3.1.80 \(\int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx\) [80]

Optimal. Leaf size=306 \[ \frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{2 d \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a+b \text {sech}^{-1}(c x)}{2 e (d+e x)^2}+\frac {b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 d^2 \sqrt {c^2 d^2-e^2}}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{2 d^2 e} \]

[Out]

1/2*(-a-b*arcsech(c*x))/e/(e*x+d)^2+1/2*b*c^2*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2))*(1/(c
*x+1))^(1/2)*(c*x+1)^(1/2)/(c^2*d^2-e^2)^(3/2)+1/2*b*arctanh((-c^2*x^2+1)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/
2)/d^2/e+1/2*b*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/d^2/
(c^2*d^2-e^2)^(1/2)+1/2*b*e*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/d/(c^2*d^2-e^2)/(e*x+d)

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Rubi [A]
time = 0.14, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6423, 975, 272, 65, 214, 745, 739, 210} \begin {gather*} -\frac {a+b \text {sech}^{-1}(c x)}{2 e (d+e x)^2}+\frac {b c^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {ArcTan}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {ArcTan}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{2 d^2 \sqrt {c^2 d^2-e^2}}+\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{2 d \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{2 d^2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(2*d*(c^2*d^2 - e^2)*(d + e*x)) - (a + b*ArcSech[c*
x])/(2*e*(d + e*x)^2) + (b*c^2*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sq
rt[1 - c^2*x^2])])/(2*(c^2*d^2 - e^2)^(3/2)) + (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTan[(e + c^2*d*x)/(Sqr
t[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(2*d^2*Sqrt[c^2*d^2 - e^2]) + (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcT
anh[Sqrt[1 - c^2*x^2]])/(2*d^2*e)

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 210

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

Rule 214

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 745

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*(d/(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0]

Rule 975

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0])) &&  !(IGtQ[m, 0] || IGtQ[n, 0])

