3.2.73 \(\int \frac {1}{(c e+d e x) (a+b \sinh ^{-1}(c+d x))^3} \, dx\) [173]
Optimal. Leaf size=27 \[ \frac {\text {Int}\left (\frac {1}{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^3},x\right )}{e} \]
[Out]
Unintegrable(1/(d*x+c)/(a+b*arcsinh(d*x+c))^3,x)/e
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Rubi [A]
time = 0.04, 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 \sinh ^{-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*ArcSinh[c + d*x])^3),x]
[Out]
Defer[Subst][Defer[Int][1/(x*(a + b*ArcSinh[x])^3), x], x, c + d*x]/(d*e)
Rubi steps
\begin {align*} \int \frac {1}{(c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{e x \left (a+b \sinh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x \left (a+b \sinh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d e}\\ \end {align*}
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Mathematica [A]
time = 0.78, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c e+d e x) \left (a+b \sinh ^{-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*ArcSinh[c + d*x])^3),x]
[Out]
Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^3), x]
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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d e x +c e \right ) \left (a +b \arcsinh \left (d x +c \right )\right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^3,x)
[Out]
int(1/(d*e*x+c*e)/(a+b*arcsinh(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*arcsinh(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 + (28*c^6*d^2 + 45*c^4*d^2 + 18*c^2*d^
2 + d^2)*b*x^2 + 2*(4*c^7*d + 9*c^5*d + 6*c^3*d + c*d)*b*x + (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^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + (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^2*x^2 + 2*c*d*x + c^2 + 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^2*x^2 + 2*c*d*x + c^2 + 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^2*x^2 + 2*
c*d*x + c^2 + 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^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d
^2*x^2 + 2*c*d*x + c^2 + 1)) + (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^2*x^2 + 2*c*
d*x + c^2 + 1))/(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 + (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^2*x^2 + 2*c*d*x + c^2 + 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^2*x^2 + 2*c*d*x + c^2 + 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^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 + (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^2*x^2 + 2*c*d*x + c^2 + 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^2*x^2 + 2*c*d*x + c^2 + 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^2*x^2 + 2*c*d*x + c^2 + 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^2*x^2 + 2*c*d*x + c^
2 + 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*x^4*e + (35*c^4*d^4 + 20*c^2*d^4 + d^4)*a*b^3*x^3*e + (21*c^5*d^3 + 20*c^3*d^3 + 3*c*d^3)*a*b^3*x^2*e +
(7*c^6*d^2 + 10*c^4*d^2 + 3*c^2*d^2)*a*b^3*x*e + (c^7*d + 2*c^5*d + c^3*d)*a*b^3*e)*sqrt(d^2*x^2 + 2*c*d*x +
c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + 3*(a^2*b^2*d^8*x^7*e + 7*a^2*b^2*c*d^7*x^6*e + (2
1*c^2*d^6 + 2*d^6)*a^2*b^2*x^5*e + 5*(7*c^3*d^5...
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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*arcsinh(d*x+c))^3,x, algorithm="fricas")
[Out]
integral(1/((b^3*d*x + b^3*c)*arcsinh(d*x + c)^3*e + 3*(a*b^2*d*x + a*b^2*c)*arcsinh(d*x + c)^2*e + 3*(a^2*b*d
*x + a^2*b*c)*arcsinh(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 {asinh}{\left (c + d x \right )} + 3 a^{2} b d x \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} c \operatorname {asinh}^{2}{\left (c + d x \right )} + 3 a b^{2} d x \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} c \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{3} d x \operatorname {asinh}^{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*asinh(d*x+c))**3,x)
[Out]
Integral(1/(a**3*c + a**3*d*x + 3*a**2*b*c*asinh(c + d*x) + 3*a**2*b*d*x*asinh(c + d*x) + 3*a*b**2*c*asinh(c +
d*x)**2 + 3*a*b**2*d*x*asinh(c + d*x)**2 + b**3*c*asinh(c + d*x)**3 + b**3*d*x*asinh(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*arcsinh(d*x+c))^3,x, algorithm="giac")
[Out]
integrate(1/((d*e*x + c*e)*(b*arcsinh(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 {asinh}\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*asinh(c + d*x))^3),x)
[Out]
int(1/((c*e + d*e*x)*(a + b*asinh(c + d*x))^3), x)
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