3.29 \(\int \frac {\text {csch}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\)

Optimal. Leaf size=55 \[ \frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} d (a+b)}-\frac {\tanh ^{-1}(\cosh (c+d x))}{d (a+b)} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4133, 481, 206, 205} \[ \frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} d (a+b)}-\frac {\tanh ^{-1}(\cosh (c+d x))}{d (a+b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b]*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(Sqrt[a]*(a + b)*d) - ArcTanh[Cosh[c + d*x]]/((a + b)*d)

Rule 205

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

Rule 206

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

Rule 481

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> -Dist[(a*e^n)/(b*c -
a*d), Int[(e*x)^(m - n)/(a + b*x^n), x], x] + Dist[(c*e^n)/(b*c - a*d), Int[(e*x)^(m - n)/(c + d*x^n), x], x]
/; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1]

Rule 4133

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

Rubi steps

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

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Mathematica [C]  time = 0.93, size = 232, normalized size = 4.22 \[ \frac {\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sinh (c) \tanh \left (\frac {d x}{2}\right ) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right )+\cosh (c) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )}{\sqrt {a}}+\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sinh (c) \tanh \left (\frac {d x}{2}\right ) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right )+\cosh (c) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )}{\sqrt {a}}+\log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )}{d (a+b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[b]*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[
a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]])/Sqrt[a] + (Sqrt[b]*ArcTan[((Sqrt[a] +
I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos
h[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]])/Sqrt[a] - Log[Cosh[(c + d*x)/2]] + Log[Sinh[(c + d*x)/2]])/((a +
b)*d)

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fricas [B]  time = 0.51, size = 533, normalized size = 9.69 \[ \left [\frac {\sqrt {-\frac {b}{a}} \log \left (\frac {a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} + 2 \, {\left (a - 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} + a - 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} + {\left (a - 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 4 \, {\left (a \cosh \left (d x + c\right )^{3} + 3 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{3} + a \cosh \left (d x + c\right ) + {\left (3 \, a \cosh \left (d x + c\right )^{2} + a\right )} \sinh \left (d x + c\right )\right )} \sqrt {-\frac {b}{a}} + a}{a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} + 2 \, {\left (a + 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} + a + 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} + {\left (a + 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a}\right ) - 2 \, \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 2 \, \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right )}{2 \, {\left (a + b\right )} d}, -\frac {\sqrt {\frac {b}{a}} \arctan \left (\frac {{\left (a \cosh \left (d x + c\right )^{3} + 3 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{3} + {\left (a + 4 \, b\right )} \cosh \left (d x + c\right ) + {\left (3 \, a \cosh \left (d x + c\right )^{2} + a + 4 \, b\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {b}{a}}}{2 \, b}\right ) - \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \sqrt {\frac {b}{a}}}{2 \, b}\right ) + \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) - \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right )}{{\left (a + b\right )} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(-b/a)*log((a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a - 2*b)*
cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a - 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a - 2*b)*cosh(d*
x + c))*sinh(d*x + c) + 4*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3 + a*cosh(
d*x + c) + (3*a*cosh(d*x + c)^2 + a)*sinh(d*x + c))*sqrt(-b/a) + a)/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sin
h(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a + 2*b)*sinh(d*x +
c)^2 + 4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) - 2*log(cosh(d*x + c) + sinh(d*x +
c) + 1) + 2*log(cosh(d*x + c) + sinh(d*x + c) - 1))/((a + b)*d), -(sqrt(b/a)*arctan(1/2*(a*cosh(d*x + c)^3 + 3
*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3 + (a + 4*b)*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 + a + 4*
b)*sinh(d*x + c))*sqrt(b/a)/b) - sqrt(b/a)*arctan(1/2*(a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt(b/a)/b) + log(c
osh(d*x + c) + sinh(d*x + c) + 1) - log(cosh(d*x + c) + sinh(d*x + c) - 1))/((a + b)*d)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(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,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b]=[31,78]Warning, need to choose a branch for the root of a polynomial with parameters. This
might be wrong.The choice was done assuming [a,b]=[-13,-93]Undef/Unsigned Inf encountered in limitLimit: Max
order reached or unable to make series expansion Error: Bad Argument Value

