\(\int (e+f x)^m (a+b \coth ^{-1}(c+d x))^2 \, dx\) [120]

Optimal result

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

[Out]

Unintegrable((f*x+e)^m*(a+b*arccoth(d*x+c))^2,x)

Rubi [N/A]

Not integrable

Time = 0.06 (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 = {} \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx \]

[In]

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

[Out]

Defer[Subst][Defer[Int][((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCoth[x])^2, x], x, c + d*x]/d

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

Mathematica [N/A]

Not integrable

Time = 2.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.36 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \left (f x +e \right )^{m} \left (a +b \,\operatorname {arccoth}\left (d x +c \right )\right )^{2}d x\]

[In]

int((f*x+e)^m*(a+b*arccoth(d*x+c))^2,x)

[Out]

int((f*x+e)^m*(a+b*arccoth(d*x+c))^2,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2} {\left (f x + e\right )}^{m} \,d x } \]

[In]

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

[Out]

integral((b^2*arccoth(d*x + c)^2 + 2*a*b*arccoth(d*x + c) + a^2)*(f*x + e)^m, x)

Sympy [N/A]

Not integrable

Time = 85.90 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )^{m}\, dx \]

[In]

integrate((f*x+e)**m*(a+b*acoth(d*x+c))**2,x)

[Out]

Integral((a + b*acoth(c + d*x))**2*(e + f*x)**m, x)

Maxima [N/A]

Not integrable

Time = 2.55 (sec) , antiderivative size = 253, normalized size of antiderivative = 12.65 \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2} {\left (f x + e\right )}^{m} \,d x } \]

[In]

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

[Out]

1/4*(b^2*f*x + b^2*e)*(f*x + e)^m*log(d*x + c + 1)^2/(f*(m + 1)) + (f*x + e)^(m + 1)*a^2/(f*(m + 1)) - integra
te(-1/4*((b^2*d*f*(m + 1)*x + (c*f*(m + 1) + f*(m + 1))*b^2)*log(d*x + c - 1)^2 - 2*(b^2*d*e - 2*(c*f*(m + 1)
+ f*(m + 1))*a*b - (2*a*b*d*f*(m + 1) - b^2*d*f)*x + (b^2*d*f*(m + 1)*x + (c*f*(m + 1) + f*(m + 1))*b^2)*log(d
*x + c - 1))*log(d*x + c + 1) - 4*(a*b*d*f*(m + 1)*x + (c*f*(m + 1) + f*(m + 1))*a*b)*log(d*x + c - 1))*(f*x +
e)^m/(d*f*(m + 1)*x + c*f*(m + 1) + f*(m + 1)), x)

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2} {\left (f x + e\right )}^{m} \,d x } \]

[In]

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

[Out]

integrate((b*arccoth(d*x + c) + a)^2*(f*x + e)^m, x)

Mupad [N/A]

Not integrable

Time = 3.87 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int {\left (e+f\,x\right )}^m\,{\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^2 \,d x \]

[In]

int((e + f*x)^m*(a + b*acoth(c + d*x))^2,x)

[Out]

int((e + f*x)^m*(a + b*acoth(c + d*x))^2, x)