Optimal. Leaf size=152 \[ \frac {e^x}{(1-a)^2}+\frac {e^x}{(1-a)^2 (1+a) \left (1+a+(-1+a) e^{4 x}\right )}-\frac {(1+4 a) \text {ArcTan}\left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{1+a}}\right )}{2 (1-a)^2 (1+a)^{3/2} \sqrt [4]{1-a^2}}-\frac {(1+4 a) \tanh ^{-1}\left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{1+a}}\right )}{2 (1-a)^2 (1+a)^{3/2} \sqrt [4]{1-a^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2320, 398, 393,
218, 214, 211} \begin {gather*} -\frac {(4 a+1) \text {ArcTan}\left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{a+1}}\right )}{2 (1-a)^2 (a+1)^{3/2} \sqrt [4]{1-a^2}}-\frac {(4 a+1) \tanh ^{-1}\left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{a+1}}\right )}{2 (1-a)^2 (a+1)^{3/2} \sqrt [4]{1-a^2}}+\frac {e^x}{(1-a)^2}+\frac {e^x}{(1-a)^2 (a+1) \left ((a-1) e^{4 x}+a+1\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 214
Rule 218
Rule 393
Rule 398
Rule 2320
Rubi steps
\begin {align*} \int \frac {e^x}{(a-\tanh (2 x))^2} \, dx &=\text {Subst}\left (\int \frac {\left (1+x^4\right )^2}{\left (1+a-(1-a) x^4\right )^2} \, dx,x,e^x\right )\\ &=\text {Subst}\left (\int \left (\frac {1}{(-1+a)^2}-\frac {4 \left (a-(1-a) x^4\right )}{(-1+a)^2 \left (1+a+(-1+a) x^4\right )^2}\right ) \, dx,x,e^x\right )\\ &=\frac {e^x}{(1-a)^2}-\frac {4 \text {Subst}\left (\int \frac {a-(1-a) x^4}{\left (1+a+(-1+a) x^4\right )^2} \, dx,x,e^x\right )}{(1-a)^2}\\ &=\frac {e^x}{(1-a)^2}+\frac {e^x}{(1-a)^2 (1+a) \left (1+a-(1-a) e^{4 x}\right )}-\frac {(1+4 a) \text {Subst}\left (\int \frac {1}{1+a+(-1+a) x^4} \, dx,x,e^x\right )}{(1-a)^2 (1+a)}\\ &=\frac {e^x}{(1-a)^2}+\frac {e^x}{(1-a)^2 (1+a) \left (1+a-(1-a) e^{4 x}\right )}-\frac {(1+4 a) \text {Subst}\left (\int \frac {1}{\sqrt {1+a}-\sqrt {1-a} x^2} \, dx,x,e^x\right )}{2 (1-a)^2 (1+a)^{3/2}}-\frac {(1+4 a) \text {Subst}\left (\int \frac {1}{\sqrt {1+a}+\sqrt {1-a} x^2} \, dx,x,e^x\right )}{2 (1-a)^2 (1+a)^{3/2}}\\ &=\frac {e^x}{(1-a)^2}+\frac {e^x}{(1-a)^2 (1+a) \left (1+a-(1-a) e^{4 x}\right )}-\frac {(1+4 a) \tan ^{-1}\left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{1+a}}\right )}{2 (1-a)^2 (1+a)^{3/2} \sqrt [4]{1-a^2}}-\frac {(1+4 a) \tanh ^{-1}\left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{1+a}}\right )}{2 (1-a)^2 (1+a)^{3/2} \sqrt [4]{1-a^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.12, size = 107, normalized size = 0.70 \begin {gather*} \frac {\frac {4 (-1+a) e^x \left (2+2 a-e^{4 x}+a^2 \left (1+e^{4 x}\right )\right )}{1+a-e^{4 x}+a e^{4 x}}+(1+4 a) \text {RootSum}\left [1+a-\text {$\#$1}^4+a \text {$\#$1}^4\&,\frac {x-\log \left (e^x-\text {$\#$1}\right )}{\text {$\#$1}^3}\&\right ]}{4 (-1+a)^3 (1+a)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.37, size = 191, normalized size = 1.26
method | result | size |
