Optimal. Leaf size=55 \[ \frac {\sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b d}-\frac {\tan ^{-1}(\sinh (c+d x))}{b d} \]
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Rubi [A] time = 0.07, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3676, 391, 203, 205} \[ \frac {\sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b d}-\frac {\tan ^{-1}(\sinh (c+d x))}{b d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 391
Rule 3676
Rubi steps
\begin {align*} \int \frac {\text {sech}^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+(a+b) x^2\right )} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{b d}+\frac {(a+b) \operatorname {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\sinh (c+d x)\right )}{b d}\\ &=-\frac {\tan ^{-1}(\sinh (c+d x))}{b d}+\frac {\sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b d}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 55, normalized size = 1.00 \[ -\frac {\frac {\sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {a+b}}\right )}{\sqrt {a}}+2 \tan ^{-1}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{b d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 540, normalized size = 9.82 \[ \left [\frac {\sqrt {-\frac {a + b}{a}} \log \left (\frac {{\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} - 2 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} - 3 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} - {\left (3 \, 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 )^{3} + 3 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{3} - a \cosh \left (d x + c\right ) + {\left (3 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )\right )} \sqrt {-\frac {a + b}{a}} + a + b}{{\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + {\left (a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a + b}\right ) - 4 \, \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}{2 \, b d}, \frac {\sqrt {\frac {a + b}{a}} \arctan \left (\frac {1}{2} \, \sqrt {\frac {a + b}{a}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}\right ) + \sqrt {\frac {a + b}{a}} \arctan \left (\frac {{\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (a + b\right )} \sinh \left (d x + c\right )^{3} + {\left (3 \, a - b\right )} \cosh \left (d x + c\right ) + {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + 3 \, a - b\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a + b}{a}}}{2 \, {\left (a + b\right )}}\right ) - 2 \, \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}{b d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 563, normalized size = 10.24 \[ \frac {\frac {{\left (2 \, \sqrt {a^{2} - b^{2} + 2 \, \sqrt {-a b} {\left (a + b\right )}} a b^{2} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} + \sqrt {a^{2} - b^{2} + 2 \, \sqrt {-a b} {\left (a + b\right )}} \sqrt {-a b} {\left (a + b\right )} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} {\left | b \right |} - {\left (a b^{2} - b^{3}\right )} \sqrt {a^{2} - b^{2} + 2 \, \sqrt {-a b} {\left (a + b\right )}} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |}\right )} \arctan \left (\frac {e^{\left (d x\right )}}{\sqrt {\frac {a b e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )} + \sqrt {{\left (a b e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )}\right )}^{2} - {\left (a b e^{\left (4 \, c\right )} + b^{2} e^{\left (4 \, c\right )}\right )} {\left (a b + b^{2}\right )}}}{a b e^{\left (4 \, c\right )} + b^{2} e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{{\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} \sqrt {-a b} {\left | b \right |}} - \frac {{\left (2 \, \sqrt {a^{2} - b^{2} - 2 \, \sqrt {-a b} {\left (a + b\right )}} a b^{2} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} - \sqrt {a^{2} - b^{2} - 2 \, \sqrt {-a b} {\left (a + b\right )}} \sqrt {-a b} {\left (a + b\right )} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} {\left | b \right |} - {\left (a b^{2} - b^{3}\right )} \sqrt {a^{2} - b^{2} - 2 \, \sqrt {-a b} {\left (a + b\right )}} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |}\right )} \arctan \left (\frac {e^{\left (d x\right )}}{\sqrt {\frac {a b e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )} - \sqrt {{\left (a b e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )}\right )}^{2} - {\left (a b e^{\left (4 \, c\right )} + b^{2} e^{\left (4 \, c\right )}\right )} {\left (a b + b^{2}\right )}}}{a b e^{\left (4 \, c\right )} + b^{2} e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{{\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} \sqrt {-a b} {\left | b \right |}} - \frac {2 \, \arctan \left (e^{\left (d x + c\right )}\right )}{b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.34, size = 494, normalized size = 8.98 \[ -\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 )}{d b \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \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 )}{d \sqrt {b \left (a +b \right )}\, \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 )}{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 )}{d \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 )}{d \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}{d \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 )}{d \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}{d \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}-\frac {2 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, \arctan \left (e^{\left (d x + c\right )}\right )}{b d} + 8 \, \int \frac {{\left (a e^{\left (3 \, c\right )} + b e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (a e^{c} + b e^{c}\right )} e^{\left (d x\right )}}{4 \, {\left (a b + b^{2} + {\left (a b e^{\left (4 \, c\right )} + b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \, {\left (a b e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.70, size = 449, normalized size = 8.16 \[ \frac {\sqrt {a+b}\,\left (2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a+b}\,\sqrt {a\,b^2\,d^2}}{2\,a\,b\,d}\right )-2\,\mathrm {atan}\left (\frac {\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {64\,\left (2\,a\,b^2\,d\,\sqrt {a+b}-6\,a^2\,b\,d\,\sqrt {a+b}\right )}{a^3\,b^3\,d^2\,{\left (a+b\right )}^2\,\left (a^2+2\,a\,b+b^2\right )}+\frac {32\,\left (3\,a^2\,\sqrt {a\,b^2\,d^2}-b^2\,\sqrt {a\,b^2\,d^2}+2\,a\,b\,\sqrt {a\,b^2\,d^2}\right )}{a^3\,b^2\,d\,{\left (a+b\right )}^{3/2}\,\left (a^2+2\,a\,b+b^2\right )\,\sqrt {a\,b^2\,d^2}}\right )-\frac {32\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (3\,a^2\,\sqrt {a\,b^2\,d^2}-b^2\,\sqrt {a\,b^2\,d^2}+2\,a\,b\,\sqrt {a\,b^2\,d^2}\right )}{a^3\,b^2\,d\,{\left (a+b\right )}^{3/2}\,\left (a^2+2\,a\,b+b^2\right )\,\sqrt {a\,b^2\,d^2}}\right )\,\left (a^4\,b\,\left (a+b\right )\,\sqrt {a\,b^2\,d^2}+a^2\,b^3\,\left (a+b\right )\,\sqrt {a\,b^2\,d^2}+2\,a^3\,b^2\,\left (a+b\right )\,\sqrt {a\,b^2\,d^2}\right )}{192\,a-64\,b}\right )\right )}{2\,\sqrt {a\,b^2\,d^2}}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (9\,a^2\,\sqrt {b^2\,d^2}+b^2\,\sqrt {b^2\,d^2}-6\,a\,b\,\sqrt {b^2\,d^2}\right )}{9\,d\,a^2\,b-6\,d\,a\,b^2+d\,b^3}\right )}{\sqrt {b^2\,d^2}} \]
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 \[ \int \frac {\operatorname {sech}^{3}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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