Optimal. Leaf size=42 \[ -\frac {2 i E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3771, 2639} \[ -\frac {2 i E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2639
Rule 3771
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {b \text {sech}(c+d x)}} \, dx &=\frac {\int \sqrt {\cosh (c+d x)} \, dx}{\sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}}\\ &=-\frac {2 i E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 42, normalized size = 1.00 \[ -\frac {2 i E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \operatorname {sech}\left (d x + c\right )}}{b \operatorname {sech}\left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \operatorname {sech}\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.37, size = 244, normalized size = 5.81 \[ \frac {\sqrt {2}}{d \sqrt {\frac {b \,{\mathrm e}^{d x +c}}{1+{\mathrm e}^{2 d x +2 c}}}}+\frac {\left (-\frac {2 \left (b \,{\mathrm e}^{2 d x +2 c}+b \right )}{b \sqrt {{\mathrm e}^{d x +c} \left (b \,{\mathrm e}^{2 d x +2 c}+b \right )}}+\frac {i \sqrt {-i \left ({\mathrm e}^{d x +c}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{d x +c}-i\right )}\, \sqrt {i {\mathrm e}^{d x +c}}\, \left (-2 i \EllipticE \left (\sqrt {-i \left ({\mathrm e}^{d x +c}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \EllipticF \left (\sqrt {-i \left ({\mathrm e}^{d x +c}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 d x +3 c} b +b \,{\mathrm e}^{d x +c}}}\right ) \sqrt {2}\, \sqrt {b \,{\mathrm e}^{d x +c} \left (1+{\mathrm e}^{2 d x +2 c}\right )}}{d \sqrt {\frac {b \,{\mathrm e}^{d x +c}}{1+{\mathrm e}^{2 d x +2 c}}}\, \left (1+{\mathrm e}^{2 d x +2 c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \operatorname {sech}\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \operatorname {sech}{\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________