3.233 \(\int e^{c (a+b x)} \cot ^{-1}(\text {sech}(a c+b c x)) \, dx\)

Optimal. Leaf size=103 \[ -\frac {\left (1-\sqrt {2}\right ) \log \left (e^{2 c (a+b x)}+3-2 \sqrt {2}\right )}{2 b c}-\frac {\left (1+\sqrt {2}\right ) \log \left (e^{2 c (a+b x)}+3+2 \sqrt {2}\right )}{2 b c}+\frac {e^{a c+b c x} \cot ^{-1}(\text {sech}(c (a+b x)))}{b c} \]

[Out]

exp(b*c*x+a*c)*arccot(sech(c*(b*x+a)))/b/c-1/2*ln(3+exp(2*c*(b*x+a))-2*2^(1/2))*(1-2^(1/2))/b/c-1/2*ln(3+exp(2
*c*(b*x+a))+2*2^(1/2))*(1+2^(1/2))/b/c

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Rubi [A]  time = 0.15, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {2194, 5208, 2282, 12, 1247, 632, 31} \[ -\frac {\left (1-\sqrt {2}\right ) \log \left (e^{2 c (a+b x)}+3-2 \sqrt {2}\right )}{2 b c}-\frac {\left (1+\sqrt {2}\right ) \log \left (e^{2 c (a+b x)}+3+2 \sqrt {2}\right )}{2 b c}+\frac {e^{a c+b c x} \cot ^{-1}(\text {sech}(c (a+b x)))}{b c} \]

Antiderivative was successfully verified.

[In]

Int[E^(c*(a + b*x))*ArcCot[Sech[a*c + b*c*x]],x]

[Out]

(E^(a*c + b*c*x)*ArcCot[Sech[c*(a + b*x)]])/(b*c) - ((1 - Sqrt[2])*Log[3 - 2*Sqrt[2] + E^(2*c*(a + b*x))])/(2*
b*c) - ((1 + Sqrt[2])*Log[3 + 2*Sqrt[2] + E^(2*c*(a + b*x))])/(2*b*c)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 5208

Int[((a_.) + ArcCot[u_]*(b_.))*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[a + b*ArcCot[u], w, x] + Dist
[b, Int[SimplifyIntegrand[(w*D[u, x])/(1 + u^2), x], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x]
&& InverseFunctionFreeQ[u, x] &&  !MatchQ[v, ((c_.) + (d_.)*x)^(m_.) /; FreeQ[{c, d, m}, x]] && FalseQ[Functi
onOfLinear[v*(a + b*ArcCot[u]), x]]

Rubi steps

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

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Mathematica [C]  time = 0.16, size = 145, normalized size = 1.41 \[ \frac {\text {RootSum}\left [\text {$\#$1}^4+6 \text {$\#$1}^2+1\& ,\frac {-7 \text {$\#$1}^2 \log \left (e^{c (a+b x)}-\text {$\#$1}\right )+7 \text {$\#$1}^2 a c+7 \text {$\#$1}^2 b c x-\log \left (e^{c (a+b x)}-\text {$\#$1}\right )+a c+b c x}{3 \text {$\#$1}^2+1}\& \right ]-4 c (a+b x)+2 e^{c (a+b x)} \cot ^{-1}\left (\frac {2 e^{c (a+b x)}}{e^{2 c (a+b x)}+1}\right )}{2 b c} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(c*(a + b*x))*ArcCot[Sech[a*c + b*c*x]],x]

[Out]

(-4*c*(a + b*x) + 2*E^(c*(a + b*x))*ArcCot[(2*E^(c*(a + b*x)))/(1 + E^(2*c*(a + b*x)))] + RootSum[1 + 6*#1^2 +
#1^4 & , (a*c + b*c*x - Log[E^(c*(a + b*x)) - #1] + 7*a*c*#1^2 + 7*b*c*x*#1^2 - 7*Log[E^(c*(a + b*x)) - #1]*#
1^2)/(1 + 3*#1^2) & ])/(2*b*c)

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fricas [B]  time = 0.45, size = 221, normalized size = 2.15 \[ \frac {2 \, {\left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right )} \arctan \left (\cosh \left (b c x + a c\right )\right ) + \sqrt {2} \log \left (-\frac {3 \, {\left (2 \, \sqrt {2} - 3\right )} \cosh \left (b c x + a c\right )^{2} - 4 \, {\left (3 \, \sqrt {2} - 4\right )} \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right ) + 3 \, {\left (2 \, \sqrt {2} - 3\right )} \sinh \left (b c x + a c\right )^{2} + 2 \, \sqrt {2} - 3}{\cosh \left (b c x + a c\right )^{2} + \sinh \left (b c x + a c\right )^{2} + 3}\right ) - \log \left (\frac {2 \, {\left (\cosh \left (b c x + a c\right )^{2} + \sinh \left (b c x + a c\right )^{2} + 3\right )}}{\cosh \left (b c x + a c\right )^{2} - 2 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )^{2}}\right )}{2 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))*arccot(sech(b*c*x+a*c)),x, algorithm="fricas")

