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

Optimal. Leaf size=55 \[ \frac{b (2 a-b) \sinh (c+d x)}{d}+\frac{(a-b)^2 \tan ^{-1}(\sinh (c+d x))}{d}+\frac{b^2 \sinh ^3(c+d x)}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.0577477, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3190, 390, 203} \[ \frac{b (2 a-b) \sinh (c+d x)}{d}+\frac{(a-b)^2 \tan ^{-1}(\sinh (c+d x))}{d}+\frac{b^2 \sinh ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a - b)^2*ArcTan[Sinh[c + d*x]])/d + ((2*a - b)*b*Sinh[c + d*x])/d + (b^2*Sinh[c + d*x]^3)/(3*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 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

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

Mathematica [A]  time = 0.227902, size = 70, normalized size = 1.27 \[ \frac{\sinh (c+d x) \left (b \left (6 a+b \left (\sinh ^2(c+d x)-3\right )\right )+\frac{3 (a-b)^2 \tanh ^{-1}\left (\sqrt{-\sinh ^2(c+d x)}\right )}{\sqrt{-\sinh ^2(c+d x)}}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sinh[c + d*x]*((3*(a - b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]])/Sqrt[-Sinh[c + d*x]^2] + b*(6*a + b*(-3 + Sinh[c
+ d*x]^2))))/(3*d)

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Maple [A]  time = 0.038, size = 89, normalized size = 1.6 \begin{align*} 2\,{\frac{{a}^{2}\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}}+2\,{\frac{ab\sinh \left ( dx+c \right ) }{d}}-4\,{\frac{ab\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}}+{\frac{{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{{b}^{2}\sinh \left ( dx+c \right ) }{d}}+2\,{\frac{{b}^{2}\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*a^2*arctan(exp(d*x+c))+2/d*a*b*sinh(d*x+c)-4/d*a*b*arctan(exp(d*x+c))+1/3*b^2*sinh(d*x+c)^3/d-b^2*sinh(d*x
+c)/d+2/d*b^2*arctan(exp(d*x+c))

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Maxima [B]  time = 1.72436, size = 180, normalized size = 3.27 \begin{align*} -\frac{1}{24} \, b^{2}{\left (\frac{{\left (15 \, e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )} e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac{15 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac{48 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d}\right )} + a b{\left (\frac{4 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac{e^{\left (d x + c\right )}}{d} - \frac{e^{\left (-d x - c\right )}}{d}\right )} + \frac{a^{2} \arctan \left (\sinh \left (d x + c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*b^2*((15*e^(-2*d*x - 2*c) - 1)*e^(3*d*x + 3*c)/d - (15*e^(-d*x - c) - e^(-3*d*x - 3*c))/d + 48*arctan(e^
(-d*x - c))/d) + a*b*(4*arctan(e^(-d*x - c))/d + e^(d*x + c)/d - e^(-d*x - c)/d) + a^2*arctan(sinh(d*x + c))/d

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Fricas [B]  time = 1.60384, size = 1143, normalized size = 20.78 \begin{align*} \frac{b^{2} \cosh \left (d x + c\right )^{6} + 6 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + b^{2} \sinh \left (d x + c\right )^{6} + 3 \,{\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 3 \,{\left (5 \, b^{2} \cosh \left (d x + c\right )^{2} + 8 \, a b - 5 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \,{\left (5 \, b^{2} \cosh \left (d x + c\right )^{3} + 3 \,{\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 3 \,{\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \,{\left (5 \, b^{2} \cosh \left (d x + c\right )^{4} + 6 \,{\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} - 8 \, a b + 5 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} - b^{2} + 48 \,{\left ({\left (a^{2} - 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} +{\left (a^{2} - 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{3}\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + 6 \,{\left (b^{2} \cosh \left (d x + c\right )^{5} + 2 \,{\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} -{\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{24 \,{\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(b^2*cosh(d*x + c)^6 + 6*b^2*cosh(d*x + c)*sinh(d*x + c)^5 + b^2*sinh(d*x + c)^6 + 3*(8*a*b - 5*b^2)*cosh
(d*x + c)^4 + 3*(5*b^2*cosh(d*x + c)^2 + 8*a*b - 5*b^2)*sinh(d*x + c)^4 + 4*(5*b^2*cosh(d*x + c)^3 + 3*(8*a*b
- 5*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 3*(8*a*b - 5*b^2)*cosh(d*x + c)^2 + 3*(5*b^2*cosh(d*x + c)^4 + 6*(8*
a*b - 5*b^2)*cosh(d*x + c)^2 - 8*a*b + 5*b^2)*sinh(d*x + c)^2 - b^2 + 48*((a^2 - 2*a*b + b^2)*cosh(d*x + c)^3
+ 3*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2 - 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^2 +
(a^2 - 2*a*b + b^2)*sinh(d*x + c)^3)*arctan(cosh(d*x + c) + sinh(d*x + c)) + 6*(b^2*cosh(d*x + c)^5 + 2*(8*a*b
- 5*b^2)*cosh(d*x + c)^3 - (8*a*b - 5*b^2)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^3 + 3*d*cosh(d*x +
c)^2*sinh(d*x + c) + 3*d*cosh(d*x + c)*sinh(d*x + c)^2 + d*sinh(d*x + c)^3)

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

[Out]

Timed out

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Giac [B]  time = 1.18227, size = 159, normalized size = 2.89 \begin{align*} \frac{2 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} \arctan \left (e^{\left (d x + c\right )}\right )}{d} - \frac{{\left (24 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 15 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} + \frac{b^{2} d^{2} e^{\left (3 \, d x + 3 \, c\right )} + 24 \, a b d^{2} e^{\left (d x + c\right )} - 15 \, b^{2} d^{2} e^{\left (d x + c\right )}}{24 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(a^2 - 2*a*b + b^2)*arctan(e^(d*x + c))/d - 1/24*(24*a*b*e^(2*d*x + 2*c) - 15*b^2*e^(2*d*x + 2*c) + b^2)*e^(
-3*d*x - 3*c)/d + 1/24*(b^2*d^2*e^(3*d*x + 3*c) + 24*a*b*d^2*e^(d*x + c) - 15*b^2*d^2*e^(d*x + c))/d^3