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

Optimal. Leaf size=23 \[ \text {Int}\left ((e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^3,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx \]

Verification is Not applicable to the result.

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

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Mathematica [A]  time = 0.35, size = 0, normalized size = 0.00 \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[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]

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fricas [A]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} \operatorname {arcoth}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname {arcoth}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname {arcoth}\left (d x + c\right ) + a^{3}\right )} {\left (f x + e\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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
)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3} {\left (f x + e\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 2.06, size = 0, normalized size = 0.00 \[ \int \left (f x +e \right )^{m} \left (a +b \,\mathrm {arccoth}\left (d x +c \right )\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (b^{3} f x + b^{3} e\right )} {\left (f x + e\right )}^{m} \log \left (d x + c + 1\right )^{3}}{8 \, f {\left (m + 1\right )}} + \frac {{\left (f x + e\right )}^{m + 1} a^{3}}{f {\left (m + 1\right )}} - \int \frac {{\left ({\left (b^{3} d f {\left (m + 1\right )} x + {\left (c f {\left (m + 1\right )} + f {\left (m + 1\right )}\right )} b^{3}\right )} \log \left (d x + c - 1\right )^{3} + 3 \, {\left (b^{3} d e - 2 \, {\left (c f {\left (m + 1\right )} + f {\left (m + 1\right )}\right )} a b^{2} - {\left (2 \, a b^{2} d f {\left (m + 1\right )} - b^{3} d f\right )} x + {\left (b^{3} d f {\left (m + 1\right )} x + {\left (c f {\left (m + 1\right )} + f {\left (m + 1\right )}\right )} b^{3}\right )} \log \left (d x + c - 1\right )\right )} \log \left (d x + c + 1\right )^{2} - 6 \, {\left (a b^{2} d f {\left (m + 1\right )} x + {\left (c f {\left (m + 1\right )} + f {\left (m + 1\right )}\right )} a b^{2}\right )} \log \left (d x + c - 1\right )^{2} - 3 \, {\left (4 \, a^{2} b d f {\left (m + 1\right )} x + 4 \, {\left (c f {\left (m + 1\right )} + f {\left (m + 1\right )}\right )} a^{2} b + {\left (b^{3} d f {\left (m + 1\right )} x + {\left (c f {\left (m + 1\right )} + f {\left (m + 1\right )}\right )} b^{3}\right )} \log \left (d x + c - 1\right )^{2} - 4 \, {\left (a b^{2} d f {\left (m + 1\right )} x + {\left (c f {\left (m + 1\right )} + f {\left (m + 1\right )}\right )} a b^{2}\right )} \log \left (d x + c - 1\right )\right )} \log \left (d x + c + 1\right ) + 12 \, {\left (a^{2} b d f {\left (m + 1\right )} x + {\left (c f {\left (m + 1\right )} + f {\left (m + 1\right )}\right )} a^{2} b\right )} \log \left (d x + c - 1\right )\right )} {\left (f x + e\right )}^{m}}{8 \, {\left (d f {\left (m + 1\right )} x + c f {\left (m + 1\right )} + f {\left (m + 1\right )}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int {\left (e+f\,x\right )}^m\,{\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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