3.1.11 \(\int \frac {a+b \sinh ^{-1}(c x)}{(d+e x)^4} \, dx\) [11]

Optimal. Leaf size=183 \[ -\frac {b c \sqrt {1+c^2 x^2}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {1+c^2 x^2}}{2 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {a+b \sinh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {b c^3 \left (2 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\frac {e-c^2 d x}{\sqrt {c^2 d^2+e^2} \sqrt {1+c^2 x^2}}\right )}{6 e \left (c^2 d^2+e^2\right )^{5/2}} \]

[Out]

1/3*(-a-b*arcsinh(c*x))/e/(e*x+d)^3-1/6*b*c^3*(2*c^2*d^2-e^2)*arctanh((-c^2*d*x+e)/(c^2*d^2+e^2)^(1/2)/(c^2*x^
2+1)^(1/2))/e/(c^2*d^2+e^2)^(5/2)-1/6*b*c*(c^2*x^2+1)^(1/2)/(c^2*d^2+e^2)/(e*x+d)^2-1/2*b*c^3*d*(c^2*x^2+1)^(1
/2)/(c^2*d^2+e^2)^2/(e*x+d)

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Rubi [A]
time = 0.10, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5828, 759, 821, 739, 212} \begin {gather*} -\frac {a+b \sinh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {b c \sqrt {c^2 x^2+1}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {c^2 x^2+1}}{2 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {b c^3 \left (2 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\frac {e-c^2 d x}{\sqrt {c^2 x^2+1} \sqrt {c^2 d^2+e^2}}\right )}{6 e \left (c^2 d^2+e^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSinh[c*x])/(d + e*x)^4,x]

[Out]

-1/6*(b*c*Sqrt[1 + c^2*x^2])/((c^2*d^2 + e^2)*(d + e*x)^2) - (b*c^3*d*Sqrt[1 + c^2*x^2])/(2*(c^2*d^2 + e^2)^2*
(d + e*x)) - (a + b*ArcSinh[c*x])/(3*e*(d + e*x)^3) - (b*c^3*(2*c^2*d^2 - e^2)*ArcTanh[(e - c^2*d*x)/(Sqrt[c^2
*d^2 + e^2]*Sqrt[1 + c^2*x^2])])/(6*e*(c^2*d^2 + e^2)^(5/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 759

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*((a + c*x^2)^(p
+ 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 5828

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*
((a + b*ArcSinh[c*x])^n/(e*(m + 1))), x] - Dist[b*c*(n/(e*(m + 1))), Int[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x
])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{(d+e x)^4} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {(b c) \int \frac {1}{(d+e x)^3 \sqrt {1+c^2 x^2}} \, dx}{3 e}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {a+b \sinh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {\left (b c^3\right ) \int \frac {-2 d+e x}{(d+e x)^2 \sqrt {1+c^2 x^2}} \, dx}{6 e \left (c^2 d^2+e^2\right )}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {1+c^2 x^2}}{2 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {a+b \sinh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {\left (b c^3 \left (2 c^2 d^2-e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {1+c^2 x^2}} \, dx}{6 e \left (c^2 d^2+e^2\right )^2}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {1+c^2 x^2}}{2 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {a+b \sinh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {\left (b c^3 \left (2 c^2 d^2-e^2\right )\right ) \text {Subst}\left (\int \frac {1}{c^2 d^2+e^2-x^2} \, dx,x,\frac {e-c^2 d x}{\sqrt {1+c^2 x^2}}\right )}{6 e \left (c^2 d^2+e^2\right )^2}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {1+c^2 x^2}}{2 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {a+b \sinh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {b c^3 \left (2 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\frac {e-c^2 d x}{\sqrt {c^2 d^2+e^2} \sqrt {1+c^2 x^2}}\right )}{6 e \left (c^2 d^2+e^2\right )^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.31, size = 205, normalized size = 1.12 \begin {gather*} \frac {1}{6} \left (-\frac {2 a}{e (d+e x)^3}-\frac {b c \sqrt {1+c^2 x^2} \left (e^2+c^2 d (4 d+3 e x)\right )}{\left (c^2 d^2+e^2\right )^2 (d+e x)^2}-\frac {2 b \sinh ^{-1}(c x)}{e (d+e x)^3}-\frac {b c^3 \left (-2 c^2 d^2+e^2\right ) \log (d+e x)}{e \left (c^2 d^2+e^2\right )^{5/2}}+\frac {b c^3 \left (-2 c^2 d^2+e^2\right ) \log \left (e-c^2 d x+\sqrt {c^2 d^2+e^2} \sqrt {1+c^2 x^2}\right )}{e \left (c^2 d^2+e^2\right )^{5/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSinh[c*x])/(d + e*x)^4,x]

