Optimal. Leaf size=81 \[ \frac{(d+e x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )^3}{e (m+1)}-\frac{3 b c \text{Unintegrable}\left (\frac{(d+e x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{c x-1} \sqrt{c x+1}},x\right )}{e (m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.383444, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (d+e x)^m \left (a+b \cosh ^{-1}(c x)\right )^3 \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int (d+e x)^m \left (a+b \cosh ^{-1}(c x)\right )^3 \, dx &=\frac{(d+e x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )^3}{e (1+m)}-\frac{(3 b c) \int \frac{(d+e x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{e (1+m)}\\ \end{align*}
Mathematica [A] time = 6.27858, size = 0, normalized size = 0. \[ \int (d+e x)^m \left (a+b \cosh ^{-1}(c x)\right )^3 \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 3.002, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{3} \operatorname{arcosh}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname{arcosh}\left (c x\right ) + a^{3}\right )}{\left (e x + d\right )}^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{3} \left (d + e x\right )^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{3}{\left (e x + d\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]