Optimal. Leaf size=77 \[ \frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{b^{5/2} d \sqrt {a+b}}-\frac {(a-b) \tanh (c+d x)}{b^2 d}-\frac {\tanh ^3(c+d x)}{3 b d} \]
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Rubi [A] time = 0.09, antiderivative size = 77, 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 = {4146, 390, 208} \[ \frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{b^{5/2} d \sqrt {a+b}}-\frac {(a-b) \tanh (c+d x)}{b^2 d}-\frac {\tanh ^3(c+d x)}{3 b d} \]
Antiderivative was successfully verified.
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Rule 208
Rule 390
Rule 4146
Rubi steps
\begin {align*} \int \frac {\text {sech}^6(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a-b}{b^2}-\frac {x^2}{b}+\frac {a^2}{b^2 \left (a+b-b x^2\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {(a-b) \tanh (c+d x)}{b^2 d}-\frac {\tanh ^3(c+d x)}{3 b d}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{b^2 d}\\ &=\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{b^{5/2} \sqrt {a+b} d}-\frac {(a-b) \tanh (c+d x)}{b^2 d}-\frac {\tanh ^3(c+d x)}{3 b d}\\ \end {align*}
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Mathematica [B] time = 2.18, size = 214, normalized size = 2.78 \[ \frac {\text {sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (3 a^2 (\cosh (2 c)-\sinh (2 c)) \tanh ^{-1}\left (\frac {(\cosh (2 c)-\sinh (2 c)) \text {sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right )+\sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4} \text {sech}(c+d x) \left (\text {sech}(c) \sinh (d x) \left (-3 a+b \text {sech}^2(c+d x)+2 b\right )+b \tanh (c) \text {sech}(c+d x)\right )\right )}{6 b^2 d \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4} \left (a+b \text {sech}^2(c+d x)\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 1905, normalized size = 24.74 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 118, normalized size = 1.53 \[ \frac {\frac {3 \, a^{2} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} b^{2}} + \frac {2 \, {\left (3 \, a e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a e^{\left (2 \, d x + 2 \, c\right )} - 6 \, b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a - 2 \, b\right )}}{b^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.28, size = 316, normalized size = 4.10 \[ -\frac {a^{2} \ln \left (-\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {a +b}\right )}{2 d \,b^{\frac {5}{2}} \sqrt {a +b}}+\frac {a^{2} \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{2 d \,b^{\frac {5}{2}} \sqrt {a +b}}-\frac {2 \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{d \,b^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {2 \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {4 \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{d \,b^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {4 \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{d \,b^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 160, normalized size = 2.08 \[ -\frac {a^{2} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{2 \, \sqrt {{\left (a + b\right )} b} b^{2} d} - \frac {2 \, {\left (6 \, {\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, a e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, a - 2 \, b\right )}}{3 \, {\left (3 \, b^{2} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, b^{2} e^{\left (-4 \, d x - 4 \, c\right )} + b^{2} e^{\left (-6 \, d x - 6 \, c\right )} + b^{2}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.99, size = 334, normalized size = 4.34 \[ \frac {8}{3\,b\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {4}{b\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}+\frac {2\,a}{b^2\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {a^2\,\ln \left (\frac {4\,a^2\,\left (2\,a\,b+a^2+a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+8\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+8\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^5\,\left (a+b\right )}-\frac {8\,a^2\,\left (a+2\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}+4\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^{9/2}\,\sqrt {a+b}}\right )}{2\,b^{5/2}\,d\,\sqrt {a+b}}+\frac {a^2\,\ln \left (\frac {8\,a^2\,\left (a+2\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}+4\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^{9/2}\,\sqrt {a+b}}+\frac {4\,a^2\,\left (2\,a\,b+a^2+a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+8\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+8\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^5\,\left (a+b\right )}\right )}{2\,b^{5/2}\,d\,\sqrt {a+b}} \]
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 \[ \int \frac {\operatorname {sech}^{6}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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