Optimal. Leaf size=109 \[ -\frac {i}{4 (1-i \sinh (x))}-\frac {15 i}{16 (1+i \sinh (x))}+\frac {i}{32 (1-i \sinh (x))^2}+\frac {9 i}{32 (1+i \sinh (x))^2}-\frac {i}{24 (1+i \sinh (x))^3}-\frac {21}{32} i \log (-\sinh (x)+i)-\frac {11}{32} i \log (\sinh (x)+i) \]
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Rubi [A] time = 0.09, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3879, 88} \[ -\frac {i}{4 (1-i \sinh (x))}-\frac {15 i}{16 (1+i \sinh (x))}+\frac {i}{32 (1-i \sinh (x))^2}+\frac {9 i}{32 (1+i \sinh (x))^2}-\frac {i}{24 (1+i \sinh (x))^3}-\frac {21}{32} i \log (-\sinh (x)+i)-\frac {11}{32} i \log (\sinh (x)+i) \]
Antiderivative was successfully verified.
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Rule 88
Rule 3879
Rubi steps
\begin {align*} \int \frac {\tanh ^5(x)}{i+\text {csch}(x)} \, dx &=\operatorname {Subst}\left (\int \frac {x^6}{(i-i x)^3 (i+i x)^4} \, dx,x,i \sinh (x)\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {i}{16 (-1+x)^3}-\frac {i}{4 (-1+x)^2}-\frac {11 i}{32 (-1+x)}+\frac {i}{8 (1+x)^4}-\frac {9 i}{16 (1+x)^3}+\frac {15 i}{16 (1+x)^2}-\frac {21 i}{32 (1+x)}\right ) \, dx,x,i \sinh (x)\right )\\ &=-\frac {21}{32} i \log (i-\sinh (x))-\frac {11}{32} i \log (i+\sinh (x))+\frac {i}{32 (1-i \sinh (x))^2}-\frac {i}{4 (1-i \sinh (x))}-\frac {i}{24 (1+i \sinh (x))^3}+\frac {9 i}{32 (1+i \sinh (x))^2}-\frac {15 i}{16 (1+i \sinh (x))}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 75, normalized size = 0.69 \[ \frac {1}{96} \left (-\frac {2 \left (33 \sinh ^4(x)+39 i \sinh ^3(x)+79 \sinh ^2(x)+29 i \sinh (x)+44\right )}{(\sinh (x)-i)^3 (\sinh (x)+i)^2}-63 i \log (-\sinh (x)+i)-33 i \log (\sinh (x)+i)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.37, size = 299, normalized size = 2.74 \[ \frac {48 i \, x e^{\left (10 \, x\right )} + 6 \, {\left (16 \, x - 11\right )} e^{\left (9 \, x\right )} + {\left (144 i \, x - 156 i\right )} e^{\left (8 \, x\right )} + 16 \, {\left (24 \, x - 23\right )} e^{\left (7 \, x\right )} + {\left (96 i \, x + 4 i\right )} e^{\left (6 \, x\right )} + 36 \, {\left (16 \, x - 15\right )} e^{\left (5 \, x\right )} + {\left (-96 i \, x - 4 i\right )} e^{\left (4 \, x\right )} + 16 \, {\left (24 \, x - 23\right )} e^{\left (3 \, x\right )} + {\left (-144 i \, x + 156 i\right )} e^{\left (2 \, x\right )} + 6 \, {\left (16 \, x - 11\right )} e^{x} + {\left (-33 i \, e^{\left (10 \, x\right )} - 66 \, e^{\left (9 \, x\right )} - 99 i \, e^{\left (8 \, x\right )} - 264 \, e^{\left (7 \, x\right )} - 66 i \, e^{\left (6 \, x\right )} - 396 \, e^{\left (5 \, x\right )} + 66 i \, e^{\left (4 \, x\right )} - 264 \, e^{\left (3 \, x\right )} + 99 i \, e^{\left (2 \, x\right )} - 66 \, e^{x} + 33 i\right )} \log \left (e^{x} + i\right ) + {\left (-63 i \, e^{\left (10 \, x\right )} - 126 \, e^{\left (9 \, x\right )} - 189 i \, e^{\left (8 \, x\right )} - 504 \, e^{\left (7 \, x\right )} - 126 i \, e^{\left (6 \, x\right )} - 756 \, e^{\left (5 \, x\right )} + 126 i \, e^{\left (4 \, x\right )} - 504 \, e^{\left (3 \, x\right )} + 189 i \, e^{\left (2 \, x\right )} - 126 \, e^{x} + 63 i\right )} \log \left (e^{x} - i\right ) - 48 i \, x}{48 \, e^{\left (10 \, x\right )} - 96 i \, e^{\left (9 \, x\right )} + 144 \, e^{\left (8 \, x\right )} - 384 i \, e^{\left (7 \, x\right )} + 96 \, e^{\left (6 \, x\right )} - 576 i \, e^{\left (5 \, x\right )} - 96 \, e^{\left (4 \, x\right )} - 384 i \, e^{\left (3 \, x\right )} - 144 \, e^{\left (2 \, x\right )} - 96 i \, e^{x} - 48} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 120, normalized size = 1.10 \[ -\frac {33 i \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 100 \, e^{\left (-x\right )} - 100 \, e^{x} - 76 i}{64 \, {\left (-i \, e^{\left (-x\right )} + i \, e^{x} - 2\right )}^{2}} - \frac {-231 i \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 1026 