3.2.47 \(\int \frac {1}{(c e+d e x) (a+b \cosh ^{-1}(c+d x))^3} \, dx\) [147]
Optimal. Leaf size=27 \[ \frac {\text {Int}\left (\frac {1}{(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^3},x\right )}{e} \]
[Out]
Unintegrable(1/(d*x+c)/(a+b*arccosh(d*x+c))^3,x)/e
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Rubi [A]
time = 0.05, 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 {1}{(c e+d e x) \left (a+b \cosh ^{-1}(c+d x)\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[1/((c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^3),x]
[Out]
Defer[Subst][Defer[Int][1/(x*(a + b*ArcCosh[x])^3), x], x, c + d*x]/(d*e)
Rubi steps
\begin {align*} \int \frac {1}{(c e+d e x) \left (a+b \cosh ^{-1}(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{e x \left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x \left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d e}\\ \end {align*}
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Mathematica [A]
time = 1.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c e+d e x) \left (a+b \cosh ^{-1}(c+d x)\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[1/((c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^3),x]
[Out]
Integrate[1/((c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^3), x]
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Maple [A]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d e x +c e \right ) \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(d*e*x+c*e)/(a+b*arccosh(d*x+c))^3,x)
[Out]
int(1/(d*e*x+c*e)/(a+b*arccosh(d*x+c))^3,x)
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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(1/(d*e*x+c*e)/(a+b*arccosh(d*x+c))^3,x, algorithm="maxima")
[Out]
-1/2*(b*d^8*x^8 + 8*b*c*d^7*x^7 + (28*c^2*d^6 - 3*d^6)*b*x^6 + 2*(28*c^3*d^5 - 9*c*d^5)*b*x^5 + (70*c^4*d^4 -
45*c^2*d^4 + 3*d^4)*b*x^4 + 4*(14*c^5*d^3 - 15*c^3*d^3 + 3*c*d^3)*b*x^3 + (b*d^5*x^5 + 5*b*c*d^4*x^4 + (2*a*d^
3 + (10*c^2*d^3 - d^3)*b)*x^3 + (6*a*c*d^2 + (10*c^3*d^2 - 3*c*d^2)*b)*x^2 + 2*(c^3 - c)*a + (c^5 - c^3)*b + (
2*(3*c^2*d - d)*a + (5*c^4*d - 3*c^2*d)*b)*x)*(d*x + c + 1)^(3/2)*(d*x + c - 1)^(3/2) + (28*c^6*d^2 - 45*c^4*d
^2 + 18*c^2*d^2 - d^2)*b*x^2 + (3*b*d^6*x^6 + 18*b*c*d^5*x^5 + (4*a*d^4 + 5*(9*c^2*d^4 - d^4)*b)*x^4 + 4*(4*a*
c*d^3 + 5*(3*c^3*d^3 - c*d^3)*b)*x^3 + ((24*c^2*d^2 - 5*d^2)*a + (45*c^4*d^2 - 30*c^2*d^2 + 2*d^2)*b)*x^2 + (4
*c^4 - 5*c^2 + 1)*a + (3*c^6 - 5*c^4 + 2*c^2)*b + 2*((8*c^3*d - 5*c*d)*a + (9*c^5*d - 10*c^3*d + 2*c*d)*b)*x)*
(d*x + c + 1)*(d*x + c - 1) + 2*(4*c^7*d - 9*c^5*d + 6*c^3*d - c*d)*b*x + (3*b*d^7*x^7 + 21*b*c*d^6*x^6 + (2*a
*d^5 + 7*(9*c^2*d^5 - d^5)*b)*x^5 + 5*(2*a*c*d^4 + 7*(3*c^3*d^4 - c*d^4)*b)*x^4 + ((20*c^2*d^3 - 3*d^3)*a + 5*
