Optimal. Leaf size=13 \[ \frac {\text {Ei}(n \cosh (a+b x))}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4341, 2178} \[ \frac {\text {Ei}(n \cosh (a+b x))}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2178
Rule 4341
Rubi steps
\begin {align*} \int e^{n \cosh (a+b x)} \tanh (a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e^{n x}}{x} \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac {\text {Ei}(n \cosh (a+b x))}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 13, normalized size = 1.00 \[ \frac {\text {Ei}(n \cosh (a+b x))}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 13, normalized size = 1.00 \[ \frac {{\rm Ei}\left (n \cosh \left (b x + a\right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (n \cosh \left (b x + a\right )\right )} \tanh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 17, normalized size = 1.31 \[ -\frac {\Ei \left (1, -n \cosh \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (n \cosh \left (b x + a\right )\right )} \tanh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.08 \[ \int {\mathrm {e}}^{n\,\mathrm {cosh}\left (a+b\,x\right )}\,\mathrm {tanh}\left (a+b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{n \cosh {\left (a + b x \right )}} \tanh {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________