[Out]

((-2*a)/(e*(d + e*x)^3) - (b*c*Sqrt[1 + c^2*x^2]*(e^2 + c^2*d*(4*d + 3*e*x)))/((c^2*d^2 + e^2)^2*(d + e*x)^2)
- (2*b*ArcSinh[c*x])/(e*(d + e*x)^3) - (b*c^3*(-2*c^2*d^2 + e^2)*Log[d + e*x])/(e*(c^2*d^2 + e^2)^(5/2)) + (b*
c^3*(-2*c^2*d^2 + e^2)*Log[e - c^2*d*x + Sqrt[c^2*d^2 + e^2]*Sqrt[1 + c^2*x^2]])/(e*(c^2*d^2 + e^2)^(5/2)))/6

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(519\) vs. \(2(168)=336\).
time = 4.50, size = 520, normalized size = 2.84

method	result	size
derivativedivides	\(\frac {-\frac {a \,c^{4}}{3 \left (c e x +c d \right )^{3} e}-\frac {b \,c^{4} \arcsinh \left (c x \right )}{3 \left (c e x +c d \right )^{3} e}-\frac {b \,c^{4} \sqrt {\left (c x +\frac {c d}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{6 e^{2} \left (c^{2} d^{2}+e^{2}\right ) \left (c x +\frac {c d}{e}\right )^{2}}-\frac {b \,c^{5} d \sqrt {\left (c x +\frac {c d}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}+e^{2}\right )^{2} \left (c x +\frac {c d}{e}\right )}-\frac {b \,c^{6} d^{2} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {c d}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {c d}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}+e^{2}\right )^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}+\frac {b \,c^{4} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {c d}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {c d}{e}}\right )}{6 e^{2} \left (c^{2} d^{2}+e^{2}\right ) \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}}{c}\)	\(520\)
default	\(\frac {-\frac {a \,c^{4}}{3 \left (c e x +c d \right )^{3} e}-\frac {b \,c^{4} \arcsinh \left (c x \right )}{3 \left (c e x +c d \right )^{3} e}-\frac {b \,c^{4} \sqrt {\left (c x +\frac {c d}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{6 e^{2} \left (c^{2} d^{2}+e^{2}\right ) \left (c x +\frac {c d}{e}\right )^{2}}-\frac {b \,c^{5} d \sqrt {\left (c x +\frac {c d}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}+e^{2}\right )^{2} \left (c x +\frac {c d}{e}\right )}-\frac {b \,c^{6} d^{2} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {c d}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {c d}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}+e^{2}\right )^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}+\frac {b \,c^{4} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {c d}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {c d}{e}}\right )}{6 e^{2} \left (c^{2} d^{2}+e^{2}\right ) \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}}{c}\)	\(520\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(c*x))/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

