3.79 \(\int \frac {\text {sech}^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4147, 391, 203, 205} \[ \frac {\tan ^{-1}(\sinh (c+d x))}{b d}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{b d \sqrt {a+b}} \]

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]^3/(a + b*Sech[c + d*x]^2),x]

[Out]

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

Rule 203

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

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 391

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

Rule 4147

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

Rubi steps

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

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Mathematica [B]  time = 0.71, size = 194, normalized size = 3.53 \[ \frac {\text {sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (2 \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tan ^{-1}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+\sqrt {a} \cosh (c) \tan ^{-1}\left (\frac {\sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text {csch}(c+d x)}{\sqrt {a}}\right )-\sqrt {a} \sinh (c) \tan ^{-1}\left (\frac {\sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text {csch}(c+d x)}{\sqrt {a}}\right )\right )}{2 b d \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \left (a+b \text {sech}^2(c+d x)\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[c + d*x]^3/(a + b*Sech[c + d*x]^2),x]

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(Sqrt[a]*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Si
nh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*Cosh[c] + 2*Sqrt[a + b]*ArcTan[Tanh[(c + d*x)/2]]*Sqrt[(Cosh[c] - Sinh
[c])^2] - Sqrt[a]*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*
Sinh[c]))/(2*b*Sqrt[a + b]*d*(a + b*Sech[c + d*x]^2)*Sqrt[(Cosh[c] - Sinh[c])^2])

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fricas [B]  time = 0.44, size = 526, normalized size = 9.56 \[ \left [\frac {\sqrt {-\frac {a}{a + b}} \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 (3 \, a + 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} - 3 \, a - 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} - {\left (3 \, a + 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (a + b\right )} \sinh \left (d x + c\right )^{3} - {\left (a + b\right )} \cosh \left (d x + c\right ) + {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} - a - b\right )} \sinh \left (d x + c\right )\right )} \sqrt {-\frac {a}{a + b}} + 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 ) + 4 \, \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}{2 \, b d}, -\frac {\sqrt {\frac {a}{a + b}} \arctan \left (\frac {1}{2} \, \sqrt {\frac {a}{a + b}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}\right ) + \sqrt {\frac {a}{a + b}} \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 (3 \, a + 4 \, b\right )} \cosh \left (d x + c\right ) + {\left (3 \, a \cosh \left (d x + c\right )^{2} + 3 \, a + 4 \, b\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a}{a + b}}}{2 \, a}\right ) - 2 \, \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}{b d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^3/(a+b*sech(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/2*(sqrt(-a/(a + b))*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*(3*a
+ 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - 3*a - 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - (3*a +
2*b)*cosh(d*x + c))*sinh(d*x + c) - 4*((a + b)*cosh(d*x + c)^3 + 3*(a + b)*cosh(d*x + c)*sinh(d*x + c)^2 + (a
+ b)*sinh(d*x + c)^3 - (a + b)*cosh(d*x + c) + (3*(a + b)*cosh(d*x + c)^2 - a - b)*sinh(d*x + c))*sqrt(-a/(a +
b)) + a)/(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))*sin
h(d*x + c) + a)) + 4*arctan(cosh(d*x + c) + sinh(d*x + c)))/(b*d), -(sqrt(a/(a + b))*arctan(1/2*sqrt(a/(a + b)
)*(cosh(d*x + c) + sinh(d*x + c))) + sqrt(a/(a + b))*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 + (3*a + 4*b)*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 + 3*a + 4*b)*sinh(d*x + c))*sq
rt(a/(a + b))/a) - 2*arctan(cosh(d*x + c) + sinh(d*x + c)))/(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(sech(d*x+c)^3/(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]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]=[-65,-82]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]=[97,-56]Undef/Unsigned Inf encountered in limitEvaluation time: 0.57Limit: Max order reached or unable
to make series expansion Error: Bad Argument Value

