\(\int \frac {\cot ^{-1}(\coth (a+b x))}{e+f x} \, dx\) [204]

Optimal result

Integrand size = 15, antiderivative size = 15 \[ \int \frac {\cot ^{-1}(\coth (a+b x))}{e+f x} \, dx=\text {Int}\left (\frac {\cot ^{-1}(\coth (a+b x))}{e+f x},x\right ) \]

[Out]

CannotIntegrate(arccot(coth(b*x+a))/(f*x+e),x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 15, 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 = {} \[ \int \frac {\cot ^{-1}(\coth (a+b x))}{e+f x} \, dx=\int \frac {\cot ^{-1}(\coth (a+b x))}{e+f x} \, dx \]

[In]

Int[ArcCot[Coth[a + b*x]]/(e + f*x),x]

[Out]

Defer[Int][ArcCot[Coth[a + b*x]]/(e + f*x), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot ^{-1}(\coth (a+b x))}{e+f x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.70 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\cot ^{-1}(\coth (a+b x))}{e+f x} \, dx=\int \frac {\cot ^{-1}(\coth (a+b x))}{e+f x} \, dx \]

[In]

Integrate[ArcCot[Coth[a + b*x]]/(e + f*x),x]

[Out]

Integrate[ArcCot[Coth[a + b*x]]/(e + f*x), x]

Maple [N/A] (verified)

Not integrable

Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00

\[\int \frac {\operatorname {arccot}\left (\coth \left (b x +a \right )\right )}{f x +e}d x\]

[In]

int(arccot(coth(b*x+a))/(f*x+e),x)

[Out]

int(arccot(coth(b*x+a))/(f*x+e),x)

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\cot ^{-1}(\coth (a+b x))}{e+f x} \, dx=\int { \frac {\operatorname {arccot}\left (\coth \left (b x + a\right )\right )}{f x + e} \,d x } \]

[In]

integrate(arccot(coth(b*x+a))/(f*x+e),x, algorithm="fricas")

[Out]

integral(arccot(coth(b*x + a))/(f*x + e), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(\coth (a+b x))}{e+f x} \, dx=\text {Timed out} \]

[In]

integrate(acot(coth(b*x+a))/(f*x+e),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 1.53 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\cot ^{-1}(\coth (a+b x))}{e+f x} \, dx=\int { \frac {\operatorname {arccot}\left (\coth \left (b x + a\right )\right )}{f x + e} \,d x } \]

[In]

integrate(arccot(coth(b*x+a))/(f*x+e),x, algorithm="maxima")

[Out]

integrate(arccot(coth(b*x + a))/(f*x + e), x)

Giac [N/A]

Not integrable

Time = 105.03 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.20 \[ \int \frac {\cot ^{-1}(\coth (a+b x))}{e+f x} \, dx=\int { \frac {\operatorname {arccot}\left (\coth \left (b x + a\right )\right )}{f x + e} \,d x } \]

[In]

integrate(arccot(coth(b*x+a))/(f*x+e),x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.82 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\cot ^{-1}(\coth (a+b x))}{e+f x} \, dx=\int \frac {\mathrm {acot}\left (\mathrm {coth}\left (a+b\,x\right )\right )}{e+f\,x} \,d x \]

[In]

int(acot(coth(a + b*x))/(e + f*x),x)

[Out]

int(acot(coth(a + b*x))/(e + f*x), x)