1/c*(-1/3*a*c^4/(c*e*x+c*d)^3/e-1/3*b*c^4/(c*e*x+c*d)^3/e*arcsinh(c*x)-1/6*b*c^4/e^2/(c^2*d^2+e^2)/(c*x+c*d/e)
^2*((c*x+c*d/e)^2-2*d*c/e*(c*x+c*d/e)+(c^2*d^2+e^2)/e^2)^(1/2)-1/2*b*c^5/e*d/(c^2*d^2+e^2)^2/(c*x+c*d/e)*((c*x
+c*d/e)^2-2*d*c/e*(c*x+c*d/e)+(c^2*d^2+e^2)/e^2)^(1/2)-1/2*b*c^6/e^2*d^2/(c^2*d^2+e^2)^2/((c^2*d^2+e^2)/e^2)^(
1/2)*ln((2*(c^2*d^2+e^2)/e^2-2*d*c/e*(c*x+c*d/e)+2*((c^2*d^2+e^2)/e^2)^(1/2)*((c*x+c*d/e)^2-2*d*c/e*(c*x+c*d/e
)+(c^2*d^2+e^2)/e^2)^(1/2))/(c*x+c*d/e))+1/6*b*c^4/e^2/(c^2*d^2+e^2)/((c^2*d^2+e^2)/e^2)^(1/2)*ln((2*(c^2*d^2+
e^2)/e^2-2*d*c/e*(c*x+c*d/e)+2*((c^2*d^2+e^2)/e^2)^(1/2)*((c*x+c*d/e)^2-2*d*c/e*(c*x+c*d/e)+(c^2*d^2+e^2)/e^2)
^(1/2))/(c*x+c*d/e)))

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Maxima [F]
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((a+b*arcsinh(c*x))/(e*x+d)^4,x, algorithm="maxima")

[Out]

1/6*(6*c*integrate(1/3/(c^3*x^6*e^4 + 3*c^3*d*x^5*e^3 + 3*c*d^2*x^2*e^2 + c*d^3*x*e + (3*c^3*d^2*e^2 + c*e^4)*
x^4 + (c^3*d^3*e + 3*c*d*e^3)*x^3 + (c^2*x^5*e^4 + 3*c^2*d*x^4*e^3 + (3*c^2*d^2*e^2 + e^4)*x^3 + 3*d^2*x*e^2 +
d^3*e + (c^2*d^3*e + 3*d*e^3)*x^2)*sqrt(c^2*x^2 + 1)), x) - 2*(c^6*d^3 - 3*c^4*d*e^2)*log(x*e + d)/(c^6*d^6*e
+ 3*c^4*d^4*e^3 + 3*c^2*d^2*e^5 + e^7) + (3*c^6*d^6 + 2*c^4*d^4*e^2 - c^2*d^2*e^4 + 2*(c^6*d^4*e^2 - c^2*e^6)
*x^2 + (5*c^6*d^5*e + 2*c^4*d^3*e^3 - 3*c^2*d*e^5)*x + (c^6*d^6 - 3*c^4*d^4*e^2 + (c^6*d^3*e^3 - 3*c^4*d*e^5)*
x^3 + 3*(c^6*d^4*e^2 - 3*c^4*d^2*e^4)*x^2 + 3*(c^6*d^5*e - 3*c^4*d^3*e^3)*x)*log(c^2*x^2 + 1) - 2*(c^6*d^6 + 3
*c^4*d^4*e^2 + 3*c^2*d^2*e^4 + e^6)*log(c*x + sqrt(c^2*x^2 + 1)))/(c^6*d^9*e + 3*c^4*d^7*e^3 + 3*c^2*d^5*e^5 +
(c^6*d^6*e^4 + 3*c^4*d^4*e^6 + 3*c^2*d^2*e^8 + e^10)*x^3 + d^3*e^7 + 3*(c^6*d^7*e^3 + 3*c^4*d^5*e^5 + 3*c^2*d
^3*e^7 + d*e^9)*x^2 + 3*(c^6*d^8*e^2 + 3*c^4*d^6*e^4 + 3*c^2*d^4*e^6 + d^2*e^8)*x) - I*(3*c^6*d^2 - c^4*e^2)*(
log(I*c*x + 1) - log(-I*c*x + 1))/((c^6*d^6 + 3*c^4*d^4*e^2 + 3*c^2*d^2*e^4 + e^6)*c))*b - 1/3*a/(x^3*e^4 + 3*
d*x^2*e^3 + 3*d^2*x*e^2 + d^3*e)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4886 vs. \(2 (164) = 328\).
time = 0.79, size = 4886, normalized size = 26.70 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))/(e*x+d)^4,x, algorithm="fricas")

[Out]