[Out]

1/2*(2*(cosh(b*c*x + a*c) + sinh(b*c*x + a*c))*arctan(cosh(b*c*x + a*c)) + sqrt(2)*log(-(3*(2*sqrt(2) - 3)*cos
h(b*c*x + a*c)^2 - 4*(3*sqrt(2) - 4)*cosh(b*c*x + a*c)*sinh(b*c*x + a*c) + 3*(2*sqrt(2) - 3)*sinh(b*c*x + a*c)
^2 + 2*sqrt(2) - 3)/(cosh(b*c*x + a*c)^2 + sinh(b*c*x + a*c)^2 + 3)) - log(2*(cosh(b*c*x + a*c)^2 + sinh(b*c*x
+ a*c)^2 + 3)/(cosh(b*c*x + a*c)^2 - 2*cosh(b*c*x + a*c)*sinh(b*c*x + a*c) + sinh(b*c*x + a*c)^2)))/(b*c)

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giac [A]  time = 0.13, size = 154, normalized size = 1.50 \[ \frac {{\left (\sqrt {2} e^{\left (-a c\right )} \log \left (-\frac {2 \, \sqrt {2} e^{\left (2 \, a c\right )} - e^{\left (2 \, b c x + 4 \, a c\right )} - 3 \, e^{\left (2 \, a c\right )}}{2 \, \sqrt {2} e^{\left (2 \, a c\right )} + e^{\left (2 \, b c x + 4 \, a c\right )} + 3 \, e^{\left (2 \, a c\right )}}\right ) + 2 \, \arctan \left (\frac {1}{2} \, e^{\left (b c x + a c\right )} + \frac {1}{2} \, e^{\left (-b c x - a c\right )}\right ) e^{\left (b c x\right )} - e^{\left (-a c\right )} \log \left (e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )\right )} e^{\left (a c\right )}}{2 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))*arccot(sech(b*c*x+a*c)),x, algorithm="giac")

[Out]

1/2*(sqrt(2)*e^(-a*c)*log(-(2*sqrt(2)*e^(2*a*c) - e^(2*b*c*x + 4*a*c) - 3*e^(2*a*c))/(2*sqrt(2)*e^(2*a*c) + e^
(2*b*c*x + 4*a*c) + 3*e^(2*a*c))) + 2*arctan(1/2*e^(b*c*x + a*c) + 1/2*e^(-b*c*x - a*c))*e^(b*c*x) - e^(-a*c)*
log(e^(4*b*c*x + 4*a*c) + 6*e^(2*b*c*x + 2*a*c) + 1))*e^(a*c)/(b*c)

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maple [C]  time = 0.76, size = 859, normalized size = 8.34 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(c*(b*x+a))*arccot(sech(b*c*x+a*c)),x)

[Out]