default | \(-\frac {2 \left (\frac {-\frac {\left (a -2\right ) \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{2 a \left (1+a \right )}-\frac {3 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{2 \left (1+a \right )}+\frac {\left (a +2\right ) \tanh \left (\frac {x}{2}\right )}{2 a \left (1+a \right )}-\frac {1}{2 \left (1+a \right )}}{a \left (\tanh ^{4}\left (\frac {x}{2}\right )\right )+6 a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-4 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )+a -4 \tanh \left (\frac {x}{2}\right )}+\frac {\left (1+4 a \right ) \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+6 a \,\textit {\_Z}^{2}-4 \textit {\_Z} +a \right )}{\sum }\frac {\left (\textit {\_R}^{2}-2 \textit {\_R} +1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{3} a -3 \textit {\_R}^{2}+3 \textit {\_R} a -1}\right )}{8+8 a}\right )}{\left (-1+a \right )^{2}}-\frac {2}{\left (-1+a \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}\) | \(191\) |
risch | \(\frac {{\mathrm e}^{x}}{a^{2}-2 a +1}+\frac {{\mathrm e}^{x}}{\left (1+a \right ) \left (a^{2}-2 a +1\right ) \left (a \,{\mathrm e}^{4 x}-{\mathrm e}^{4 x}+a +1\right )}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (256 a^{16}-512 a^{15}-1536 a^{14}+3584 a^{13}+3584 a^{12}-10752 a^{11}-3584 a^{10}+17920 a^{9}-17920 a^{7}+3584 a^{6}+10752 a^{5}-3584 a^{4}-3584 a^{3}+1536 a^{2}+512 a -256\right ) \textit {\_Z}^{4}+256 a^{4}+256 a^{3}+96 a^{2}+16 a +1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{x}+\frac {\left (-\frac {4 a^{4}}{1+4 a}+\frac {8 a^{2}}{1+4 a}-\frac {4}{1+4 a}\right ) \textit {\_R}}{\frac {4 a}{1+4 a}+\frac {1}{1+4 a}}\right )\right )\) | \(215\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 1163 vs.
\(2 (119) = 238\).
time = 0.40, size = 1163, normalized size = 7.65 \begin {gather*} -\frac {4 \, {\left (a^{4} - 2 \, a^{2} + {\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )} e^{\left (4 \, x\right )} + 1\right )} \left (-\frac {256 \, a^{4} + 256 \, a^{3} + 96 \, a^{2} + 16 \, a + 1}{a^{16} - 2 \, a^{15} - 6 \, a^{14} + 14 \, a^{13} + 14 \, a^{12} - 42 \, a^{11} - 14 \, a^{10} + 70 \, a^{9} - 70 \, a^{7} + 14 \, a^{6} + 42 \, a^{5} - 14 \, a^{4} - 14 \, a^{3} + 6 \, a^{2} + 2 \, a - 1}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (4 \, a^{13} - 7 \, a^{12} - 18 \, a^{11} + 36 \, a^{10} + 30 \, a^{9} - 75 \, a^{8} - 20 \, a^{7} + 80 \, a^{6} - 45 \, a^{4} + 6 \, a^{3} + 12 \, a^{2} - 2 \, a - 1\right )} \left (-\frac {256 \, a^{4} + 256 \, a^{3} + 96 \, a^{2} + 16 \, a + 1}{a^{16} - 2 \, a^{15} - 6 \, a^{14} + 14 \, a^{13} + 14 \, a^{12} - 42 \, a^{11} - 14 \, a^{10} + 70 \, a^{9} - 70 \, a^{7} + 14 \, a^{6} + 42 \, a^{5} - 14 \, a^{4} - 14 \, a^{3} + 6 \, a^{2} + 2 \, a - 1}\right )^{\frac {3}{4}} e^{x} - {\left (a^{12} - 2 \, a^{11} - 4 \, a^{10} + 10 \, a^{9} + 5 \, a^{8} - 20 \, a^{7} + 20 \, a^{5} - 5 \, a^{4} - 10 \, a^{3} + 4 \, a^{2} + 2 \, a - 1\right )} \sqrt {{\left (16 \, a^{2} + 8 \, a + 1\right )} e^{\left (2 \, x\right )} + {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} \sqrt {-\frac {256 \, a^{4} + 256 \, a^{3} + 96 \, a^{2} + 16 \, a + 1}{a^{16} - 2 \, a^{15} - 6 \, a^{14} + 14 \, a^{13} + 14 \, a^{12} - 42 \, a^{11} - 14 \, a^{10} + 70 \, a^{9} - 70 \, a^{7} + 14 \, a^{6} + 42 \, a^{5} - 14 \, a^{4} - 14 \, a^{3} + 6 \, a^{2} + 2 \, a - 1}}} \left (-\frac {256 \, a^{4} + 256 \, a^{3} + 96 \, a^{2} + 16 \, a + 1}{a^{16} - 2 \, a^{15} - 6 \, a^{14} + 14 \, a^{13} + 14 \, a^{12} - 42 \, a^{11} - 14 \, a^{10} + 70 \, a^{9} - 70 \, a^{7} + 14 \, a^{6} + 42 \, a^{5} - 14 \, a^{4} - 14 \, a^{3} + 6 \, a^{2} + 2 \, a - 1}\right )^{\frac {3}{4}}}{256 \, a^{4} + 256 \, a^{3} + 96 \, a^{2} + 16 \, a + 1}\right ) + {\left (a^{4} - 2 \, a^{2} + {\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )} e^{\left (4 \, x\right )} + 1\right )} \left (-\frac {256 \, a^{4} + 256 \, a^{3} + 96 \, a^{2} + 16 \, a + 1}{a^{16} - 2 \, a^{15} - 6 \, a^{14} + 14 \, a^{13} + 14 \, a^{12} - 42 \, a^{11} - 14 \, a^{10} + 70 \, a^{9} - 70 \, a^{7} + 14 \, a^{6} + 42 \, a^{5} - 14 \, a^{4} - 14 \, a^{3} + 6 \, a^{2} + 2 \, a - 1}\right )^{\frac {1}{4}} \log \left ({\left (4 \, a + 1\right )} e^{x} + {\left (a^{4} - 2 \, a^{2} + 1\right )} \left (-\frac {256 \, a^{4} + 256 \, a^{3} + 96 \, a^{2} + 16 \, a + 1}{a^{16} - 2 \, a^{15} - 6 \, a^{14} + 14 \, a^{13} + 14 \, a^{12} - 42 \, a^{11} - 14 \, a^{10} + 70 \, a^{9} - 70 \, a^{7} + 14 \, a^{6} + 42 \, a^{5} - 14 \, a^{4} - 14 \, a^{3} + 6 \, a^{2} + 2 \, a - 1}\right )^{\frac {1}{4}}\right ) - {\left (a^{4} - 2 \, a^{2} + {\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )} e^{\left (4 \, x\right )} + 1\right )} \left (-\frac {256 \, a^{4} + 256 \, a^{3} + 96 \, a^{2} + 16 \, a + 1}{a^{16} - 2 \, a^{15} - 6 \, a^{14} + 14 \, a^{13} + 14 \, a^{12} - 42 \, a^{11} - 14 \, a^{10} + 70 \, a^{9} - 70 \, a^{7} + 14 \, a^{6} + 42 \, a^{5} - 14 \, a^{4} - 14 \, a^{3} + 6 \, a^{2} + 2 \, a - 1}\right )^{\frac {1}{4}} \log \left ({\left (4 \, a + 1\right )} e^{x} - {\left (a^{4} - 2 \, a^{2} + 1\right )} \left (-\frac {256 \, a^{4} + 256 \, a^{3} + 96 \, a^{2} + 16 \, a + 1}{a^{16} - 2 \, a^{15} - 6 \, a^{14} + 14 \, a^{13} + 14 \, a^{12} - 42 \, a^{11} - 14 \, a^{10} + 70 \, a^{9} - 70 \, a^{7} + 14 \, a^{6} + 42 \, a^{5} - 14 \, a^{4} - 14 \, a^{3} + 6 \, a^{2} + 2 \, a - 1}\right )^{\frac {1}{4}}\right ) - 4 \, {\left (a^{2} - 1\right )} e^{\left (5 \, x\right )} - 4 \, {\left (a^{2} + 2 \, a + 2\right )} e^{x}}{4 \, {\left (a^{4} - 2 \, a^{2} + {\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )} e^{\left (4 \, x\right )} + 1\right )}} \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 \frac {e^{x}}{\left (a - \tanh {\left (2 x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 456 vs.
