Optimal. Leaf size=50 \[ \frac {x}{a}-\frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{a \sqrt {a-b} d} \]
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Rubi [A]
time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4212, 3260,
211} \begin {gather*} \frac {x}{a}-\frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{a d \sqrt {a-b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 3260
Rule 4212
Rubi steps
\begin {align*} \int \frac {1}{a+b \text {csch}^2(c+d x)} \, dx &=\frac {x}{a}+\frac {b \int \frac {1}{-b-a \sinh ^2(c+d x)} \, dx}{a}\\ &=\frac {x}{a}+\frac {b \text {Subst}\left (\int \frac {1}{-b-(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{a d}\\ &=\frac {x}{a}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{a \sqrt {a-b} d}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 52, normalized size = 1.04 \begin {gather*} \frac {\frac {c}{d}+x-\frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{\sqrt {a-b} d}}{a} \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. \(207\) vs.
\(2(42)=84\).
time = 1.63, size = 208, normalized size = 4.16
method | result | size |
risch | \(\frac {x}{a}+\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}+a -2 b}{a}\right )}{2 \left (a -b \right ) d a}-\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}-a +2 b}{a}\right )}{2 \left (a -b \right ) d a}\) | \(124\) |
derivativedivides | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {2 b^{2} \left (-\frac {\left (\sqrt {a \left (a -b \right )}-a \right ) \arctanh \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{2 \sqrt {a \left (a -b \right )}\, b \sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}+\frac {\left (\sqrt {a \left (a -b \right )}+a \right ) \arctan \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{2 \sqrt {a \left (a -b \right )}\, b \sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a}}{d}\) | \(208\) |
default | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {2 b^{2} \left (-\frac {\left (\sqrt {a \left (a -b \right )}-a \right ) \arctanh \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{2 \sqrt {a \left (a -b \right )}\, b \sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}+\frac {\left (\sqrt {a \left (a -b \right )}+a \right ) \arctan \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{2 \sqrt {a \left (a -b \right )}\, b \sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a}}{d}\) | \(208\) |
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 [A]
time = 0.38, size = 457, normalized size = 9.14 \begin {gather*} \left [\frac {2 \, d x + \sqrt {-\frac {b}{a - b}} \log \left (\frac {a^{2} \cosh \left (d x + c\right )^{4} + 4 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{2} \sinh \left (d x + c\right )^{4} - 2 \, {\left (a^{2} - 2 \, a b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{2} \cosh \left (d x + c\right )^{2} - a^{2} + 2 \, a b\right )} \sinh \left (d x + c\right )^{2} + a^{2} - 8 \, a b + 8 \, b^{2} + 4 \, {\left (a^{2} \cosh \left (d x + c\right )^{3} - {\left (a^{2} - 2 \, a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 4 \, {\left ({\left (a^{2} - a b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} - a b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} - a b\right )} \sinh \left (d x + c\right )^{2} - a^{2} + 3 \, a b - 2 \, b^{2}\right )} \sqrt {-\frac {b}{a - b}}}{a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} - 2 \, {\left (a - 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} - a + 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} - {\left (a - 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a}\right )}{2 \, a d}, \frac {d x - \sqrt {\frac {b}{a - b}} \arctan \left (\frac {{\left (a \cosh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a \sinh \left (d x + c\right )^{2} - a + 2 \, b\right )} \sqrt {\frac {b}{a - b}}}{2 \, b}\right )}{a d}\right ] \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}{a + b \operatorname {csch}^{2}{\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 64, normalized size = 1.28 \begin {gather*} -\frac {\frac {b \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}} a} - \frac {d x + c}{a}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.93, size = 471, normalized size = 9.42 \begin {gather*} \frac {x}{a}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\left (a^5\,\sqrt {a^3\,d^2-a^2\,b\,d^2}-a^4\,b\,\sqrt {a^3\,d^2-a^2\,b\,d^2}\right )\,\left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (\frac {2\,\left (a^2-8\,a\,b+8\,b^2\right )\,\left (8\,b^{5/2}\,\sqrt {a^3\,d^2-a^2\,b\,d^2}-8\,a\,b^{3/2}\,\sqrt {a^3\,d^2-a^2\,b\,d^2}+a^2\,\sqrt {b}\,\sqrt {a^3\,d^2-a^2\,b\,d^2}\right )}{a^8\,d\,{\left (a-b\right )}^2\,\sqrt {a^3\,d^2-a^2\,b\,d^2}}+\frac {4\,\sqrt {b}\,\left (2\,a-4\,b\right )\,\left (4\,d\,a^3\,b-12\,d\,a^2\,b^2+8\,d\,a\,b^3\right )}{a^7\,\left (a-b\right )\,\sqrt {a^3\,d^2-a^2\,b\,d^2}\,\sqrt {a^2\,d^2\,\left (a-b\right )}}\right )+\frac {2\,\left (2\,a\,b^{3/2}\,\sqrt {a^3\,d^2-a^2\,b\,d^2}-a^2\,\sqrt {b}\,\sqrt {a^3\,d^2-a^2\,b\,d^2}\right )\,\left (a^2-8\,a\,b+8\,b^2\right )}{a^8\,d\,{\left (a-b\right )}^2\,\sqrt {a^3\,d^2-a^2\,b\,d^2}}+\frac {4\,\sqrt {b}\,\left (2\,a^2\,b^2\,d-2\,a^3\,b\,d\right )\,\left (2\,a-4\,b\right )}{a^7\,\left (a-b\right )\,\sqrt {a^3\,d^2-a^2\,b\,d^2}\,\sqrt {a^2\,d^2\,\left (a-b\right )}}\right )}{4\,b}\right )}{\sqrt {a^3\,d^2-a^2\,b\,d^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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