(21*c^4*d^3 - 14*c^2*d^3 + d^3)*b)*x^3 + ((20*c^3*d^2 - 9*c*d^2)*a + (63*c^5*d^2 - 70*c^3*d^2 + 15*c*d^2)*b)*x
^2 + (2*c^5 - 3*c^3 + c)*a + (3*c^7 - 7*c^5 + 5*c^3 - c)*b + ((10*c^4*d - 9*c^2*d + d)*a + (21*c^6*d - 35*c^4*
d + 15*c^2*d - d)*b)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (c^8 - 3*c^6 + 3*c^4 - c^2)*b + (2*(b*d^3*x^3 +
3*b*c*d^2*x^2 + (3*c^2*d - d)*b*x + (c^3 - c)*b)*(d*x + c + 1)^(3/2)*(d*x + c - 1)^(3/2) + (4*b*d^4*x^4 + 16*b
*c*d^3*x^3 + (24*c^2*d^2 - 5*d^2)*b*x^2 + 2*(8*c^3*d - 5*c*d)*b*x + (4*c^4 - 5*c^2 + 1)*b)*(d*x + c + 1)*(d*x
+ c - 1) + (2*b*d^5*x^5 + 10*b*c*d^4*x^4 + (20*c^2*d^3 - 3*d^3)*b*x^3 + (20*c^3*d^2 - 9*c*d^2)*b*x^2 + (10*c^4
*d - 9*c^2*d + d)*b*x + (2*c^5 - 3*c^3 + c)*b)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1
)*sqrt(d*x + c - 1) + c))/(a^2*b^2*d^9*x^8*e + 8*a^2*b^2*c*d^8*x^7*e + (28*c^2*d^7 - 3*d^7)*a^2*b^2*x^6*e + 2*
(28*c^3*d^6 - 9*c*d^6)*a^2*b^2*x^5*e + (70*c^4*d^5 - 45*c^2*d^5 + 3*d^5)*a^2*b^2*x^4*e + 4*(14*c^5*d^4 - 15*c^
3*d^4 + 3*c*d^4)*a^2*b^2*x^3*e + (28*c^6*d^3 - 45*c^4*d^3 + 18*c^2*d^3 - d^3)*a^2*b^2*x^2*e + 2*(4*c^7*d^2 - 9
*c^5*d^2 + 6*c^3*d^2 - c*d^2)*a^2*b^2*x*e + (c^8*d - 3*c^6*d + 3*c^4*d - c^2*d)*a^2*b^2*e + (a^2*b^2*d^6*x^5*e
+ 5*a^2*b^2*c*d^5*x^4*e + 10*a^2*b^2*c^2*d^4*x^3*e + 10*a^2*b^2*c^3*d^3*x^2*e + 5*a^2*b^2*c^4*d^2*x*e + a^2*b
^2*c^5*d*e)*(d*x + c + 1)^(3/2)*(d*x + c - 1)^(3/2) + 3*(a^2*b^2*d^7*x^6*e + 6*a^2*b^2*c*d^6*x^5*e + (15*c^2*d
^5 - d^5)*a^2*b^2*x^4*e + 4*(5*c^3*d^4 - c*d^4)*a^2*b^2*x^3*e + 3*(5*c^4*d^3 - 2*c^2*d^3)*a^2*b^2*x^2*e + 2*(3
*c^5*d^2 - 2*c^3*d^2)*a^2*b^2*x*e + (c^6*d - c^4*d)*a^2*b^2*e)*(d*x + c + 1)*(d*x + c - 1) + (b^4*d^9*x^8*e +
8*b^4*c*d^8*x^7*e + (28*c^2*d^7 - 3*d^7)*b^4*x^6*e + 2*(28*c^3*d^6 - 9*c*d^6)*b^4*x^5*e + (70*c^4*d^5 - 45*c^2
*d^5 + 3*d^5)*b^4*x^4*e + 4*(14*c^5*d^4 - 15*c^3*d^4 + 3*c*d^4)*b^4*x^3*e + (28*c^6*d^3 - 45*c^4*d^3 + 18*c^2*
d^3 - d^3)*b^4*x^2*e + 2*(4*c^7*d^2 - 9*c^5*d^2 + 6*c^3*d^2 - c*d^2)*b^4*x*e + (c^8*d - 3*c^6*d + 3*c^4*d - c^
2*d)*b^4*e + (b^4*d^6*x^5*e + 5*b^4*c*d^5*x^4*e + 10*b^4*c^2*d^4*x^3*e + 10*b^4*c^3*d^3*x^2*e + 5*b^4*c^4*d^2*
x*e + b^4*c^5*d*e)*(d*x + c + 1)^(3/2)*(d*x + c - 1)^(3/2) + 3*(b^4*d^7*x^6*e + 6*b^4*c*d^6*x^5*e + (15*c^2*d^
5 - d^5)*b^4*x^4*e + 4*(5*c^3*d^4 - c*d^4)*b^4*x^3*e + 3*(5*c^4*d^3 - 2*c^2*d^3)*b^4*x^2*e + 2*(3*c^5*d^2 - 2*
c^3*d^2)*b^4*x*e + (c^6*d - c^4*d)*b^4*e)*(d*x + c + 1)*(d*x + c - 1) + 3*(b^4*d^8*x^7*e + 7*b^4*c*d^7*x^6*e +
(21*c^2*d^6 - 2*d^6)*b^4*x^5*e + 5*(7*c^3*d^5 - 2*c*d^5)*b^4*x^4*e + (35*c^4*d^4 - 20*c^2*d^4 + d^4)*b^4*x^3*
e + (21*c^5*d^3 - 20*c^3*d^3 + 3*c*d^3)*b^4*x^2*e + (7*c^6*d^2 - 10*c^4*d^2 + 3*c^2*d^2)*b^4*x*e + (c^7*d - 2*
c^5*d + c^3*d)*b^4*e)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^
