Optimal. Leaf size=89 \[ \frac {(a+b) \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{2 a^{3/2} f}-\frac {\sqrt {a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text {csch}(e+f x)}{2 a f} \]
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Rubi [A]
time = 0.08, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3265, 390, 385,
212} \begin {gather*} \frac {(a+b) \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )}{2 a^{3/2} f}-\frac {\coth (e+f x) \text {csch}(e+f x) \sqrt {a+b \cosh ^2(e+f x)-b}}{2 a f} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 385
Rule 390
Rule 3265
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \sqrt {a-b+b x^2}} \, dx,x,\cosh (e+f x)\right )}{f}\\ &=-\frac {\sqrt {a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text {csch}(e+f x)}{2 a f}+\frac {(a+b) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\cosh (e+f x)\right )}{2 a f}\\ &=-\frac {\sqrt {a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text {csch}(e+f x)}{2 a f}+\frac {(a+b) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{2 a f}\\ &=\frac {(a+b) \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{2 a^{3/2} f}-\frac {\sqrt {a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text {csch}(e+f x)}{2 a f}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 102, normalized size = 1.15 \begin {gather*} \frac {2 (a+b) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \cosh (e+f x)}{\sqrt {2 a-b+b \cosh (2 (e+f x))}}\right )-\sqrt {2} \sqrt {a} \sqrt {2 a-b+b \cosh (2 (e+f x))} \coth (e+f x) \text {csch}(e+f x)}{4 a^{3/2} f} \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. \(233\) vs.
\(2(77)=154\).
time = 2.14, size = 234, normalized size = 2.63
method | result | size |
default | \(-\frac {\sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\, \left (-\ln \left (\frac {\left (a +b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {b \left (\cosh ^{4}\left (f x +e \right )\right )+\left (a -b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}+a -b}{\sinh \left (f x +e \right )^{2}}\right ) \left (\sinh ^{2}\left (f x +e \right )\right ) a^{2}-\ln \left (\frac {\left (a +b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {b \left (\cosh ^{4}\left (f x +e \right )\right )+\left (a -b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}+a -b}{\sinh \left (f x +e \right )^{2}}\right ) b \left (\sinh ^{2}\left (f x +e \right )\right ) a +2 a^{\frac {3}{2}} \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\right )}{4 \sinh \left (f x +e \right )^{2} a^{\frac {5}{2}} \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\) | \(234\) |
risch | \(\text {Expression too large to display}\) | \(567767\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 591 vs.
\(2 (77) = 154\).
time = 0.47, size = 1285, normalized size = 14.44 \begin {gather*} \left [\frac {{\left ({\left (a + b\right )} \cosh \left (f x + e\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + {\left (a + b\right )} \sinh \left (f x + e\right )^{4} - 2 \, {\left (a + b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (f x + e\right )^{2} - a - b\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (f x + e\right )^{3} - {\left (a + b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + a + b\right )} \sqrt {a} \log \left (-\frac {{\left (a + b\right )} \cosh \left (f x + e\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + {\left (a + b\right )} \sinh \left (f x + e\right )^{4} + 2 \, {\left (3 \, a - b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (f x + e\right )^{2} + 3 \, a - b\right )} \sinh \left (f x + e\right )^{2} + 2 \, \sqrt {2} {\left (\cosh \left (f x + e\right )^{2} + 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2} + 1\right )} \sqrt {a} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}} + 4 \, {\left ({\left (a + b\right )} \cosh \left (f x + e\right )^{3} + {\left (3 \, a - b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + a + b}{\cosh \left (f x + e\right )^{4} + 4 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + \sinh \left (f x + e\right )^{4} + 2 \, {\left (3 \, \cosh \left (f x + e\right )^{2} - 1\right )} \sinh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right )^{2} + 4 \, {\left (\cosh \left (f x + e\right )^{3} - \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + 1}\right ) - 2 \, \sqrt {2} {\left (a \cosh \left (f x + e\right )^{2} + 2 \, a \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + a \sinh \left (f x + e\right )^{2} + a\right )} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{4 \, {\left (a^{2} f \cosh \left (f x + e\right )^{4} + 4 \, a^{2} f \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + a^{2} f \sinh \left (f x + e\right )^{4} - 2 \, a^{2} f \cosh \left (f x + e\right )^{2} + a^{2} f + 2 \, {\left (3 \, a^{2} f \cosh \left (f x + e\right )^{2} - a^{2} f\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left (a^{2} f \cosh \left (f x + e\right )^{3} - a^{2} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )\right )}}, -\frac {{\left ({\left (a + b\right )} \cosh \left (f x + e\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + {\left (a + b\right )} \sinh \left (f x + e\right )^{4} - 2 \, {\left (a + b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (f x + e\right )^{2} - a - b\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (f x + e\right )^{3} - {\left (a + b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + a + b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {2} {\left (\cosh \left (f x + e\right )^{2} + 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2} + 1\right )} \sqrt {-a} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{b \cosh \left (f x + e\right )^{4} + 4 \, b \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + b \sinh \left (f x + e\right )^{4} + 2 \, {\left (2 \, a - b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, b \cosh \left (f x + e\right )^{2} + 2 \, a - b\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left (b \cosh \left (f x + e\right )^{3} + {\left (2 \, a - b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + b}\right ) + \sqrt {2} {\left (a \cosh \left (f x + e\right )^{2} + 2 \, a \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + a \sinh \left (f x + e\right )^{2} + a\right )} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{2 \, {\left (a^{2} f \cosh \left (f x + e\right )^{4} + 4 \, a^{2} f \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + a^{2} f \sinh \left (f x + e\right )^{4} - 2 \, a^{2} f \cosh \left (f x + e\right )^{2} + a^{2} f + 2 \, {\left (3 \, a^{2} f \cosh \left (f x + e\right )^{2} - a^{2} f\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left (a^{2} f \cosh \left (f x + e\right )^{3} - a^{2} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )\right )}}\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 {\operatorname {csch}^{3}{\left (e + f x \right )}}{\sqrt {a + b \sinh ^{2}{\left (e + f x \right )}}}\, 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. 669 vs.
\(2 (77) = 154\).
time = 0.56, size = 669, normalized size = 7.52 \begin {gather*} -\frac {{\left (\frac {{\left (a + b\right )} \arctan \left (-\frac {\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b} - \sqrt {b}}{2 \, \sqrt {-a}}\right ) e^{\left (-4 \, e\right )}}{\sqrt {-a} a} - \frac {2 \, {\left ({\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}\right )}^{3} a + {\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}\right )}^{3} b + 5 \, {\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}\right )}^{2} a \sqrt {b} - 3 \, {\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}\right )}^{2} b^{\frac {3}{2}} + 4 \, {\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}\right )} a^{2} - 9 \, {\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}\right )} a b + 3 \, {\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}\right )} b^{2} - 4 \, a^{2} \sqrt {b} + 3 \, a b^{\frac {3}{2}} - b^{\frac {5}{2}}\right )} e^{\left (-4 \, e\right )}}{{\left ({\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}\right )}^{2} - 2 \, {\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}\right )} \sqrt {b} - 4 \, a + b\right )}^{2} a}\right )} e^{\left (4 \, e\right )}}{f} \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 {sinh}\left (e+f\,x\right )}^3\,\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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