3.1.9 \(\int \frac {a+b \sinh ^{-1}(c x)}{(d+e x)^2} \, dx\) [9]

Optimal. Leaf size=82 \[ -\frac {a+b \sinh ^{-1}(c x)}{e (d+e x)}-\frac {b c \tanh ^{-1}\left (\frac {e-c^2 d x}{\sqrt {c^2 d^2+e^2} \sqrt {1+c^2 x^2}}\right )}{e \sqrt {c^2 d^2+e^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5828, 739, 212} \begin {gather*} -\frac {a+b \sinh ^{-1}(c x)}{e (d+e x)}-\frac {b c \tanh ^{-1}\left (\frac {e-c^2 d x}{\sqrt {c^2 x^2+1} \sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSinh[c*x])/(d + e*x)^2,x]

[Out]

-((a + b*ArcSinh[c*x])/(e*(d + e*x))) - (b*c*ArcTanh[(e - c^2*d*x)/(Sqrt[c^2*d^2 + e^2]*Sqrt[1 + c^2*x^2])])/(
e*Sqrt[c^2*d^2 + e^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 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)^2} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{e (d+e x)}+\frac {(b c) \int \frac {1}{(d+e x) \sqrt {1+c^2 x^2}} \, dx}{e}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{e (d+e x)}-\frac {(b c) \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 )}{e}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{e (d+e x)}-\frac {b c \tanh ^{-1}\left (\frac {e-c^2 d x}{\sqrt {c^2 d^2+e^2} \sqrt {1+c^2 x^2}}\right )}{e \sqrt {c^2 d^2+e^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.07, size = 79, normalized size = 0.96 \begin {gather*} -\frac {\frac {a+b \sinh ^{-1}(c x)}{d+e x}+\frac {b c \tanh ^{-1}\left (\frac {e-c^2 d x}{\sqrt {c^2 d^2+e^2} \sqrt {1+c^2 x^2}}\right )}{\sqrt {c^2 d^2+e^2}}}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSinh[c*x])/(d + e*x)^2,x]

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(187\) vs. \(2(78)=156\).
time = 4.02, size = 188, normalized size = 2.29

method	result	size
derivativedivides	\(\frac {-\frac {a \,c^{2}}{\left (c e x +c d \right ) e}-\frac {b \,c^{2} \arcsinh \left (c x \right )}{\left (c e x +c d \right ) e}-\frac {b \,c^{2} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {c d}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {c d}{e}}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}}{c}\)	\(188\)
default	\(\frac {-\frac {a \,c^{2}}{\left (c e x +c d \right ) e}-\frac {b \,c^{2} \arcsinh \left (c x \right )}{\left (c e x +c d \right ) e}-\frac {b \,c^{2} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {c d}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {c d}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {c d}{e}}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}}{c}\)	\(188\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(c*x))/(e*x+d)^2,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 91, normalized size = 1.11 \begin {gather*} {\left (\frac {c \operatorname {arsinh}\left (\frac {c d x e}{{\left | x e^{2} + d e \right |}} - \frac {e^{2}}{c {\left | x e^{2} + d e \right |}}\right ) e^{\left (-2\right )}}{\sqrt {c^{2} d^{2} e^{\left (-2\right )} + 1}} - \frac {\operatorname {arsinh}\left (c x\right )}{x e^{2} + d e}\right )} b - \frac {a}{x e^{2} + d e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))/(e*x+d)^2,x, algorithm="maxima")

[Out]

(c*arcsinh(c*d*x*e/abs(x*e^2 + d*e) - e^2/(c*abs(x*e^2 + d*e)))*e^(-2)/sqrt(c^2*d^2*e^(-2) + 1) - arcsinh(c*x)
/(x*e^2 + d*e))*b - a/(x*e^2 + d*e)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 565 vs. \(2 (76) = 152\).
time = 0.38, size = 565, normalized size = 6.89 \begin {gather*} -\frac {a c^{2} d^{3} + a d \cosh \left (1\right )^{2} + 2 \, a d \cosh \left (1\right ) \sinh \left (1\right ) + a d \sinh \left (1\right )^{2} - {\left (b c d x \cosh \left (1\right ) + b c d x \sinh \left (1\right ) + b c d^{2}\right )} \sqrt {\frac {{\left (c^{2} d^{2} + 1\right )} \cosh \left (1\right ) - {\left (c^{2} d^{2} - 1\right )} \sinh \left (1\right )}{\cosh \left (1\right ) - \sinh \left (1\right )}} \log \left (-\frac {c^{3} d^{2} x - c d \cosh \left (1\right ) - c d \sinh \left (1\right ) + {\left (c^{2} d^{2} + c d \sqrt {\frac {{\left (c^{2} d^{2} + 1\right )} \cosh \left (1\right ) - {\left (c^{2} d^{2} - 1\right )} \sinh \left (1\right )}{\cosh \left (1\right ) - \sinh \left (1\right )}} + \cosh \left (1\right )^{2} + 2 \, \cosh \left (1\right ) \sinh \left (1\right ) + \sinh \left (1\right )^{2}\right )} \sqrt {c^{2} x^{2} + 1} + {\left (c^{2} d x - \cosh \left (1\right ) - \sinh \left (1\right )\right )} \sqrt {\frac {{\left (c^{2} d^{2} + 1\right )} \cosh \left (1\right ) - {\left (c^{2} d^{2} - 1\right )} \sinh \left (1\right )}{\cosh \left (1\right ) - \sinh \left (1\right )}}}{x \cosh \left (1\right ) + x \sinh \left (1\right ) + d}\right ) - {\left (b c^{2} d^{2} x \cosh \left (1\right ) + b x \cosh \left (1\right )^{3} + 3 \, b x \cosh \left (1\right ) \sinh \left (1\right )^{2} + b x \sinh \left (1\right )^{3} + {\left (b c^{2} d^{2} x + 3 \, b x \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (b c^{2} d^{2} x \cosh \left (1\right ) + b c^{2} d^{3} + b x \cosh \left (1\right )^{3} + b x \sinh \left (1\right )^{3} + b d \cosh \left (1\right )^{2} + {\left (3 \, b x \cosh \left (1\right ) + b d\right )} \sinh \left (1\right )^{2} + {\left (b c^{2} d^{2} x + 3 \, b x \cosh \left (1\right )^{2} + 2 \, b d \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right )}{c^{2} d^{3} x \cosh \left (1\right )^{2} + c^{2} d^{4} \cosh \left (1\right ) + d x \cosh \left (1\right )^{4} + d x \sinh \left (1\right )^{4} + d^{2} \cosh \left (1\right )^{3} + {\left (4 \, d x \cosh \left (1\right ) + d^{2}\right )} \sinh \left (1\right )^{3} + {\left (c^{2} d^{3} x + 6 \, d x \cosh \left (1\right )^{2} + 3 \, d^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + {\left (2 \, c^{2} d^{3} x \cosh \left (1\right ) + c^{2} d^{4} + 4 \, d x \cosh \left (1\right )^{3} + 3 \, d^{2} \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))/(e*x+d)^2,x, algorithm="fricas")

