Optimal. Leaf size=73 \[ \frac {\cosh ^2(e+f x) \sinh (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {2 \sinh (e+f x)}{3 a^2 f \sqrt {a+b \sinh ^2(e+f x)}} \]
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Rubi [A]
time = 0.06, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3269, 386, 197}
\begin {gather*} \frac {2 \sinh (e+f x)}{3 a^2 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\sinh (e+f x) \cosh ^2(e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 386
Rule 3269
Rubi steps
\begin {align*} \int \frac {\cosh ^3(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1+x^2}{\left (a+b x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac {\cosh ^2(e+f x) \sinh (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {2 \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 a f}\\ &=\frac {\cosh ^2(e+f x) \sinh (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {2 \sinh (e+f x)}{3 a^2 f \sqrt {a+b \sinh ^2(e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 50, normalized size = 0.68 \begin {gather*} \frac {3 a \sinh (e+f x)+(a+2 b) \sinh ^3(e+f x)}{3 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.36, size = 65, normalized size = 0.89
method | result | size |
default | \(\frac {\mathit {`\,int/indef0`\,}\left (\frac {\cosh ^{2}\left (f x +e \right )}{\left (b^{2} \left (\sinh ^{4}\left (f x +e \right )\right )+2 a b \left (\sinh ^{2}\left (f x +e \right )\right )+a^{2}\right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 955 vs.
\(2 (70) = 140\).
time = 0.52, size = 955, normalized size = 13.08 \begin {gather*} -\frac {b^{4} e^{\left (-10 \, f x - 10 \, e\right )} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - b^{4} - {\left (16 \, a^{4} - 32 \, a^{3} b + 6 \, a^{2} b^{2} + 10 \, a b^{3} - 5 \, b^{4}\right )} e^{\left (-2 \, f x - 2 \, e\right )} + 10 \, {\left (2 \, a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right )} e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, {\left (3 \, a^{2} b^{2} - 3 \, a b^{3} + b^{4}\right )} e^{\left (-6 \, f x - 6 \, e\right )} + 5 \, {\left (2 \, a b^{3} - b^{4}\right )} e^{\left (-8 \, f x - 8 \, e\right )}}{12 \, {\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right )} {\left (2 \, {\left (2 \, a - b\right )} e^{\left (-2 \, f x - 2 \, e\right )} + b e^{\left (-4 \, f x - 4 \, e\right )} + b\right )}^{\frac {5}{2}} f} + \frac {2 \, a^{2} b^{2} - 2 \, a b^{3} + b^{4} + 5 \, {\left (4 \, a^{3} b - 6 \, a^{2} b^{2} + 4 \, a b^{3} - b^{4}\right )} e^{\left (-2 \, f x - 2 \, e\right )} + 2 \, {\left (24 \, a^{4} - 48 \, a^{3} b + 49 \, a^{2} b^{2} - 25 \, a b^{3} + 5 \, b^{4}\right )} e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, {\left (6 \, a^{3} b - 9 \, a^{2} b^{2} + 5 \, a b^{3} - b^{4}\right )} e^{\left (-6 \, f x - 6 \, e\right )} + 5 \, {\left (4 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} e^{\left (-8 \, f x - 8 \, e\right )} + {\left (2 \, a b^{3} - b^{4}\right )} e^{\left (-10 \, f x - 10 \, e\right )}}{4 \, {\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right )} {\left (2 \, {\left (2 \, a - b\right )} e^{\left (-2 \, f x - 2 \, e\right )} + b e^{\left (-4 \, f x - 4 \, e\right )} + b\right )}^{\frac {5}{2}} f} - \frac {2 \, a b^{3} - b^{4} + 5 \, {\left (4 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} e^{\left (-2 \, f x - 2 \, e\right )} + 10 \, {\left (6 \, a^{3} b - 9 \, a^{2} b^{2} + 5 \, a b^{3} - b^{4}\right )} e^{\left (-4 \, f x - 