-1/6*(9*b*c^6*d^8*x*cosh(1) + 3*b*c^4*d^4*x^3*cosh(1)^5 + (2*a + 3*b)*c^6*d^9 + 2*a*d^3*cosh(1)^6 + 2*a*d^3*si
nh(1)^6 + 3*(b*c^4*d^4*x^3 + 4*a*d^3*cosh(1))*sinh(1)^5 + 3*(3*b*c^4*d^5*x^2 + 2*a*c^2*d^5)*cosh(1)^4 + 3*(5*b
*c^4*d^4*x^3*cosh(1) + 3*b*c^4*d^5*x^2 + 2*a*c^2*d^5 + 10*a*d^3*cosh(1)^2)*sinh(1)^4 + 3*(b*c^6*d^6*x^3 + 3*b*
c^4*d^6*x)*cosh(1)^3 + (3*b*c^6*d^6*x^3 + 30*b*c^4*d^4*x^3*cosh(1)^2 + 9*b*c^4*d^6*x + 40*a*d^3*cosh(1)^3 + 12
*(3*b*c^4*d^5*x^2 + 2*a*c^2*d^5)*cosh(1))*sinh(1)^3 + 3*(3*b*c^6*d^7*x^2 + (2*a + b)*c^4*d^7)*cosh(1)^2 + 3*(3
*b*c^6*d^7*x^2 + 10*b*c^4*d^4*x^3*cosh(1)^3 + (2*a + b)*c^4*d^7 + 10*a*d^3*cosh(1)^4 + 6*(3*b*c^4*d^5*x^2 + 2*
a*c^2*d^5)*cosh(1)^2 + 3*(b*c^6*d^6*x^3 + 3*b*c^4*d^6*x)*cosh(1))*sinh(1)^2 + (6*b*c^5*d^7*x*cosh(1) - b*c^3*d
^3*x^3*cosh(1)^5 - b*c^3*d^3*x^3*sinh(1)^5 + 2*b*c^5*d^8 - 3*b*c^3*d^4*x^2*cosh(1)^4 - (5*b*c^3*d^3*x^3*cosh(1
) + 3*b*c^3*d^4*x^2)*sinh(1)^4 + (2*b*c^5*d^5*x^3 - 3*b*c^3*d^5*x)*cosh(1)^3 + (2*b*c^5*d^5*x^3 - 10*b*c^3*d^3
*x^3*cosh(1)^2 - 12*b*c^3*d^4*x^2*cosh(1) - 3*b*c^3*d^5*x)*sinh(1)^3 + (6*b*c^5*d^6*x^2 - b*c^3*d^6)*cosh(1)^2
+ (6*b*c^5*d^6*x^2 - 10*b*c^3*d^3*x^3*cosh(1)^3 - 18*b*c^3*d^4*x^2*cosh(1)^2 - b*c^3*d^6 + 3*(2*b*c^5*d^5*x^3
- 3*b*c^3*d^5*x)*cosh(1))*sinh(1)^2 + (6*b*c^5*d^7*x - 5*b*c^3*d^3*x^3*cosh(1)^4 - 12*b*c^3*d^4*x^2*cosh(1)^3
+ 3*(2*b*c^5*d^5*x^3 - 3*b*c^3*d^5*x)*cosh(1)^2 + 2*(6*b*c^5*d^6*x^2 - b*c^3*d^6)*cosh(1))*sinh(1))*sqrt(((c^
2*d^2 + 1)*cosh(1) - (c^2*d^2 - 1)*sinh(1))/(cosh(1) - sinh(1)))*log(-(c^3*d^2*x - c*d*cosh(1) - c*d*sinh(1) +
(c^2*d^2 - c*d*sqrt(((c^2*d^2 + 1)*cosh(1) - (c^2*d^2 - 1)*sinh(1))/(cosh(1) - sinh(1))) + cosh(1)^2 + 2*cosh
(1)*sinh(1) + sinh(1)^2)*sqrt(c^2*x^2 + 1) - (c^2*d*x - cosh(1) - sinh(1))*sqrt(((c^2*d^2 + 1)*cosh(1) - (c^2*
d^2 - 1)*sinh(1))/(cosh(1) - sinh(1))))/(x*cosh(1) + x*sinh(1) + d)) - 2*(3*b*c^6*d^7*x^2*cosh(1)^2 + 3*b*c^6*
d^8*x*cosh(1) + 9*b*c^4*d^5*x^2*cosh(1)^4 + 9*b*c^2*d^3*x^2*cosh(1)^6 + b*x^3*cosh(1)^9 + b*x^3*sinh(1)^9 + 3*
b*d*x^2*cosh(1)^8 + 3*(3*b*x^3*cosh(1) + b*d*x^2)*sinh(1)^8 + 3*(b*c^2*d^2*x^3 + b*d^2*x)*cosh(1)^7 + 3*(b*c^2
*d^2*x^3 + 12*b*x^3*cosh(1)^2 + 8*b*d*x^2*cosh(1) + b*d^2*x)*sinh(1)^7 + 3*(3*b*c^2*d^3*x^2 + 28*b*x^3*cosh(1)
^3 + 28*b*d*x^2*cosh(1)^2 + 7*(b*c^2*d^2*x^3 + b*d^2*x)*cosh(1))*sinh(1)^6 + 3*(b*c^4*d^4*x^3 + 3*b*c^2*d^4*x)
*cosh(1)^5 + 3*(b*c^4*d^4*x^3 + 18*b*c^2*d^3*x^2*cosh(1) + 3*b*c^2*d^4*x + 42*b*x^3*cosh(1)^4 + 56*b*d*x^2*cos
h(1)^3 + 21*(b*c^2*d^2*x^3 + b*d^2*x)*cosh(1)^2)*sinh(1)^5 + 3*(3*b*c^4*d^5*x^2 + 45*b*c^2*d^3*x^2*cosh(1)^2 +
42*b*x^3*cosh(1)^5 + 70*b*d*x^2*cosh(1)^4 + 35*(b*c^2*d^2*x^3 + b*d^2*x)*cosh(1)^3 + 5*(b*c^4*d^4*x^3 + 3*b*c
^2*d^4*x)*cosh(1))*sinh(1)^4 + (b*c^6*d^6*x^3 + 9*b*c^4*d^6*x)*cosh(1)^3 + (b*c^6*d^6*x^3 + 36*b*c^4*d^5*x^2*c
osh(1) + 9*b*c^4*d^6*x + 180*b*c^2*d^3*x^2*cosh(1)^3 + 84*b*x^3*cosh(1)^6 + 168*b*d*x^2*cosh(1)^5 + 105*(b*c^2
*d^2*x^3 + b*d^2*x)*cosh(1)^4 + 30*(b*c^4*d^4*x^3 + 3*b*c^2*d^4*x)*cosh(1)^2)*sinh(1)^3 + 3*(b*c^6*d^7*x^2 + 1
8*b*c^4*d^5*x^2*cosh(1)^2 + 45*b*c^2*d^3*x^2*cosh(1)^4 + 12*b*x^3*cosh(1)^7 + 28*b*d*x^2*cosh(1)^6 + 21*(b*c^2
*d^2*x^3 + b*d^2*x)*cosh(1)^5 + 10*(b*c^4*d^4*x^3 + 3*b*c^2*d^4*x)*cosh(1)^3 + (b*c^6*d^6*x^3 + 9*b*c^4*d^6*x)
*cosh(1))*sinh(1)^2 + 3*(2*b*c^6*d^7*x^2*cosh(1) + b*c^6*d^8*x + 12*b*c^4*d^5*x^2*cosh(1)^3 + 18*b*c^2*d^3*x^2
*cosh(1)^5 + 3*b*x^3*cosh(1)^8 + 8*b*d*x^2*cosh(1)^7 + 7*(b*c^2*d^2*x^3 + b*d^2*x)*cosh(1)^6 + 5*(b*c^4*d^4*x^
3 + 3*b*c^2*d^4*x)*cosh(1)^4 + (b*c^6*d^6*x^3 + 9*b*c^4*d^6*x)*cosh(1)^2)*sinh(1))*log(c*x + sqrt(c^2*x^2 + 1)
) - 2*(3*b*c^6*d^8*x*cosh(1) + b*c^6*d^9 + b*x^3*cosh(1)^9 + b*x^3*sinh(1)^9 + 3*b*d*x^2*cosh(1)^8 + 3*(3*b*x^
3*cosh(1) + b*d*x^2)*sinh(1)^8 + 3*(b*c^2*d^2*x^3 + b*d^2*x)*cosh(1)^7 + 3*(b*c^2*d^2*x^3 + 12*b*x^3*cosh(1)^2