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 1548 i \, e^{\left (-x\right )} - 1548 i \, e^{x} - 776}{192 \, {\left (e^{\left (-x\right )} - e^{x} + 2 i\right )}^{3}} - \frac {11}{32} i \, \log \left (-e^{\left (-x\right )} + e^{x} + 2 i\right ) - \frac {21}{32} i \, \log \left (-e^{\left (-x\right )} + e^{x} - 2 i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 155, normalized size = 1.42 \[ i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\frac {11 i \ln \left (\tanh \left (\frac {x}{2}\right )+i\right )}{16}+\frac {i}{8 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{4}}+\frac {i}{4 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )+i\right )}-\frac {21 i \ln \left (\tanh \left (\frac {x}{2}\right )-i\right )}{16}+\frac {3 i}{8 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}}+\frac {i}{3 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{6}}-\frac {3 i}{8 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{4}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{5}}+\frac {11}{12 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{3}}+\frac {1}{\tanh \left (\frac {x}{2}\right )-i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 144, normalized size = 1.32 \[ -i \, x + \frac {33 \, e^{\left (-x\right )} + 78 i \, e^{\left (-2 \, x\right )} + 184 \, e^{\left (-3 \, x\right )} - 2 i \, e^{\left (-4 \, x\right )} + 270 \, e^{\left (-5 \, x\right )} + 2 i \, e^{\left (-6 \, x\right )} + 184 \, e^{\left (-7 \, x\right )} - 78 i \, e^{\left (-8 \, x\right )} + 33 \, e^{\left (-9 \, x\right )}}{48 i \, e^{\left (-x\right )} - 72 \, e^{\left (-2 \, x\right )} + 192 i \, e^{\left (-3 \, x\right )} - 48 \, e^{\left (-4 \, x\right )} + 288 i \, e^{\left (-5 \, x\right )} + 48 \, e^{\left (-6 \, x\right )} + 192 i \, e^{\left (-7 \, x\right )} + 72 \, e^{\left (-8 \, x\right )} + 48 i \, e^{\left (-9 \, x\right )} + 24 \, e^{\left (-10 \, x\right )} - 24} - \frac {11}{16} i \, \log \left (e^{\left (-x\right )} - i\right ) - \frac {21}{16} i \, \log \left (i \, e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.11, size = 274, normalized size = 2.51 \[ x\,1{}\mathrm {i}-\ln \left (\left (\frac {5\,{\mathrm {e}}^x}{8}-\frac {5}{8}{}\mathrm {i}\right )\,\left (\frac {5\,{\mathrm {e}}^x}{8}+\frac {5}{8}{}\mathrm {i}\right )\right )\,1{}\mathrm {i}+\frac {5\,\mathrm {atan}\left ({\mathrm {e}}^x\right )}{8}+\frac {1{}\mathrm {i}}{3\,\left (15\,{\mathrm {e}}^{4\,x}-15\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{6\,x}+1-{\mathrm {e}}^{3\,x}\,20{}\mathrm {i}+{\mathrm {e}}^{5\,x}\,6{}\mathrm {i}+{\mathrm {e}}^x\,6{}\mathrm {i}\right )}-\frac {1}{{\mathrm {e}}^{2\,x}\,10{}\mathrm {i}-10\,{\mathrm {e}}^{3\,x}-{\mathrm {e}}^{4\,x}\,5{}\mathrm {i}+{\mathrm {e}}^{5\,x}+5\,{\mathrm {e}}^x-\mathrm {i}}-\frac {31}{12\,\left ({\mathrm {e}}^{2\,x}\,3{}\mathrm {i}-{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x-\mathrm {i}\right )}-\frac {5{}\mathrm {i}}{8\,\left ({\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}+\frac {17{}\mathrm {i}}{8\,\left ({\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1-{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}+{\mathrm {e}}^x\,4{}\mathrm {i}\right )}+\frac {1{}\mathrm {i}}{8\,\left ({\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1+{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}-{\mathrm {e}}^x\,4{}\mathrm {i}\right )}+\frac {3{}\mathrm {i}}{1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,2{}\mathrm {i}}-\frac {15}{8\,\left ({\mathrm {e}}^x-\mathrm {i}\right )}+\frac {1}{2\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}-\frac {1}{4\,\left ({\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}\right )} \]
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 \[ \int \frac {\tanh ^{5}{\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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