Optimal. Leaf size=79 \[ \frac {\cosh (e+f x)}{3 (a-b) f \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}+\frac {2 \cosh (e+f x)}{3 (a-b)^2 f \sqrt {a-b+b \cosh ^2(e+f x)}} \]
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Rubi [A]
time = 0.05, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3265, 198, 197}
\begin {gather*} \frac {2 \cosh (e+f x)}{3 f (a-b)^2 \sqrt {a+b \cosh ^2(e+f x)-b}}+\frac {\cosh (e+f x)}{3 f (a-b) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 3265
Rubi steps
\begin {align*} \int \frac {\sinh (e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (a-b+b x^2\right )^{5/2}} \, dx,x,\cosh (e+f x)\right )}{f}\\ &=\frac {\cosh (e+f x)}{3 (a-b) f \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}+\frac {2 \text {Subst}\left (\int \frac {1}{\left (a-b+b x^2\right )^{3/2}} \, dx,x,\cosh (e+f x)\right )}{3 (a-b) f}\\ &=\frac {\cosh (e+f x)}{3 (a-b) f \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}+\frac {2 \cosh (e+f x)}{3 (a-b)^2 f \sqrt {a-b+b \cosh ^2(e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 63, normalized size = 0.80 \begin {gather*} \frac {2 \sqrt {2} \cosh (e+f x) (3 a-2 b+b \cosh (2 (e+f x)))}{3 (a-b)^2 f (2 a-b+b \cosh (2 (e+f x)))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.39, size = 57, normalized size = 0.72
method | result | size |
default | \(\frac {\left (2 b \left (\sinh ^{2}\left (f x +e \right )\right )+3 a -b \right ) \cosh \left (f x +e \right )}{3 \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} \left (a^{2}-2 a b +b^{2}\right ) f}\) | \(57\) |
risch | \(\text {Expression too large to display}\) | \(394373\) |
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. 499 vs.
\(2 (75) = 150\).
time = 0.52, size = 499, normalized size = 6.32 \begin {gather*} \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 )}}{3 \, {\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 )}}{3 \, {\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. 1186 vs.
\(2 (71) = 142\).
time = 0.71, size = 1186, normalized size = 15.01 \begin {gather*} \frac {2 \, \sqrt {2} {\left (b \cosh \left (f x + e\right )^{6} + 6 \, b \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{5} + b \sinh \left (f x + e\right )^{6} + 3 \, {\left (2 \, a - b\right )} \cosh \left (f x + e\right )^{4} + 3 \, {\left (5 \, b \cosh \left (f x + e\right )^{2} + 2 \, a - b\right )} \sinh \left (f x + e\right )^{4} + 4 \, {\left (5 \, b \cosh \left (f x + e\right )^{3} + 3 \, {\left (2 \, a - b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )^{3} + 3 \, {\left (2 \, a - b\right )} \cosh \left (f x + e\right )^{2} + 3 \, {\left (5 \, b \cosh \left (f x + e\right )^{4} + 6 \, {\left (2 \, a - b\right )} \cosh \left (f x + e\right )^{2} + 2 \, a - b\right )} \sinh \left (f x + e\right )^{2} + 6 \, {\left (b \cosh \left (f x + e\right )^{5} + 2 \, {\left (2 \, a - b\right )} \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 {\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 ({\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} f \cosh \left (f x + e\right )^{8} + 8 \, {\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} f \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{7} + {\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} f \sinh \left (f x + e\right )^{8} + 4 \, {\left (2 \, a^{3} b - 5 \, a^{2} b^{2} + 4 \, a b^{3} - b^{4}\right )} f \cosh \left (f x + e\right )^{6} + 4 \, {\left (7 \, {\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} f \cosh \left (f x + e\right )^{2} + {\left (2 \, a^{3} b - 5 \, a^{2} b^{2} + 4 \, a b^{3} - b^{4}\right )} f\right )} \sinh \left (f x + e\right )^{6} + 2 \, {\left (8 \, a^{4} - 24 \, a^{3} b + 27 \, a^{2} b^{2} - 14 \, a b^{3} + 3 \, b^{4}\right )} f \cosh \left (f x + e\right )^{4} + 8 \, {\left (7 \, {\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} f \cosh \left (f x + e\right )^{3} + 3 \, {\left (2 \, a^{3} b - 5 \, a^{2} b^{2} + 4 \, a b^{3} - b^{4}\right )} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )^{5} + 2 \, {\left (35 \, {\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} f \cosh \left (f x + e\right )^{4} + 30 \, {\left (2 \, a^{3} b - 5 \, a^{2} b^{2} + 4 \, a b^{3} - b^{4}\right )} f \cosh \left (f x + e\right )^{2} + {\left (8 \, a^{4} - 24 \, a^{3} b + 27 \, a^{2} b^{2} - 14 \, a b^{3} + 3 \, b^{4}\right )} f\right )} \sinh \left (f x + e\right )^{4} + 4 \, {\left (2 \, a^{3} b - 5 \, a^{2} b^{2} + 4 \, a b^{3} - b^{4}\right )} f \cosh \left (f x + e\right )^{2} + 8 \, {\left (7 \, {\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} f \cosh \left (f x + e\right )^{5} + 10 \, {\left (2 \, a^{3} b - 5 \, a^{2} b^{2} + 4 \, a b^{3} - b^{4}\right )} f \cosh \left (f x + e\right )^{3} + {\left (8 \, a^{4} - 24 \, a^{3} b + 27 \, a^{2} b^{2} - 14 \, a b^{3} + 3 \, b^{4}\right )} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )^{3} + 4 \, {\left (7 \, {\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} f \cosh \left (f x + e\right )^{6} + 15 \, {\left (2 \, a^{3} b - 5 \, a^{2} b^{2} + 4 \, a b^{3} - b^{4}\right )} f \cosh \left (f x + e\right )^{4} + 3 \, {\left (8 \, a^{4} - 24 \, a^{3} b + 27 \, a^{2} b^{2} - 14 \, a b^{3} + 3 \, b^{4}\right )} f \cosh \left (f x + e\right )^{2} + {\left (2 \, a^{3} b - 5 \, a^{2} b^{2} + 4 \, a b^{3} - b^{4}\right )} f\right )} \sinh \left (f x + e\right )^{2} + {\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} f + 8 \, {\left ({\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} f \cosh \left (f x + e\right )^{7} + 3 \, {\left (2 \, a^{3} b - 5 \, a^{2} b^{2} + 4 \, a b^{3} - b^{4}\right )} f \cosh \left (f x + e\right )^{5} + {\left (8 \, a^{4} - 24 \, a^{3} b + 27 \, a^{2} b^{2} - 14 \, a b^{3} + 3 \, b^{4}\right )} f \cosh \left (f x + e\right )^{3} + {\left (2 \, a^{3} b - 5 \, a^{2} b^{2} + 4 \, a b^{3} - b^{4}\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(-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 [B] Leaf count of result is larger than twice the leaf count of optimal. 254 vs.
\(2 (71) = 142\).
time = 0.89, size = 254, normalized size = 3.22 \begin {gather*} \frac {2 \, {\left (\frac {a^{2} b 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 )}} + {\left ({\left (\frac {a^{2} b e^{\left (2 \, f x + 12 \, 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 )}} + \frac {3 \, {\left (2 \, a^{3} e^{\left (10 \, e\right )} - a^{2} b 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 (2 \, a^{3} e^{\left (8 \, e\right )} - a^{2} b 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 )}\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 = 1.29, size = 133, normalized size = 1.68 \begin {gather*} \frac {4\,{\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 (b+6\,a\,{\mathrm {e}}^{2\,e+2\,f\,x}-4\,b\,{\mathrm {e}}^{2\,e+2\,f\,x}+b\,{\mathrm {e}}^{4\,e+4\,f\,x}\right )}{3\,f\,{\left (a-b\right )}^2\,{\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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