Optimal. Leaf size=85 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{(a-b)^{3/2} f}-\frac {b \sinh (e+f x)}{a (a-b) f \sqrt {a+b \sinh ^2(e+f x)}} \]
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Rubi [A]
time = 0.07, antiderivative size = 85, 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 = {3269, 390, 385,
209} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{f (a-b)^{3/2}}-\frac {b \sinh (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 385
Rule 390
Rule 3269
Rubi steps
\begin {align*} \int \frac {\text {sech}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac {b \sinh (e+f x)}{a (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{(a-b) f}\\ &=-\frac {b \sinh (e+f x)}{a (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{1-(-a+b) x^2} \, dx,x,\frac {\sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{(a-b) f}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{(a-b)^{3/2} f}-\frac {b \sinh (e+f x)}{a (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 7.51, size = 315, normalized size = 3.71 \begin {gather*} \frac {\text {sech}^7(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x) \left (4 (a-b)^2 \, _2F_1\left (2,2;\frac {7}{2};\frac {(a-b) \tanh ^2(e+f x)}{a}\right ) \sinh ^4(e+f x) \left (a+b \sinh ^2(e+f x)\right ) \sqrt {\frac {\text {sech}^2(e+f x) \left (a^2-b^2 \sinh ^2(e+f x)+a b \left (-1+\sinh ^2(e+f x)\right )\right ) \tanh ^2(e+f x)}{a^2}}+15 a \cosh ^2(e+f x) \left (3 a+2 b \sinh ^2(e+f x)\right ) \left (-\text {ArcSin}\left (\sqrt {\frac {(a-b) \tanh ^2(e+f x)}{a}}\right ) \left (a+b \sinh ^2(e+f x)\right )+a \cosh ^2(e+f x) \sqrt {\frac {\text {sech}^2(e+f x) \left (a^2-b^2 \sinh ^2(e+f x)+a b \left (-1+\sinh ^2(e+f x)\right )\right ) \tanh ^2(e+f x)}{a^2}}\right )\right )}{15 a^5 f \left (\frac {(a-b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right ) \tanh ^2(e+f x)}{a^2}\right )^{3/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.96, size = 101, normalized size = 1.19
method | result | size |
default | \(\frac {\mathit {`\,int/indef0`\,}\left (\frac {-b \left (\sinh ^{2}\left (f x +e \right )\right )-a}{\left (-b^{2} \left (\sinh ^{6}\left (f x +e \right )\right )+\left (-2 a b -b^{2}\right ) \left (\sinh ^{4}\left (f x +e \right )\right )+\left (-a^{2}-2 a b \right ) \left (\sinh ^{2}\left (f x +e \right )\right )-a^{2}\right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f}\) | \(101\) |
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. 800 vs.
\(2 (77) = 154\).
time = 0.46, size = 1717, normalized size = 20.20 \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}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, 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. 294 vs.
\(2 (77) = 154\).
time = 0.54, size = 294, normalized size = 3.46 \begin {gather*} -\frac {{\left (\frac {\frac {{\left (a^{2} b e^{\left (4 \, e\right )} - 2 \, a b^{2} e^{\left (4 \, e\right )} + b^{3} e^{\left (4 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a^{4} e^{\left (6 \, e\right )} - 3 \, a^{3} b e^{\left (6 \, e\right )} + 3 \, a^{2} b^{2} e^{\left (6 \, e\right )} - a b^{3} e^{\left (6 \, e\right )}} - \frac {a^{2} b e^{\left (2 \, e\right )} - 2 \, a b^{2} e^{\left (2 \, e\right )} + b^{3} e^{\left (2 \, e\right )}}{a^{4} e^{\left (6 \, e\right )} - 3 \, a^{3} b e^{\left (6 \, e\right )} + 3 \, a^{2} b^{2} e^{\left (6 \, e\right )} - a b^{3} e^{\left (6 \, 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}} - \frac {2 \, \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 - b}}\right )}{{\left (a e^{\left (4 \, e\right )} - b e^{\left (4 \, e\right )}\right )} \sqrt {a - b}}\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 {cosh}\left (e+f\,x\right )\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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