Rule 6423

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

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.44, size = 342, normalized size = 1.12 \begin {gather*} \frac {1}{2} \left (-\frac {a}{e (d+e x)^2}+\frac {b \sqrt {\frac {1-c x}{1+c x}} (e+c e x)}{d (c d-e) (c d+e) (d+e x)}-\frac {b \text {sech}^{-1}(c x)}{e (d+e x)^2}-\frac {b \log (x)}{d^2 e}+\frac {b \log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )}{d^2 e}-\frac {i b \left (2 c^2 d^2-e^2\right ) \log \left (\frac {4 d^2 e \sqrt {c^2 d^2-e^2} \left (i e+i c^2 d x+\sqrt {c^2 d^2-e^2} \sqrt {\frac {1-c x}{1+c x}}+c \sqrt {c^2 d^2-e^2} x \sqrt {\frac {1-c x}{1+c x}}\right )}{b \left (2 c^2 d^2-e^2\right ) (d+e x)}\right )}{d^2 (c d-e) (c d+e) \sqrt {c^2 d^2-e^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a/(e*(d + e*x)^2)) + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(e + c*e*x))/(d*(c*d - e)*(c*d + e)*(d + e*x)) - (b*ArcSe
ch[c*x])/(e*(d + e*x)^2) - (b*Log[x])/(d^2*e) + (b*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 +
c*x)]])/(d^2*e) - (I*b*(2*c^2*d^2 - e^2)*Log[(4*d^2*e*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e
^2]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d^2 - e^2]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/(b*(2*c^2*d^2 - e^2)*(d +
e*x))])/(d^2*(c*d - e)*(c*d + e)*Sqrt[c^2*d^2 - e^2]))/2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1093\) vs. \(2(267)=534\).
time = 1.47, size = 1094, normalized size = 3.58 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsech(c*x))/(e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2141 vs. \(2 (198) = 396\).
time = 0.72, size = 4375, normalized size = 14.30 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(a*c^4*d^6 - 2*b*c^2*d^3*x*cosh(1)^3 - (2*a + b)*c^2*d^4*cosh(1)^2 + b*x^2*cosh(1)^6 + b*x^2*sinh(1)^6 +
2*b*d*x*cosh(1)^5 + 2*(3*b*x^2*cosh(1) + b*d*x)*sinh(1)^5 - (b*c^2*d^2*x^2 - (a + b)*d^2)*cosh(1)^4 - (b*c^2*
d^2*x^2 - 15*b*x^2*cosh(1)^2 - 10*b*d*x*cosh(1) - (a + b)*d^2)*sinh(1)^4 - 2*(b*c^2*d^3*x - 10*b*x^2*cosh(1)^3
- 10*b*d*x*cosh(1)^2 + 2*(b*c^2*d^2*x^2 - (a + b)*d^2)*cosh(1))*sinh(1)^3 - (6*b*c^2*d^3*x*cosh(1) + (2*a + b
)*c^2*d^4 - 15*b*x^2*cosh(1)^4 - 20*b*d*x*cosh(1)^3 + 6*(b*c^2*d^2*x^2 - (a + b)*d^2)*cosh(1)^2)*sinh(1)^2 + (
4*b*c^2*d^3*x*cosh(1)^2 + 2*b*c^2*d^4*cosh(1) - b*x^2*cosh(1)^5 - b*x^2*sinh(1)^5 - 2*b*d*x*cosh(1)^4 - (5*b*x