\(2 (119) = 238\).
time = 0.39, size = 456, normalized size = 3.00 \begin {gather*} -\frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} {\left (4 \, a + 1\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}} + 2 \, e^{x}\right )}}{2 \, \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a^{5} - \sqrt {2} a^{4} - 2 \, \sqrt {2} a^{3} + 2 \, \sqrt {2} a^{2} + \sqrt {2} a - \sqrt {2}\right )}} - \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} {\left (4 \, a + 1\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}} - 2 \, e^{x}\right )}}{2 \, \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a^{5} - \sqrt {2} a^{4} - 2 \, \sqrt {2} a^{3} + 2 \, \sqrt {2} a^{2} + \sqrt {2} a - \sqrt {2}\right )}} - \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} {\left (4 \, a + 1\right )} \log \left (\sqrt {2} \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}} e^{x} + \sqrt {\frac {a + 1}{a - 1}} + e^{\left (2 \, x\right )}\right )}{4 \, {\left (\sqrt {2} a^{5} - \sqrt {2} a^{4} - 2 \, \sqrt {2} a^{3} + 2 \, \sqrt {2} a^{2} + \sqrt {2} a - \sqrt {2}\right )}} + \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} {\left (4 \, a + 1\right )} \log \left (-\sqrt {2} \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}} e^{x} + \sqrt {\frac {a + 1}{a - 1}} + e^{\left (2 \, x\right )}\right )}{4 \, {\left (\sqrt {2} a^{5} - \sqrt {2} a^{4} - 2 \, \sqrt {2} a^{3} + 2 \, \sqrt {2} a^{2} + \sqrt {2} a - \sqrt {2}\right )}} + \frac {e^{x}}{a^{2} - 2 \, a + 1} + \frac {e^{x}}{{\left (a^{3} - a^{2} - a + 1\right )} {\left (a e^{\left (4 \, x\right )} + a - e^{\left (4 \, x\right )} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 23.35, size = 280, normalized size = 1.84 \begin {gather*} \frac {{\mathrm {e}}^x}{{\left (a-1\right )}^2}+\frac {\ln \left (\frac {4\,a+1}{{\left (a-1\right )}^{13/4}\,{\left (-a-1\right )}^{3/4}}+\frac {{\mathrm {e}}^x\,\left (4\,a+1\right )}{a^4-2\,a^3+2\,a-1}\right )\,\left (4\,a+1\right )}{4\,{\left (a-1\right )}^{9/4}\,{\left (-a-1\right )}^{7/4}}-\frac {\ln \left (\frac {{\mathrm {e}}^x\,\left (4\,a+1\right )}{a^4-2\,a^3+2\,a-1}-\frac {4\,a+1}{{\left (a-1\right )}^{13/4}\,{\left (-a-1\right )}^{3/4}}\right )\,\left (4\,a+1\right )}{4\,{\left (a-1\right )}^{9/4}\,{\left (-a-1\right )}^{7/4}}+\frac {{\mathrm {e}}^x}{{\left (a-1\right )}^2\,\left (a+1\right )\,\left (a+{\mathrm {e}}^{4\,x}\,\left (a-1\right )+1\right )}-\frac {\ln \left (\frac {{\mathrm {e}}^x\,\left (4\,a+1\right )}{{\left (a-1\right )}^3\,\left (a+1\right )}-\frac {\left (4\,a+1\right )\,1{}\mathrm {i}}{{\left (a-1\right )}^{13/4}\,{\left (-a-1\right )}^{3/4}}\right )\,\left (4\,a+1\right )\,1{}\mathrm {i}}{4\,{\left (a-1\right )}^{9/4}\,{\left (-a-1\right )}^{7/4}}+\frac {\ln \left (\frac {{\mathrm {e}}^x\,\left (4\,a+1\right )}{{\left (a-1\right )}^3\,\left (a+1\right )}+\frac {\left (4\,a+1\right )\,1{}\mathrm {i}}{{\left (a-1\right )}^{13/4}\,{\left (-a-1\right )}^{3/4}}\right )\,\left (4\,a+1\right )\,1{}\mathrm {i}}{4\,{\left (a-1\right )}^{9/4}\,{\left (-a-1\right )}^{7/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________