\(\int (d x)^m (a+b \text {sech}^{-1}(c x))^3 \, dx\) [69]
Optimal result
Integrand size = 16, antiderivative size = 16 \[
\int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\text {Int}\left ((d x)^m \left (a+b \text {sech}^{-1}(c x)\right )^3,x\right )
\]
[Out]
Unintegrable((d*x)^m*(a+b*arcsech(c*x))^3,x)
Rubi [N/A]
Not integrable
Time = 0.02 (sec) , antiderivative size = 16, 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 = {} \[
\int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx
\]
[In]
Int[(d*x)^m*(a + b*ArcSech[c*x])^3,x]
[Out]
Defer[Int][(d*x)^m*(a + b*ArcSech[c*x])^3, x]
Rubi steps \begin{align*}
\text {integral}& = \int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx \\
\end{align*}
Mathematica [N/A]
Not integrable
Time = 5.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12
\[
\int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx
\]
[In]
Integrate[(d*x)^m*(a + b*ArcSech[c*x])^3,x]
[Out]
Integrate[(d*x)^m*(a + b*ArcSech[c*x])^3, x]
Maple [N/A] (verified)
Not integrable
Time = 0.65 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00
\[\int \left (d x \right )^{m} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )^{3}d x\]
[In]
int((d*x)^m*(a+b*arcsech(c*x))^3,x)
[Out]
int((d*x)^m*(a+b*arcsech(c*x))^3,x)
Fricas [N/A]
Not integrable
Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.75
\[
\int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} \left (d x\right )^{m} \,d x }
\]
[In]
integrate((d*x)^m*(a+b*arcsech(c*x))^3,x, algorithm="fricas")
[Out]
integral((b^3*arcsech(c*x)^3 + 3*a*b^2*arcsech(c*x)^2 + 3*a^2*b*arcsech(c*x) + a^3)*(d*x)^m, x)
Sympy [N/A]
Not integrable
Time = 5.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94
\[
\int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\int \left (d x\right )^{m} \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}\, dx
\]
[In]
integrate((d*x)**m*(a+b*asech(c*x))**3,x)
[Out]
Integral((d*x)**m*(a + b*asech(c*x))**3, x)
Maxima [N/A]
Not integrable
Time = 10.45 (sec) , antiderivative size = 1450, normalized size of antiderivative = 90.62
\[
\int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} \left (d x\right )^{m} \,d x }
\]
[In]
integrate((d*x)^m*(a+b*arcsech(c*x))^3,x, algorithm="maxima")
[Out]
b^3*d^m*x*x^m*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)^3/(m + 1) + (d*x)^(m + 1)*a^3/(d*(m + 1)) - integrate(((b^
3*c^2*d^m*(m + 1)*x^2 - b^3*d^m*(m + 1))*x^m*log(x)^3 - 3*(b^3*d^m*(m + 1)*log(c) - a*b^2*d^m*(m + 1) - (b^3*c
^2*d^m*(m + 1)*log(c) - a*b^2*c^2*d^m*(m + 1))*x^2)*x^m*log(x)^2 + 3*((b^3*c^2*d^m*(m + 1)*x^2 - b^3*d^m*(m +
1))*x^m*log(x) + ((b^3*c^2*d^m*(m + 1)*x^2 - b^3*d^m*(m + 1))*x^m*log(x) - (b^3*d^m*(m + 1)*log(c) - a*b^2*d^m
