\(\int \frac {1}{(c+d x)^2 (a+b \tanh (e+f x))^2} \, dx\) [77]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{(c+d x)^2 (a+b \tanh (e+f x))^2} \, dx=\text {Int}\left (\frac {1}{(c+d x)^2 (a+b \tanh (e+f x))^2},x\right ) \]

[Out]

Unintegrable(1/(d*x+c)^2/(a+b*tanh(f*x+e))^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 = {} \[ \int \frac {1}{(c+d x)^2 (a+b \tanh (e+f x))^2} \, dx=\int \frac {1}{(c+d x)^2 (a+b \tanh (e+f x))^2} \, dx \]

[In]

Int[1/((c + d*x)^2*(a + b*Tanh[e + f*x])^2),x]

[Out]

Defer[Int][1/((c + d*x)^2*(a + b*Tanh[e + f*x])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(c+d x)^2 (a+b \tanh (e+f x))^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 28.49 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \tanh (e+f x))^2} \, dx=\int \frac {1}{(c+d x)^2 (a+b \tanh (e+f x))^2} \, dx \]

[In]

Integrate[1/((c + d*x)^2*(a + b*Tanh[e + f*x])^2),x]

[Out]

Integrate[1/((c + d*x)^2*(a + b*Tanh[e + f*x])^2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (d x +c \right )^{2} \left (a +b \tanh \left (f x +e \right )\right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 4.80 \[ \int \frac {1}{(c+d x)^2 (a+b \tanh (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \tanh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 3.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c+d x)^2 (a+b \tanh (e+f x))^2} \, dx=\int \frac {1}{\left (a + b \tanh {\left (e + f x \right )}\right )^{2} \left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 2.11 (sec) , antiderivative size = 784, normalized size of antiderivative = 39.20 \[ \int \frac {1}{(c+d x)^2 (a+b \tanh (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \tanh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-(a^2*c*f - 2*a*b*c*f + (c*f - 2*d)*b^2 + (a^2*d*f - 2*a*b*d*f + b^2*d*f)*x + (a^2*c*f*e^(2*e) - b^2*c*f*e^(2*
e) + (a^2*d*f*e^(2*e) - b^2*d*f*e^(2*e))*x)*e^(2*f*x))/(a^4*c^2*d*f - 2*a^2*b^2*c^2*d*f + b^4*c^2*d*f + (a^4*d
^3*f - 2*a^2*b^2*d^3*f + b^4*d^3*f)*x^2 + 2*(a^4*c*d^2*f - 2*a^2*b^2*c*d^2*f + b^4*c*d^2*f)*x + (a^4*c^2*d*f*e
^(2*e) + 2*a^3*b*c^2*d*f*e^(2*e) - 2*a*b^3*c^2*d*f*e^(2*e) - b^4*c^2*d*f*e^(2*e) + (a^4*d^3*f*e^(2*e) + 2*a^3*
b*d^3*f*e^(2*e) - 2*a*b^3*d^3*f*e^(2*e) - b^4*d^3*f*e^(2*e))*x^2 + 2*(a^4*c*d^2*f*e^(2*e) + 2*a^3*b*c*d^2*f*e^
(2*e) - 2*a*b^3*c*d^2*f*e^(2*e) - b^4*c*d^2*f*e^(2*e))*x)*e^(2*f*x)) + integrate(4*(a*b*d*f*x + a*b*c*f + b^2*
d)/(a^4*c^3*f - 2*a^2*b^2*c^3*f + b^4*c^3*f + (a^4*d^3*f - 2*a^2*b^2*d^3*f + b^4*d^3*f)*x^3 + 3*(a^4*c*d^2*f -
2*a^2*b^2*c*d^2*f + b^4*c*d^2*f)*x^2 + 3*(a^4*c^2*d*f - 2*a^2*b^2*c^2*d*f + b^4*c^2*d*f)*x + (a^4*c^3*f*e^(2*
e) + 2*a^3*b*c^3*f*e^(2*e) - 2*a*b^3*c^3*f*e^(2*e) - b^4*c^3*f*e^(2*e) + (a^4*d^3*f*e^(2*e) + 2*a^3*b*d^3*f*e^
(2*e) - 2*a*b^3*d^3*f*e^(2*e) - b^4*d^3*f*e^(2*e))*x^3 + 3*(a^4*c*d^2*f*e^(2*e) + 2*a^3*b*c*d^2*f*e^(2*e) - 2*
a*b^3*c*d^2*f*e^(2*e) - b^4*c*d^2*f*e^(2*e))*x^2 + 3*(a^4*c^2*d*f*e^(2*e) + 2*a^3*b*c^2*d*f*e^(2*e) - 2*a*b^3*
c^2*d*f*e^(2*e) - b^4*c^2*d*f*e^(2*e))*x)*e^(2*f*x)), x)

Giac [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \tanh (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \tanh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \tanh (e+f x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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