Optimal. Leaf size=109 \[ \frac {x}{a^2}-\frac {2 b \left (2 a^2-b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 (a-b)^{3/2} (a+b)^{3/2} d}+\frac {b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d (a+b \text {sech}(c+d x))} \]
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Rubi [A]
time = 0.12, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3870, 4004,
3916, 2738, 214} \begin {gather*} -\frac {2 b \left (2 a^2-b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac {b^2 \tanh (c+d x)}{a d \left (a^2-b^2\right ) (a+b \text {sech}(c+d x))}+\frac {x}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2738
Rule 3870
Rule 3916
Rule 4004
Rubi steps
\begin {align*} \int \frac {1}{(a+b \text {sech}(c+d x))^2} \, dx &=\frac {b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d (a+b \text {sech}(c+d x))}-\frac {\int \frac {-a^2+b^2+a b \text {sech}(c+d x)}{a+b \text {sech}(c+d x)} \, dx}{a \left (a^2-b^2\right )}\\ &=\frac {x}{a^2}+\frac {b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d (a+b \text {sech}(c+d x))}-\frac {\left (b \left (2 a^2-b^2\right )\right ) \int \frac {\text {sech}(c+d x)}{a+b \text {sech}(c+d x)} \, dx}{a^2 \left (a^2-b^2\right )}\\ &=\frac {x}{a^2}+\frac {b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d (a+b \text {sech}(c+d x))}-\frac {\left (2 a^2-b^2\right ) \int \frac {1}{1+\frac {a \cosh (c+d x)}{b}} \, dx}{a^2 \left (a^2-b^2\right )}\\ &=\frac {x}{a^2}+\frac {b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d (a+b \text {sech}(c+d x))}+\frac {\left (2 i \left (2 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{a^2 \left (a^2-b^2\right ) d}\\ &=\frac {x}{a^2}-\frac {2 b \left (2 a^2-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 (a-b)^{3/2} (a+b)^{3/2} d}+\frac {b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d (a+b \text {sech}(c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 203, normalized size = 1.86 \begin {gather*} \frac {a \left (\left (a^2-b^2\right )^{3/2} (c+d x)+\left (4 a^2 b-2 b^3\right ) \text {ArcTan}\left (\frac {(-a+b) \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )\right ) \cosh (c+d x)+b \left (\left (a^2-b^2\right )^{3/2} (c+d x)+\left (4 a^2 b-2 b^3\right ) \text {ArcTan}\left (\frac {(-a+b) \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+a b \sqrt {a^2-b^2} \sinh (c+d x)\right )}{a^2 (a-b) (a+b) \sqrt {a^2-b^2} d (b+a \cosh (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.50, size = 164, normalized size = 1.50
method | result | size |
derivativedivides | \(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}-\frac {2 b \left (-\frac {a b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}+\frac {\left (2 a^{2}-b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{2}}}{d}\) | \(164\) |
default | \(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}-\frac {2 b \left (-\frac {a b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}+\frac {\left (2 a^{2}-b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{2}}}{d}\) | \(164\) |
risch | \(\frac {x}{a^{2}}-\frac {2 b^{2} \left (b \,{\mathrm e}^{d x +c}+a \right )}{d \,a^{2} \left (a^{2}-b^{2}\right ) \left (a \,{\mathrm e}^{2 d x +2 c}+2 b \,{\mathrm e}^{d x +c}+a \right )}-\frac {2 b \ln \left ({\mathrm e}^{d x +c}+\frac {b \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}+\frac {b^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {b \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}+\frac {2 b \ln \left ({\mathrm e}^{d x +c}+\frac {b \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}-\frac {b^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {b \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}\) | \(368\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 544 vs.
