3.81 \(\int \frac{\text{sech}^5(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx\)

Optimal. Leaf size=86 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{b^2 d \sqrt{a+b}}-\frac{(2 a-b) \tan ^{-1}(\sinh (c+d x))}{2 b^2 d}+\frac{\tanh (c+d x) \text{sech}(c+d x)}{2 b d} \]

[Out]

-((2*a - b)*ArcTan[Sinh[c + d*x]])/(2*b^2*d) + (a^(3/2)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(b^2*Sqrt
[a + b]*d) + (Sech[c + d*x]*Tanh[c + d*x])/(2*b*d)

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Rubi [A]  time = 0.102759, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4147, 414, 522, 203, 205} \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{b^2 d \sqrt{a+b}}-\frac{(2 a-b) \tan ^{-1}(\sinh (c+d x))}{2 b^2 d}+\frac{\tanh (c+d x) \text{sech}(c+d x)}{2 b d} \]

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]^5/(a + b*Sech[c + d*x]^2),x]

[Out]

-((2*a - b)*ArcTan[Sinh[c + d*x]])/(2*b^2*d) + (a^(3/2)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(b^2*Sqrt
[a + b]*d) + (Sech[c + d*x]*Tanh[c + d*x])/(2*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 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 522

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

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}^5(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^2 \left (a+b+a x^2\right )} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\text{sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac{\operatorname{Subst}\left (\int \frac{a-b-a x^2}{\left (1+x^2\right ) \left (a+b+a x^2\right )} \, dx,x,\sinh (c+d x)\right )}{2 b d}\\ &=\frac{\text{sech}(c+d x) \tanh (c+d x)}{2 b d}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{a+b+a x^2} \, dx,x,\sinh (c+d x)\right )}{b^2 d}-\frac{(2 a-b) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 b^2 d}\\ &=-\frac{(2 a-b) \tan ^{-1}(\sinh (c+d x))}{2 b^2 d}+\frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{b^2 \sqrt{a+b} d}+\frac{\text{sech}(c+d x) \tanh (c+d x)}{2 b d}\\ \end{align*}

Mathematica [B]  time = 1.86462, size = 213, normalized size = 2.48 \[ \frac{\cosh (c) \text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (2 a^{3/2} (\tanh (c)-1) \tan ^{-1}\left (\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right )+b \sqrt{a+b} \text{sech}^2(c) \sqrt{(\cosh (c)-\sinh (c))^2} \sinh (d x) \text{sech}^2(c+d x)-\sqrt{a+b} \text{sech}(c) \sqrt{(\cosh (c)-\sinh (c))^2} \left (2 (2 a-b) \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )-b \tanh (c) \text{sech}(c+d x)\right )\right )}{4 b^2 d \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \left (a+b \text{sech}^2(c+d x)\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[c + d*x]^5/(a + b*Sech[c + d*x]^2),x]

[Out]

(Cosh[c]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(b*Sqrt[a + b]*Sech[c]^2*Sech[c + d*x]^2*Sqrt[(Cosh[c
] - Sinh[c])^2]*Sinh[d*x] + 2*a^(3/2)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] +
Sinh[c]))/Sqrt[a]]*(-1 + Tanh[c]) - Sqrt[a + b]*Sech[c]*Sqrt[(Cosh[c] - Sinh[c])^2]*(2*(2*a - b)*ArcTan[Tanh[
(c + d*x)/2]] - b*Sech[c + d*x]*Tanh[c])))/(4*b^2*Sqrt[a + b]*d*(a + b*Sech[c + d*x]^2)*Sqrt[(Cosh[c] - Sinh[c
])^2])

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Maple [B]  time = 0.047, size = 189, normalized size = 2.2 \begin{align*}{\frac{1}{d{b}^{2}}{a}^{{\frac{3}{2}}}\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}{d{b}^{2}}{a}^{{\frac{3}{2}}}\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} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+1 \right ) ^{-2}}+{\frac{1}{bd}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+1 \right ) ^{-2}}+{\frac{1}{bd}\arctan \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-2\,{\frac{\arctan \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) a}{d{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^5/(a+b*sech(d*x+c)^2),x)

[Out]

