Optimal. Leaf size=184 \[ \frac {b^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{3/2} \sqrt {\sqrt {a}-\sqrt {b}} d}+\frac {\tanh ^{-1}(\cosh (c+d x))}{2 a d}+\frac {b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{3/2} \sqrt {\sqrt {a}+\sqrt {b}} d}+\frac {1}{4 a d (1-\cosh (c+d x))}-\frac {1}{4 a d (1+\cosh (c+d x))} \]
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Rubi [A]
time = 0.16, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3294, 1184,
213, 1107, 211, 214} \begin {gather*} \frac {b^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{3/2} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{3/2} d \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {1}{4 a d (1-\cosh (c+d x))}-\frac {1}{4 a d (\cosh (c+d x)+1)}+\frac {\tanh ^{-1}(\cosh (c+d x))}{2 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 213
Rule 214
Rule 1107
Rule 1184
Rule 3294
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \left (a-b+2 b x^2-b x^4\right )} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{4 a (-1+x)^2}+\frac {1}{4 a (1+x)^2}-\frac {1}{2 a \left (-1+x^2\right )}+\frac {b}{a \left (a-b+2 b x^2-b x^4\right )}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {1}{4 a d (1-\cosh (c+d x))}-\frac {1}{4 a d (1+\cosh (c+d x))}-\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\cosh (c+d x)\right )}{2 a d}+\frac {b \text {Subst}\left (\int \frac {1}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{a d}\\ &=\frac {\tanh ^{-1}(\cosh (c+d x))}{2 a d}+\frac {1}{4 a d (1-\cosh (c+d x))}-\frac {1}{4 a d (1+\cosh (c+d x))}-\frac {b^{3/2} \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 a^{3/2} d}+\frac {b^{3/2} \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 a^{3/2} d}\\ &=\frac {b^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{3/2} \sqrt {\sqrt {a}-\sqrt {b}} d}+\frac {\tanh ^{-1}(\cosh (c+d x))}{2 a d}+\frac {b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{3/2} \sqrt {\sqrt {a}+\sqrt {b}} d}+\frac {1}{4 a d (1-\cosh (c+d x))}-\frac {1}{4 a d (1+\cosh (c+d x))}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.28, size = 265, normalized size = 1.44 \begin {gather*} -\frac {\text {csch}^2\left (\frac {1}{2} (c+d x)\right )+4 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+4 b \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-c \text {$\#$1}-d x \text {$\#$1}-2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}+c \text {$\#$1}^3+d x \text {$\#$1}^3+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^3}{-b-8 a \text {$\#$1}^2+3 b \text {$\#$1}^2-3 b \text {$\#$1}^4+b \text {$\#$1}^6}\&\right ]+\text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.16, size = 181, normalized size = 0.98
method | result | size |
derivativedivides | \(\frac {\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}+2 b \left (\frac {\arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {\sqrt {a b}\, a -a b}}-\frac {\arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {-\sqrt {a b}\, a -a b}}\right )-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{d}\) | \(181\) |
default | \(\frac {\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}+2 b \left (\frac {\arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {\sqrt {a b}\, a -a b}}-\frac {\arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {-\sqrt {a b}\, a -a b}}\right )-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{d}\) | \(181\) |
risch | \(-\frac {{\mathrm e}^{d x +c} \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d a}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d a}+8 \left (\munderset {\textit {\_R} =\RootOf \left (\left (1048576 a^{7} d^{4}-1048576 a^{6} b \,d^{4}\right ) \textit {\_Z}^{4}+2048 a^{3} b^{2} d^{2} \textit {\_Z}^{2}-b^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (\frac {65536 d^{3} a^{5}}{b^{2}}-\frac {65536 d^{3} a^{4}}{b}\right ) \textit {\_R}^{3}+\left (\frac {64 a^{2} d}{b}+64 a d \right ) \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\right )\) | \(187\) |
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. 1954 vs.
\(2 (136) = 272\).
time = 0.44, size = 1954, normalized size = 10.62 \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 467 vs.
\(2 (136) = 272\).