\(2 (100) = 200\).
time = 0.41, size = 1207, normalized size = 11.07 \begin {gather*} \left [-\frac {2 \, a^{3} b^{2} - 2 \, a b^{4} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x \cosh \left (d x + c\right )^{2} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x \sinh \left (d x + c\right )^{2} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x + {\left (2 \, a^{3} b - a b^{3} + {\left (2 \, a^{3} b - a b^{3}\right )} \cosh \left (d x + c\right )^{2} + {\left (2 \, a^{3} b - a b^{3}\right )} \sinh \left (d x + c\right )^{2} + 2 \, {\left (2 \, a^{2} b^{2} - b^{4}\right )} \cosh \left (d x + c\right ) + 2 \, {\left (2 \, a^{2} b^{2} - b^{4} + {\left (2 \, a^{3} b - a b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) - a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + b\right )}}{a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) + 2 \, {\left (a \cosh \left (d x + c\right ) + b\right )} \sinh \left (d x + c\right ) + a}\right ) + 2 \, {\left (a^{2} b^{3} - b^{5} - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d x\right )} \cosh \left (d x + c\right ) + 2 \, {\left (a^{2} b^{3} - b^{5} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x \cosh \left (d x + c\right ) - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d x\right )} \sinh \left (d x + c\right )}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cosh \left (d x + c\right )^{2} + {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \sinh \left (d x + c\right )^{2} + 2 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d \cosh \left (d x + c\right ) + {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d + 2 \, {\left ({\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cosh \left (d x + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d\right )} \sinh \left (d x + c\right )}, -\frac {2 \, a^{3} b^{2} - 2 \, a b^{4} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x \cosh \left (d x + c\right )^{2} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x \sinh \left (d x + c\right )^{2} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x - 2 \, {\left (2 \, a^{3} b - a b^{3} + {\left (2 \, a^{3} b - a b^{3}\right )} \cosh \left (d x + c\right )^{2} + {\left (2 \, a^{3} b - a b^{3}\right )} \sinh \left (d x + c\right )^{2} + 2 \, {\left (2 \, a^{2} b^{2} - b^{4}\right )} \cosh \left (d x + c\right ) + 2 \, {\left (2 \, a^{2} b^{2} - b^{4} + {\left (2 \, a^{3} b - a b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right ) + 2 \, {\left (a^{2} b^{3} - b^{5} - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d x\right )} \cosh \left (d x + c\right ) + 2 \, {\left (a^{2} b^{3} - b^{5} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x \cosh \left (d x + c\right ) - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d x\right )} \sinh \left (d x + c\right )}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cosh \left (d x + c\right )^{2} + {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \sinh \left (d x + c\right )^{2} + 2 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d \cosh \left (d x + c\right ) + {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d + 2 \, {\left ({\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cosh \left (d x + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d\right )} \sinh \left (d x + c\right )}\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 {1}{\left (a + b \operatorname {sech}{\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 134, normalized size = 1.23 \begin {gather*} -\frac {\frac {2 \, {\left (2 \, a^{2} b - b^{3}\right )} \arctan \left (\frac {a e^{\left (d x + c\right )} + b}{\sqrt {a^{2} - b^{2}}}\right )}{{\left (a^{4} - a^{2} b^{2}\right )} \sqrt {a^{2} - b^{2}}} + \frac {2 \, {\left (b^{3} e^{\left (d x + c\right )} + a b^{2}\right )}}{{\left (a^{4} - a^{2} b^{2}\right )} {\left (a e^{\left (2 \, d x + 2 \, c\right )} + 2 \, b e^{\left (d x + c\right )} + a\right )}} - \frac {d x + c}{a^{2}}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.85, size = 296, normalized size = 2.72 \begin {gather*} \frac {\frac {2\,b^2}{d\,\left (a\,b^2-a^3\right )}+\frac {2\,b^3\,{\mathrm {e}}^{c+d\,x}}{a\,d\,\left (a\,b^2-a^3\right )}}{a+2\,b\,{\mathrm {e}}^{c+d\,x}+a\,{\mathrm {e}}^{2\,c+2\,d\,x}}+\frac {x}{a^2}+\frac {b\,\ln \left (\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (2\,a^2\,b-b^3\right )}{a^3\,\left (a^2-b^2\right )}-\frac {2\,b\,\left (2\,a^2-b^2\right )\,\left (a+b\,{\mathrm {e}}^{c+d\,x}\right )}{a^3\,{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}}\right )\,\left (2\,a^2-b^2\right )}{a^2\,d\,{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}}-\frac {b\,\ln \left (\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (2\,a^2\,b-b^3\right )}{a^3\,\left (a^2-b^2\right )}+\frac {2\,b\,\left (2\,a^2-b^2\right )\,\left (a+b\,{\mathrm {e}}^{c+d\,x}\right )}{a^3\,{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}}\right )\,\left (2\,a^2-b^2\right )}{a^2\,d\,{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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