\(\int (e+f x)^m (a+b \coth ^{-1}(c+d x))^3 \, dx\) [121]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\text {Int}\left ((e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^3,x\right ) \]

[Out]

Unintegrable((f*x+e)^m*(a+b*arccoth(d*x+c))^3,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 = {} \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx \]

[In]

Int[(e + f*x)^m*(a + b*ArcCoth[c + d*x])^3,x]

[Out]

Defer[Subst][Defer[Int][((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCoth[x])^3, 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 )^3 \, dx,x,c+d x\right )}{d} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx \]

[In]

Integrate[(e + f*x)^m*(a + b*ArcCoth[c + d*x])^3,x]

[Out]

Integrate[(e + f*x)^m*(a + b*ArcCoth[c + d*x])^3, x]

Maple [N/A] (verified)

Not integrable

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

\[\int \left (f x +e \right )^{m} \left (a +b \,\operatorname {arccoth}\left (d x +c \right )\right )^{3}d x\]

[In]

int((f*x+e)^m*(a+b*arccoth(d*x+c))^3,x)

[Out]

int((f*x+e)^m*(a+b*arccoth(d*x+c))^3,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.60 \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3} {\left (f x + e\right )}^{m} \,d x } \]

[In]

integrate((f*x+e)^m*(a+b*arccoth(d*x+c))^3,x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\text {Timed out} \]

[In]

integrate((f*x+e)**m*(a+b*acoth(d*x+c))**3,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 4.13 (sec) , antiderivative size = 418, normalized size of antiderivative = 20.90 \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3} {\left (f x + e\right )}^{m} \,d x } \]

[In]

integrate((f*x+e)^m*(a+b*arccoth(d*x+c))^3,x, algorithm="maxima")

[Out]

1/8*(b^3*f*x + b^3*e)*(f*x + e)^m*log(d*x + c + 1)^3/(f*(m + 1)) + (f*x + e)^(m + 1)*a^3/(f*(m + 1)) - integra
te(1/8*((b^3*d*f*(m + 1)*x + (c*f*(m + 1) + f*(m + 1))*b^3)*log(d*x + c - 1)^3 + 3*(b^3*d*e - 2*(c*f*(m + 1) +
f*(m + 1))*a*b^2 - (2*a*b^2*d*f*(m + 1) - b^3*d*f)*x + (b^3*d*f*(m + 1)*x + (c*f*(m + 1) + f*(m + 1))*b^3)*lo
g(d*x + c - 1))*log(d*x + c + 1)^2 - 6*(a*b^2*d*f*(m + 1)*x + (c*f*(m + 1) + f*(m + 1))*a*b^2)*log(d*x + c - 1
)^2 - 3*(4*a^2*b*d*f*(m + 1)*x + 4*(c*f*(m + 1) + f*(m + 1))*a^2*b + (b^3*d*f*(m + 1)*x + (c*f*(m + 1) + f*(m
+ 1))*b^3)*log(d*x + c - 1)^2 - 4*(a*b^2*d*f*(m + 1)*x + (c*f*(m + 1) + f*(m + 1))*a*b^2)*log(d*x + c - 1))*lo
g(d*x + c + 1) + 12*(a^2*b*d*f*(m + 1)*x + (c*f*(m + 1) + f*(m + 1))*a^2*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 = 0.37 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3} {\left (f x + e\right )}^{m} \,d x } \]

[In]

integrate((f*x+e)^m*(a+b*arccoth(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((b*arccoth(d*x + c) + a)^3*(f*x + e)^m, x)

Mupad [N/A]

Not integrable

Time = 3.86 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int {\left (e+f\,x\right )}^m\,{\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^3 \,d x \]

[In]

int((e + f*x)^m*(a + b*acoth(c + d*x))^3,x)

[Out]

int((e + f*x)^m*(a + b*acoth(c + d*x))^3, x)