3.57 \(\int \frac{a+b \tanh (e+f x)}{(c+d x)^2} \, dx\)

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{a+b \tanh (e+f x)}{(c+d x)^2},x\right ) \]

[Out]

Unintegrable[(a + b*Tanh[e + f*x])/(c + d*x)^2, x]

________________________________________________________________________________________

Rubi [A]  time = 0.0292781, 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 = {} \[ \int \frac{a+b \tanh (e+f x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Tanh[e + f*x])/(c + d*x)^2,x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{a+b \tanh (e+f x)}{(c+d x)^2} \, dx &=\int \frac{a+b \tanh (e+f x)}{(c+d x)^2} \, dx\\ \end{align*}

Mathematica [A]  time = 10.9549, size = 0, normalized size = 0. \[ \int \frac{a+b \tanh (e+f x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.121, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\tanh \left ( fx+e \right ) }{ \left ( dx+c \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \tanh \left (f x + e\right ) + a}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*tanh(f*x + e) + a)/(d^2*x^2 + 2*c*d*x + c^2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*tanh(e + f*x))/(c + d*x)**2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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