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

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Rubi [A]  time = 0.0691455, antiderivative size = 55, normalized size of antiderivative = 1., 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 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]

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 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]

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*}

Mathematica [B]  time = 0.681698, 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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Maple [B]  time = 0.048, size = 107, normalized size = 2. \begin{align*} -{\frac{1}{bd}\sqrt{a}\arctan \left ({\frac{1}{2} \left ( 2\,\tanh \left ( 1/2\,dx+c/2 \right ) \sqrt{a+b}+2\,\sqrt{b} \right ){\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a+b}}}}+{\frac{1}{bd}\sqrt{a}\arctan \left ({\frac{1}{2} \left ( -2\,\tanh \left ( 1/2\,dx+c/2 \right ) \sqrt{a+b}+2\,\sqrt{b} \right ){\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a+b}}}}+2\,{\frac{\arctan \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) }{bd}} \end{align*}

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*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)+2*b^(1/2))/a^(1/2))+1/d*a^(1/2)/b/(a+
b)^(1/2)*arctan(1/2*(-2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)+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., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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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Fricas [B]  time = 2.30188, size = 1431, normalized size = 26.02 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{3}{\left (c + d x \right )}}{a + b \operatorname{sech}^{2}{\left (c + d x \right )}}\, dx \end{align*}

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

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