Optimal. Leaf size=33 \[ \frac {1}{32} e^x \sqrt {16 e^{2 x}+25}-\frac {25}{128} \sinh ^{-1}\left (\frac {4 e^x}{5}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2248, 321, 215} \[ \frac {1}{32} e^x \sqrt {16 e^{2 x}+25}-\frac {25}{128} \sinh ^{-1}\left (\frac {4 e^x}{5}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 215
Rule 321
Rule 2248
Rubi steps
\begin {align*} \int \frac {e^{3 x}}{\sqrt {25+16 e^{2 x}}} \, dx &=\operatorname {Subst}\left (\int \frac {x^2}{\sqrt {25+16 x^2}} \, dx,x,e^x\right )\\ &=\frac {1}{32} e^x \sqrt {25+16 e^{2 x}}-\frac {25}{32} \operatorname {Subst}\left (\int \frac {1}{\sqrt {25+16 x^2}} \, dx,x,e^x\right )\\ &=\frac {1}{32} e^x \sqrt {25+16 e^{2 x}}-\frac {25}{128} \sinh ^{-1}\left (\frac {4 e^x}{5}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 33, normalized size = 1.00 \[ \frac {1}{32} e^x \sqrt {16 e^{2 x}+25}-\frac {25}{128} \sinh ^{-1}\left (\frac {4 e^x}{5}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 33, normalized size = 1.00 \[ \frac {1}{32} \, \sqrt {16 \, e^{\left (2 \, x\right )} + 25} e^{x} + \frac {25}{128} \, \log \left (\sqrt {16 \, e^{\left (2 \, x\right )} + 25} - 4 \, e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 33, normalized size = 1.00 \[ \frac {1}{32} \, \sqrt {16 \, e^{\left (2 \, x\right )} + 25} e^{x} + \frac {25}{128} \, \log \left (\sqrt {16 \, e^{\left (2 \, x\right )} + 25} - 4 \, e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 23, normalized size = 0.70 \[ -\frac {25 \arcsinh \left (\frac {4 \,{\mathrm e}^{x}}{5}\right )}{128}+\frac {\sqrt {16 \,{\mathrm e}^{2 x}+25}\, {\mathrm e}^{x}}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.05, size = 74, normalized size = 2.24 \[ \frac {25 \, \sqrt {16 \, e^{\left (2 \, x\right )} + 25} e^{\left (-x\right )}}{32 \, {\left ({\left (16 \, e^{\left (2 \, x\right )} + 25\right )} e^{\left (-2 \, x\right )} - 16\right )}} - \frac {25}{256} \, \log \left (\sqrt {16 \, e^{\left (2 \, x\right )} + 25} e^{\left (-x\right )} + 4\right ) + \frac {25}{256} \, \log \left (\sqrt {16 \, e^{\left (2 \, x\right )} + 25} e^{\left (-x\right )} - 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\mathrm {e}}^{3\,x}}{\sqrt {16\,{\mathrm {e}}^{2\,x}+25}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{3 x}}{\sqrt {16 e^{2 x} + 25}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________