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maple [A]  time = 0.25, size = 67, normalized size = 1.22 \[ \frac {b \arctan \left (\frac {2 \left (a +b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a -2 b}{4 \sqrt {a b}}\right )}{d \left (a +b \right ) \sqrt {a b}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a +b \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)/(a+b*sech(d*x+c)^2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {\log \left ({\left (e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{a d + b d} + \frac {\log \left ({\left (e^{\left (d x + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{a d + b d} + 2 \, \int \frac {b e^{\left (3 \, d x + 3 \, c\right )} - b e^{\left (d x + c\right )}}{a^{2} + a b + {\left (a^{2} e^{\left (4 \, c\right )} + a b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \, {\left (a^{2} e^{\left (2 \, c\right )} + 3 \, a b e^{\left (2 \, c\right )} + 2 \, b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log((e^(d*x + c) + 1)*e^(-c))/(a*d + b*d) + log((e^(d*x + c) - 1)*e^(-c))/(a*d + b*d) + 2*integrate((b*e^(3*d
*x + 3*c) - b*e^(d*x + c))/(a^2 + a*b + (a^2*e^(4*c) + a*b*e^(4*c))*e^(4*d*x) + 2*(a^2*e^(2*c) + 3*a*b*e^(2*c)
+ 2*b^2*e^(2*c))*e^(2*d*x)), x)

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mupad [B]  time = 2.18, size = 616, normalized size = 11.20 \[ -\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (b^4\,\sqrt {-a^2\,d^2-2\,a\,b\,d^2-b^2\,d^2}+16\,a^2\,b^2\,\sqrt {-a^2\,d^2-2\,a\,b\,d^2-b^2\,d^2}+8\,a\,b^3\,\sqrt {-a^2\,d^2-2\,a\,b\,d^2-b^2\,d^2}\right )}{16\,d\,a^3\,b^2+24\,d\,a^2\,b^3+9\,d\,a\,b^4+d\,b^5}\right )}{\sqrt {-a^2\,d^2-2\,a\,b\,d^2-b^2\,d^2}}-\frac {\sqrt {b}\,\left (2\,\mathrm {atan}\left (\frac {\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {64\,\left (2\,b^{7/2}\,d+8\,a^2\,b^{3/2}\,d+10\,a\,b^{5/2}\,d\right )}{a^5\,\left (a+b\right )\,\sqrt {a\,d^2\,{\left (a+b\right )}^2}\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}}+\frac {32\,\left (b^2\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}+4\,a\,b\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}\right )}{a^5\,\sqrt {b}\,d\,{\left (a+b\right )}^2\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}}\right )+\frac {32\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (b^2\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}+4\,a\,b\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}\right )}{a^5\,\sqrt {b}\,d\,{\left (a+b\right )}^2\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}}\right )\,\left (a^6\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}+a^5\,b\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}\right )}{64\,b^2+256\,a\,b}\right )-2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a\,d^2\,{\left (a+b\right )}^2}}{2\,\sqrt {b}\,d\,\left (a+b\right )}\right )\right )}{2\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (2*atan((exp(d*x)*exp(c)*(b^4*(- a^2*d^2 - b^2*d^2 - 2*a*b*d^2)^(1/2) + 16*a^2*b^2*(- a^2*d^2 - b^2*d^2 - 2*
a*b*d^2)^(1/2) + 8*a*b^3*(- a^2*d^2 - b^2*d^2 - 2*a*b*d^2)^(1/2)))/(b^5*d + 24*a^2*b^3*d + 16*a^3*b^2*d + 9*a*
b^4*d)))/(- a^2*d^2 - b^2*d^2 - 2*a*b*d^2)^(1/2) - (b^(1/2)*(2*atan(((exp(d*x)*exp(c)*((64*(2*b^(7/2)*d + 8*a^
2*b^(3/2)*d + 10*a*b^(5/2)*d))/(a^5*(a + b)*(a*d^2*(a + b)^2)^(1/2)*(a^3*d^2 + a*b^2*d^2 + 2*a^2*b*d^2)^(1/2))
+ (32*(b^2*(a^3*d^2 + a*b^2*d^2 + 2*a^2*b*d^2)^(1/2) + 4*a*b*(a^3*d^2 + a*b^2*d^2 + 2*a^2*b*d^2)^(1/2)))/(a^5
*b^(1/2)*d*(a + b)^2*(a^3*d^2 + a*b^2*d^2 + 2*a^2*b*d^2)^(1/2))) + (32*exp(3*c)*exp(3*d*x)*(b^2*(a^3*d^2 + a*b
^2*d^2 + 2*a^2*b*d^2)^(1/2) + 4*a*b*(a^3*d^2 + a*b^2*d^2 + 2*a^2*b*d^2)^(1/2)))/(a^5*b^(1/2)*d*(a + b)^2*(a^3*
d^2 + a*b^2*d^2 + 2*a^2*b*d^2)^(1/2)))*(a^6*(a^3*d^2 + a*b^2*d^2 + 2*a^2*b*d^2)^(1/2) + a^5*b*(a^3*d^2 + a*b^2
*d^2 + 2*a^2*b*d^2)^(1/2)))/(256*a*b + 64*b^2)) - 2*atan((exp(d*x)*exp(c)*(a*d^2*(a + b)^2)^(1/2))/(2*b^(1/2)*
d*(a + b)))))/(2*(a^3*d^2 + a*b^2*d^2 + 2*a^2*b*d^2)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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