\(\int \frac {\text {sech}(a+b x)}{c+d x} \, dx\) [29]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {\text {sech}(a+b x)}{c+d x} \, dx=\text {Int}\left (\frac {\text {sech}(a+b x)}{c+d x},x\right ) \]

[Out]

Unintegrable(sech(b*x+a)/(d*x+c),x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\text {sech}(a+b x)}{c+d x} \, dx=\int \frac {\text {sech}(a+b x)}{c+d x} \, dx \]

[In]

Int[Sech[a + b*x]/(c + d*x),x]

[Out]

Defer[Int][Sech[a + b*x]/(c + d*x), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\text {sech}(a+b x)}{c+d x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.82 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\text {sech}(a+b x)}{c+d x} \, dx=\int \frac {\text {sech}(a+b x)}{c+d x} \, dx \]

[In]

Integrate[Sech[a + b*x]/(c + d*x),x]

[Out]

Integrate[Sech[a + b*x]/(c + d*x), x]

Maple [N/A] (verified)

Not integrable

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00

\[\int \frac {\operatorname {sech}\left (b x +a \right )}{d x +c}d x\]

[In]

int(sech(b*x+a)/(d*x+c),x)

[Out]

int(sech(b*x+a)/(d*x+c),x)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\text {sech}(a+b x)}{c+d x} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right )}{d x + c} \,d x } \]

[In]

integrate(sech(b*x+a)/(d*x+c),x, algorithm="fricas")

[Out]

integral(sech(b*x + a)/(d*x + c), x)

Sympy [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {\text {sech}(a+b x)}{c+d x} \, dx=\int \frac {\operatorname {sech}{\left (a + b x \right )}}{c + d x}\, dx \]

[In]

integrate(sech(b*x+a)/(d*x+c),x)

[Out]

Integral(sech(a + b*x)/(c + d*x), x)

Maxima [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\text {sech}(a+b x)}{c+d x} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right )}{d x + c} \,d x } \]

[In]

integrate(sech(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

integrate(sech(b*x + a)/(d*x + c), x)

Giac [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\text {sech}(a+b x)}{c+d x} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right )}{d x + c} \,d x } \]

[In]

integrate(sech(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

integrate(sech(b*x + a)/(d*x + c), x)

Mupad [N/A]

Not integrable

Time = 1.73 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int \frac {\text {sech}(a+b x)}{c+d x} \, dx=\int \frac {1}{\mathrm {cosh}\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]

[In]

int(1/(cosh(a + b*x)*(c + d*x)),x)

[Out]

int(1/(cosh(a + b*x)*(c + d*x)), x)