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

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

[Out]

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

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Rubi [A]  time = 0.0470149, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3190, 385, 199, 203} \[ \frac{(3 a+b) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac{(a-b) \tanh (c+d x) \text{sech}^3(c+d x)}{4 d}+\frac{(3 a+b) \tanh (c+d x) \text{sech}(c+d x)}{8 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3190

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

Rule 385

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

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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])

Rubi steps

\begin{align*} \int \text{sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b x^2}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{(a-b) \text{sech}^3(c+d x) \tanh (c+d x)}{4 d}+\frac{(3 a+b) \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 d}\\ &=\frac{(3 a+b) \text{sech}(c+d x) \tanh (c+d x)}{8 d}+\frac{(a-b) \text{sech}^3(c+d x) \tanh (c+d x)}{4 d}+\frac{(3 a+b) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{8 d}\\ &=\frac{(3 a+b) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac{(3 a+b) \text{sech}(c+d x) \tanh (c+d x)}{8 d}+\frac{(a-b) \text{sech}^3(c+d x) \tanh (c+d x)}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.149494, size = 60, normalized size = 0.86 \[ \frac{(3 a+b) \tan ^{-1}(\sinh (c+d x))+2 (a-b) \tanh (c+d x) \text{sech}^3(c+d x)+(3 a+b) \tanh (c+d x) \text{sech}(c+d x)}{8 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.072, size = 124, normalized size = 1.8 \begin{align*}{\frac{\tanh \left ( dx+c \right ) a \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{4\,d}}+{\frac{3\,a{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{8\,d}}+{\frac{3\,a\arctan \left ({{\rm e}^{dx+c}} \right ) }{4\,d}}-{\frac{b\sinh \left ( dx+c \right ) }{3\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}+{\frac{b\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{12\,d}}+{\frac{b{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{8\,d}}+{\frac{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)^5*(a+b*sinh(d*x+c)^2),x)

[Out]

1/4/d*a*tanh(d*x+c)*sech(d*x+c)^3+3/8/d*a*sech(d*x+c)*tanh(d*x+c)+3/4/d*a*arctan(exp(d*x+c))-1/3/d*b*sinh(d*x+
c)/cosh(d*x+c)^4+1/12/d*b*tanh(d*x+c)*sech(d*x+c)^3+1/8/d*b*sech(d*x+c)*tanh(d*x+c)+1/4/d*b*arctan(exp(d*x+c))

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Maxima [B]  time = 1.53705, size = 308, normalized size = 4.4 \begin{align*} -\frac{1}{4} \, a{\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 )} - \frac{1}{4} \, b{\left (\frac{\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{e^{\left (-d x - c\right )} - 7 \, e^{\left (-3 \, d x - 3 \, c\right )} + 7 \, e^{\left (-5 \, d x - 5 \, c\right )} - 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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^5*(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

-1/4*a*(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))) - 1/4*b*(arc
tan(e^(-d*x - c))/d - (e^(-d*x - c) - 7*e^(-3*d*x - 3*c) + 7*e^(-5*d*x - 5*c) - 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)))

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Fricas [B]  time = 1.60049, size = 2838, normalized size = 40.54 \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*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

1/4*((3*a + b)*cosh(d*x + c)^7 + 7*(3*a + b)*cosh(d*x + c)*sinh(d*x + c)^6 + (3*a + b)*sinh(d*x + c)^7 + (11*a
- 7*b)*cosh(d*x + c)^5 + (21*(3*a + b)*cosh(d*x + c)^2 + 11*a - 7*b)*sinh(d*x + c)^5 + 5*(7*(3*a + b)*cosh(d*
x + c)^3 + (11*a - 7*b)*cosh(d*x + c))*sinh(d*x + c)^4 - (11*a - 7*b)*cosh(d*x + c)^3 + (35*(3*a + b)*cosh(d*x
+ c)^4 + 10*(11*a - 7*b)*cosh(d*x + c)^2 - 11*a + 7*b)*sinh(d*x + c)^3 + (21*(3*a + b)*cosh(d*x + c)^5 + 10*(
11*a - 7*b)*cosh(d*x + c)^3 - 3*(11*a - 7*b)*cosh(d*x + c))*sinh(d*x + c)^2 + ((3*a + b)*cosh(d*x + c)^8 + 8*(
3*a + b)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a + b)*sinh(d*x + c)^8 + 4*(3*a + b)*cosh(d*x + c)^6 + 4*(7*(3*a +
b)*cosh(d*x + c)^2 + 3*a + b)*sinh(d*x + c)^6 + 8*(7*(3*a + b)*cosh(d*x + c)^3 + 3*(3*a + b)*cosh(d*x + c))*s
inh(d*x + c)^5 + 6*(3*a + b)*cosh(d*x + c)^4 + 2*(35*(3*a + b)*cosh(d*x + c)^4 + 30*(3*a + b)*cosh(d*x + c)^2
+ 9*a + 3*b)*sinh(d*x + c)^4 + 8*(7*(3*a + b)*cosh(d*x + c)^5 + 10*(3*a + b)*cosh(d*x + c)^3 + 3*(3*a + b)*cos
h(d*x + c))*sinh(d*x + c)^3 + 4*(3*a + b)*cosh(d*x + c)^2 + 4*(7*(3*a + b)*cosh(d*x + c)^6 + 15*(3*a + b)*cosh
(d*x + c)^4 + 9*(3*a + b)*cosh(d*x + c)^2 + 3*a + b)*sinh(d*x + c)^2 + 8*((3*a + b)*cosh(d*x + c)^7 + 3*(3*a +
b)*cosh(d*x + c)^5 + 3*(3*a + b)*cosh(d*x + c)^3 + (3*a + b)*cosh(d*x + c))*sinh(d*x + c) + 3*a + b)*arctan(c
osh(d*x + c) + sinh(d*x + c)) - (3*a + b)*cosh(d*x + c) + (7*(3*a + b)*cosh(d*x + c)^6 + 5*(11*a - 7*b)*cosh(d
*x + c)^4 - 3*(11*a - 7*b)*cosh(d*x + c)^2 - 3*a - b)*sinh(d*x + c))/(d*cosh(d*x + c)^8 + 8*d*cosh(d*x + c)*si
nh(d*x + c)^7 + d*sinh(d*x + c)^8 + 4*d*cosh(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))*sinh(d*x + c)^5 + 6*d*cosh(d*x + c)^4 + 2*(35*d*cosh(d*x + c)^4 + 30*d*c
osh(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))*sin
h(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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)**5*(a+b*sinh(d*x+c)**2),x)

[Out]

Timed out

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Giac [B]  time = 1.15173, size = 209, normalized size = 2.99 \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 (3 \, a + b\right )}}{16 \, d} + \frac{3 \, a{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 20 \, a{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 4 \, 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)^5*(a+b*sinh(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)))*(3*a + b)/d + 1/4*(3*a*(e^(d*x + c) - e^(-d*x - c
))^3 + b*(e^(d*x + c) - e^(-d*x - c))^3 + 20*a*(e^(d*x + c) - e^(-d*x - c)) - 4*b*(e^(d*x + c) - e^(-d*x - c))
)/(((e^(d*x + c) - e^(-d*x - c))^2 + 4)^2*d)