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

Optimal. Leaf size=49 \[ \frac{a^2 \sinh ^3(c+d x)}{3 d}+\frac{a (a+2 b) \sinh (c+d x)}{d}+\frac{b^2 \tan ^{-1}(\sinh (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0592035, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {4147, 390, 203} \[ \frac{a^2 \sinh ^3(c+d x)}{3 d}+\frac{a (a+2 b) \sinh (c+d x)}{d}+\frac{b^2 \tan ^{-1}(\sinh (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0260091, size = 72, normalized size = 1.47 \[ \frac{a^2 \sinh ^3(c+d x)}{3 d}+\frac{a^2 \sinh (c+d x)}{d}+\frac{2 a b \sinh (c) \cosh (d x)}{d}+\frac{2 a b \cosh (c) \sinh (d x)}{d}+\frac{b^2 \tan ^{-1}(\sinh (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.041, size = 66, normalized size = 1.4 \begin{align*}{\frac{2\,{a}^{2}\sinh \left ( dx+c \right ) }{3\,d}}+{\frac{{a}^{2}\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+2\,{\frac{ab\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(cosh(d*x+c)^3*(a+b*sech(d*x+c)^2)^2,x)

[Out]

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

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Maxima [B]  time = 1.739, size = 142, normalized size = 2.9 \begin{align*} \frac{1}{24} \, a^{2}{\left (\frac{e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac{9 \, e^{\left (d x + c\right )}}{d} - \frac{9 \, e^{\left (-d x - c\right )}}{d} - \frac{e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + a b{\left (\frac{e^{\left (d x + c\right )}}{d} - \frac{e^{\left (-d x - c\right )}}{d}\right )} - \frac{2 \, b^{2} \arctan \left (e^{\left (-d x - c\right )}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^3*(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

1/24*a^2*(e^(3*d*x + 3*c)/d + 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d - e^(-3*d*x - 3*c)/d) + a*b*(e^(d*x + c)/d -
e^(-d*x - c)/d) - 2*b^2*arctan(e^(-d*x - c))/d

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Fricas [B]  time = 2.2106, size = 1057, normalized size = 21.57 \begin{align*} \frac{a^{2} \cosh \left (d x + c\right )^{6} + 6 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + a^{2} \sinh \left (d x + c\right )^{6} + 3 \,{\left (3 \, a^{2} + 8 \, a b\right )} \cosh \left (d x + c\right )^{4} + 3 \,{\left (5 \, a^{2} \cosh \left (d x + c\right )^{2} + 3 \, a^{2} + 8 \, a b\right )} \sinh \left (d x + c\right )^{4} + 4 \,{\left (5 \, a^{2} \cosh \left (d x + c\right )^{3} + 3 \,{\left (3 \, a^{2} + 8 \, a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 3 \,{\left (3 \, a^{2} + 8 \, a b\right )} \cosh \left (d x + c\right )^{2} + 3 \,{\left (5 \, a^{2} \cosh \left (d x + c\right )^{4} + 6 \,{\left (3 \, a^{2} + 8 \, a b\right )} \cosh \left (d x + c\right )^{2} - 3 \, a^{2} - 8 \, a b\right )} \sinh \left (d x + c\right )^{2} - a^{2} + 48 \,{\left (b^{2} \cosh \left (d x + c\right )^{3} + 3 \, b^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b^{2} \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 (a^{2} \cosh \left (d x + c\right )^{5} + 2 \,{\left (3 \, a^{2} + 8 \, a b\right )} \cosh \left (d x + c\right )^{3} -{\left (3 \, a^{2} + 8 \, a b\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(cosh(d*x+c)^3*(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [B]  time = 1.20217, size = 149, normalized size = 3.04 \begin{align*} \frac{2 \, b^{2} \arctan \left (e^{\left (d x + c\right )}\right )}{d} - \frac{{\left (9 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 24 \, a b e^{\left (2 \, d x + 2 \, c\right )} + a^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} + \frac{a^{2} d^{2} e^{\left (3 \, d x + 3 \, c\right )} + 9 \, a^{2} d^{2} e^{\left (d x + c\right )} + 24 \, a b 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(cosh(d*x+c)^3*(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")

[Out]

2*b^2*arctan(e^(d*x + c))/d - 1/24*(9*a^2*e^(2*d*x + 2*c) + 24*a*b*e^(2*d*x + 2*c) + a^2)*e^(-3*d*x - 3*c)/d +
1/24*(a^2*d^2*e^(3*d*x + 3*c) + 9*a^2*d^2*e^(d*x + c) + 24*a*b*d^2*e^(d*x + c))/d^3