Optimal. Leaf size=89 \[ -\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d e (e (c+d x))^{5/2}}+\frac {6 b \text {Int}\left (\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{\sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}},x\right )}{5 e} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.21, 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 = {}
\begin {gather*} \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{7/2}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{7/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^3}{(e x)^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d e (e (c+d x))^{5/2}}+\frac {(6 b) \text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt {-1+x} (e x)^{5/2} \sqrt {1+x}} \, dx,x,c+d x\right )}{5 d e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 95.18, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{7/2}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{3}}{\left (d e x +c e \right )^{\frac {7}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )^{3}}{\left (e \left (c + d x\right )\right )^{\frac {7}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________