Optimal. Leaf size=19 \[ \frac {\text {Ei}(n \cosh (a c+b x c))}{b c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4341, 2178} \[ \frac {\text {Ei}(n \cosh (a c+b x c))}{b c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2178
Rule 4341
Rubi steps
\begin {align*} \int e^{n \cosh (c (a+b x))} \tanh (a c+b c x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e^{n x}}{x} \, dx,x,\cosh (a c+b c x)\right )}{b c}\\ &=\frac {\text {Ei}(n \cosh (a c+b c x))}{b c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 18, normalized size = 0.95 \[ \frac {\text {Ei}(n \cosh (c (a+b x)))}{b c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 19, normalized size = 1.00 \[ \frac {{\rm Ei}\left (n \cosh \left (b c x + a c\right )\right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (n \cosh \left ({\left (b x + a\right )} c\right )\right )} \tanh \left (b c x + a c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.34, size = 31, normalized size = 1.63 \[ \frac {\Shi \left (n \cosh \left (c \left (b x +a \right )\right )\right )+\Chi \left (n \cosh \left (c \left (b x +a \right )\right )\right )}{c b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (n \cosh \left ({\left (b x + a\right )} c\right )\right )} \tanh \left (b c x + a c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int {\mathrm {e}}^{n\,\mathrm {cosh}\left (c\,\left (a+b\,x\right )\right )}\,\mathrm {tanh}\left (a\,c+b\,c\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{n \cosh {\left (a c + b c x \right )}} \tanh {\left (a c + b c x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________