1/d*a^(3/2)/b^2/(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^(3/2)/b^2/
(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/b/(tanh(1/2*d*x+1/2*c)^2+1)^
2*tanh(1/2*d*x+1/2*c)^3+1/d/b/(tanh(1/2*d*x+1/2*c)^2+1)^2*tanh(1/2*d*x+1/2*c)+1/d/b*arctan(tanh(1/2*d*x+1/2*c)
)-2/d/b^2*arctan(tanh(1/2*d*x+1/2*c))*a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{e^{\left (3 \, d x + 3 \, c\right )} - e^{\left (d x + c\right )}}{b d e^{\left (4 \, d x + 4 \, c\right )} + 2 \, b d e^{\left (2 \, d x + 2 \, c\right )} + b d} - \frac{{\left (2 \, a e^{c} - b e^{c}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) e^{\left (-c\right )}}{b^{2} d} + 32 \, \int \frac{a^{2} e^{\left (3 \, d x + 3 \, c\right )} + a^{2} e^{\left (d x + c\right )}}{16 \,{\left (a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + a b^{2} + 2 \,{\left (a b^{2} e^{\left (2 \, c\right )} + 2 \, b^{3} 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)^5/(a+b*sech(d*x+c)^2),x, algorithm="maxima")

[Out]

(e^(3*d*x + 3*c) - e^(d*x + c))/(b*d*e^(4*d*x + 4*c) + 2*b*d*e^(2*d*x + 2*c) + b*d) - (2*a*e^c - b*e^c)*arctan
(e^(d*x + c))*e^(-c)/(b^2*d) + 32*integrate(1/16*(a^2*e^(3*d*x + 3*c) + a^2*e^(d*x + c))/(a*b^2*e^(4*d*x + 4*c
) + a*b^2 + 2*(a*b^2*e^(2*c) + 2*b^3*e^(2*c))*e^(2*d*x)), x)

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Fricas [B]  time = 2.52429, size = 3997, normalized size = 46.48 \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)^5/(a+b*sech(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/2*(2*b*cosh(d*x + c)^3 + 6*b*cosh(d*x + c)*sinh(d*x + c)^2 + 2*b*sinh(d*x + c)^3 + (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*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a)*sinh(d*
x + c)^2 + 4*(a*cosh(d*x + c)^3 + a*cosh(d*x + c))*sinh(d*x + c) + a)*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))*sinh(d*x + c) + a)) - 2*((2*a - b)*cosh(d*x + c)^
4 + 4*(2*a - b)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a - b)*sinh(d*x + c)^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3
*(2*a - b)*cosh(d*x + c)^2 + 2*a - b)*sinh(d*x + c)^2 + 4*((2*a - b)*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c)
)*sinh(d*x + c) + 2*a - b)*arctan(cosh(d*x + c) + sinh(d*x + c)) - 2*b*cosh(d*x + c) + 2*(3*b*cosh(d*x + c)^2
- b)*sinh(d*x + c))/(b^2*d*cosh(d*x + c)^4 + 4*b^2*d*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*d*sinh(d*x + c)^4 + 2
*b^2*d*cosh(d*x + c)^2 + b^2*d + 2*(3*b^2*d*cosh(d*x + c)^2 + b^2*d)*sinh(d*x + c)^2 + 4*(b^2*d*cosh(d*x + c)^
3 + b^2*d*cosh(d*x + c))*sinh(d*x + c)), (b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)*sinh(d*x + c)^2 + b*sinh(d*x +
c)^3 + (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*cosh(d*x + c)^2 + 2*(
3*a*cosh(d*x + c)^2 + a)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + a*cosh(d*x + c))*sinh(d*x + c) + a)*sqrt(a/(
a + b))*arctan(1/2*sqrt(a/(a + b))*(cosh(d*x + c) + sinh(d*x + c))) + (a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*s
inh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*a*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a)*sinh(d*x + c)^2 + 4*(a*
cosh(d*x + c)^3 + a*cosh(d*x + c))*sinh(d*x + c) + a)*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))*sqrt(a/(a + b))/a) - ((2*a - b)*cosh(d*x + c)^4 + 4*(2*a - b)*cosh(d*x + c)*sinh(d*x + c)^3 + (
2*a - b)*sinh(d*x + c)^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*(2*a - b)*cosh(d*x + c)^2 + 2*a - b)*sinh(d*x +
c)^2 + 4*((2*a - b)*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + 2*a - b)*arctan(cosh(d*x + c) +
sinh(d*x + c)) - b*cosh(d*x + c) + (3*b*cosh(d*x + c)^2 - b)*sinh(d*x + c))/(b^2*d*cosh(d*x + c)^4 + 4*b^2*d*
cosh(d*x + c)*sinh(d*x + c)^3 + b^2*d*sinh(d*x + c)^4 + 2*b^2*d*cosh(d*x + c)^2 + b^2*d + 2*(3*b^2*d*cosh(d*x
+ c)^2 + b^2*d)*sinh(d*x + c)^2 + 4*(b^2*d*cosh(d*x + c)^3 + b^2*d*cosh(d*x + c))*sinh(d*x + c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{5}{\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)**5/(a+b*sech(d*x+c)**2),x)

[Out]

Integral(sech(c + d*x)**5/(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)^5/(a+b*sech(d*x+c)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError