3.1.36 \(\int (d+e x)^m (a+b \cosh ^{-1}(c x))^3 \, dx\) [36]

Optimal. Leaf size=82 \[ \frac {(d+e x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )^3}{e (1+m)}-\frac {3 b c \text {Int}\left (\frac {(d+e x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}},x\right )}{e (1+m)} \]

[Out]

(e*x+d)^(1+m)*(a+b*arccosh(c*x))^3/e/(1+m)-3*b*c*Unintegrable((e*x+d)^(1+m)*(a+b*arccosh(c*x))^2/(c*x-1)^(1/2)
/(c*x+1)^(1/2),x)/e/(1+m)

________________________________________________________________________________________

Rubi [A]
time = 0.26, 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 (d+e x)^m \left (a+b \cosh ^{-1}(c x)\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

((d + e*x)^(1 + m)*(a + b*ArcCosh[c*x])^3)/(e*(1 + m)) - (3*b*c*Defer[Int][((d + e*x)^(1 + m)*(a + b*ArcCosh[c
*x])^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x])/(e*(1 + m))

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 = 9.37, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^m \left (a+b \cosh ^{-1}(c x)\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{m} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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]

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

[Out]

(x*e + d)^(m + 1)*a^3*e^(-1)/(m + 1) + (b^3*x*e + b^3*d)*e^(m*log(x*e + d) - 1)*log(c*x + sqrt(c*x + 1)*sqrt(c
*x - 1))^3/(m + 1) + integrate(-3*(((b^3*c^2*d*x + a*b^2*(m + 1)*e - (a*b^2*c^2*(m + 1) - b^3*c^2)*x^2*e)*sqrt
(c*x + 1)*sqrt(c*x - 1)*(x*e + d)^m + (b^3*c^3*d*x^2 - b^3*c*d - (a*b^2*c^3*(m + 1) - b^3*c^3)*x^3*e + (a*b^2*
c*(m + 1) - b^3*c)*x*e)*(x*e + d)^m)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2 - ((a^2*b*c^2*(m + 1)*x^2*e - a^
2*b*(m + 1)*e)*sqrt(c*x + 1)*sqrt(c*x - 1)*(x*e + d)^m + (a^2*b*c^3*(m + 1)*x^3*e - a^2*b*c*(m + 1)*x*e)*(x*e
+ d)^m)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^3*(m + 1)*x^3*e - c*(m + 1)*x*e + (c^2*(m + 1)*x^2*e - (m +
1)*e)*sqrt(c*x + 1)*sqrt(c*x - 1)), x)

________________________________________________________________________________________

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]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{3} \left (d + e x\right )^{m}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*acosh(c*x))**3*(d + e*x)**m, x)

________________________________________________________________________________________

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]

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

[Out]

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

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^3\,{\left (d+e\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________