Optimal. Leaf size=44 \[ \frac {a}{5 f \left (a \cosh ^2(e+f x)\right )^{5/2}}-\frac {1}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3255, 3284, 16,
45} \begin {gather*} \frac {a}{5 f \left (a \cosh ^2(e+f x)\right )^{5/2}}-\frac {1}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 16
Rule 45
Rule 3255
Rule 3284
Rubi steps
\begin {align*} \int \frac {\tanh ^3(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac {\tanh ^3(e+f x)}{\left (a \cosh ^2(e+f x)\right )^{3/2}} \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {1-x}{x^2 (a x)^{3/2}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=-\frac {a^2 \text {Subst}\left (\int \frac {1-x}{(a x)^{7/2}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=-\frac {a^2 \text {Subst}\left (\int \left (\frac {1}{(a x)^{7/2}}-\frac {1}{a (a x)^{5/2}}\right ) \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=\frac {a}{5 f \left (a \cosh ^2(e+f x)\right )^{5/2}}-\frac {1}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.09, size = 34, normalized size = 0.77 \begin {gather*} \frac {a \left (3-5 \cosh ^2(e+f x)\right )}{15 f \left (a \cosh ^2(e+f x)\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.18, size = 44, normalized size = 1.00
method | result | size |
default | \(\frac {\mathit {`\,int/indef0`\,}\left (\frac {\sinh ^{3}\left (f x +e \right )}{\cosh \left (f x +e \right )^{6} a \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f}\) | \(44\) |
risch | \(-\frac {8 \left (5 \,{\mathrm e}^{4 f x +4 e}-2 \,{\mathrm e}^{2 f x +2 e}+5\right ) {\mathrm e}^{2 f x +2 e}}{15 f \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, \left ({\mathrm e}^{2 f x +2 e}+1\right )^{4} a}\) | \(81\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 286 vs.
\(2 (38) = 76\).
time = 0.52, size = 286, normalized size = 6.50 \begin {gather*} -\frac {8 \, e^{\left (-3 \, f x - 3 \, e\right )}}{3 \, {\left (5 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + 5 \, a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + a^{\frac {3}{2}} e^{\left (-10 \, f x - 10 \, e\right )} + a^{\frac {3}{2}}\right )} f} + \frac {16 \, e^{\left (-5 \, f x - 5 \, e\right )}}{15 \, {\left (5 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + 5 \, a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + a^{\frac {3}{2}} e^{\left (-10 \, f x - 10 \, e\right )} + a^{\frac {3}{2}}\right )} f} - \frac {8 \, e^{\left (-7 \, f x - 7 \, e\right )}}{3 \, {\left (5 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + 5 \, a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + a^{\frac {3}{2}} e^{\left (-10 \, f x - 10 \, e\right )} + a^{\frac {3}{2}}\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1400 vs.
\(2 (36) = 72\).
time = 0.43, size = 1400, normalized size = 31.82 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh ^{3}{\left (e + f x \right )}}{\left (a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.93, size = 305, normalized size = 6.93 \begin {gather*} \frac {272\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{15\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^3\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {16\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{3\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^2\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {128\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{5\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^4\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}+\frac {64\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{5\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^5\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________