Optimal. Leaf size=46 \[ \frac {x}{a}-\frac {\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {b} d} \]
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Rubi [A]
time = 0.09, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4226, 2000,
492, 212, 214} \begin {gather*} \frac {x}{a}-\frac {\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {b} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 214
Rule 492
Rule 2000
Rule 4226
Rubi steps
\begin {align*} \int \frac {\tanh ^2(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {x^2}{\left (1-x^2\right ) \left (a+b \left (1-x^2\right )\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{a d}-\frac {(a+b) \text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{a d}\\ &=\frac {x}{a}-\frac {\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {b} d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(174\) vs. \(2(46)=92\).
time = 0.24, size = 174, normalized size = 3.78 \begin {gather*} \frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^2(c+d x) \left (\sqrt {a+b} d x \sqrt {b (\cosh (c)-\sinh (c))^4}+(a+b) \tanh ^{-1}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (-\cosh (2 c)+\sinh (2 c))\right )}{2 a \sqrt {a+b} d \left (a+b \text {sech}^2(c+d x)\right ) \sqrt {b (\cosh (c)-\sinh (c))^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(143\) vs.
\(2(38)=76\).
time = 2.45, size = 144, normalized size = 3.13
method | result | size |
risch | \(\frac {x}{a}+\frac {\sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {b \left (a +b \right )}+a +2 b}{a}\right )}{2 b d a}-\frac {\sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {b \left (a +b \right )}-a -2 b}{a}\right )}{2 b d a}\) | \(104\) |
derivativedivides | \(\frac {\frac {2 \left (a +b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a}}{d}\) | \(144\) |
default | \(\frac {\frac {2 \left (a +b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a}}{d}\) | \(144\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 291 vs.
\(2 (38) = 76\).
time = 0.50, size = 291, normalized size = 6.33 \begin {gather*} -\frac {{\left (a + 2 \, b\right )} \log \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{8 \, \sqrt {{\left (a + b\right )} b} a d} + \frac {\log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{4 \, \sqrt {{\left (a + b\right )} b} d} + \frac {{\left (a + 2 \, b\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{8 \, \sqrt {{\left (a + b\right )} b} a d} + \frac {\log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a + 2 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{4 \, a d} - \frac {\log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{4 \, a d} \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. 81 vs.
\(2 (38) = 76\).
time = 0.44, size = 419, normalized size = 9.11 \begin {gather*} \left [\frac {2 \, d x + \sqrt {\frac {a + b}{b}} \log \left (\frac {a^{2} \cosh \left (d x + c\right )^{4} + 4 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{2} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{2} \cosh \left (d x + c\right )^{2} + a^{2} + 2 \, a b\right )} \sinh \left (d x + c\right )^{2} + a^{2} + 8 \, a b + 8 \, b^{2} + 4 \, {\left (a^{2} \cosh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 4 \, {\left (a b \cosh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b \sinh \left (d x + c\right )^{2} + a b + 2 \, b^{2}\right )} \sqrt {\frac {a + b}{b}}}{a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} + 2 \, {\left (a + 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} + a + 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} + {\left (a + 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a}\right )}{2 \, a d}, \frac {d x - \sqrt {-\frac {a + b}{b}} \arctan \left (\frac {{\left (a \cosh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a \sinh \left (d x + c\right )^{2} + a + 2 \, b\right )} \sqrt {-\frac {a + b}{b}}}{2 \, {\left (a + b\right )}}\right )}{a d}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\tanh ^{2}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.16, size = 66, normalized size = 1.43 \begin {gather*} -\frac {\frac {{\left (a + b\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} a} - \frac {d x + c}{a}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.43, size = 105, normalized size = 2.28 \begin {gather*} \frac {x}{a}+\frac {\mathrm {atan}\left (\frac {\sqrt {-a^2\,b\,d^2}}{a\,d\,\sqrt {a+b}}+\frac {\sqrt {-a^2\,b\,d^2}}{2\,b\,d\,\sqrt {a+b}}+\frac {{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\sqrt {-a^2\,b\,d^2}}{2\,b\,d\,\sqrt {a+b}}\right )\,\sqrt {a+b}}{\sqrt {-a^2\,b\,d^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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