[Out]

-(a*c^2*d^3 + a*d*cosh(1)^2 + 2*a*d*cosh(1)*sinh(1) + a*d*sinh(1)^2 - (b*c*d*x*cosh(1) + b*c*d*x*sinh(1) + b*c
*d^2)*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))) + cos
h(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)) - (b*c^2*d^2*x*cosh(1) + b
*x*cosh(1)^3 + 3*b*x*cosh(1)*sinh(1)^2 + b*x*sinh(1)^3 + (b*c^2*d^2*x + 3*b*x*cosh(1)^2)*sinh(1))*log(c*x + sq
rt(c^2*x^2 + 1)) - (b*c^2*d^2*x*cosh(1) + b*c^2*d^3 + b*x*cosh(1)^3 + b*x*sinh(1)^3 + b*d*cosh(1)^2 + (3*b*x*c
osh(1) + b*d)*sinh(1)^2 + (b*c^2*d^2*x + 3*b*x*cosh(1)^2 + 2*b*d*cosh(1))*sinh(1))*log(-c*x + sqrt(c^2*x^2 + 1
)))/(c^2*d^3*x*cosh(1)^2 + c^2*d^4*cosh(1) + d*x*cosh(1)^4 + d*x*sinh(1)^4 + d^2*cosh(1)^3 + (4*d*x*cosh(1) +
d^2)*sinh(1)^3 + (c^2*d^3*x + 6*d*x*cosh(1)^2 + 3*d^2*cosh(1))*sinh(1)^2 + (2*c^2*d^3*x*cosh(1) + c^2*d^4 + 4*
d*x*cosh(1)^3 + 3*d^2*cosh(1)^2)*sinh(1))

________________________________________________________________________________________

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 )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asinh(c*x))/(e*x+d)**2,x)

[Out]

Integral((a + b*asinh(c*x))/(d + e*x)**2, x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (78) = 156\).
time = 0.58, size = 232, normalized size = 2.83 \begin {gather*} {\left (\frac {c \log \left (-c^{2} d e + \sqrt {c^{2} d^{2} + e^{2}} {\left | c \right |} {\left | e \right |}\right ) \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{\sqrt {c^{2} d^{2} + e^{2}} {\left | e \right |}} - \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{{\left (e x + d\right )} e} - \frac {c \log \left (-c^{2} d e + \sqrt {c^{2} d^{2} + e^{2}} {\left (\sqrt {c^{2} - \frac {2 \, c^{2} d}{e x + d} + \frac {c^{2} d^{2}}{{\left (e x + d\right )}^{2}} + \frac {e^{2}}{{\left (e x + d\right )}^{2}}} + \frac {\sqrt {c^{2} d^{2} e^{2} + e^{4}}}{{\left (e x + d\right )} e}\right )} {\left | e \right |}\right )}{\sqrt {c^{2} d^{2} + e^{2}} {\left | e \right |} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}\right )} b - \frac {a}{{\left (e x + d\right )} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))/(e*x+d)^2,x, algorithm="giac")

[Out]

(c*log(-c^2*d*e + sqrt(c^2*d^2 + e^2)*abs(c)*abs(e))*sgn(1/(e*x + d))*sgn(e)/(sqrt(c^2*d^2 + e^2)*abs(e)) - lo
g(c*x + sqrt(c^2*x^2 + 1))/((e*x + d)*e) - c*log(-c^2*d*e + sqrt(c^2*d^2 + e^2)*(sqrt(c^2 - 2*c^2*d/(e*x + d)
+ c^2*d^2/(e*x + d)^2 + e^2/(e*x + d)^2) + sqrt(c^2*d^2*e^2 + e^4)/((e*x + d)*e))*abs(e))/(sqrt(c^2*d^2 + e^2)
*abs(e)*sgn(1/(e*x + d))*sgn(e)))*b - a/((e*x + d)*e)

________________________________________________________________________________________

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 )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(c*x))/(d + e*x)^2,x)

[Out]

int((a + b*asinh(c*x))/(d + e*x)^2, x)

________________________________________________________________________________________