\(\int \sec (\coth (a+b x)) \, dx\) [224]

Optimal result

Integrand size = 7, antiderivative size = 7 \[ \int \sec (\coth (a+b x)) \, dx=\frac {1}{2} \text {Int}\left (\frac {\text {csch}^2(a+b x) \sec (\coth (a+b x))}{-1+\coth (a+b x)},x\right )-\frac {1}{2} \text {Int}\left (\frac {\text {csch}^2(a+b x) \sec (\coth (a+b x))}{1+\coth (a+b x)},x\right ) \]

[Out]

1/2*Unintegrable(csch(b*x+a)^2*sec(coth(b*x+a))/(-1+coth(b*x+a)),x)-1/2*Unintegrable(csch(b*x+a)^2*sec(coth(b*
x+a))/(1+coth(b*x+a)),x)

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 7, 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 = {} \[ \int \sec (\coth (a+b x)) \, dx=\int \sec (\coth (a+b x)) \, dx \]

[In]

Int[Sec[Coth[a + b*x]],x]

[Out]

-1/2*Defer[Subst][Defer[Int][Sec[x]/(-1 + x), x], x, Coth[a + b*x]]/b + Defer[Subst][Defer[Int][Sec[x]/(1 + x)
, x], x, Coth[a + b*x]]/(2*b)

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sec (x)}{1-x^2} \, dx,x,\coth (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {\sec (x)}{2 (-1+x)}+\frac {\sec (x)}{2 (1+x)}\right ) \, dx,x,\coth (a+b x)\right )}{b} \\ & = -\frac {\text {Subst}\left (\int \frac {\sec (x)}{-1+x} \, dx,x,\coth (a+b x)\right )}{2 b}+\frac {\text {Subst}\left (\int \frac {\sec (x)}{1+x} \, dx,x,\coth (a+b x)\right )}{2 b} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 6.18 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.29 \[ \int \sec (\coth (a+b x)) \, dx=\int \sec (\coth (a+b x)) \, dx \]

[In]

Integrate[Sec[Coth[a + b*x]],x]

[Out]

Integrate[Sec[Coth[a + b*x]], x]

Maple [N/A] (verified)

Not integrable

Time = 0.25 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00

\[\int \sec \left (\coth \left (b x +a \right )\right )d x\]

[In]

int(sec(coth(b*x+a)),x)

[Out]

int(sec(coth(b*x+a)),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.29 \[ \int \sec (\coth (a+b x)) \, dx=\int { \sec \left (\coth \left (b x + a\right )\right ) \,d x } \]

[In]

integrate(sec(coth(b*x+a)),x, algorithm="fricas")

[Out]

integral(sec(coth(b*x + a)), x)

Sympy [N/A]

Not integrable

Time = 3.95 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.14 \[ \int \sec (\coth (a+b x)) \, dx=\int \sec {\left (\coth {\left (a + b x \right )} \right )}\, dx \]

[In]

integrate(sec(coth(b*x+a)),x)

[Out]

Integral(sec(coth(a + b*x)), x)

Maxima [N/A]

Not integrable

Time = 0.64 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.29 \[ \int \sec (\coth (a+b x)) \, dx=\int { \sec \left (\coth \left (b x + a\right )\right ) \,d x } \]

[In]

integrate(sec(coth(b*x+a)),x, algorithm="maxima")

[Out]

integrate(sec(coth(b*x + a)), x)

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.29 \[ \int \sec (\coth (a+b x)) \, dx=\int { \sec \left (\coth \left (b x + a\right )\right ) \,d x } \]

[In]

integrate(sec(coth(b*x+a)),x, algorithm="giac")

[Out]

integrate(sec(coth(b*x + a)), x)

Mupad [N/A]

Not integrable

Time = 2.31 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.57 \[ \int \sec (\coth (a+b x)) \, dx=\int \frac {1}{\cos \left (\mathrm {coth}\left (a+b\,x\right )\right )} \,d x \]

[In]

int(1/cos(coth(a + b*x)),x)

[Out]

int(1/cos(coth(a + b*x)), x)