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

Optimal. Leaf size=70 \[ \frac{(4 a+3 b) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac{(4 a+3 b) \tanh (c+d x) \text{sech}(c+d x)}{8 d}+\frac{b \tanh (c+d x) \text{sech}^3(c+d x)}{4 d} \]

[Out]

((4*a + 3*b)*ArcTan[Sinh[c + d*x]])/(8*d) + ((4*a + 3*b)*Sech[c + d*x]*Tanh[c + d*x])/(8*d) + (b*Sech[c + d*x]
^3*Tanh[c + d*x])/(4*d)

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Rubi [A]  time = 0.0502423, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4046, 3768, 3770} \[ \frac{(4 a+3 b) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac{(4 a+3 b) \tanh (c+d x) \text{sech}(c+d x)}{8 d}+\frac{b \tanh (c+d x) \text{sech}^3(c+d x)}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*a + 3*b)*ArcTan[Sinh[c + d*x]])/(8*d) + ((4*a + 3*b)*Sech[c + d*x]*Tanh[c + d*x])/(8*d) + (b*Sech[c + d*x]
^3*Tanh[c + d*x])/(4*d)

Rule 4046

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> -Simp[(C*Cot[
e + f*x]*(b*Csc[e + f*x])^m)/(f*(m + 1)), x] + Dist[(C*m + A*(m + 1))/(m + 1), Int[(b*Csc[e + f*x])^m, x], x]
/; FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] &&  !LeQ[m, -1]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \text{sech}^3(c+d x) \left (a+b \text{sech}^2(c+d x)\right ) \, dx &=\frac{b \text{sech}^3(c+d x) \tanh (c+d x)}{4 d}+\frac{1}{4} (4 a+3 b) \int \text{sech}^3(c+d x) \, dx\\ &=\frac{(4 a+3 b) \text{sech}(c+d x) \tanh (c+d x)}{8 d}+\frac{b \text{sech}^3(c+d x) \tanh (c+d x)}{4 d}+\frac{1}{8} (4 a+3 b) \int \text{sech}(c+d x) \, dx\\ &=\frac{(4 a+3 b) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac{(4 a+3 b) \text{sech}(c+d x) \tanh (c+d x)}{8 d}+\frac{b \text{sech}^3(c+d x) \tanh (c+d x)}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.104215, size = 60, normalized size = 0.86 \[ \frac{(4 a+3 b) \tan ^{-1}(\sinh (c+d x))+(4 a+3 b) \tanh (c+d x) \text{sech}(c+d x)+2 b \tanh (c+d x) \text{sech}^3(c+d x)}{8 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*a + 3*b)*ArcTan[Sinh[c + d*x]] + (4*a + 3*b)*Sech[c + d*x]*Tanh[c + d*x] + 2*b*Sech[c + d*x]^3*Tanh[c + d*
x])/(8*d)

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Maple [A]  time = 0.021, size = 83, normalized size = 1.2 \begin{align*}{\frac{a{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{2\,d}}+{\frac{a\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}}+{\frac{b \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}\tanh \left ( dx+c \right ) }{4\,d}}+{\frac{3\,b{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{8\,d}}+{\frac{3\,b\arctan \left ({{\rm e}^{dx+c}} \right ) }{4\,d}} \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]

1/2/d*a*sech(d*x+c)*tanh(d*x+c)+1/d*a*arctan(exp(d*x+c))+1/4*b*sech(d*x+c)^3*tanh(d*x+c)/d+3/8*b*sech(d*x+c)*t
anh(d*x+c)/d+3/4/d*b*arctan(exp(d*x+c))

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Maxima [B]  time = 1.68479, size = 248, normalized size = 3.54 \begin{align*} -\frac{1}{4} \, b{\left (\frac{3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{3 \, e^{\left (-d x - c\right )} + 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} - 3 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d{\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} - a{\left (\frac{\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} \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]

-1/4*b*(3*arctan(e^(-d*x - c))/d - (3*e^(-d*x - c) + 11*e^(-3*d*x - 3*c) - 11*e^(-5*d*x - 5*c) - 3*e^(-7*d*x -
7*c))/(d*(4*e^(-2*d*x - 2*c) + 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) + 1))) - a*(arctan(
e^(-d*x - c))/d - (e^(-d*x - c) - e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) + 1)))

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Fricas [B]  time = 2.15486, size = 2931, normalized size = 41.87 \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]

