Optimal. Leaf size=75 \[ \frac {4 e^{5 x}}{3 \left (1-e^{4 x}\right )^3}-\frac {5 e^{5 x}}{6 \left (1-e^{4 x}\right )^2}+\frac {3 e^x}{8 \left (1-e^{4 x}\right )}-\frac {3 \text {ArcTan}\left (e^x\right )}{16}-\frac {3}{16} \tanh ^{-1}\left (e^x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2320, 12, 474,
468, 294, 218, 212, 209} \begin {gather*} -\frac {3}{16} \text {ArcTan}\left (e^x\right )+\frac {3 e^x}{8 \left (1-e^{4 x}\right )}-\frac {5 e^{5 x}}{6 \left (1-e^{4 x}\right )^2}+\frac {4 e^{5 x}}{3 \left (1-e^{4 x}\right )^3}-\frac {3}{16} \tanh ^{-1}\left (e^x\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 209
Rule 212
Rule 218
Rule 294
Rule 468
Rule 474
Rule 2320
Rubi steps
\begin {align*} \int e^x \coth ^2(2 x) \text {csch}^2(2 x) \, dx &=\text {Subst}\left (\int \frac {4 x^4 \left (1+x^4\right )^2}{\left (1-x^4\right )^4} \, dx,x,e^x\right )\\ &=4 \text {Subst}\left (\int \frac {x^4 \left (1+x^4\right )^2}{\left (1-x^4\right )^4} \, dx,x,e^x\right )\\ &=\frac {4 e^{5 x}}{3 \left (1-e^{4 x}\right )^3}-\frac {1}{3} \text {Subst}\left (\int \frac {x^4 \left (8+12 x^4\right )}{\left (1-x^4\right )^3} \, dx,x,e^x\right )\\ &=\frac {4 e^{5 x}}{3 \left (1-e^{4 x}\right )^3}-\frac {5 e^{5 x}}{6 \left (1-e^{4 x}\right )^2}+\frac {3}{2} \text {Subst}\left (\int \frac {x^4}{\left (1-x^4\right )^2} \, dx,x,e^x\right )\\ &=\frac {4 e^{5 x}}{3 \left (1-e^{4 x}\right )^3}-\frac {5 e^{5 x}}{6 \left (1-e^{4 x}\right )^2}+\frac {3 e^x}{8 \left (1-e^{4 x}\right )}-\frac {3}{8} \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,e^x\right )\\ &=\frac {4 e^{5 x}}{3 \left (1-e^{4 x}\right )^3}-\frac {5 e^{5 x}}{6 \left (1-e^{4 x}\right )^2}+\frac {3 e^x}{8 \left (1-e^{4 x}\right )}-\frac {3}{16} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,e^x\right )-\frac {3}{16} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^x\right )\\ &=\frac {4 e^{5 x}}{3 \left (1-e^{4 x}\right )^3}-\frac {5 e^{5 x}}{6 \left (1-e^{4 x}\right )^2}+\frac {3 e^x}{8 \left (1-e^{4 x}\right )}-\frac {3}{16} \tan ^{-1}\left (e^x\right )-\frac {3}{16} \tanh ^{-1}\left (e^x\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 3.85, size = 310, normalized size = 4.13 \begin {gather*} \frac {e^{-7 x} \left (-1070609085-946471617 e^{4 x}+369641285 e^{8 x}+351173641 e^{12 x}-23818496 e^{16 x}+1070609085 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};e^{4 x}\right )+732349800 e^{4 x} \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};e^{4 x}\right )-635067810 e^{8 x} \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};e^{4 x}\right )-384831720 e^{12 x} \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};e^{4 x}\right )+60913125 e^{16 x} \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};e^{4 x}\right )+1280 e^{16 x} \left (821+1346 e^{4 x}+557 e^{8 x}\right ) \, _4F_3\left (2,2,2,\frac {9}{4};1,1,\frac {21}{4};e^{4 x}\right )+10240 e^{16 x} \left (23+42 e^{4 x}+19 e^{8 x}\right ) \, _5F_4\left (2,2,2,2,\frac {9}{4};1,1,1,\frac {21}{4};e^{4 x}\right )+20480 e^{16 x} \, _6F_5\left (2,2,2,2,2,\frac {9}{4};1,1,1,1,\frac {21}{4};e^{4 x}\right )+40960 e^{20 x} \, _6F_5\left (2,2,2,2,2,\frac {9}{4};1,1,1,1,\frac {21}{4};e^{4 x}\right )+20480 e^{24 x} \, _6F_5\left (2,2,2,2,2,\frac {9}{4};1,1,1,1,\frac {21}{4};e^{4 x}\right )\right )}{3818880} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains complex when optimal does not.