4 \, e\right )} + 2 \, {\left (24 \, a^{4} - 48 \, a^{3} b + 49 \, a^{2} b^{2} - 25 \, a b^{3} + 5 \, b^{4}\right )} e^{\left (-6 \, f x - 6 \, e\right )} + 5 \, {\left (4 \, a^{3} b - 6 \, a^{2} b^{2} + 4 \, a b^{3} - b^{4}\right )} e^{\left (-8 \, f x - 8 \, e\right )} + {\left (2 \, a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} e^{\left (-10 \, f x - 10 \, e\right )}}{4 \, {\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right )} {\left (2 \, {\left (2 \, a - b\right )} e^{\left (-2 \, f x - 2 \, e\right )} + b e^{\left (-4 \, f x - 4 \, e\right )} + b\right )}^{\frac {5}{2}} f} + \frac {b^{4} + 5 \, {\left (2 \, a b^{3} - b^{4}\right )} e^{\left (-2 \, f x - 2 \, e\right )} + 10 \, {\left (3 \, a^{2} b^{2} - 3 \, a b^{3} + b^{4}\right )} e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, {\left (2 \, a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right )} e^{\left (-6 \, f x - 6 \, e\right )} - {\left (16 \, a^{4} - 32 \, a^{3} b + 6 \, a^{2} b^{2} + 10 \, a b^{3} - 5 \, b^{4}\right )} e^{\left (-8 \, f x - 8 \, e\right )} - {\left (4 \, a^{3} b - 6 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-10 \, f x - 10 \, e\right )}}{12 \, {\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right )} {\left (2 \, {\left (2 \, a - b\right )} e^{\left (-2 \, f x - 2 \, e\right )} + b e^{\left (-4 \, f x - 4 \, e\right )} + b\right )}^{\frac {5}{2}} f} \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. 945 vs.
\(2 (65) = 130\).
time = 0.49, size = 945, normalized size = 12.95 \begin {gather*} \frac {\sqrt {2} {\left ({\left (a + 2 \, b\right )} \cosh \left (f x + e\right )^{6} + 6 \, {\left (a + 2 \, b\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{5} + {\left (a + 2 \, b\right )} \sinh \left (f x + e\right )^{6} + 3 \, {\left (3 \, a - 2 \, b\right )} \cosh \left (f x + e\right )^{4} + 3 \, {\left (5 \, {\left (a + 2 \, b\right )} \cosh \left (f x + e\right )^{2} + 3 \, a - 2 \, b\right )} \sinh \left (f x + e\right )^{4} + 4 \, {\left (5 \, {\left (a + 2 \, b\right )} \cosh \left (f x + e\right )^{3} + 3 \, {\left (3 \, a - 2 \, b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )^{3} - 3 \, {\left (3 \, a - 2 \, b\right )} \cosh \left (f x + e\right )^{2} + 3 \, {\left (5 \, {\left (a + 2 \, b\right )} \cosh \left (f x + e\right )^{4} + 6 \, {\left (3 \, a - 2 \, b\right )} \cosh \left (f x + e\right )^{2} - 3 \, a + 2 \, b\right )} \sinh \left (f x + e\right )^{2} + 6 \, {\left ({\left (a + 2 \, b\right )} \cosh \left (f x + e\right )^{5} + 2 \, {\left (3 \, a - 2 \, b\right )} \cosh \left (f x + e\right )^{3} - {\left (3 \, a - 2 \, b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) - a - 2 \, b\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}}}}{3 \, {\left (a^{2} b^{2} f \cosh \left (f x + e\right )^{8} + 8 \, a^{2} b^{2} f \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{7} + a^{2} b^{2} f \sinh \left (f x + e\right )^{8} + 4 \, {\left (2 \, a^{3} b - a^{2} b^{2}\right )} f \cosh \left (f x + e\right )^{6} + 4 \, {\left (7 \, a^{2} b^{2} f \cosh \left (f x + e\right )^{2} + {\left (2 \, a^{3} b - a^{2} b^{2}\right )} f\right )} \sinh \left (f x + e\right )^{6} + 2 \, {\left (8 \, a^{4} - 8 \, a^{3} b + 3 \, a^{2} b^{2}\right )} f \cosh \left (f x + e\right )^{4} + 8 \, {\left (7 \, a^{2} b^{2} f \cosh \left (f x + e\right )^{3} + 3 \, {\left (2 \, a^{3} b - a^{2} b^{2}\right )} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )^{5} + a^{2} b^{2} f + 2 \, {\left (35 \, a^{2} b^{2} f \cosh \left (f x + e\right )^{4} + 30 \, {\left (2 \, a^{3} b - a^{2} b^{2}\right )} f \cosh \left (f x + e\right )^{2} + {\left (8 \, a^{4} - 8 \, a^{3} b + 3 \, a^{2} b^{2}\right )} f\right )} \sinh \left (f x + e\right )^{4} + 4 \, {\left (2 \, a^{3} b - a^{2} b^{2}\right )} f \cosh \left (f x + e\right )^{2} + 8 \, {\left (7 \, a^{2} b^{2} f \cosh \left (f x + e\right )^{5} + 10 \, {\left (2 \, a^{3} b - a^{2} b^{2}\right )} f \cosh \left (f x + e\right )^{3} + {\left (8 \, a^{4} - 8 \, a^{3} b + 3 \, a^{2} b^{2}\right )} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )^{3} + 4 \, {\left (7 \, a^{2} b^{2} f \cosh \left (f x + e\right )^{6} + 15 \, {\left (2 \, a^{3} b - a^{2} b^{2}\right )} f \cosh \left (f x + e\right )^{4} + 3 \, {\left (8 \, a^{4} - 8 \, a^{3} b + 3 \, a^{2} b^{2}\right )} f \cosh \left (f x + e\right )^{2} + {\left (2 \, a^{3} b - a^{2} b^{2}\right )} f\right )} \sinh \left (f x + e\right )^{2} + 8 \, {\left (a^{2} b^{2} f \cosh \left (f x + e\right )^{7} + 3 \, {\left (2 \, a^{3} b - a^{2} b^{2}\right )} f \cosh \left (f x + e\right )^{5} + {\left (8 \, a^{4} - 8 \, a^{3} b + 3 \, a^{2} b^{2}\right )} f \cosh \left (f x + e\right )^{3} + {\left (2 \, a^{3} b - a^{2} b^{2}\right )} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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. 333 vs.
\(2 (65) = 130\).
time = 1.13, size = 333, normalized size = 4.56 \begin {gather*} \frac {{\left ({\left (\frac {{\left (a^{3} e^{\left (12 \, e\right )} - 3 \, a b^{2} e^{\left (12 \, e\right )} + 2 \, b^{3} e^{\left (12 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a^{4} e^{\left (6 \, e\right )} - 2 \, a^{3} b e^{\left (6 \, e\right )} + a^{2} b^{2} e^{\left (6 \, e\right )}} + \frac {3 \, {\left (3 \, a^{3} e^{\left (10 \, e\right )} - 8 \, a^{2} b e^{\left (10 \, e\right )} + 7 \, a b^{2} e^{\left (10 \, e\right )} - 2 \, b^{3} e^{\left (10 \, e\right )}\right )}}{a^{4} e^{\left (6 \, e\right )} - 2 \, a^{3} b e^{\left (6 \, e\right )} + a^{2} b^{2} e^{\left (6 \, e\right )}}\right )} e^{\left (2 \, f x\right )} - \frac {3 \, {\left (3 \, a^{3} e^{\left (8 \, e\right )} - 8 \, a^{2} b e^{\left (8 \, e\right )} + 7 \, a b^{2} e^{\left (8 \, e\right )} - 2 \, b^{3} e^{\left (8 \, e\right )}\right )}}{a^{4} e^{\left (6 \, e\right )} - 2 \, a^{3} b e^{\left (6 \, e\right )} + a^{2} b^{2} e^{\left (6 \, e\right )}}\right )} e^{\left (2 \, f x\right )} - \frac {a^{3} e^{\left (6 \, e\right )} - 3 \, a b^{2} e^{\left (6 \, e\right )} + 2 \, b^{3} e^{\left (6 \, e\right )}}{a^{4} e^{\left (6 \, e\right )} - 2 \, a^{3} b e^{\left (6 \, e\right )} + a^{2} b^{2} e^{\left (6 \, e\right )}}}{3 \, {\left (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 )}^{\frac {3}{2}} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.83, size = 144, normalized size = 1.97 \begin {gather*} \frac {2\,{\mathrm {e}}^{e+f\,x}\,\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )\,\sqrt {a+b\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}\,\left (a+2\,b+10\,a\,{\mathrm {e}}^{2\,e+2\,f\,x}+a\,{\mathrm {e}}^{4\,e+4\,f\,x}-4\,b\,{\mathrm {e}}^{2\,e+2\,f\,x}+2\,b\,{\mathrm {e}}^{4\,e+4\,f\,x}\right )}{3\,a^2\,f\,{\left (b+4\,a\,{\mathrm {e}}^{2\,e+2\,f\,x}-2\,b\,{\mathrm {e}}^{2\,e+2\,f\,x}+b\,{\mathrm {e}}^{4\,e+4\,f\,x}\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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