Optimal. Leaf size=52 \[ \frac {\tanh (c+d x)}{b d}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{b^{3/2} d \sqrt {a+b}} \]
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Rubi [A] time = 0.07, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4146, 388, 208} \[ \frac {\tanh (c+d x)}{b d}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{b^{3/2} d \sqrt {a+b}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 388
Rule 4146
Rubi steps
\begin {align*} \int \frac {\text {sech}^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\tanh (c+d x)}{b d}-\frac {a \operatorname {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{b d}\\ &=-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{b^{3/2} \sqrt {a+b} d}+\frac {\tanh (c+d x)}{b d}\\ \end {align*}
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Mathematica [B] time = 0.63, size = 182, normalized size = 3.50 \[ \frac {\text {sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (\sqrt {a+b} \text {sech}(c) \sinh (d x) \sqrt {b (\cosh (c)-\sinh (c))^4} \text {sech}(c+d x)+a (\sinh (2 c)-\cosh (2 c)) \tanh ^{-1}\left (\frac {(\cosh (2 c)-\sinh (2 c)) \text {sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right )\right )}{2 b d \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4} \left (a+b \text {sech}^2(c+d x)\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 645, normalized size = 12.40 \[ \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\right )} \sqrt {a b + b^{2}} \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 \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 {a b + b^{2}}}{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 ) - 4 \, a b - 4 \, b^{2}}{2 \, {\left ({\left (a b^{2} + b^{3}\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (a b^{2} + b^{3}\right )} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a b^{2} + b^{3}\right )} d \sinh \left (d x + c\right )^{2} + {\left (a b^{2} + b^{3}\right )} d\right )}}, -\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\right )} \sqrt {-a b - b^{2}} \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 {-a b - b^{2}}}{2 \, {\left (a b + b^{2}\right )}}\right ) + 2 \, a b + 2 \, b^{2}}{{\left (a b^{2} + b^{3}\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (a b^{2} + b^{3}\right )} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a b^{2} + b^{3}\right )} d \sinh \left (d x + c\right )^{2} + {\left (a b^{2} + b^{3}\right )} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 72, normalized size = 1.38 \[ -\frac {\frac {a \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} b} + \frac {2}{b {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 141, normalized size = 2.71 \[ \frac {a \ln \left (-\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {a +b}\right )}{2 d \,b^{\frac {3}{2}} \sqrt {a +b}}-\frac {a \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{2 d \,b^{\frac {3}{2}} \sqrt {a +b}}+\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 91, normalized size = 1.75 \[ \frac {a \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{2 \, \sqrt {{\left (a + b\right )} b} b d} + \frac {2}{{\left (b e^{\left (-2 \, d x - 2 \, c\right )} + b\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 166, normalized size = 3.19 \[ \frac {a\,\ln \left (\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}}{b}-\frac {2\,\left (a\,d+a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+2\,b\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^{3/2}\,d\,\sqrt {a+b}}\right )}{2\,b^{3/2}\,d\,\sqrt {a+b}}-\frac {2}{b\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {a\,\ln \left (\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}}{b}+\frac {2\,\left (a\,d+a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+2\,b\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^{3/2}\,d\,\sqrt {a+b}}\right )}{2\,b^{3/2}\,d\,\sqrt {a+b}} \]
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 \[ \int \frac {\operatorname {sech}^{4}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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