3.1.10 \(\int (a+b \text {sech}^2(c+d x))^2 \sinh ^3(c+d x) \, dx\) [10]

Optimal. Leaf size=72 \[ -\frac {a (a-2 b) \cosh (c+d x)}{d}+\frac {a^2 \cosh ^3(c+d x)}{3 d}+\frac {(2 a-b) b \text {sech}(c+d x)}{d}+\frac {b^2 \text {sech}^3(c+d x)}{3 d} \]

[Out]

-a*(a-2*b)*cosh(d*x+c)/d+1/3*a^2*cosh(d*x+c)^3/d+(2*a-b)*b*sech(d*x+c)/d+1/3*b^2*sech(d*x+c)^3/d

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Rubi [A]
time = 0.06, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {4218, 459} \begin {gather*} \frac {a^2 \cosh ^3(c+d x)}{3 d}-\frac {a (a-2 b) \cosh (c+d x)}{d}+\frac {b (2 a-b) \text {sech}(c+d x)}{d}+\frac {b^2 \text {sech}^3(c+d x)}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rule 4218

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

Rubi steps

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

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Mathematica [A]
time = 0.36, size = 83, normalized size = 1.15 \begin {gather*} \frac {\left (-26 a^2+168 a b-16 b^2-3 \left (11 a^2-64 a b+16 b^2\right ) \cosh (2 (c+d x))-6 a (a-4 b) \cosh (4 (c+d x))+a^2 \cosh (6 (c+d x))\right ) \text {sech}^3(c+d x)}{96 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-26*a^2 + 168*a*b - 16*b^2 - 3*(11*a^2 - 64*a*b + 16*b^2)*Cosh[2*(c + d*x)] - 6*a*(a - 4*b)*Cosh[4*(c + d*x)
] + a^2*Cosh[6*(c + d*x)])*Sech[c + d*x]^3)/(96*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(173\) vs. \(2(68)=136\).
time = 1.73, size = 174, normalized size = 2.42

method	result	size
risch	\(\frac {a^{2} {\mathrm e}^{3 d x +3 c}}{24 d}-\frac {3 a^{2} {\mathrm e}^{d x +c}}{8 d}+\frac {a b \,{\mathrm e}^{d x +c}}{d}-\frac {3 a^{2} {\mathrm e}^{-d x -c}}{8 d}+\frac {a \,{\mathrm e}^{-d x -c} b}{d}+\frac {a^{2} {\mathrm e}^{-3 d x -3 c}}{24 d}+\frac {2 \,{\mathrm e}^{d x +c} b \left (6 a \,{\mathrm e}^{4 d x +4 c}-3 b \,{\mathrm e}^{4 d x +4 c}+12 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+6 a -3 b \right )}{3 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{3}}\)	\(174\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sech(d*x+c)^2)^2*sinh(d*x+c)^3,x,method=_RETURNVERBOSE)

[Out]