*(m + 1) + (a*b^2*c^2*d^m*(m + 1) - (d^m*(m + 1)*log(c) + d^m)*b^3*c^2)*x^2)*x^m)*sqrt(c*x + 1)*sqrt(-c*x + 1)
- (b^3*d^m*(m + 1)*log(c) - a*b^2*d^m*(m + 1) - (b^3*c^2*d^m*(m + 1)*log(c) - a*b^2*c^2*d^m*(m + 1))*x^2)*x^m
)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)^2 - 3*(b^3*d^m*(m + 1)*log(c)^2 - 2*a*b^2*d^m*(m + 1)*log(c) + a^2*b*d
^m*(m + 1) - (b^3*c^2*d^m*(m + 1)*log(c)^2 - 2*a*b^2*c^2*d^m*(m + 1)*log(c) + a^2*b*c^2*d^m*(m + 1))*x^2)*x^m*
log(x) + ((b^3*c^2*d^m*(m + 1)*x^2 - b^3*d^m*(m + 1))*x^m*log(x)^3 - 3*(b^3*d^m*(m + 1)*log(c) - a*b^2*d^m*(m
+ 1) - (b^3*c^2*d^m*(m + 1)*log(c) - a*b^2*c^2*d^m*(m + 1))*x^2)*x^m*log(x)^2 - 3*(b^3*d^m*(m + 1)*log(c)^2 -
2*a*b^2*d^m*(m + 1)*log(c) + a^2*b*d^m*(m + 1) - (b^3*c^2*d^m*(m + 1)*log(c)^2 - 2*a*b^2*c^2*d^m*(m + 1)*log(c
) + a^2*b*c^2*d^m*(m + 1))*x^2)*x^m*log(x) - (b^3*d^m*(m + 1)*log(c)^3 - 3*a*b^2*d^m*(m + 1)*log(c)^2 + 3*a^2*
b*d^m*(m + 1)*log(c) - (b^3*c^2*d^m*(m + 1)*log(c)^3 - 3*a*b^2*c^2*d^m*(m + 1)*log(c)^2 + 3*a^2*b*c^2*d^m*(m +
1)*log(c))*x^2)*x^m)*sqrt(c*x + 1)*sqrt(-c*x + 1) - (b^3*d^m*(m + 1)*log(c)^3 - 3*a*b^2*d^m*(m + 1)*log(c)^2
+ 3*a^2*b*d^m*(m + 1)*log(c) - (b^3*c^2*d^m*(m + 1)*log(c)^3 - 3*a*b^2*c^2*d^m*(m + 1)*log(c)^2 + 3*a^2*b*c^2*
d^m*(m + 1)*log(c))*x^2)*x^m - 3*((b^3*c^2*d^m*(m + 1)*x^2 - b^3*d^m*(m + 1))*x^m*log(x)^2 - 2*(b^3*d^m*(m + 1
)*log(c) - a*b^2*d^m*(m + 1) - (b^3*c^2*d^m*(m + 1)*log(c) - a*b^2*c^2*d^m*(m + 1))*x^2)*x^m*log(x) + ((b^3*c^
2*d^m*(m + 1)*x^2 - b^3*d^m*(m + 1))*x^m*log(x)^2 - 2*(b^3*d^m*(m + 1)*log(c) - a*b^2*d^m*(m + 1) - (b^3*c^2*d
^m*(m + 1)*log(c) - a*b^2*c^2*d^m*(m + 1))*x^2)*x^m*log(x) - (b^3*d^m*(m + 1)*log(c)^2 - 2*a*b^2*d^m*(m + 1)*l
og(c) + a^2*b*d^m*(m + 1) - (b^3*c^2*d^m*(m + 1)*log(c)^2 - 2*a*b^2*c^2*d^m*(m + 1)*log(c) + a^2*b*c^2*d^m*(m
+ 1))*x^2)*x^m)*sqrt(c*x + 1)*sqrt(-c*x + 1) - (b^3*d^m*(m + 1)*log(c)^2 - 2*a*b^2*d^m*(m + 1)*log(c) + a^2*b*
d^m*(m + 1) - (b^3*c^2*d^m*(m + 1)*log(c)^2 - 2*a*b^2*c^2*d^m*(m + 1)*log(c) + a^2*b*c^2*d^m*(m + 1))*x^2)*x^m
)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1))/(c^2*(m + 1)*x^2 + (c^2*(m + 1)*x^2 - m - 1)*sqrt(c*x + 1)*sqrt(-c*x
+ 1) - m - 1), x)
Giac [N/A]
Not integrable
Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12
\[
\int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} \left (d x\right )^{m} \,d x }
\]
[In]
integrate((d*x)^m*(a+b*arcsech(c*x))^3,x, algorithm="giac")
[Out]
integrate((b*arcsech(c*x) + a)^3*(d*x)^m, x)
Mupad [N/A]
Not integrable
Time = 4.59 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38
\[
\int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\int {\left (d\,x\right )}^m\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x
\]
[In]
int((d*x)^m*(a + b*acosh(1/(c*x)))^3,x)
[Out]
int((d*x)^m*(a + b*acosh(1/(c*x)))^3, x)