time = 0.51, size = 467, normalized size = 2.54 \begin {gather*} \frac {\frac {2 \, {\left ({\left (4 \, \sqrt {-b^{2} - \sqrt {a b} b} a b + 5 \, \sqrt {-b^{2} - \sqrt {a b} b} b^{2}\right )} {\left | a \right |} {\left | b \right |} - {\left (4 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b + 5 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b^{2}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a b + \sqrt {a^{2} b^{2} + {\left (a^{2} - a b\right )} a b}}{a b}}}\right )}{{\left (4 \, a^{3} b^{2} + a^{2} b^{3} - 5 \, a b^{4}\right )} {\left | a \right |}} + \frac {2 \, {\left ({\left (4 \, \sqrt {-b^{2} + \sqrt {a b} b} a b + 5 \, \sqrt {-b^{2} + \sqrt {a b} b} b^{2}\right )} {\left | a \right |} {\left | b \right |} + {\left (4 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b + 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b^{2}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a b - \sqrt {a^{2} b^{2} + {\left (a^{2} - a b\right )} a b}}{a b}}}\right )}{{\left (4 \, a^{3} b^{2} + a^{2} b^{3} - 5 \, a b^{4}\right )} {\left | a \right |}} + \frac {\log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right )}{a} - \frac {\log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right )}{a} - \frac {4 \, {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )} a}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 12.21, size = 1517, normalized size = 8.24 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (256\,a^6\,\sqrt {-a^2\,d^2}+b^6\,\sqrt {-a^2\,d^2}+96\,a^2\,b^4\,\sqrt {-a^2\,d^2}-288\,a^3\,b^3\,\sqrt {-a^2\,d^2}+512\,a^4\,b^2\,\sqrt {-a^2\,d^2}-16\,a\,b^5\,\sqrt {-a^2\,d^2}-512\,a^5\,b\,\sqrt {-a^2\,d^2}\right )}{256\,d\,a^7-512\,d\,a^6\,b+512\,d\,a^5\,b^2-288\,d\,a^4\,b^3+96\,d\,a^3\,b^4-16\,d\,a^2\,b^5+d\,a\,b^6}\right )}{\sqrt {-a^2\,d^2}}-\ln \left (\frac {\left (\frac {\left (\frac {8589934592\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (8\,a^2-7\,a\,b+3\,b^2\right )\,\sqrt {-\frac {\sqrt {a^7\,b^3}+a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{b^5\,{\left (a-b\right )}^2}-\frac {4294967296\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (12\,a^3-7\,a^2\,b+2\,a\,b^2+b^3\right )}{a^2\,b^4\,{\left (a-b\right )}^3}\right )\,\sqrt {-\frac {\sqrt {a^7\,b^3}+a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{4}-\frac {4294967296\,d\,{\mathrm {e}}^{c+d\,x}\,\left (2\,a^2-2\,a\,b+b^2\right )}{a^3\,b^4\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {\sqrt {a^7\,b^3}+a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{4}+\frac {268435456\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (4\,a^3-a\,b^2+b^3\right )}{a^5\,b^3\,{\left (a-b\right )}^3}\right )\,\sqrt {-\frac {\sqrt {a^7\,b^3}+a^3\,b^2}{16\,\left (a^7\,d^2-a^6\,b\,d^2\right )}}+\ln \left (\frac {268435456\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (4\,a^3-a\,b^2+b^3\right )}{a^5\,b^3\,{\left (a-b\right )}^3}-\frac {\left (\frac {\left (\frac {8589934592\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (8\,a^2-7\,a\,b+3\,b^2\right )\,\sqrt {-\frac {\sqrt {a^7\,b^3}+a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{b^5\,{\left (a-b\right )}^2}+\frac {4294967296\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (12\,a^3-7\,a^2\,b+2\,a\,b^2+b^3\right )}{a^2\,b^4\,{\left (a-b\right )}^3}\right )\,\sqrt {-\frac {\sqrt {a^7\,b^3}+a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{4}-\frac {4294967296\,d\,{\mathrm {e}}^{c+d\,x}\,\left (2\,a^2-2\,a\,b+b^2\right )}{a^3\,b^4\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {\sqrt {a^7\,b^3}+a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{4}\right )\,\sqrt {-\frac {\sqrt {a^7\,b^3}+a^3\,b^2}{16\,\left (a^7\,d^2-a^6\,b\,d^2\right )}}-\ln \left (\frac {\left (\frac {\left (\frac {8589934592\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (8\,a^2-7\,a\,b+3\,b^2\right )\,\sqrt {\frac {\sqrt {a^7\,b^3}-a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{b^5\,{\left (a-b\right )}^2}-\frac {4294967296\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (12\,a^3-7\,a^2\,b+2\,a\,b^2+b^3\right )}{a^2\,b^4\,{\left (a-b\right )}^3}\right )\,\sqrt {\frac {\sqrt {a^7\,b^3}-a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{4}-\frac {4294967296\,d\,{\mathrm {e}}^{c+d\,x}\,\left (2\,a^2-2\,a\,b+b^2\right )}{a^3\,b^4\,{\left (a-b\right )}^2}\right )\,\sqrt {\frac {\sqrt {a^7\,b^3}-a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{4}+\frac {268435456\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (4\,a^3-a\,b^2+b^3\right )}{a^5\,b^3\,{\left (a-b\right )}^3}\right )\,\sqrt {\frac {\sqrt {a^7\,b^3}-a^3\,b^2}{16\,\left (a^7\,d^2-a^6\,b\,d^2\right )}}+\ln \left (\frac {268435456\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (4\,a^3-a\,b^2+b^3\right )}{a^5\,b^3\,{\left (a-b\right )}^3}-\frac {\left (\frac {\left (\frac {8589934592\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (8\,a^2-7\,a\,b+3\,b^2\right )\,\sqrt {\frac {\sqrt {a^7\,b^3}-a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{b^5\,{\left (a-b\right )}^2}+\frac {4294967296\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (12\,a^3-7\,a^2\,b+2\,a\,b^2+b^3\right )}{a^2\,b^4\,{\left (a-b\right )}^3}\right )\,\sqrt {\frac {\sqrt {a^7\,b^3}-a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{4}-\frac {4294967296\,d\,{\mathrm {e}}^{c+d\,x}\,\left (2\,a^2-2\,a\,b+b^2\right )}{a^3\,b^4\,{\left (a-b\right )}^2}\right )\,\sqrt {\frac {\sqrt {a^7\,b^3}-a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{4}\right )\,\sqrt {\frac {\sqrt {a^7\,b^3}-a^3\,b^2}{16\,\left (a^7\,d^2-a^6\,b\,d^2\right )}}-\frac {{\mathrm {e}}^{c+d\,x}}{a\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}}{a\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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