Optimal. Leaf size=131 \[ \frac {\left (3 a^2-2 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} b^{5/2} d}+\frac {(a+b) \text {sech}^2(c+d x) \tanh (c+d x)}{4 a b d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {3 \left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \tanh (c+d x)}{8 d \left (a+b \tanh ^2(c+d x)\right )} \]
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Rubi [A]
time = 0.09, antiderivative size = 131, 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 = {3756, 424, 393,
211} \begin {gather*} \frac {3 \left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \tanh (c+d x)}{8 d \left (a+b \tanh ^2(c+d x)\right )}+\frac {\left (3 a^2-2 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} b^{5/2} d}+\frac {(a+b) \tanh (c+d x) \text {sech}^2(c+d x)}{4 a b d \left (a+b \tanh ^2(c+d x)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 393
Rule 424
Rule 3756
Rubi steps
\begin {align*} \int \frac {\text {sech}^6(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{\left (a+b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {(a+b) \text {sech}^2(c+d x) \tanh (c+d x)}{4 a b d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {-a+3 b+(3 a-b) x^2}{\left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 a b d}\\ &=\frac {(a+b) \text {sech}^2(c+d x) \tanh (c+d x)}{4 a b d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {3 \left (a^2-b^2\right ) \tanh (c+d x)}{8 a^2 b^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac {\left (3 a^2-2 a b+3 b^2\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^2 b^2 d}\\ &=\frac {\left (3 a^2-2 a b+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} b^{5/2} d}+\frac {(a+b) \text {sech}^2(c+d x) \tanh (c+d x)}{4 a b d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {3 \left (a^2-b^2\right ) \tanh (c+d x)}{8 a^2 b^2 d \left (a+b \tanh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.68, size = 128, normalized size = 0.98 \begin {gather*} \frac {\left (3 a^2-2 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )-\frac {\sqrt {a} \sqrt {b} (a+b) \left (3 a^2-10 a b+3 b^2+3 \left (a^2-b^2\right ) \cosh (2 (c+d x))\right ) \sinh (2 (c+d x))}{(a-b+(a+b) \cosh (2 (c+d x)))^2}}{8 a^{5/2} 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. \(375\) vs.
\(2(117)=234\).
time = 2.82, size = 376, normalized size = 2.87
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (\frac {\left (3 a^{2}-2 a b -5 b^{2}\right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a \,b^{2}}+\frac {\left (9 a^{3}+14 a^{2} b -7 a \,b^{2}-12 b^{3}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2} b^{2}}+\frac {\left (9 a^{3}+14 a^{2} b -7 a \,b^{2}-12 b^{3}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2} b^{2}}+\frac {\left (3 a^{2}-2 a b -5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a \,b^{2}}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}-\frac {\left (3 a^{2}-2 a b +3 b^{2}\right ) \left (-\frac {\left (-a +\sqrt {b \left (a +b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (a +\sqrt {b \left (a +b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{4 a \,b^{2}}}{d}\) | \(376\) |
default | \(\frac {-\frac {2 \left (\frac {\left (3 a^{2}-2 a b -5 b^{2}\right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a \,b^{2}}+\frac {\left (9 a^{3}+14 a^{2} b -7 a \,b^{2}-12 b^{3}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2} b^{2}}+\frac {\left (9 a^{3}+14 a^{2} b -7 a \,b^{2}-12 b^{3}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2} b^{2}}+\frac {\left (3 a^{2}-2 a b -5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a \,b^{2}}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}-\frac {\left (3 a^{2}-2 a b +3 b^{2}\right ) \left (-\frac {\left (-a +\sqrt {b \left (a +b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (a +\sqrt {b \left (a +b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{4 a \,b^{2}}}{d}\) | \(376\) |
risch | \(\frac {3 a^{3} {\mathrm e}^{6 d x +6 c}+a^{2} b \,{\mathrm e}^{6 d x +6 c}+a \,b^{2} {\mathrm e}^{6 d x +6 c}+3 b^{3} {\mathrm e}^{6 d x +6 c}+9 a^{3} {\mathrm e}^{4 d x +4 c}-15 a^{2} b \,{\mathrm e}^{4 d x +4 c}+15 a \,b^{2} {\mathrm e}^{4 d x +4 c}-9 b^{3} {\mathrm e}^{4 d x +4 c}+9 a^{3} {\mathrm e}^{2 d x +2 c}-13 a^{2} b \,{\mathrm e}^{2 d x +2 c}-13 a \,b^{2} {\mathrm e}^{2 d x +2 c}+9 b^{3} {\mathrm e}^{2 d x +2 c}+3 a^{3}+3 a^{2} b -3 a \,b^{2}-3 b^{3}}{4 \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )^{2} d \,a^{2} b^{2}}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}-2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{16 \sqrt {-a b}\, d \,b^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}-2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{8 \sqrt {-a b}\, d b a}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}-2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{16 \sqrt {-a b}\, d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}+2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{16 \sqrt {-a b}\, d \,b^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}+2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{8 \sqrt {-a b}\, d b a}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}+2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{16 \sqrt {-a b}\, d \,a^{2}}\) | \(619\) |
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. 332 vs.
\(2 (117) = 234\).
time = 0.66, size = 332, normalized size = 2.53 \begin {gather*} -\frac {3 \, a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - 3 \, b^{3} + {\left (9 \, a^{3} - 13 \, a^{2} b - 13 \, a b^{2} + 9 \, b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, {\left (3 \, a^{3} - 5 \, a^{2} b + 5 \, a b^{2} - 3 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (3 \, a^{3} + a^{2} b + a b^{2} + 3 \, b^{3}\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} - a^{2} b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{4} b^{2} - 2 \, a^{3} b^{3} + 3 \, a^{2} b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (a^{4} b^{2} - a^{2} b^{4}\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} - \frac {{\left (3 \, a^{2} - 2 \, a b + 3 \, b^{2}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b^{2} 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. 2464 vs.
\(2 (117) = 234\).
time = 0.46, size = 5233, normalized size = 39.95 \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 \tanh ^{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. 322 vs.
\(2 (117) = 234\).
time = 0.91, size = 322, normalized size = 2.46 \begin {gather*} \frac {\frac {{\left (3 \, a^{2} - 2 \, a b + 3 \, b^{2}\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 )}{\sqrt {a b} a^{2} b^{2}} + \frac {2 \, {\left (3 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 3 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 9 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 15 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 15 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 9 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 9 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 13 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 13 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - 3 \, b^{3}\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 )}^{2} a^{2} b^{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 (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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