\(\int \frac {1}{(c+d x) \sqrt {b \tanh (e+f x)}} \, dx\) [27]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{(c+d x) \sqrt {b \tanh (e+f x)}} \, dx=\text {Int}\left (\frac {1}{(c+d x) \sqrt {b \tanh (e+f x)}},x\right ) \]

[Out]

Unintegrable(1/(d*x+c)/(b*tanh(f*x+e))^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {1}{(c+d x) \sqrt {b \tanh (e+f x)}} \, dx=\int \frac {1}{(c+d x) \sqrt {b \tanh (e+f x)}} \, dx \]

[In]

Int[1/((c + d*x)*Sqrt[b*Tanh[e + f*x]]),x]

[Out]

Defer[Int][1/((c + d*x)*Sqrt[b*Tanh[e + f*x]]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(c+d x) \sqrt {b \tanh (e+f x)}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) \sqrt {b \tanh (e+f x)}} \, dx=\int \frac {1}{(c+d x) \sqrt {b \tanh (e+f x)}} \, dx \]

[In]

Integrate[1/((c + d*x)*Sqrt[b*Tanh[e + f*x]]),x]

[Out]

Integrate[1/((c + d*x)*Sqrt[b*Tanh[e + f*x]]), x]

Maple [N/A] (verified)

Not integrable

Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

\[\int \frac {1}{\left (d x +c \right ) \sqrt {b \tanh \left (f x +e \right )}}d x\]

[In]

int(1/(d*x+c)/(b*tanh(f*x+e))^(1/2),x)

[Out]

int(1/(d*x+c)/(b*tanh(f*x+e))^(1/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{(c+d x) \sqrt {b \tanh (e+f x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(d*x+c)/(b*tanh(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 0.95 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(c+d x) \sqrt {b \tanh (e+f x)}} \, dx=\int \frac {1}{\sqrt {b \tanh {\left (e + f x \right )}} \left (c + d x\right )}\, dx \]

[In]

integrate(1/(d*x+c)/(b*tanh(f*x+e))**(1/2),x)

[Out]

Integral(1/(sqrt(b*tanh(e + f*x))*(c + d*x)), x)

Maxima [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c+d x) \sqrt {b \tanh (e+f x)}} \, dx=\int { \frac {1}{{\left (d x + c\right )} \sqrt {b \tanh \left (f x + e\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)/(b*tanh(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((d*x + c)*sqrt(b*tanh(f*x + e))), x)

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c+d x) \sqrt {b \tanh (e+f x)}} \, dx=\int { \frac {1}{{\left (d x + c\right )} \sqrt {b \tanh \left (f x + e\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)/(b*tanh(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(1/((d*x + c)*sqrt(b*tanh(f*x + e))), x)

Mupad [N/A]

Not integrable

Time = 1.76 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c+d x) \sqrt {b \tanh (e+f x)}} \, dx=\int \frac {1}{\sqrt {b\,\mathrm {tanh}\left (e+f\,x\right )}\,\left (c+d\,x\right )} \,d x \]

[In]

int(1/((b*tanh(e + f*x))^(1/2)*(c + d*x)),x)

[Out]

int(1/((b*tanh(e + f*x))^(1/2)*(c + d*x)), x)