1/2*I/b/c*exp(c*(b*x+a))*ln(exp(2*c*(b*x+a))+1+2*I*exp(c*(b*x+a)))-1/4/b/c*Pi*csgn(I*(exp(2*c*(b*x+a))+1+2*I*e
xp(c*(b*x+a))))*csgn(I*(exp(2*c*(b*x+a))+1+2*I*exp(c*(b*x+a)))/(1+exp(2*c*(b*x+a))))^2*exp(c*(b*x+a))+1/4/b/c*
Pi*csgn(I*(exp(2*c*(b*x+a))+1+2*I*exp(c*(b*x+a))))*csgn(I/(1+exp(2*c*(b*x+a))))*csgn(I*(exp(2*c*(b*x+a))+1+2*I
*exp(c*(b*x+a)))/(1+exp(2*c*(b*x+a))))*exp(c*(b*x+a))+1/4/b/c*Pi*csgn(I*(exp(2*c*(b*x+a))+1+2*I*exp(c*(b*x+a))
)/(1+exp(2*c*(b*x+a))))^3*exp(c*(b*x+a))-1/4/b/c*Pi*csgn(I/(1+exp(2*c*(b*x+a))))*csgn(I*(exp(2*c*(b*x+a))+1+2*
I*exp(c*(b*x+a)))/(1+exp(2*c*(b*x+a))))^2*exp(c*(b*x+a))-1/4/b/c*Pi*csgn(I*(-exp(2*c*(b*x+a))-1+2*I*exp(c*(b*x
+a))))*csgn(I/(1+exp(2*c*(b*x+a))))*csgn(I*(-exp(2*c*(b*x+a))-1+2*I*exp(c*(b*x+a)))/(1+exp(2*c*(b*x+a))))*exp(
c*(b*x+a))-1/4/b/c*Pi*csgn(I*(-exp(2*c*(b*x+a))-1+2*I*exp(c*(b*x+a))))*csgn(I*(-exp(2*c*(b*x+a))-1+2*I*exp(c*(
b*x+a)))/(1+exp(2*c*(b*x+a))))^2*exp(c*(b*x+a))+1/4/b/c*Pi*csgn(I/(1+exp(2*c*(b*x+a))))*csgn(I*(-exp(2*c*(b*x+
a))-1+2*I*exp(c*(b*x+a)))/(1+exp(2*c*(b*x+a))))^2*exp(c*(b*x+a))+1/4/b/c*Pi*csgn(I*(-exp(2*c*(b*x+a))-1+2*I*ex
p(c*(b*x+a)))/(1+exp(2*c*(b*x+a))))^3*exp(c*(b*x+a))-1/2*I/b/c*exp(c*(b*x+a))*ln(exp(2*c*(b*x+a))+1-2*I*exp(c*
(b*x+a)))+1/2/b/c*exp(c*(b*x+a))*Pi+1/2/b/c*ln(exp(2*c*(b*x+a))+(2^(1/2)-1)^2)*2^(1/2)-1/2/b/c*ln(exp(2*c*(b*x
+a))+(1+2^(1/2))^2)*2^(1/2)+2*a/b-1/2/b/c*ln(exp(2*c*(b*x+a))+(2^(1/2)-1)^2)-1/2/b/c*ln(exp(2*c*(b*x+a))+(1+2^
(1/2))^2)

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maxima [A]  time = 0.48, size = 169, normalized size = 1.64 \[ \frac {\operatorname {arccot}\left (\operatorname {sech}\left (b c x + a c\right )\right ) e^{\left ({\left (b x + a\right )} c\right )}}{b c} + \frac {3 \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (2 \, b c x + 2 \, a c\right )} - 3}{2 \, \sqrt {2} + e^{\left (2 \, b c x + 2 \, a c\right )} + 3}\right )}{8 \, b c} - \frac {\sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (-2 \, b c x - 2 \, a c\right )} - 3}{2 \, \sqrt {2} + e^{\left (-2 \, b c x - 2 \, a c\right )} + 3}\right )}{8 \, b c} - \frac {\log \left (e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}{2 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))*arccot(sech(b*c*x+a*c)),x, algorithm="maxima")

[Out]

arccot(sech(b*c*x + a*c))*e^((b*x + a)*c)/(b*c) + 3/8*sqrt(2)*log(-(2*sqrt(2) - e^(2*b*c*x + 2*a*c) - 3)/(2*sq
rt(2) + e^(2*b*c*x + 2*a*c) + 3))/(b*c) - 1/8*sqrt(2)*log(-(2*sqrt(2) - e^(-2*b*c*x - 2*a*c) - 3)/(2*sqrt(2) +
e^(-2*b*c*x - 2*a*c) + 3))/(b*c) - 1/2*log(e^(4*b*c*x + 4*a*c) + 6*e^(2*b*c*x + 2*a*c) + 1)/(b*c)

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mupad [B]  time = 0.83, size = 135, normalized size = 1.31 \[ \frac {{\mathrm {e}}^{a\,c+b\,c\,x}\,\mathrm {acot}\left (\frac {1}{\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}+\frac {{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}}\right )}{b\,c}+\frac {\ln \left (-8\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}-2\,\sqrt {2}-6\,\sqrt {2}\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}\right )\,\left (\sqrt {2}-1\right )}{2\,b\,c}-\frac {\ln \left (2\,\sqrt {2}-8\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}+6\,\sqrt {2}\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}\right )\,\left (\sqrt {2}+1\right )}{2\,b\,c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(1/cosh(a*c + b*c*x))*exp(c*(a + b*x)),x)

[Out]

(exp(a*c + b*c*x)*acot(1/((exp(b*c*x)*exp(a*c))/2 + (exp(-b*c*x)*exp(-a*c))/2)))/(b*c) + (log(- 8*exp(2*c*(a +
b*x)) - 2*2^(1/2) - 6*2^(1/2)*exp(2*c*(a + b*x)))*(2^(1/2) - 1))/(2*b*c) - (log(2*2^(1/2) - 8*exp(2*c*(a + b*
x)) + 6*2^(1/2)*exp(2*c*(a + b*x)))*(2^(1/2) + 1))/(2*b*c)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))*acot(sech(b*c*x+a*c)),x)

[Out]

Timed out

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