1/24*a^2/d*exp(3*d*x+3*c)-3/8*a^2/d*exp(d*x+c)+a*b/d*exp(d*x+c)-3/8*a^2/d*exp(-d*x-c)+a/d*exp(-d*x-c)*b+1/24*a
^2/d*exp(-3*d*x-3*c)+2/3*exp(d*x+c)*b*(6*a*exp(4*d*x+4*c)-3*b*exp(4*d*x+4*c)+12*a*exp(2*d*x+2*c)-2*b*exp(2*d*x
+2*c)+6*a-3*b)/d/(1+exp(2*d*x+2*c))^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (68) = 136\).
time = 0.30, size = 266, normalized size = 3.69 \begin {gather*} \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 {5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1}{d {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}\right )} - \frac {2}{3} \, b^{2} {\left (\frac {3 \, e^{\left (-d x - c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac {2 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sech(d*x+c)^2)^2*sinh(d*x+c)^3,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 +
(5*e^(-2*d*x - 2*c) + 1)/(d*(e^(-d*x - c) + e^(-3*d*x - 3*c)))) - 2/3*b^2*(3*e^(-d*x - c)/(d*(3*e^(-2*d*x - 2
*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1)) + 2*e^(-3*d*x - 3*c)/(d*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x -
4*c) + e^(-6*d*x - 6*c) + 1)) + 3*e^(-5*d*x - 5*c)/(d*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6
*c) + 1)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (68) = 136\).
time = 0.41, size = 212, normalized size = 2.94 \begin {gather*} \frac {a^{2} \cosh \left (d x + c\right )^{6} + a^{2} \sinh \left (d x + c\right )^{6} - 6 \, {\left (a^{2} - 4 \, a b\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, a^{2} \cosh \left (d x + c\right )^{2} - 2 \, a^{2} + 8 \, a b\right )} \sinh \left (d x + c\right )^{4} - 3 \, {\left (11 \, a^{2} - 64 \, a b + 16 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, a^{2} \cosh \left (d x + c\right )^{4} - 12 \, {\left (a^{2} - 4 \, a b\right )} \cosh \left (d x + c\right )^{2} - 11 \, a^{2} + 64 \, a b - 16 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} - 26 \, a^{2} + 168 \, a b - 16 \, b^{2}}{24 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(a^2*cosh(d*x + c)^6 + a^2*sinh(d*x + c)^6 - 6*(a^2 - 4*a*b)*cosh(d*x + c)^4 + 3*(5*a^2*cosh(d*x + c)^2 -
2*a^2 + 8*a*b)*sinh(d*x + c)^4 - 3*(11*a^2 - 64*a*b + 16*b^2)*cosh(d*x + c)^2 + 3*(5*a^2*cosh(d*x + c)^4 - 12
*(a^2 - 4*a*b)*cosh(d*x + c)^2 - 11*a^2 + 64*a*b - 16*b^2)*sinh(d*x + c)^2 - 26*a^2 + 168*a*b - 16*b^2)/(d*cos
h(d*x + c)^3 + 3*d*cosh(d*x + c)*sinh(d*x + c)^2 + 3*d*cosh(d*x + c))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2} \sinh ^{3}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sech(d*x+c)**2)**2*sinh(d*x+c)**3,x)

[Out]

Integral((a + b*sech(c + d*x)**2)**2*sinh(c + d*x)**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (68) = 136\).
time = 0.42, size = 140, normalized size = 1.94 \begin {gather*} \frac {a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 12 \, a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 24 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + \frac {16 \, {\left (6 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 3 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 4 \, b^{2}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3}}}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(a^2*(e^(d*x + c) + e^(-d*x - c))^3 - 12*a^2*(e^(d*x + c) + e^(-d*x - c)) + 24*a*b*(e^(d*x + c) + e^(-d*x
- c)) + 16*(6*a*b*(e^(d*x + c) + e^(-d*x - c))^2 - 3*b^2*(e^(d*x + c) + e^(-d*x - c))^2 + 4*b^2)/(e^(d*x + c)
+ e^(-d*x - c))^3)/d

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Mupad [B]
time = 1.50, size = 201, normalized size = 2.79 \begin {gather*} \frac {{\mathrm {e}}^{c+d\,x}\,\left (8\,a\,b-3\,a^2\right )}{8\,d}+\frac {{\mathrm {e}}^{-c-d\,x}\,\left (8\,a\,b-3\,a^2\right )}{8\,d}+\frac {a^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}+\frac {a^2\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {8\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (2\,a\,b-b^2\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {8\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^3*(a + b/cosh(c + d*x)^2)^2,x)

[Out]

(exp(c + d*x)*(8*a*b - 3*a^2))/(8*d) + (exp(- c - d*x)*(8*a*b - 3*a^2))/(8*d) + (a^2*exp(- 3*c - 3*d*x))/(24*d
) + (a^2*exp(3*c + 3*d*x))/(24*d) - (8*b^2*exp(c + d*x))/(3*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6
*c + 6*d*x) + 1)) + (2*exp(c + d*x)*(2*a*b - b^2))/(d*(exp(2*c + 2*d*x) + 1)) + (8*b^2*exp(c + d*x))/(3*d*(2*e
xp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1))

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