Optimal. Leaf size=156 \[ \frac {x}{a^3}-\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{8 a^3 (a-b)^{5/2} d}+\frac {b \coth (c+d x)}{4 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^2}+\frac {(7 a-4 b) b \coth (c+d x)}{8 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )} \]
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Rubi [A]
time = 0.15, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4213, 425, 541,
536, 212, 211} \begin {gather*} \frac {x}{a^3}+\frac {b (7 a-4 b) \coth (c+d x)}{8 a^2 d (a-b)^2 \left (a+b \coth ^2(c+d x)-b\right )}-\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{8 a^3 d (a-b)^{5/2}}+\frac {b \coth (c+d x)}{4 a d (a-b) \left (a+b \coth ^2(c+d x)-b\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 212
Rule 425
Rule 536
Rule 541
Rule 4213
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-b+b x^2\right )^3} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac {b \coth (c+d x)}{4 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-4 a+b+3 b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )^2} \, dx,x,\coth (c+d x)\right )}{4 a (a-b) d}\\ &=\frac {b \coth (c+d x)}{4 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^2}+\frac {(7 a-4 b) b \coth (c+d x)}{8 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {8 a^2-9 a b+4 b^2-(7 a-4 b) b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )} \, dx,x,\coth (c+d x)\right )}{8 a^2 (a-b)^2 d}\\ &=\frac {b \coth (c+d x)}{4 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^2}+\frac {(7 a-4 b) b \coth (c+d x)}{8 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\coth (c+d x)\right )}{a^3 d}+\frac {\left (b \left (15 a^2-20 a b+8 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\coth (c+d x)\right )}{8 a^3 (a-b)^2 d}\\ &=\frac {x}{a^3}-\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{8 a^3 (a-b)^{5/2} d}+\frac {b \coth (c+d x)}{4 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^2}+\frac {(7 a-4 b) b \coth (c+d x)}{8 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 1.05, size = 210, normalized size = 1.35 \begin {gather*} \frac {(-a+2 b+a \cosh (2 (c+d x))) \text {csch}^6(c+d x) \left (8 (c+d x) (a-2 b-a \cosh (2 (c+d x)))^2-\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right ) (a-2 b-a \cosh (2 (c+d x)))^2}{(a-b)^{5/2}}-\frac {4 a b^2 \sinh (2 (c+d x))}{a-b}+\frac {3 a (3 a-2 b) b (-a+2 b+a \cosh (2 (c+d x))) \sinh (2 (c+d x))}{(a-b)^2}\right )}{64 a^3 d \left (a+b \text {csch}^2(c+d x)\right )^3} \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. \(441\) vs.
\(2(142)=284\).
time = 2.70, size = 442, normalized size = 2.83
method | result | size |
derivativedivides | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3}}+\frac {2 b \left (\frac {\frac {16 a b \left (7 a -4 b \right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a^{2}-256 a b +128 b^{2}}+\frac {16 \left (36 a^{2}-31 a b +4 b^{2}\right ) a \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a^{2}-256 a b +128 b^{2}}+\frac {16 \left (36 a^{2}-31 a b +4 b^{2}\right ) a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a^{2}-256 a b +128 b^{2}}+\frac {16 a b \left (7 a -4 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{128 a^{2}-256 a b +128 b^{2}}}{\left (b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \right )^{2}}+\frac {2 \left (15 a^{2}-20 a b +8 b^{2}\right ) b \left (-\frac {\left (\sqrt {a \left (a -b \right )}-a \right ) \arctanh \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{2 \sqrt {a \left (a -b \right )}\, b \sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}+\frac {\left (\sqrt {a \left (a -b \right )}+a \right ) \arctan \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{2 \sqrt {a \left (a -b \right )}\, b \sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{16 a^{2}-32 a b +16 b^{2}}\right )}{a^{3}}}{d}\) | \(442\) |
default | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3}}+\frac {2 b \left (\frac {\frac {16 a b \left (7 a -4 b \right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a^{2}-256 a b +128 b^{2}}+\frac {16 \left (36 a^{2}-31 a b +4 b^{2}\right ) a \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a^{2}-256 a b +128 b^{2}}+\frac {16 \left (36 a^{2}-31 a b +4 b^{2}\right ) a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a^{2}-256 a b +128 b^{2}}+\frac {16 a b \left (7 a -4 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{128 a^{2}-256 a b +128 b^{2}}}{\left (b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \right )^{2}}+\frac {2 \left (15 a^{2}-20 a b +8 b^{2}\right ) b \left (-\frac {\left (\sqrt {a \left (a -b \right )}-a \right ) \arctanh \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{2 \sqrt {a \left (a -b \right )}\, b \sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}+\frac {\left (\sqrt {a \left (a -b \right )}+a \right ) \arctan \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{2 \sqrt {a \left (a -b \right )}\, b \sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{16 a^{2}-32 a b +16 b^{2}}\right )}{a^{3}}}{d}\) | \(442\) |
risch | \(\frac {x}{a^{3}}+\frac {b \left (9 a^{3} {\mathrm e}^{6 d x +6 c}-28 a^{2} b \,{\mathrm e}^{6 d x +6 c}+16 a \,b^{2} {\mathrm e}^{6 d x +6 c}-27 a^{3} {\mathrm e}^{4 d x +4 c}+90 a^{2} b \,{\mathrm e}^{4 d x +4 c}-120 a \,b^{2} {\mathrm e}^{4 d x +4 c}+48 b^{3} {\mathrm e}^{4 d x +4 c}+27 a^{3} {\mathrm e}^{2 d x +2 c}-68 a^{2} b \,{\mathrm e}^{2 d x +2 c}+32 a \,b^{2} {\mathrm e}^{2 d x +2 c}-9 a^{3}+6 a^{2} b \right )}{4 a^{3} \left (a -b \right )^{2} d \left (a \,{\mathrm e}^{4 d x +4 c}-2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}+\frac {15 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}+a -2 b}{a}\right )}{16 \left (a -b \right )^{3} d a}-\frac {5 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}+a -2 b}{a}\right ) b}{4 \left (a -b \right )^{3} d \,a^{2}}+\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}+a -2 b}{a}\right ) b^{2}}{2 \left (a -b \right )^{3} d \,a^{3}}-\frac {15 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}-a +2 b}{a}\right )}{16 \left (a -b \right )^{3} d a}+\frac {5 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}-a +2 b}{a}\right ) b}{4 \left (a -b \right )^{3} d \,a^{2}}-\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}-a +2 b}{a}\right ) b^{2}}{2 \left (a -b \right )^{3} d \,a^{3}}\) | \(579\) |
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. 3140 vs.
\(2 (142) = 284\).
time = 0.44, size = 6569, normalized size = 42.11 \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 {csch}^{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. 327 vs.
\(2 (142) = 284\).
time = 0.66, size = 327, normalized size = 2.10 \begin {gather*} -\frac {\frac {{\left (15 \, a^{2} b - 20 \, a b^{2} + 8 \, b^{3}\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^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} \sqrt {a b - b^{2}}} - \frac {2 \, {\left (9 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} - 28 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 27 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 90 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 120 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 27 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} - 68 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 32 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 9 \, a^{3} b + 6 \, a^{2} b^{2}\right )}}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\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}} - \frac {8 \, {\left (d x + c\right )}}{a^{3}}}{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}{{\left (a+\frac {b}{{\mathrm {sinh}\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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