1/4*((4*a + 3*b)*cosh(d*x + c)^7 + 7*(4*a + 3*b)*cosh(d*x + c)*sinh(d*x + c)^6 + (4*a + 3*b)*sinh(d*x + c)^7 +
(4*a + 11*b)*cosh(d*x + c)^5 + (21*(4*a + 3*b)*cosh(d*x + c)^2 + 4*a + 11*b)*sinh(d*x + c)^5 + 5*(7*(4*a + 3*
b)*cosh(d*x + c)^3 + (4*a + 11*b)*cosh(d*x + c))*sinh(d*x + c)^4 - (4*a + 11*b)*cosh(d*x + c)^3 + (35*(4*a + 3
*b)*cosh(d*x + c)^4 + 10*(4*a + 11*b)*cosh(d*x + c)^2 - 4*a - 11*b)*sinh(d*x + c)^3 + (21*(4*a + 3*b)*cosh(d*x
+ c)^5 + 10*(4*a + 11*b)*cosh(d*x + c)^3 - 3*(4*a + 11*b)*cosh(d*x + c))*sinh(d*x + c)^2 + ((4*a + 3*b)*cosh(
d*x + c)^8 + 8*(4*a + 3*b)*cosh(d*x + c)*sinh(d*x + c)^7 + (4*a + 3*b)*sinh(d*x + c)^8 + 4*(4*a + 3*b)*cosh(d*
x + c)^6 + 4*(7*(4*a + 3*b)*cosh(d*x + c)^2 + 4*a + 3*b)*sinh(d*x + c)^6 + 8*(7*(4*a + 3*b)*cosh(d*x + c)^3 +
3*(4*a + 3*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 6*(4*a + 3*b)*cosh(d*x + c)^4 + 2*(35*(4*a + 3*b)*cosh(d*x + c)
^4 + 30*(4*a + 3*b)*cosh(d*x + c)^2 + 12*a + 9*b)*sinh(d*x + c)^4 + 8*(7*(4*a + 3*b)*cosh(d*x + c)^5 + 10*(4*a
+ 3*b)*cosh(d*x + c)^3 + 3*(4*a + 3*b)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(4*a + 3*b)*cosh(d*x + c)^2 + 4*(7*
(4*a + 3*b)*cosh(d*x + c)^6 + 15*(4*a + 3*b)*cosh(d*x + c)^4 + 9*(4*a + 3*b)*cosh(d*x + c)^2 + 4*a + 3*b)*sinh
(d*x + c)^2 + 8*((4*a + 3*b)*cosh(d*x + c)^7 + 3*(4*a + 3*b)*cosh(d*x + c)^5 + 3*(4*a + 3*b)*cosh(d*x + c)^3 +
(4*a + 3*b)*cosh(d*x + c))*sinh(d*x + c) + 4*a + 3*b)*arctan(cosh(d*x + c) + sinh(d*x + c)) - (4*a + 3*b)*cos
h(d*x + c) + (7*(4*a + 3*b)*cosh(d*x + c)^6 + 5*(4*a + 11*b)*cosh(d*x + c)^4 - 3*(4*a + 11*b)*cosh(d*x + c)^2
- 4*a - 3*b)*sinh(d*x + c))/(d*cosh(d*x + c)^8 + 8*d*cosh(d*x + c)*sinh(d*x + c)^7 + d*sinh(d*x + c)^8 + 4*d*c
osh(d*x + c)^6 + 4*(7*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^6 + 8*(7*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sin
h(d*x + c)^5 + 6*d*cosh(d*x + c)^4 + 2*(35*d*cosh(d*x + c)^4 + 30*d*cosh(d*x + c)^2 + 3*d)*sinh(d*x + c)^4 + 8
*(7*d*cosh(d*x + c)^5 + 10*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*d*cosh(d*x + c)^2 + 4*(7
*d*cosh(d*x + c)^6 + 15*d*cosh(d*x + c)^4 + 9*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^2 + 8*(d*cosh(d*x + c)^7 +
3*d*cosh(d*x + c)^5 + 3*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c) + d)

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

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Giac [B]  time = 1.20099, size = 213, normalized size = 3.04 \begin{align*} \frac{{\left (\pi + 2 \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )}{\left (4 \, a + 3 \, b\right )}}{16 \, d} + \frac{4 \, a{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 3 \, b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 16 \, a{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 20 \, b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{4 \,{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{2} d} \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]

1/16*(pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(4*a + 3*b)/d + 1/4*(4*a*(e^(d*x + c) - e^(-d*x -
c))^3 + 3*b*(e^(d*x + c) - e^(-d*x - c))^3 + 16*a*(e^(d*x + c) - e^(-d*x - c)) + 20*b*(e^(d*x + c) - e^(-d*x
- c)))/(((e^(d*x + c) - e^(-d*x - c))^2 + 4)^2*d)