3.71 \(\int \frac{1}{(c+d x) (a+b \tanh (e+f x))} \, dx\)

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{1}{(c+d x) (a+b \tanh (e+f x))},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0627407, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{(c+d x) (a+b \tanh (e+f x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 13.9082, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x) (a+b \tanh (e+f x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.146, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) \left ( a+b\tanh \left ( fx+e \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} 2 \, b \int \frac{1}{a^{2} c - b^{2} c +{\left (a^{2} d - b^{2} d\right )} x +{\left (a^{2} c e^{\left (2 \, e\right )} + 2 \, a b c e^{\left (2 \, e\right )} + b^{2} c e^{\left (2 \, e\right )} +{\left (a^{2} d e^{\left (2 \, e\right )} + 2 \, a b d e^{\left (2 \, e\right )} + b^{2} d e^{\left (2 \, e\right )}\right )} x\right )} e^{\left (2 \, f x\right )}}\,{d x} + \frac{\log \left (d x + c\right )}{a d + b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*integrate(1/(a^2*c - b^2*c + (a^2*d - b^2*d)*x + (a^2*c*e^(2*e) + 2*a*b*c*e^(2*e) + b^2*c*e^(2*e) + (a^2*d
*e^(2*e) + 2*a*b*d*e^(2*e) + b^2*d*e^(2*e))*x)*e^(2*f*x)), x) + log(d*x + c)/(a*d + b*d)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a d x + a c +{\left (b d x + b c\right )} \tanh \left (f x + e\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/(a*d*x + a*c + (b*d*x + b*c)*tanh(f*x + e)), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \tanh{\left (e + f x \right )}\right ) \left (c + d x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x + c\right )}{\left (b \tanh \left (f x + e\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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