3.1.79 \(\int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^2} \, dx\) [79]

Optimal. Leaf size=147 \[ -\frac {a+b \text {sech}^{-1}(c x)}{e (d+e x)}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{d \sqrt {c^2 d^2-e^2}}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d e} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6423, 975, 272, 65, 214, 739, 210} \begin {gather*} -\frac {a+b \text {sech}^{-1}(c x)}{e (d+e x)}+\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 )}{d \sqrt {c^2 d^2-e^2}}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.15, size = 222, normalized size = 1.51 \begin {gather*} -\frac {a}{e (d+e x)}-\frac {b \text {sech}^{-1}(c x)}{e (d+e x)}-\frac {b \log (x)}{d e}+\frac {b \log (d+e x)}{d \sqrt {-c^2 d^2+e^2}}+\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 e}-\frac {b \log \left (e+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 )}{d \sqrt {-c^2 d^2+e^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a/(e*(d + e*x))) - (b*ArcSech[c*x])/(e*(d + e*x)) - (b*Log[x])/(d*e) + (b*Log[d + e*x])/(d*Sqrt[-(c^2*d^2) +
e^2]) + (b*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/(d*e) - (b*Log[e + c^2*d*x + S
qrt[-(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)]])/(d*Sq
rt[-(c^2*d^2) + e^2])

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Maple [A]
time = 1.55, size = 243, normalized size = 1.65

method	result	size
derivativedivides	\(\frac {-\frac {a \,c^{2}}{\left (c e x +c d \right ) e}-\frac {b \,c^{2} \mathrm {arcsech}\left (c x \right )}{\left (c e x +c d \right ) e}+\frac {b \,c^{2} \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{e d \sqrt {-c^{2} x^{2}+1}}-\frac {b \,c^{2} \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right )}{e \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, d \sqrt {-c^{2} x^{2}+1}}}{c}\)	\(243\)
default	\(\frac {-\frac {a \,c^{2}}{\left (c e x +c d \right ) e}-\frac {b \,c^{2} \mathrm {arcsech}\left (c x \right )}{\left (c e x +c d \right ) e}+\frac {b \,c^{2} \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{e d \sqrt {-c^{2} x^{2}+1}}-\frac {b \,c^{2} \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right )}{e \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, d \sqrt {-c^{2} x^{2}+1}}}{c}\)	\(243\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsech(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*arcsech(c*x)+b*c^2/e*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/
d/(-c^2*x^2+1)^(1/2)*arctanh(1/(-c^2*x^2+1)^(1/2))-b*c^2/e*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(-(c^2*d
^2-e^2)/e^2)^(1/2)/d/(-c^2*x^2+1)^(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]
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*arcsech(c*x))/(e*x+d)^2,x, algorithm="maxima")

[Out]

(c^2*integrate(x^2/(c^2*d^2*x^2 + (c^2*d^2*x^2 - d^2 + (c^2*d*x^2*e - d*e)*x)*sqrt(c*x + 1)*sqrt(-c*x + 1) - d
^2 + (c^2*d*x^2*e - d*e)*x), x) + (x*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1) - x*log(c) - x*log(x))/(d*x*e + d^2
) - integrate(1/(c^2*d^2*x^2 - d^2 + (c^2*d*x^2*e - d*e)*x), x))*b - a/(x*e^2 + d*e)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (97) = 194\).
time = 0.40, size = 1161, normalized size = 7.90 \begin {gather*} \left [-\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 x \cosh \left (1\right )^{2} + b x \sinh \left (1\right )^{2} + b d \cosh \left (1\right ) + {\left (2 \, b x \cosh \left (1\right ) + b d\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 )}} \log \left (\frac {c^{2} d x \cosh \left (1\right ) + \cosh \left (1\right )^{2} + {\left (c^{2} d x + 2 \, \cosh \left (1\right )\right )} \sinh \left (1\right ) + \sinh \left (1\right )^{2} - {\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 )}} - {\left (c^{3} d^{2} x - c x \cosh \left (1\right )^{2} - 2 \, c x \cosh \left (1\right ) \sinh \left (1\right ) - c x \sinh \left (1\right )^{2} + {\left (c x \cosh \left (1\right ) + c x \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 )}}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{x \cosh \left (1\right ) + x \sinh \left (1\right ) + d}\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 (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + {\left (b c^{2} d^{3} - b d \cosh \left (1\right )^{2} - 2 \, b d \cosh \left (1\right ) \sinh \left (1\right ) - b d \sinh \left (1\right )^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\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 )}, -\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} - 2 \, {\left (b x \cosh \left (1\right )^{2} + b x \sinh \left (1\right )^{2} + b d \cosh \left (1\right ) + {\left (2 \, b x \cosh \left (1\right ) + b d\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 )}} \arctan \left (-\frac {{\left (c d x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - x \cosh \left (1\right ) - x \sinh \left (1\right ) - d\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 )}}}{c^{2} d^{2} x - x \cosh \left (1\right )^{2} - 2 \, x \cosh \left (1\right ) \sinh \left (1\right ) - x \sinh \left (1\right )^{2}}\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 (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + {\left (b c^{2} d^{3} - b d \cosh \left (1\right )^{2} - 2 \, b d \cosh \left (1\right ) \sinh \left (1\right ) - b d \sinh \left (1\right )^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\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 )}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(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*x*cosh(1)^2 + b*x*sinh(1)^2 + b*d*co
sh(1) + (2*b*x*cosh(1) + b*d)*sinh(1))*sqrt(-((c^2*d^2 - 1)*cosh(1) - (c^2*d^2 + 1)*sinh(1))/(cosh(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 + cosh(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)) + (b*c^2*d^2*x*c
osh(1) + b*c^2*d^3 - b*x*cosh(1)^3 - b*x*sinh(1)^3 - b*d*cosh(1)^2 - (3*b*x*cosh(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*x^2)) - 1)/x) + (b*c^2*d^3
- b*d*cosh(1)^2 - 2*b*d*cosh(1)*sinh(1) - b*d*sinh(1)^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)))
/(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)), -(a*c^2*d^3 - a*d*cosh(1)^2 - 2*a*d*cosh(1)*sinh(1) - a*d*sinh(1)^2 -
2*(b*x*cosh(1)^2 + b*x*sinh(1)^2 + b*d*cosh(1) + (2*b*x*cosh(1) + b*d)*sinh(1))*sqrt(((c^2*d^2 - 1)*cosh(1) -
(c^2*d^2 + 1)*sinh(1))/(cosh(1) - sinh(1)))*arctan(-(c*d*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - x*cosh(1) - x*sinh
(1) - d)*sqrt(((c^2*d^2 - 1)*cosh(1) - (c^2*d^2 + 1)*sinh(1))/(cosh(1) - sinh(1)))/(c^2*d^2*x - x*cosh(1)^2 -
2*x*cosh(1)*sinh(1) - x*sinh(1)^2)) + (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*c
osh(1)^2 - (3*b*x*cosh(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*x^2)) - 1)/x) + (b*c^2*d^3 - b*d*cosh(1)^2 - 2*b*d*cosh(1)*sinh(1) - b*d*sinh(1)^2)*
log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)))/(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))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{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*acosh(1/(c*x)))/(d + e*x)^2,x)

[Out]

int((a + b*acosh(1/(c*x)))/(d + e*x)^2, x)

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