time = 1.56, size = 60, normalized size = 0.80
method | result | size |
risch | \(-\frac {{\mathrm e}^{x} \left (29 \,{\mathrm e}^{8 x}-6 \,{\mathrm e}^{4 x}+9\right )}{24 \left ({\mathrm e}^{4 x}-1\right )^{3}}+\frac {3 i \ln \left ({\mathrm e}^{x}-i\right )}{32}-\frac {3 i \ln \left ({\mathrm e}^{x}+i\right )}{32}+\frac {3 \ln \left ({\mathrm e}^{x}-1\right )}{32}-\frac {3 \ln \left ({\mathrm e}^{x}+1\right )}{32}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.48, size = 59, normalized size = 0.79 \begin {gather*} -\frac {29 \, e^{\left (9 \, x\right )} - 6 \, e^{\left (5 \, x\right )} + 9 \, e^{x}}{24 \, {\left (e^{\left (12 \, x\right )} - 3 \, e^{\left (8 \, x\right )} + 3 \, e^{\left (4 \, x\right )} - 1\right )}} - \frac {3}{16} \, \arctan \left (e^{x}\right ) - \frac {3}{32} \, \log \left (e^{x} + 1\right ) + \frac {3}{32} \, \log \left (e^{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 992 vs.
\(2 (51) = 102\).
time = 0.40, size = 992, normalized size = 13.23 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int e^{x} \coth ^{2}{\left (2 x \right )} \operatorname {csch}^{2}{\left (2 x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.41, size = 48, normalized size = 0.64 \begin {gather*} -\frac {29 \, e^{\left (9 \, x\right )} - 6 \, e^{\left (5 \, x\right )} + 9 \, e^{x}}{24 \, {\left (e^{\left (4 \, x\right )} - 1\right )}^{3}} - \frac {3}{16} \, \arctan \left (e^{x}\right ) - \frac {3}{32} \, \log \left (e^{x} + 1\right ) + \frac {3}{32} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.18, size = 114, normalized size = 1.52 \begin {gather*} \frac {3\,\ln \left (\frac {3}{8}-\frac {3\,{\mathrm {e}}^x}{8}\right )}{32}-\frac {3\,\ln \left (-\frac {3\,{\mathrm {e}}^x}{8}-\frac {3}{8}\right )}{32}-\frac {7\,{\mathrm {e}}^x}{8\,\left ({\mathrm {e}}^{4\,x}-1\right )}-\frac {\frac {2\,{\mathrm {e}}^{5\,x}}{3}+\frac {{\mathrm {e}}^{9\,x}}{3}+\frac {{\mathrm {e}}^x}{3}}{3\,{\mathrm {e}}^{4\,x}-3\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{12\,x}-1}-\frac {5\,{\mathrm {e}}^x}{6\,\left ({\mathrm {e}}^{8\,x}-2\,{\mathrm {e}}^{4\,x}+1\right )}-\frac {\ln \left (-\frac {3\,{\mathrm {e}}^x}{8}-\frac {3}{8}{}\mathrm {i}\right )\,3{}\mathrm {i}}{32}+\frac {\ln \left (-\frac {3\,{\mathrm {e}}^x}{8}+\frac {3}{8}{}\mathrm {i}\right )\,3{}\mathrm {i}}{32} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________