\(\int \frac {\text {sech}(a+b x)}{(c+d x)^2} \, dx\) [30]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {\text {sech}(a+b x)}{(c+d x)^2} \, dx=\text {Int}\left (\frac {\text {sech}(a+b x)}{(c+d x)^2},x\right ) \]

[Out]

Unintegrable(sech(b*x+a)/(d*x+c)^2,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 = {} \[ \int \frac {\text {sech}(a+b x)}{(c+d x)^2} \, dx=\int \frac {\text {sech}(a+b x)}{(c+d x)^2} \, dx \]

[In]

Int[Sech[a + b*x]/(c + d*x)^2,x]

[Out]

Defer[Int][Sech[a + b*x]/(c + d*x)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\text {sech}(a+b x)}{(c+d x)^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 5.74 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\text {sech}(a+b x)}{(c+d x)^2} \, dx=\int \frac {\text {sech}(a+b x)}{(c+d x)^2} \, dx \]

[In]

Integrate[Sech[a + b*x]/(c + d*x)^2,x]

[Out]

Integrate[Sech[a + b*x]/(c + d*x)^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00

\[\int \frac {\operatorname {sech}\left (b x +a \right )}{\left (d x +c \right )^{2}}d x\]

[In]

int(sech(b*x+a)/(d*x+c)^2,x)

[Out]

int(sech(b*x+a)/(d*x+c)^2,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.93 \[ \int \frac {\text {sech}(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate(sech(b*x+a)/(d*x+c)^2,x, algorithm="fricas")

[Out]

integral(sech(b*x + a)/(d^2*x^2 + 2*c*d*x + c^2), x)

Sympy [N/A]

Not integrable

Time = 0.66 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}(a+b x)}{(c+d x)^2} \, dx=\int \frac {\operatorname {sech}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

integrate(sech(b*x+a)/(d*x+c)**2,x)

[Out]

Integral(sech(a + b*x)/(c + d*x)**2, x)

Maxima [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\text {sech}(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate(sech(b*x+a)/(d*x+c)^2,x, algorithm="maxima")

[Out]

integrate(sech(b*x + a)/(d*x + c)^2, x)

Giac [N/A]

Not integrable

Time = 1.91 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\text {sech}(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate(sech(b*x+a)/(d*x+c)^2,x, algorithm="giac")

[Out]

integrate(sech(b*x + a)/(d*x + c)^2, x)

Mupad [N/A]

Not integrable

Time = 1.80 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int \frac {\text {sech}(a+b x)}{(c+d x)^2} \, dx=\int \frac {1}{\mathrm {cosh}\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

int(1/(cosh(a + b*x)*(c + d*x)^2),x)

[Out]

int(1/(cosh(a + b*x)*(c + d*x)^2), x)