Optimal. Leaf size=173 \[ \frac {x}{a^3}-\frac {b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 (a-b)^{5/2} (a+b)^{5/2} d}+\frac {b^2 \tanh (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \text {sech}(c+d x))^2}+\frac {b^2 \left (5 a^2-2 b^2\right ) \tanh (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \text {sech}(c+d x))} \]
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Rubi [A]
time = 0.24, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3870, 4145,
4004, 3916, 2738, 214} \begin {gather*} \frac {x}{a^3}+\frac {b^2 \left (5 a^2-2 b^2\right ) \tanh (c+d x)}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \text {sech}(c+d x))}+\frac {b^2 \tanh (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \text {sech}(c+d x))^2}-\frac {b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 d (a-b)^{5/2} (a+b)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2738
Rule 3870
Rule 3916
Rule 4004
Rule 4145
Rubi steps
\begin {align*} \int \frac {1}{(a+b \text {sech}(c+d x))^3} \, dx &=\frac {b^2 \tanh (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \text {sech}(c+d x))^2}-\frac {\int \frac {-2 \left (a^2-b^2\right )+2 a b \text {sech}(c+d x)-b^2 \text {sech}^2(c+d x)}{(a+b \text {sech}(c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac {b^2 \tanh (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \text {sech}(c+d x))^2}+\frac {b^2 \left (5 a^2-2 b^2\right ) \tanh (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \text {sech}(c+d x))}+\frac {\int \frac {2 \left (a^2-b^2\right )^2-a b \left (4 a^2-b^2\right ) \text {sech}(c+d x)}{a+b \text {sech}(c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {x}{a^3}+\frac {b^2 \tanh (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \text {sech}(c+d x))^2}+\frac {b^2 \left (5 a^2-2 b^2\right ) \tanh (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \text {sech}(c+d x))}-\frac {\left (b \left (6 a^4-5 a^2 b^2+2 b^4\right )\right ) \int \frac {\text {sech}(c+d x)}{a+b \text {sech}(c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )^2}\\ &=\frac {x}{a^3}+\frac {b^2 \tanh (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \text {sech}(c+d x))^2}+\frac {b^2 \left (5 a^2-2 b^2\right ) \tanh (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \text {sech}(c+d x))}-\frac {\left (6 a^4-5 a^2 b^2+2 b^4\right ) \int \frac {1}{1+\frac {a \cosh (c+d x)}{b}} \, dx}{2 a^3 \left (a^2-b^2\right )^2}\\ &=\frac {x}{a^3}+\frac {b^2 \tanh (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \text {sech}(c+d x))^2}+\frac {b^2 \left (5 a^2-2 b^2\right ) \tanh (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \text {sech}(c+d x))}+\frac {\left (i \left (6 a^4-5 a^2 b^2+2 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{a^3 \left (a^2-b^2\right )^2 d}\\ &=\frac {x}{a^3}-\frac {b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 (a-b)^{5/2} (a+b)^{5/2} d}+\frac {b^2 \tanh (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \text {sech}(c+d x))^2}+\frac {b^2 \left (5 a^2-2 b^2\right ) \tanh (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \text {sech}(c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.48, size = 205, normalized size = 1.18 \begin {gather*} \frac {(b+a \cosh (c+d x)) \text {sech}^3(c+d x) \left (2 (c+d x) (b+a \cosh (c+d x))^2+\frac {2 b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \text {ArcTan}\left (\frac {(-a+b) \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cosh (c+d x))^2}{\left (a^2-b^2\right )^{5/2}}+\frac {a b^3 \sinh (c+d x)}{(-a+b) (a+b)}+\frac {3 a b^2 \left (2 a^2-b^2\right ) (b+a \cosh (c+d x)) \sinh (c+d x)}{(a-b)^2 (a+b)^2}\right )}{2 a^3 d (a+b \text {sech}(c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.78, size = 251, normalized size = 1.45
method | result | size |
derivativedivides | \(\frac {-\frac {2 b \left (\frac {-\frac {\left (6 a^{2}+a b -2 b^{2}\right ) a b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (6 a^{2}-a b -2 b^{2}\right ) a b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{2}-2 a b +b^{2}\right )}}{\left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {\left (6 a^{4}-5 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3}}}{d}\) | \(251\) |
default | \(\frac {-\frac {2 b \left (\frac {-\frac {\left (6 a^{2}+a b -2 b^{2}\right ) a b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (6 a^{2}-a b -2 b^{2}\right ) a b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{2}-2 a b +b^{2}\right )}}{\left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {\left (6 a^{4}-5 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3}}}{d}\) | \(251\) |
risch | \(\frac {x}{a^{3}}-\frac {b^{2} \left (7 a^{3} b \,{\mathrm e}^{3 d x +3 c}-4 a \,b^{3} {\mathrm e}^{3 d x +3 c}+6 a^{4} {\mathrm e}^{2 d x +2 c}+9 a^{2} b^{2} {\mathrm e}^{2 d x +2 c}-6 b^{4} {\mathrm e}^{2 d x +2 c}+17 a^{3} b \,{\mathrm e}^{d x +c}-8 b^{3} {\mathrm e}^{d x +c} a +6 a^{4}-3 a^{2} b^{2}\right )}{a^{3} \left (a^{2}-b^{2}\right )^{2} d \left (a \,{\mathrm e}^{2 d x +2 c}+2 b \,{\mathrm e}^{d x +c}+a \right )^{2}}-\frac {3 b a \ln \left ({\mathrm e}^{d x +c}+\frac {b \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}+\frac {5 b^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {b \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d a}-\frac {b^{5} \ln \left ({\mathrm e}^{d x +c}+\frac {b \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{3}}+\frac {3 b a \ln \left ({\mathrm e}^{d x +c}+\frac {b \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {5 b^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {b \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d a}+\frac {b^{5} \ln \left ({\mathrm e}^{d x +c}+\frac {b \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{3}}\) | \(631\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 2000 vs.
\(2 (160) = 320\).
time = 0.44, size = 4125, normalized size = 23.84 \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 {1}{\left (a + b \operatorname {sech}{\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 [A]
time = 0.41, size = 261, normalized size = 1.51 \begin {gather*} -\frac {\frac {{\left (6 \, a^{4} b - 5 \, a^{2} b^{3} + 2 \, b^{5}\right )} \arctan \left (\frac {a e^{\left (d x + c\right )} + b}{\sqrt {a^{2} - b^{2}}}\right )}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {a^{2} - b^{2}}} + \frac {7 \, a^{3} b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 4 \, a b^{5} e^{\left (3 \, d x + 3 \, c\right )} + 6 \, a^{4} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a^{2} b^{4} e^{\left (2 \, d x + 2 \, c\right )} - 6 \, b^{6} e^{\left (2 \, d x + 2 \, c\right )} + 17 \, a^{3} b^{3} e^{\left (d x + c\right )} - 8 \, a b^{5} e^{\left (d x + c\right )} + 6 \, a^{4} b^{2} - 3 \, a^{2} b^{4}}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} {\left (a e^{\left (2 \, d x + 2 \, c\right )} + 2 \, b e^{\left (d x + c\right )} + a\right )}^{2}} - \frac {d x + c}{a^{3}}}{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}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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