Optimal. Leaf size=55 \[ \frac {\tan ^{-1}(\sinh (c+d x))}{b d}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{b d \sqrt {a+b}} \]
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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 = {4147, 391, 203, 205} \[ \frac {\tan ^{-1}(\sinh (c+d x))}{b d}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{b d \sqrt {a+b}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 391
Rule 4147
Rubi steps
\begin {align*} \int \frac {\text {sech}^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+b+a 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 \operatorname {Subst}\left (\int \frac {1}{a+b+a x^2} \, dx,x,\sinh (c+d x)\right )}{b d}\\ &=\frac {\tan ^{-1}(\sinh (c+d x))}{b d}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{b \sqrt {a+b} d}\\ \end {align*}
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Mathematica [B] time = 0.71, size = 194, normalized size = 3.53 \[ \frac {\text {sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (2 \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tan ^{-1}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+\sqrt {a} \cosh (c) \tan ^{-1}\left (\frac {\sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text {csch}(c+d x)}{\sqrt {a}}\right )-\sqrt {a} \sinh (c) \tan ^{-1}\left (\frac {\sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text {csch}(c+d x)}{\sqrt {a}}\right )\right )}{2 b d \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \left (a+b \text {sech}^2(c+d x)\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 526, normalized size = 9.56 \[ \left [\frac {\sqrt {-\frac {a}{a + b}} \log \left (\frac {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 (3 \, a + 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} - 3 \, a - 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} - {\left (3 \, a + 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 4 \, {\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 (a + b\right )} \cosh \left (d x + c\right ) + {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} - a - b\right )} \sinh \left (d x + c\right )\right )} \sqrt {-\frac {a}{a + b}} + a}{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 \, \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}{2 \, b d}, -\frac {\sqrt {\frac {a}{a + b}} \arctan \left (\frac {1}{2} \, \sqrt {\frac {a}{a + b}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}\right ) + \sqrt {\frac {a}{a + b}} \arctan \left (\frac {{\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} + {\left (3 \, a + 4 \, b\right )} \cosh \left (d x + c\right ) + {\left (3 \, a \cosh \left (d x + c\right )^{2} + 3 \, a + 4 \, b\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a}{a + b}}}{2 \, a}\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 [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.25, size = 108, normalized size = 1.96 \[ -\frac {\sqrt {a}\, \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{d b \sqrt {a +b}}-\frac {\sqrt {a}\, \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{d b \sqrt {a +b}}+\frac {2 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d b} \]
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 \[ \frac {2 \, \arctan \left (e^{\left (d x + c\right )}\right )}{b d} - 8 \, \int \frac {a e^{\left (3 \, d x + 3 \, c\right )} + a e^{\left (d x + c\right )}}{4 \, {\left (a b e^{\left (4 \, d x + 4 \, c\right )} + a b + 2 \, {\left (a b e^{\left (2 \, c\right )} + 2 \, b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.81, size = 307, normalized size = 5.58 \[ \frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (9\,a^2\,\sqrt {b^2\,d^2}+16\,b^2\,\sqrt {b^2\,d^2}+24\,a\,b\,\sqrt {b^2\,d^2}\right )}{9\,d\,a^2\,b+24\,d\,a\,b^2+16\,d\,b^3}\right )}{\sqrt {b^2\,d^2}}-\frac {\sqrt {a}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {a}\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {b^2\,d^2\,\left (a+b\right )}}{2\,b\,d\,\left (a+b\right )}\right )+2\,\mathrm {atan}\left (\frac {4\,b^4\,d^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+4\,a^2\,b^2\,d^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {b^3\,d^2+a\,b^2\,d^2}\,\sqrt {b^2\,d^2\,\left (a+b\right )}+8\,a\,b^3\,d^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+a\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\sqrt {b^3\,d^2+a\,b^2\,d^2}\,\sqrt {b^2\,d^2\,\left (a+b\right )}}{\sqrt {a}\,d\,\left (2\,b^2+2\,a\,b\right )\,\sqrt {b^2\,d^2\,\left (a+b\right )}}\right )\right )}{2\,\sqrt {b^3\,d^2+a\,b^2\,d^2}} \]
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 \[ \int \frac {\operatorname {sech}^{3}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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