Optimal. Leaf size=77 \[ \frac {(a+b) \tanh (c+d x)}{2 a b d \left (a+b \tanh ^2(c+d x)\right )}-\frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2} d} \]
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Rubi [A] time = 0.08, antiderivative size = 77, 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 = {3675, 385, 205} \[ \frac {(a+b) \tanh (c+d x)}{2 a b d \left (a+b \tanh ^2(c+d x)\right )}-\frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2} d} \]
Antiderivative was successfully verified.
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Rule 205
Rule 385
Rule 3675
Rubi steps
\begin {align*} \int \frac {\text {sech}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{\left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {(a+b) \tanh (c+d x)}{2 a b d \left (a+b \tanh ^2(c+d x)\right )}-\frac {(a-b) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{2 a b d}\\ &=-\frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2} d}+\frac {(a+b) \tanh (c+d x)}{2 a b d \left (a+b \tanh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 83, normalized size = 1.08 \[ \frac {(b-a) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )+\frac {\sqrt {a} \sqrt {b} (a+b) \sinh (2 (c+d x))}{(a+b) \cosh (2 (c+d x))+a-b}}{2 a^{3/2} b^{3/2} d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 1443, normalized size = 18.74 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.55, size = 156, normalized size = 2.03 \[ -\frac {\frac {{\left (a e^{\left (2 \, c\right )} - b e^{\left (2 \, c\right )}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right ) e^{\left (-2 \, c\right )}}{\sqrt {a b} a b} + \frac {2 \, {\left (a e^{\left (2 \, d x + 2 \, c\right )} - b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}}{{\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )} a b}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.37, size = 746, normalized size = 9.69 \[ \frac {\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a \right ) b}+\frac {\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a \right ) a}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a \right ) b}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a \right ) a}+\frac {a \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 d b \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}-\frac {\arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 d b \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {a \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 d b \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}+\frac {\arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 d b \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}+\frac {\arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 d a \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}-\frac {\arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right ) b}{2 d \sqrt {b \left (a +b \right )}\, a \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}-\frac {\arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 d a \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}-\frac {\arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right ) b}{2 d \sqrt {b \left (a +b \right )}\, a \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 127, normalized size = 1.65 \[ \frac {{\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a + b}{{\left (a^{2} b + a b^{2} + 2 \, {\left (a^{2} b - a b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{2} b + a b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac {{\left (a - b\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{2 \, \sqrt {a b} a b d} \]
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 \[ \int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^4\,{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]
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 \[ \int \frac {\operatorname {sech}^{4}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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