Optimal. Leaf size=36 \[ \frac {\cosh (e+f x)}{f (a-b) \sqrt {a+b \cosh ^2(e+f x)-b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3186, 191} \[ \frac {\cosh (e+f x)}{f (a-b) \sqrt {a+b \cosh ^2(e+f x)-b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 191
Rule 3186
Rubi steps
\begin {align*} \int \frac {\sinh (e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (a-b+b x^2\right )^{3/2}} \, dx,x,\cosh (e+f x)\right )}{f}\\ &=\frac {\cosh (e+f x)}{(a-b) f \sqrt {a-b+b \cosh ^2(e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 43, normalized size = 1.19 \[ \frac {\sqrt {2} \cosh (e+f x)}{f (a-b) \sqrt {2 a+b \cosh (2 (e+f x))-b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.98, size = 296, normalized size = 8.22 \[ \frac {\sqrt {2} {\left (\cosh \left (f x + e\right )^{2} + 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2} + 1\right )} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{{\left (a b - b^{2}\right )} f \cosh \left (f x + e\right )^{4} + 4 \, {\left (a b - b^{2}\right )} f \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + {\left (a b - b^{2}\right )} f \sinh \left (f x + e\right )^{4} + 2 \, {\left (2 \, a^{2} - 3 \, a b + b^{2}\right )} f \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, {\left (a b - b^{2}\right )} f \cosh \left (f x + e\right )^{2} + {\left (2 \, a^{2} - 3 \, a b + b^{2}\right )} f\right )} \sinh \left (f x + e\right )^{2} + {\left (a b - b^{2}\right )} f + 4 \, {\left ({\left (a b - b^{2}\right )} f \cosh \left (f x + e\right )^{3} + {\left (2 \, a^{2} - 3 \, a b + b^{2}\right )} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 32, normalized size = 0.89 \[ \frac {\cosh \left (f x +e \right )}{\left (a -b \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.45, size = 236, normalized size = 6.56 \[ \frac {b^{2} e^{\left (-6 \, f x - 6 \, e\right )} + 2 \, a b - b^{2} + {\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} e^{\left (-2 \, f x - 2 \, e\right )} + 3 \, {\left (2 \, a b - b^{2}\right )} e^{\left (-4 \, f x - 4 \, e\right )}}{2 \, {\left (a^{2} - a b\right )} {\left (2 \, {\left (2 \, a - b\right )} e^{\left (-2 \, f x - 2 \, e\right )} + b e^{\left (-4 \, f x - 4 \, e\right )} + b\right )}^{\frac {3}{2}} f} + \frac {b^{2} + 3 \, {\left (2 \, a b - b^{2}\right )} e^{\left (-2 \, f x - 2 \, e\right )} + {\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} e^{\left (-4 \, f x - 4 \, e\right )} + {\left (2 \, a b - b^{2}\right )} e^{\left (-6 \, f x - 6 \, e\right )}}{2 \, {\left (a^{2} - a b\right )} {\left (2 \, {\left (2 \, a - b\right )} e^{\left (-2 \, f x - 2 \, e\right )} + b e^{\left (-4 \, f x - 4 \, e\right )} + b\right )}^{\frac {3}{2}} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.92, size = 191, normalized size = 5.31 \[ -\frac {{\mathrm {e}}^{e+f\,x}\,\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}\,\left (\frac {2\,{\mathrm {e}}^{e+f\,x}\,\mathrm {sinh}\left (e+f\,x\right )\,\left (b\,\left (2\,a-b\right )-b\,\left (4\,a-2\,b\right )\right )}{f\,\left (a\,b^2-a^2\,b\right )}+\frac {2\,b^2\,\mathrm {cosh}\left (e+f\,x\right )\,{\mathrm {e}}^{e+f\,x}}{f\,\left (a\,b^2-a^2\,b\right )}+\frac {b\,{\mathrm {e}}^{2\,e+2\,f\,x}\,\left (4\,a-2\,b\right )}{f\,\left (a\,b^2-a^2\,b\right )}\right )}{4\,a\,{\mathrm {e}}^{2\,e+2\,f\,x}-2\,b\,{\mathrm {e}}^{2\,e+2\,f\,x}+2\,b\,{\mathrm {e}}^{2\,e+2\,f\,x}\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh {\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________