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maple [B]  time = 0.25, size = 108, normalized size = 1.96 \[ -\frac {\sqrt {a}\, \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{d b \sqrt {a +b}}-\frac {\sqrt {a}\, \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{d b \sqrt {a +b}}+\frac {2 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^3/(a+b*sech(d*x+c)^2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, \arctan \left (e^{\left (d x + c\right )}\right )}{b d} - 8 \, \int \frac {a e^{\left (3 \, d x + 3 \, c\right )} + a e^{\left (d x + c\right )}}{4 \, {\left (a b e^{\left (4 \, d x + 4 \, c\right )} + a b + 2 \, {\left (a b e^{\left (2 \, c\right )} + 2 \, b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^3/(a+b*sech(d*x+c)^2),x, algorithm="maxima")

[Out]

2*arctan(e^(d*x + c))/(b*d) - 8*integrate(1/4*(a*e^(3*d*x + 3*c) + a*e^(d*x + c))/(a*b*e^(4*d*x + 4*c) + a*b +
2*(a*b*e^(2*c) + 2*b^2*e^(2*c))*e^(2*d*x)), x)

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mupad [B]  time = 1.81, size = 307, normalized size = 5.58 \[ \frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (9\,a^2\,\sqrt {b^2\,d^2}+16\,b^2\,\sqrt {b^2\,d^2}+24\,a\,b\,\sqrt {b^2\,d^2}\right )}{9\,d\,a^2\,b+24\,d\,a\,b^2+16\,d\,b^3}\right )}{\sqrt {b^2\,d^2}}-\frac {\sqrt {a}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {a}\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {b^2\,d^2\,\left (a+b\right )}}{2\,b\,d\,\left (a+b\right )}\right )+2\,\mathrm {atan}\left (\frac {4\,b^4\,d^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+4\,a^2\,b^2\,d^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {b^3\,d^2+a\,b^2\,d^2}\,\sqrt {b^2\,d^2\,\left (a+b\right )}+8\,a\,b^3\,d^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+a\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\sqrt {b^3\,d^2+a\,b^2\,d^2}\,\sqrt {b^2\,d^2\,\left (a+b\right )}}{\sqrt {a}\,d\,\left (2\,b^2+2\,a\,b\right )\,\sqrt {b^2\,d^2\,\left (a+b\right )}}\right )\right )}{2\,\sqrt {b^3\,d^2+a\,b^2\,d^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(c + d*x)^3*(a + b/cosh(c + d*x)^2)),x)

[Out]

(2*atan((exp(d*x)*exp(c)*(9*a^2*(b^2*d^2)^(1/2) + 16*b^2*(b^2*d^2)^(1/2) + 24*a*b*(b^2*d^2)^(1/2)))/(16*b^3*d
+ 24*a*b^2*d + 9*a^2*b*d)))/(b^2*d^2)^(1/2) - (a^(1/2)*(2*atan((a^(1/2)*exp(d*x)*exp(c)*(b^2*d^2*(a + b))^(1/2
))/(2*b*d*(a + b))) + 2*atan((4*b^4*d^2*exp(d*x)*exp(c) + 4*a^2*b^2*d^2*exp(d*x)*exp(c) - a*exp(d*x)*exp(c)*(b
^3*d^2 + a*b^2*d^2)^(1/2)*(b^2*d^2*(a + b))^(1/2) + 8*a*b^3*d^2*exp(d*x)*exp(c) + a*exp(3*c)*exp(3*d*x)*(b^3*d
^2 + a*b^2*d^2)^(1/2)*(b^2*d^2*(a + b))^(1/2))/(a^(1/2)*d*(2*a*b + 2*b^2)*(b^2*d^2*(a + b))^(1/2)))))/(2*(b^3*
d^2 + a*b^2*d^2)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}^{3}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)**3/(a+b*sech(d*x+c)**2),x)

[Out]

Integral(sech(c + d*x)**3/(a + b*sech(c + d*x)**2), x)

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