Optimal. Leaf size=91 \[ \frac {\cosh (e+f x) E\left (\text {ArcTan}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{\sqrt {a} \sqrt {b} f \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}} \]
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Rubi [A]
time = 0.07, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {3271, 422}
\begin {gather*} \frac {\cosh (e+f x) E\left (\text {ArcTan}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{\sqrt {a} \sqrt {b} f \sqrt {a+b \sinh ^2(e+f x)} \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 3271
Rubi steps
\begin {align*} \int \frac {\cosh ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac {\cosh (e+f x) E\left (\tan ^{-1}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{\sqrt {a} \sqrt {b} f \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.23, size = 143, normalized size = 1.57 \begin {gather*} \frac {i \sqrt {2} a \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-i \sqrt {2} a \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )+b \sinh (2 (e+f x))}{a b f \sqrt {4 a-2 b+2 b \cosh (2 (e+f x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.47, size = 181, normalized size = 1.99
method | result | size |
default | \(\frac {\sqrt {-\frac {b}{a}}\, \left (\cosh ^{2}\left (f x +e \right )\right ) \sinh \left (f x +e \right )+\sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-\sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )}{\sqrt {-\frac {b}{a}}\, a \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\) | \(181\) |
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. 1068 vs.
\(2 (97) = 194\).
time = 0.16, size = 1068, normalized size = 11.74 \begin {gather*} -\frac {4 \, {\left (b^{2} \cosh \left (f x + e\right )^{4} + 4 \, b^{2} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + b^{2} \sinh \left (f x + e\right )^{4} + 2 \, {\left (2 \, a b - b^{2}\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (f x + e\right )^{2} + 2 \, a b - b^{2}\right )} \sinh \left (f x + e\right )^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \left (f x + e\right )^{3} + {\left (2 \, a b - b^{2}\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )\right )} \sqrt {b} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} \sqrt {\frac {a^{2} - a b}{b^{2}}} F(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} - 8 \, a b + b^{2} + 4 \, {\left (2 \, a b - b^{2}\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}}{b^{2}}) + {\left ({\left (2 \, a b - b^{2}\right )} \cosh \left (f x + e\right )^{4} + 4 \, {\left (2 \, a b - b^{2}\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + {\left (2 \, a b - b^{2}\right )} \sinh \left (f x + e\right )^{4} + 2 \, {\left (4 \, a^{2} - 4 \, a b + b^{2}\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, {\left (2 \, a b - b^{2}\right )} \cosh \left (f x + e\right )^{2} + 4 \, a^{2} - 4 \, a b + b^{2}\right )} \sinh \left (f x + e\right )^{2} + 2 \, a b - b^{2} + 4 \, {\left ({\left (2 \, a b - b^{2}\right )} \cosh \left (f x + e\right )^{3} + {\left (4 \, a^{2} - 4 \, a b + b^{2}\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) - 2 \, {\left (b^{2} \cosh \left (f x + e\right )^{4} + 4 \, b^{2} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + b^{2} \sinh \left (f x + e\right )^{4} + 2 \, {\left (2 \, a b - b^{2}\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (f x + e\right )^{2} + 2 \, a b - b^{2}\right )} \sinh \left (f x + e\right )^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \left (f x + e\right )^{3} + {\left (2 \, a b - b^{2}\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {b} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} E(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} - 8 \, a b + b^{2} + 4 \, {\left (2 \, a b - b^{2}\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}}{b^{2}}) - \sqrt {2} {\left (b^{2} \cosh \left (f x + e\right )^{3} + 3 \, b^{2} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + b^{2} \sinh \left (f x + e\right )^{3} + {\left (2 \, a b - b^{2}\right )} \cosh \left (f x + e\right ) + {\left (3 \, b^{2} \cosh \left (f x + e\right )^{2} + 2 \, a b - b^{2}\right )} \sinh \left (f x + e\right )\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}}}}{a b^{3} f \cosh \left (f x + e\right )^{4} + 4 \, a b^{3} f \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + a b^{3} f \sinh \left (f x + e\right )^{4} + a b^{3} f + 2 \, {\left (2 \, a^{2} b^{2} - a b^{3}\right )} f \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, a b^{3} f \cosh \left (f x + e\right )^{2} + {\left (2 \, a^{2} b^{2} - a b^{3}\right )} f\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left (a b^{3} f \cosh \left (f x + e\right )^{3} + {\left (2 \, a^{2} b^{2} - a b^{3}\right )} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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 {{\mathrm {cosh}\left (e+f\,x\right )}^2}{{\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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