Optimal. Leaf size=108 \[ \frac {3 \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 \sqrt {b} (a+b)^{5/2} d}+\frac {\tanh (c+d x)}{4 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {3 \tanh (c+d x)}{8 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )} \]
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Rubi [A]
time = 0.06, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4231, 205, 214}
\begin {gather*} \frac {3 \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 \sqrt {b} d (a+b)^{5/2}}+\frac {3 \tanh (c+d x)}{8 d (a+b)^2 \left (a-b \tanh ^2(c+d x)+b\right )}+\frac {\tanh (c+d x)}{4 d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 214
Rule 4231
Rubi steps
\begin {align*} \int \frac {\text {sech}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (a+b-b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\tanh (c+d x)}{4 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {3 \text {Subst}\left (\int \frac {1}{\left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 (a+b) d}\\ &=\frac {\tanh (c+d x)}{4 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {3 \tanh (c+d x)}{8 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {3 \text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{8 (a+b)^2 d}\\ &=\frac {3 \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 \sqrt {b} (a+b)^{5/2} d}+\frac {\tanh (c+d x)}{4 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {3 \tanh (c+d x)}{8 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(258\) vs. \(2(108)=216\).
time = 1.52, size = 258, normalized size = 2.39 \begin {gather*} \frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^6(c+d x) \left (\frac {3 \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 ) (a+2 b+a \cosh (2 (c+d x)))^2 (\cosh (2 c)-\sinh (2 c))}{\sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}+\frac {4 b (a+b) \text {sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{a^2}-\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}(2 c) \left (\left (5 a^2+16 a b+8 b^2\right ) \sinh (2 c)-a (5 a+2 b) \sinh (2 d x)\right )}{a^2}\right )}{64 (a+b)^2 d \left (a+b \text {sech}^2(c+d x)\right )^3} \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. \(259\) vs.
\(2(94)=188\).
time = 2.13, size = 260, normalized size = 2.41
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (-\frac {5 \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )}-\frac {3 \left (5 a +b \right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{2}}-\frac {3 \left (5 a +b \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{2}}-\frac {5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a +b \right )}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}-\frac {3 \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{4 \left (a^{2}+2 a b +b^{2}\right )}}{d}\) | \(260\) |
default | \(\frac {-\frac {2 \left (-\frac {5 \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )}-\frac {3 \left (5 a +b \right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{2}}-\frac {3 \left (5 a +b \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{2}}-\frac {5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a +b \right )}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}-\frac {3 \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{4 \left (a^{2}+2 a b +b^{2}\right )}}{d}\) | \(260\) |
risch | \(-\frac {5 a^{3} {\mathrm e}^{6 d x +6 c}+16 a^{2} b \,{\mathrm e}^{6 d x +6 c}+8 a \,b^{2} {\mathrm e}^{6 d x +6 c}+15 a^{3} {\mathrm e}^{4 d x +4 c}+46 a^{2} b \,{\mathrm e}^{4 d x +4 c}+56 a \,b^{2} {\mathrm e}^{4 d x +4 c}+16 b^{3} {\mathrm e}^{4 d x +4 c}+15 a^{3} {\mathrm e}^{2 d x +2 c}+32 a^{2} b \,{\mathrm e}^{2 d x +2 c}+8 a \,b^{2} {\mathrm e}^{2 d x +2 c}+5 a^{3}+2 a^{2} b}{4 a^{2} d \left (a +b \right )^{2} \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{16 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{16 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d}\) | \(364\) |
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. 353 vs.
\(2 (100) = 200\).
time = 0.53, size = 353, normalized size = 3.27 \begin {gather*} \frac {5 \, a^{3} + 2 \, a^{2} b + {\left (15 \, a^{3} + 32 \, a^{2} b + 8 \, a b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (15 \, a^{3} + 46 \, a^{2} b + 56 \, a b^{2} + 16 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (5 \, a^{3} + 16 \, a^{2} b + 8 \, a b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{4 \, {\left (a^{6} + 2 \, a^{5} b + a^{4} b^{2} + 4 \, {\left (a^{6} + 4 \, a^{5} b + 5 \, a^{4} b^{2} + 2 \, a^{3} b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{6} + 14 \, a^{5} b + 27 \, a^{4} b^{2} + 24 \, a^{3} b^{3} + 8 \, a^{2} b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (a^{6} + 4 \, a^{5} b + 5 \, a^{4} b^{2} + 2 \, a^{3} b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{6} + 2 \, a^{5} b + a^{4} b^{2}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} - \frac {3 \, \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 )}{16 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {{\left (a + b\right )} b} 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. 2434 vs.
\(2 (100) = 200\).
time = 0.40, size = 5109, normalized size = 47.31 \begin {gather*} \text {Too large to display} \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 {\operatorname {sech}^{2}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 282 vs.
\(2 (100) = 200\).
time = 0.81, size = 282, normalized size = 2.61 \begin {gather*} \frac {\frac {3 \, \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {-a b - b^{2}}} - \frac {2 \, {\left (5 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 8 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 15 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 46 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 56 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 16 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 15 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 32 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, a^{3} + 2 \, a^{2} b\right )}}{{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{2}}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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