2 + 3*(a^2*b^2*d^8*x^7*e + 7*a^2*b^2*c*d^7*x^6*e + (21*c^2*d^6 - 2*d^6)*a^2*b^2*x^5*e + 5*(7*c^3*d^5 - 2*c*d^5
)*a^2*b^2*x^4*e + (35*c^4*d^4 - 20*c^2*d^4 + d^4)*a^2*b^2*x^3*e + (21*c^5*d^3 - 20*c^3*d^3 + 3*c*d^3)*a^2*b^2*
x^2*e + (7*c^6*d^2 - 10*c^4*d^2 + 3*c^2*d^2)*a^2*b^2*x*e + (c^7*d - 2*c^5*d + c^3*d)*a^2*b^2*e)*sqrt(d*x + c +
1)*sqrt(d*x + c - 1) + 2*(a*b^3*d^9*x^8*e + 8*a*b^3*c*d^8*x^7*e + (28*c^2*d^7 - 3*d^7)*a*b^3*x^6*e + 2*(28*c^
3*d^6 - 9*c*d^6)*a*b^3*x^5*e + (70*c^4*d^5 - 45*c^2*d^5 + 3*d^5)*a*b^3*x^4*e + 4*(14*c^5*d^4 - 15*c^3*d^4 + 3*
c*d^4)*a*b^3*x^3*e + (28*c^6*d^3 - 45*c^4*d^3 + 18*c^2*d^3 - d^3)*a*b^3*x^2*e + 2*(4*c^7*d^2 - 9*c^5*d^2 + 6*c
^3*d^2 - c*d^2)*a*b^3*x*e + (c^8*d - 3*c^6*d + 3*c^4*d - c^2*d)*a*b^3*e + (a*b^3*d^6*x^5*e + 5*a*b^3*c*d^5*x^4
*e + 10*a*b^3*c^2*d^4*x^3*e + 10*a*b^3*c^3*d^3*x^2*e + 5*a*b^3*c^4*d^2*x*e + a*b^3*c^5*d*e)*(d*x + c + 1)^(3/2
)*(d*x + c - 1)^(3/2) + 3*(a*b^3*d^7*x^6*e + 6*a*b^3*c*d^6*x^5*e + (15*c^2*d^5 - d^5)*a*b^3*x^4*e + 4*(5*c^3*d
^4 - c*d^4)*a*b^3*x^3*e + 3*(5*c^4*d^3 - 2*c^2*d^3)*a*b^3*x^2*e + 2*(3*c^5*d^2 - 2*c^3*d^2)*a*b^3*x*e + (c^6*d
- c^4*d)*a*b^3*e)*(d*x + c + 1)*(d*x + c - 1) + 3*(a*b^3*d^8*x^7*e + 7*a*b^3*c*d^7*x^6*e + (21*c^2*d^6 - 2*d^
6)*a*b^3*x^5*e + 5*(7*c^3*d^5 - 2*c*d^5)*a*b^3*...
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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(1/(d*e*x+c*e)/(a+b*arccosh(d*x+c))^3,x, algorithm="fricas")
[Out]
integral(1/((b^3*d*x + b^3*c)*arccosh(d*x + c)^3*e + 3*(a*b^2*d*x + a*b^2*c)*arccosh(d*x + c)^2*e + 3*(a^2*b*d
*x + a^2*b*c)*arccosh(d*x + c)*e + (a^3*d*x + a^3*c)*e), x)
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{a^{3} c + a^{3} d x + 3 a^{2} b c \operatorname {acosh}{\left (c + d x \right )} + 3 a^{2} b d x \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} c \operatorname {acosh}^{2}{\left (c + d x \right )} + 3 a b^{2} d x \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} c \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{3} d x \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(d*e*x+c*e)/(a+b*acosh(d*x+c))**3,x)
[Out]
Integral(1/(a**3*c + a**3*d*x + 3*a**2*b*c*acosh(c + d*x) + 3*a**2*b*d*x*acosh(c + d*x) + 3*a*b**2*c*acosh(c +
d*x)**2 + 3*a*b**2*d*x*acosh(c + d*x)**2 + b**3*c*acosh(c + d*x)**3 + b**3*d*x*acosh(c + d*x)**3), x)/e
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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(1/(d*e*x+c*e)/(a+b*arccosh(d*x+c))^3,x, algorithm="giac")
[Out]
integrate(1/((d*e*x + c*e)*(b*arccosh(d*x + c) + a)^3), x)
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{\left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((c*e + d*e*x)*(a + b*acosh(c + d*x))^3),x)
[Out]
int(1/((c*e + d*e*x)*(a + b*acosh(c + d*x))^3), x)
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