\(\int \frac {(a+b \coth (e+f x))^3}{(c+d x)^2} \, dx\) [51]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {(a+b \coth (e+f x))^3}{(c+d x)^2} \, dx=\text {Int}\left (\frac {(a+b \coth (e+f x))^3}{(c+d x)^2},x\right ) \]

[Out]

Unintegrable((a+b*coth(f*x+e))^3/(d*x+c)^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 = {} \[ \int \frac {(a+b \coth (e+f x))^3}{(c+d x)^2} \, dx=\int \frac {(a+b \coth (e+f x))^3}{(c+d x)^2} \, dx \]

[In]

Int[(a + b*Coth[e + f*x])^3/(c + d*x)^2,x]

[Out]

Defer[Int][(a + b*Coth[e + f*x])^3/(c + d*x)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b \coth (e+f x))^3}{(c+d x)^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 56.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \coth (e+f x))^3}{(c+d x)^2} \, dx=\int \frac {(a+b \coth (e+f x))^3}{(c+d x)^2} \, dx \]

[In]

Integrate[(a + b*Coth[e + f*x])^3/(c + d*x)^2,x]

[Out]

Integrate[(a + b*Coth[e + f*x])^3/(c + d*x)^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.32 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \frac {\left (a +b \coth \left (f x +e \right )\right )^{3}}{\left (d x +c \right )^{2}}d x\]

[In]

int((a+b*coth(f*x+e))^3/(d*x+c)^2,x)

[Out]

int((a+b*coth(f*x+e))^3/(d*x+c)^2,x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.15 \[ \int \frac {(a+b \coth (e+f x))^3}{(c+d x)^2} \, dx=\int { \frac {{\left (b \coth \left (f x + e\right ) + a\right )}^{3}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate((a+b*coth(f*x+e))^3/(d*x+c)^2,x, algorithm="fricas")

[Out]

integral((b^3*coth(f*x + e)^3 + 3*a*b^2*coth(f*x + e)^2 + 3*a^2*b*coth(f*x + e) + a^3)/(d^2*x^2 + 2*c*d*x + c^
2), x)

Sympy [N/A]

Not integrable

Time = 3.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b \coth (e+f x))^3}{(c+d x)^2} \, dx=\int \frac {\left (a + b \coth {\left (e + f x \right )}\right )^{3}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

Integral((a + b*coth(e + f*x))**3/(c + d*x)**2, x)

Maxima [N/A]

Not integrable

Time = 0.95 (sec) , antiderivative size = 1144, normalized size of antiderivative = 57.20 \[ \int \frac {(a+b \coth (e+f x))^3}{(c+d x)^2} \, dx=\int { \frac {{\left (b \coth \left (f x + e\right ) + a\right )}^{3}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate((a+b*coth(f*x+e))^3/(d*x+c)^2,x, algorithm="maxima")

[Out]

-a^3/(d^2*x + c*d) - (3*a^2*b*c^2*f^2 + 3*(c^2*f^2 - 2*c*d*f)*a*b^2 + (c^2*f^2 + 2*d^2)*b^3 + (3*a^2*b*d^2*f^2
+ 3*a*b^2*d^2*f^2 + b^3*d^2*f^2)*x^2 + 2*(3*a^2*b*c*d*f^2 + b^3*c*d*f^2 + 3*(c*d*f^2 - d^2*f)*a*b^2)*x + (3*a
^2*b*c^2*f^2*e^(4*e) + 3*a*b^2*c^2*f^2*e^(4*e) + b^3*c^2*f^2*e^(4*e) + (3*a^2*b*d^2*f^2*e^(4*e) + 3*a*b^2*d^2*
f^2*e^(4*e) + b^3*d^2*f^2*e^(4*e))*x^2 + 2*(3*a^2*b*c*d*f^2*e^(4*e) + 3*a*b^2*c*d*f^2*e^(4*e) + b^3*c*d*f^2*e^
(4*e))*x)*e^(4*f*x) - 2*(3*a^2*b*c^2*f^2*e^(2*e) + 3*(c^2*f^2*e^(2*e) - c*d*f*e^(2*e))*a*b^2 + (c^2*f^2*e^(2*e
) - c*d*f*e^(2*e) + d^2*e^(2*e))*b^3 + (3*a^2*b*d^2*f^2*e^(2*e) + 3*a*b^2*d^2*f^2*e^(2*e) + b^3*d^2*f^2*e^(2*e
))*x^2 + (6*a^2*b*c*d*f^2*e^(2*e) + 3*(2*c*d*f^2*e^(2*e) - d^2*f*e^(2*e))*a*b^2 + (2*c*d*f^2*e^(2*e) - d^2*f*e
^(2*e))*b^3)*x)*e^(2*f*x))/(d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2 + (d^4*f^2*x^3*e^(4*e)
+ 3*c*d^3*f^2*x^2*e^(4*e) + 3*c^2*d^2*f^2*x*e^(4*e) + c^3*d*f^2*e^(4*e))*e^(4*f*x) - 2*(d^4*f^2*x^3*e^(2*e) +
3*c*d^3*f^2*x^2*e^(2*e) + 3*c^2*d^2*f^2*x*e^(2*e) + c^3*d*f^2*e^(2*e))*e^(2*f*x)) - integrate((3*a^2*b*c^2*f^
2 - 6*a*b^2*c*d*f + (c^2*f^2 + 3*d^2)*b^3 + (3*a^2*b*d^2*f^2 + b^3*d^2*f^2)*x^2 + 2*(3*a^2*b*c*d*f^2 + b^3*c*d
*f^2 - 3*a*b^2*d^2*f)*x)/(d^4*f^2*x^4 + 4*c*d^3*f^2*x^3 + 6*c^2*d^2*f^2*x^2 + 4*c^3*d*f^2*x + c^4*f^2 + (d^4*f
^2*x^4*e^e + 4*c*d^3*f^2*x^3*e^e + 6*c^2*d^2*f^2*x^2*e^e + 4*c^3*d*f^2*x*e^e + c^4*f^2*e^e)*e^(f*x)), x) + int
egrate(-(3*a^2*b*c^2*f^2 - 6*a*b^2*c*d*f + (c^2*f^2 + 3*d^2)*b^3 + (3*a^2*b*d^2*f^2 + b^3*d^2*f^2)*x^2 + 2*(3*
a^2*b*c*d*f^2 + b^3*c*d*f^2 - 3*a*b^2*d^2*f)*x)/(d^4*f^2*x^4 + 4*c*d^3*f^2*x^3 + 6*c^2*d^2*f^2*x^2 + 4*c^3*d*f
^2*x + c^4*f^2 - (d^4*f^2*x^4*e^e + 4*c*d^3*f^2*x^3*e^e + 6*c^2*d^2*f^2*x^2*e^e + 4*c^3*d*f^2*x*e^e + c^4*f^2*
e^e)*e^(f*x)), x)

Giac [N/A]

Not integrable

Time = 0.80 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \coth (e+f x))^3}{(c+d x)^2} \, dx=\int { \frac {{\left (b \coth \left (f x + e\right ) + a\right )}^{3}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate((a+b*coth(f*x+e))^3/(d*x+c)^2,x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \coth (e+f x))^3}{(c+d x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {coth}\left (e+f\,x\right )\right )}^3}{{\left (c+d\,x\right )}^2} \,d x \]

[In]

int((a + b*coth(e + f*x))^3/(c + d*x)^2,x)

[Out]

int((a + b*coth(e + f*x))^3/(c + d*x)^2, x)