \text {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 ) \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^2 d}+\frac {\left (4 b^{4/3}\right ) \text {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 ) \text {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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 3.91, size = 370, normalized size = 1.17 \begin {gather*} \frac {8 a \coth \left (\frac {1}{2} (c+d x)\right )-24 b \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+4 b^2 \text {RootSum}\left [-b+3 b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {c+d x+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-2 c \text {$\#$1}^2-2 d x \text {$\#$1}^2-4 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+c \text {$\#$1}^4+d x \text {$\#$1}^4+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}+4 a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\&\right ]+8 a \text {csch}^3(c+d x) \sinh ^4\left (\frac {1}{2} (c+d x)\right )-\frac {1}{2} a \text {csch}^4\left (\frac {1}{2} (c+d x)\right ) \sinh (c+d x)+8 a \tanh \left (\frac {1}{2} (c+d x)\right )}{24 a^2 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 = 2.80, size = 164, normalized size = 0.52
method | result | size |
derivativedivides | \(\frac {-\frac {\frac {\left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}-\frac {1}{24 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {3}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {4 b^{2} \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a^{2}}}{d}\) | \(164\) |
default | \(\frac {-\frac {\frac {\left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}-\frac {1}{24 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {3}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {4 b^{2} \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a^{2}}}{d}\) | \(164\) |
risch | \(-\frac {4 \left (3 \,{\mathrm e}^{2 d x +2 c}-1\right )}{3 a d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}+\frac {b \ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d}-\frac {b \ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d}+16 \left (\munderset {\textit {\_R} =\RootOf \left (\left (12230590464 a^{14} d^{6}+12230590464 a^{12} b^{2} d^{6}\right ) \textit {\_Z}^{6}-15925248 a^{8} b^{4} d^{4} \textit {\_Z}^{4}+6912 a^{4} d^{2} \textit {\_Z}^{2} b^{6}-b^{8}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\left (-\frac {254803968 a^{12} d^{5}}{b^{7}}-\frac {254803968 a^{10} d^{5}}{b^{5}}\right ) \textit {\_R}^{5}+\left (\frac {5308416 d^{4} a^{9}}{b^{5}}+\frac {5308416 d^{4} a^{7}}{b^{3}}\right ) \textit {\_R}^{4}+\frac {331776 a^{6} d^{3} \textit {\_R}^{3}}{b^{3}}+\left (\frac {2304 a^{5} d^{2}}{b^{3}}-\frac {4608 a^{3} d^{2}}{b}\right ) \textit {\_R}^{2}-\frac {144 d \,a^{2} \textit {\_R}}{b}+\frac {b}{a}\right )\right )\) | \(255\) |
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. 30233 vs. \(2 (234) = 468\).
time = 6.68, size = 30233, normalized size = 95.37 \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 = 59.69, size = 2500, normalized size = 7.89 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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