^2*cosh(1) + 2*b*d*x)*sinh(1)^4 + (2*b*c^2*d^2*x^2 - b*d^2)*cosh(1)^3 + (2*b*c^2*d^2*x^2 - 10*b*x^2*cosh(1)^2
- 8*b*d*x*cosh(1) - b*d^2)*sinh(1)^3 + (4*b*c^2*d^3*x - 10*b*x^2*cosh(1)^3 - 12*b*d*x*cosh(1)^2 + 3*(2*b*c^2*d
^2*x^2 - b*d^2)*cosh(1))*sinh(1)^2 + (8*b*c^2*d^3*x*cosh(1) + 2*b*c^2*d^4 - 5*b*x^2*cosh(1)^4 - 8*b*d*x*cosh(1
)^3 + 3*(2*b*c^2*d^2*x^2 - b*d^2)*cosh(1)^2)*sinh(1))*sqrt(-((c^2*d^2 - 1)*cosh(1) - (c^2*d^2 + 1)*sinh(1))/(c
osh(1) - sinh(1)))*log((c^2*d*x*cosh(1) + cosh(1)^2 + (c^2*d*x + 2*cosh(1))*sinh(1) + sinh(1)^2 - (c^2*d*x + c
osh(1) + sinh(1))*sqrt(-((c^2*d^2 - 1)*cosh(1) - (c^2*d^2 + 1)*sinh(1))/(cosh(1) - sinh(1))) - (c^3*d^2*x - c*
x*cosh(1)^2 - 2*c*x*cosh(1)*sinh(1) - c*x*sinh(1)^2 + (c*x*cosh(1) + c*x*sinh(1))*sqrt(-((c^2*d^2 - 1)*cosh(1)
- (c^2*d^2 + 1)*sinh(1))/(cosh(1) - sinh(1))))*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/(x*cosh(1) + x*sinh(1) + d)) +
(2*b*c^4*d^5*x*cosh(1) + b*c^4*d^6 - 4*b*c^2*d^3*x*cosh(1)^3 + b*x^2*cosh(1)^6 + b*x^2*sinh(1)^6 + 2*b*d*x*co
sh(1)^5 + 2*(3*b*x^2*cosh(1) + b*d*x)*sinh(1)^5 - (2*b*c^2*d^2*x^2 - b*d^2)*cosh(1)^4 - (2*b*c^2*d^2*x^2 - 15*
b*x^2*cosh(1)^2 - 10*b*d*x*cosh(1) - b*d^2)*sinh(1)^4 - 4*(b*c^2*d^3*x - 5*b*x^2*cosh(1)^3 - 5*b*d*x*cosh(1)^2
+ (2*b*c^2*d^2*x^2 - b*d^2)*cosh(1))*sinh(1)^3 + (b*c^4*d^4*x^2 - 2*b*c^2*d^4)*cosh(1)^2 + (b*c^4*d^4*x^2 - 1
2*b*c^2*d^3*x*cosh(1) - 2*b*c^2*d^4 + 15*b*x^2*cosh(1)^4 + 20*b*d*x*cosh(1)^3 - 6*(2*b*c^2*d^2*x^2 - b*d^2)*co
sh(1)^2)*sinh(1)^2 + 2*(b*c^4*d^5*x - 6*b*c^2*d^3*x*cosh(1)^2 + 3*b*x^2*cosh(1)^5 + 5*b*d*x*cosh(1)^4 - 2*(2*b
*c^2*d^2*x^2 - b*d^2)*cosh(1)^3 + (b*c^4*d^4*x^2 - 2*b*c^2*d^4)*cosh(1))*sinh(1))*log((c*x*sqrt(-(c^2*x^2 - 1)
/(c^2*x^2)) - 1)/x) + (b*c^4*d^6 - 2*b*c^2*d^4*cosh(1)^2 + b*d^2*cosh(1)^4 + 4*b*d^2*cosh(1)*sinh(1)^3 + b*d^2
*sinh(1)^4 - 2*(b*c^2*d^4 - 3*b*d^2*cosh(1)^2)*sinh(1)^2 - 4*(b*c^2*d^4*cosh(1) - b*d^2*cosh(1)^3)*sinh(1))*lo
g((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) - 2*(3*b*c^2*d^3*x*cosh(1)^2 + (2*a + b)*c^2*d^4*cosh(1) - 3
*b*x^2*cosh(1)^5 - 5*b*d*x*cosh(1)^4 + 2*(b*c^2*d^2*x^2 - (a + b)*d^2)*cosh(1)^3)*sinh(1) - (b*c^3*d^3*x^2*cos
h(1)^3 + b*c^3*d^4*x*cosh(1)^2 - b*c*d*x^2*cosh(1)^5 - b*c*d*x^2*sinh(1)^5 - b*c*d^2*x*cosh(1)^4 - (5*b*c*d*x^
2*cosh(1) + b*c*d^2*x)*sinh(1)^4 + (b*c^3*d^3*x^2 - 10*b*c*d*x^2*cosh(1)^2 - 4*b*c*d^2*x*cosh(1))*sinh(1)^3 +
(3*b*c^3*d^3*x^2*cosh(1) + b*c^3*d^4*x - 10*b*c*d*x^2*cosh(1)^3 - 6*b*c*d^2*x*cosh(1)^2)*sinh(1)^2 + (3*b*c^3*