+ 8*b*d*x^2*cosh(1) + b*d^2*x)*sinh(1)^7 + (9*b*c^2*d^3*x^2 + b*d^3)*cosh(1)^6 + (9*b*c^2*d^3*x^2 + 84*b*x^3*
cosh(1)^3 + 84*b*d*x^2*cosh(1)^2 + b*d^3 + 21*(b*c^2*d^2*x^3 + b*d^2*x)*cosh(1))*sinh(1)^6 + 3*(b*c^4*d^4*x^3
+ 3*b*c^2*d^4*x)*cosh(1)^5 + 3*(b*c^4*d^4*x^3 + 3*b*c^2*d^4*x + 42*b*x^3*cosh(1)^4 + 56*b*d*x^2*cosh(1)^3 + 21
*(b*c^2*d^2*x^3 + b*d^2*x)*cosh(1)^2 + 2*(9*b*c^2*d^3*x^2 + b*d^3)*cosh(1))*sinh(1)^5 + 3*(3*b*c^4*d^5*x^2 + b
*c^2*d^5)*cosh(1)^4 + 3*(3*b*c^4*d^5*x^2 + 42*b*x^3*cosh(1)^5 + b*c^2*d^5 + 70*b*d*x^2*cosh(1)^4 + 35*(b*c^2*d
^2*x^3 + b*d^2*x)*cosh(1)^3 + 5*(9*b*c^2*d^3*x^2 + b*d^3)*cosh(1)^2 + 5*(b*c^4*d^4*x^3 + 3*b*c^2*d^4*x)*cosh(1
))*sinh(1)^4 + (b*c^6*d^6*x^3 + 9*b*c^4*d^6*x)*cosh(1)^3 + (b*c^6*d^6*x^3 + 9*b*c^4*d^6*x + 84*b*x^3*cosh(1)^6
+ 168*b*d*x^2*cosh(1)^5 + 105*(b*c^2*d^2*x^3 + b*d^2*x)*cosh(1)^4 + 20*(9*b*c^2*d^3*x^2 + b*d^3)*cosh(1)^3 +
30*(b*c^4*d^4*x^3 + 3*b*c^2*d^4*x)*cosh(1)^2 + 12*(3*b*c^4*d^5*x^2 + b*c^2*d^5)*cosh(1))*sinh(1)^3 + 3*(b*c^6*
d^7*x^2 + b*c^4*d^7)*cosh(1)^2 + 3*(b*c^6*d^7*x^2 + b*c^4*d^7 + 12*b*x^3*cosh(1)^7 + 28*b*d*x^2*cosh(1)^6 + 21
*(b*c^2*d^2*x^3 + b*d^2*x)*cosh(1)^5 + 5*(9*b*c^2*d^3*x^2 + b*d^3)*cosh(1)^4 + 10*(b*c^4*d^4*x^3 + 3*b*c^2*d^4
*x)*cosh(1)^3 + 6*(3*b*c^4*d^5*x^2 + b*c^2*d^5)...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\left (d + e x\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asinh(c*x))/(e*x+d)**4,x)

[Out]

Integral((a + b*asinh(c*x))/(d + e*x)**4, x)

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Giac [F]
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((a+b*arcsinh(c*x))/(e*x+d)^4,x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)/(e*x + d)^4, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (d+e\,x\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(c*x))/(d + e*x)^4,x)

[Out]

int((a + b*asinh(c*x))/(d + e*x)^4, x)

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