3.2.44 \(\int \frac {\coth (c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\) [144]

Optimal. Leaf size=46 \[ \frac {b \log \left (b+a \cosh ^2(c+d x)\right )}{2 a (a+b) d}+\frac {\log (\sinh (c+d x))}{(a+b) d} \]

[Out]

1/2*b*ln(b+a*cosh(d*x+c)^2)/a/(a+b)/d+ln(sinh(d*x+c))/(a+b)/d

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Rubi [A]
time = 0.06, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4223, 457, 78} \begin {gather*} \frac {\log (\sinh (c+d x))}{d (a+b)}+\frac {b \log \left (a \cosh ^2(c+d x)+b\right )}{2 a d (a+b)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*Log[b + a*Cosh[c + d*x]^2])/(2*a*(a + b)*d) + Log[Sinh[c + d*x]]/((a + b)*d)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4223

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> Module[{ff =
FreeFactors[Cos[e + f*x], x]}, Dist[-(f*ff^(m + n*p - 1))^(-1), Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*
(ff*x)^n)^p/x^(m + n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[(m - 1)/2] &&
IntegerQ[n] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\coth (c+d x)}{a+b \text {sech}^2(c+d x)} \, dx &=-\frac {\text {Subst}\left (\int \frac {x^3}{\left (1-x^2\right ) \left (b+a x^2\right )} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {x}{(1-x) (b+a x)} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\text {Subst}\left (\int \left (\frac {1}{(-a-b) (-1+x)}-\frac {b}{(a+b) (b+a x)}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {b \log \left (b+a \cosh ^2(c+d x)\right )}{2 a (a+b) d}+\frac {\log (\sinh (c+d x))}{(a+b) d}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 42, normalized size = 0.91 \begin {gather*} \frac {2 a \log (\sinh (c+d x))+b \log \left (a+b+a \sinh ^2(c+d x)\right )}{2 a^2 d+2 a b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*Log[Sinh[c + d*x]] + b*Log[a + b + a*Sinh[c + d*x]^2])/(2*a^2*d + 2*a*b*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(124\) vs. \(2(44)=88\).
time = 2.20, size = 125, normalized size = 2.72

method	result	size
risch	\(\frac {x}{a}-\frac {2 x}{a +b}-\frac {2 c}{d \left (a +b \right )}-\frac {2 b x}{a \left (a +b \right )}-\frac {2 b c}{a d \left (a +b \right )}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left (a +b \right )}+\frac {b \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (2 b +a \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a d \left (a +b \right )}\)	\(119\)
derivativedivides	\(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a +b}+\frac {b \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}{2 a \left (a +b \right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a}}{d}\)	\(125\)
default	\(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a +b}+\frac {b \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}{2 a \left (a +b \right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a}}{d}\)	\(125\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-1/a*ln(tanh(1/2*d*x+1/2*c)-1)+1/(a+b)*ln(tanh(1/2*d*x+1/2*c))+1/2/a*b/(a+b)*ln(a*tanh(1/2*d*x+1/2*c)^4+b
*tanh(1/2*d*x+1/2*c)^4+2*a*tanh(1/2*d*x+1/2*c)^2-2*b*tanh(1/2*d*x+1/2*c)^2+a+b)-1/a*ln(tanh(1/2*d*x+1/2*c)+1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (44) = 88\).
time = 0.28, size = 100, normalized size = 2.17 \begin {gather*} \frac {b \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \, {\left (a^{2} + a b\right )} d} + \frac {d x + c}{a d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{{\left (a + b\right )} d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{{\left (a + b\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/((a^2 + a*b)*d) + (d*x + c)/(a*d) + log(e^(-d
*x - c) + 1)/((a + b)*d) + log(e^(-d*x - c) - 1)/((a + b)*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (44) = 88\).
time = 0.42, size = 115, normalized size = 2.50 \begin {gather*} -\frac {2 \, {\left (a + b\right )} d x - b \log \left (\frac {2 \, {\left (a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + a + 2 \, b\right )}}{\cosh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2}}\right ) - 2 \, a \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{2 \, {\left (a^{2} + a b\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*(a + b)*d*x - b*log(2*(a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 + a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x
+ c)*sinh(d*x + c) + sinh(d*x + c)^2)) - 2*a*log(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))))/((a^2 + a*b
)*d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth {\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)/(a+b*sech(d*x+c)**2),x)

[Out]

Integral(coth(c + d*x)/(a + b*sech(c + d*x)**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [B]
time = 1.82, size = 228, normalized size = 4.96 \begin {gather*} \frac {\ln \left (8\,a\,b^5-b^6-24\,a^2\,b^4+32\,a^3\,b^3-16\,a^4\,b^2+b^6\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-8\,a\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+24\,a^2\,b^4\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-32\,a^3\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+16\,a^4\,b^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )}{a\,d+b\,d}-\frac {x}{a}+\frac {b\,\ln \left (2\,a^2-a\,b+4\,a^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+2\,a^2\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}-4\,b^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+6\,a\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-a\,b\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}\right )}{2\,d\,a^2+2\,b\,d\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(c + d*x)/(a + b/cosh(c + d*x)^2),x)

[Out]

log(8*a*b^5 - b^6 - 24*a^2*b^4 + 32*a^3*b^3 - 16*a^4*b^2 + b^6*exp(2*c)*exp(2*d*x) - 8*a*b^5*exp(2*c)*exp(2*d*
x) + 24*a^2*b^4*exp(2*c)*exp(2*d*x) - 32*a^3*b^3*exp(2*c)*exp(2*d*x) + 16*a^4*b^2*exp(2*c)*exp(2*d*x))/(a*d +
b*d) - x/a + (b*log(2*a^2 - a*b + 4*a^2*exp(2*c)*exp(2*d*x) + 2*a^2*exp(4*c)*exp(4*d*x) - 4*b^2*exp(2*c)*exp(2
*d*x) + 6*a*b*exp(2*c)*exp(2*d*x) - a*b*exp(4*c)*exp(4*d*x)))/(2*a^2*d + 2*a*b*d)

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