d^3*x^2*cosh(1)^2 + 2*b*c^3*d^4*x*cosh(1) - 5*b*c*d*x^2*cosh(1)^4 - 4*b*c*d^2*x*cosh(1)^3)*sinh(1))*sqrt(-(c^2
*x^2 - 1)/(c^2*x^2)))/(2*c^4*d^7*x*cosh(1)^2 + c^4*d^8*cosh(1) - 4*c^2*d^5*x*cosh(1)^4 + d^2*x^2*cosh(1)^7 + d
^2*x^2*sinh(1)^7 + 2*d^3*x*cosh(1)^6 + (7*d^2*x^2*cosh(1) + 2*d^3*x)*sinh(1)^6 - (2*c^2*d^4*x^2 - d^4)*cosh(1)
^5 - (2*c^2*d^4*x^2 - 21*d^2*x^2*cosh(1)^2 - 12*d^3*x*cosh(1) - d^4)*sinh(1)^5 - (4*c^2*d^5*x - 35*d^2*x^2*cos
h(1)^3 - 30*d^3*x*cosh(1)^2 + 5*(2*c^2*d^4*x^2 - d^4)*cosh(1))*sinh(1)^4 + (c^4*d^6*x^2 - 2*c^2*d^6)*cosh(1)^3
+ (c^4*d^6*x^2 - 16*c^2*d^5*x*cosh(1) - 2*c^2*d^6 + 35*d^2*x^2*cosh(1)^4 + 40*d^3*x*cosh(1)^3 - 10*(2*c^2*d^4
*x^2 - d^4)*cosh(1)^2)*sinh(1)^3 + (2*c^4*d^7*x - 24*c^2*d^5*x*cosh(1)^2 + 21*d^2*x^2*cosh(1)^5 + 30*d^3*x*cos
h(1)^4 - 10*(2*c^2*d^4*x^2 - d^4)*cosh(1)^3 + 3*(c^4*d^6*x^2 - 2*c^2*d^6)*cosh(1))*sinh(1)^2 + (4*c^4*d^7*x*co
sh(1) + c^4*d^8 - 16*c^2*d^5*x*cosh(1)^3 + 7*d^2*x^2*cosh(1)^6 + 12*d^3*x*cosh(1)^5 - 5*(2*c^2*d^4*x^2 - d^4)*
cosh(1)^4 + 3*(c^4*d^6*x^2 - 2*c^2*d^6)*cosh(1)^2)*sinh(1)), -1/2*(a*c^4*d^6 - 2*b*c^2*d^3*x*cosh(1)^3 - (2*a
+ b)*c^2*d^4*cosh(1)^2 + b*x^2*cosh(1)^6 + b*x^2*sinh(1)^6 + 2*b*d*x*cosh(1)^5 + 2*(3*b*x^2*cosh(1) + b*d*x)*s
inh(1)^5 - (b*c^2*d^2*x^2 - (a + b)*d^2)*cosh(1)^4 - (b*c^2*d^2*x^2 - 15*b*x^2*cosh(1)^2 - 10*b*d*x*cosh(1) -
(a + b)*d^2)*sinh(1)^4 - 2*(b*c^2*d^3*x - 10*b*x^2*cosh(1)^3 - 10*b*d*x*cosh(1)^2 + 2*(b*c^2*d^2*x^2 - (a + b)
*d^2)*cosh(1))*sinh(1)^3 - (6*b*c^2*d^3*x*cosh(1) + (2*a + b)*c^2*d^4 - 15*b*x^2*cosh(1)^4 - 20*b*d*x*cosh(1)^
3 + 6*(b*c^2*d^2*x^2 - (a + b)*d^2)*cosh(1)^2)*sinh(1)^2 - 2*(4*b*c^2*d^3*x*cosh(1)^2 + 2*b*c^2*d^4*cosh(1) -
b*x^2*cosh(1)^5 - b*x^2*sinh(1)^5 - 2*b*d*x*cosh(1)^4 - (5*b*x^2*cosh(1) + 2*b*d*x)*sinh(1)^4 + (2*b*c^2*d^2*x
^2 - b*d^2)*cosh(1)^3 + (2*b*c^2*d^2*x^2 - 10*b*x^2*cosh(1)^2 - 8*b*d*x*cosh(1) - b*d^2)*sinh(1)^3 + (4*b*c^2*
d^3*x - 10*b*x^2*cosh(1)^3 - 12*b*d*x*cosh(1)^2...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asech(c*x))/(e*x+d)**3,x)

[Out]

Integral((a + b*asech(c*x))/(d + e*x)**3, 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*arcsech(c*x))/(e*x+d)^3,x, algorithm="giac")

[Out]

integrate((b*arcsech(c*x) + a)/(e*x + d)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(1/(c*x)))/(d + e*x)^3,x)

[Out]

int((a + b*acosh(1/(c*x)))/(d + e*x)^3, x)

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