Optimal. Leaf size=144 \[ \frac {\left (3 a^2+8 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 b^{5/2} (a+b)^{5/2} d}-\frac {a \text {sech}^2(c+d x) \tanh (c+d x)}{4 b (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {3 a (a+2 b) \tanh (c+d x)}{8 b^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )} \]
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Rubi [A]
time = 0.11, antiderivative size = 144, 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 = {4231, 424, 393,
214} \begin {gather*} \frac {\left (3 a^2+8 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 b^{5/2} d (a+b)^{5/2}}-\frac {3 a (a+2 b) \tanh (c+d x)}{8 b^2 d (a+b)^2 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {a \tanh (c+d x) \text {sech}^2(c+d x)}{4 b 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 214
Rule 393
Rule 424
Rule 4231
Rubi steps
\begin {align*} \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{\left (a+b-b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {a \text {sech}^2(c+d x) \tanh (c+d x)}{4 b (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-a-4 b+(3 a+4 b) x^2}{\left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 b (a+b) d}\\ &=-\frac {a \text {sech}^2(c+d x) \tanh (c+d x)}{4 b (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {3 a (a+2 b) \tanh (c+d x)}{8 b^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\left (3 a^2+8 a b+8 b^2\right ) \text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{8 b^2 (a+b)^2 d}\\ &=\frac {\left (3 a^2+8 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 b^{5/2} (a+b)^{5/2} d}-\frac {a \text {sech}^2(c+d x) \tanh (c+d x)}{4 b (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {3 a (a+2 b) \tanh (c+d x)}{8 b^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.68, size = 125, normalized size = 0.87 \begin {gather*} \frac {\frac {\left (3 a^2+8 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}-\frac {a \sqrt {b} \left (3 a^2+16 a b+16 b^2+3 a (a+2 b) \cosh (2 (c+d x))\right ) \sinh (2 (c+d x))}{(a+b)^2 (a+2 b+a \cosh (2 (c+d x)))^2}}{8 b^{5/2} d} \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. \(327\) vs.
\(2(130)=260\).
time = 1.96, size = 328, normalized size = 2.28
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (\frac {a \left (3 a +8 b \right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right ) b^{2}}+\frac {a \left (9 a^{2}+13 a b -8 b^{2}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{2} b^{2}}+\frac {a \left (9 a^{2}+13 a b -8 b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{2} b^{2}}+\frac {a \left (3 a +8 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a +b \right ) b^{2}}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}-\frac {\left (3 a^{2}+8 a b +8 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{4 b^{2} \left (a^{2}+2 a b +b^{2}\right )}}{d}\) | \(328\) |
default | \(\frac {-\frac {2 \left (\frac {a \left (3 a +8 b \right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right ) b^{2}}+\frac {a \left (9 a^{2}+13 a b -8 b^{2}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{2} b^{2}}+\frac {a \left (9 a^{2}+13 a b -8 b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{2} b^{2}}+\frac {a \left (3 a +8 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a +b \right ) b^{2}}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}-\frac {\left (3 a^{2}+8 a b +8 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{4 b^{2} \left (a^{2}+2 a b +b^{2}\right )}}{d}\) | \(328\) |
risch | \(\frac {3 a^{3} {\mathrm e}^{6 d x +6 c}+8 a^{2} b \,{\mathrm e}^{6 d x +6 c}+8 a \,b^{2} {\mathrm e}^{6 d x +6 c}+9 a^{3} {\mathrm e}^{4 d x +4 c}+42 a^{2} b \,{\mathrm e}^{4 d x +4 c}+72 a \,b^{2} {\mathrm e}^{4 d x +4 c}+48 b^{3} {\mathrm e}^{4 d x +4 c}+9 a^{3} {\mathrm e}^{2 d x +2 c}+40 a^{2} b \,{\mathrm e}^{2 d x +2 c}+40 a \,b^{2} {\mathrm e}^{2 d x +2 c}+3 a^{3}+6 a^{2} b}{4 d \,b^{2} \left (a +b \right )^{2} \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) a^{2}}{16 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d \,b^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) a}{2 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d b}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{2 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) a^{2}}{16 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d \,b^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) a}{2 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d b}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{2 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d}\) | \(688\) |
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. 395 vs.
\(2 (136) = 272\).
time = 0.61, size = 395, normalized size = 2.74 \begin {gather*} -\frac {{\left (3 \, a^{2} + 8 \, a b + 8 \, b^{2}\right )} \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} b^{2} + 2 \, a b^{3} + b^{4}\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {3 \, a^{3} + 6 \, a^{2} b + {\left (9 \, a^{3} + 40 \, a^{2} b + 40 \, a b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, {\left (3 \, a^{3} + 14 \, a^{2} b + 24 \, a b^{2} + 16 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (3 \, a^{3} + 8 \, a^{2} b + 8 \, a b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{4 \, {\left (a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4} + 4 \, {\left (a^{4} b^{2} + 4 \, a^{3} b^{3} + 5 \, a^{2} b^{4} + 2 \, a b^{5}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{4} b^{2} + 14 \, a^{3} b^{3} + 27 \, a^{2} b^{4} + 24 \, a b^{5} + 8 \, b^{6}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (a^{4} b^{2} + 4 \, a^{3} b^{3} + 5 \, a^{2} b^{4} + 2 \, a b^{5}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} 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. 2823 vs.
\(2 (136) = 272\).
time = 0.50, size = 5887, normalized size = 40.88 \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}^{6}{\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. 302 vs.
\(2 (136) = 272\).
time = 0.99, size = 302, normalized size = 2.10 \begin {gather*} \frac {\frac {{\left (3 \, a^{2} + 8 \, a b + 8 \, b^{2}\right )} \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} b^{2} + 2 \, a b^{3} + b^{4}\right )} \sqrt {-a b - b^{2}}} + \frac {2 \, {\left (3 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 8 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 8 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 9 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 42 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 72 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 9 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 40 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 40 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{3} + 6 \, a^{2} b\right )}}{{\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\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 )}^6\,{\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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