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

Optimal. Leaf size=44 \[ -\frac {(a-b) \cosh (c+d x)}{d}+\frac {a \cosh ^3(c+d x)}{3 d}+\frac {b \text {sech}(c+d x)}{d} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4218, 459} \begin {gather*} -\frac {(a-b) \cosh (c+d x)}{d}+\frac {a \cosh ^3(c+d x)}{3 d}+\frac {b \text {sech}(c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.04, size = 53, normalized size = 1.20 \begin {gather*} -\frac {3 a \cosh (c+d x)}{4 d}+\frac {b \cosh (c+d x)}{d}+\frac {a \cosh (3 (c+d x))}{12 d}+\frac {b \text {sech}(c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*Cosh[c + d*x])/(4*d) + (b*Cosh[c + d*x])/d + (a*Cosh[3*(c + d*x)])/(12*d) + (b*Sech[c + d*x])/d

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(110\) vs. \(2(42)=84\).
time = 1.38, size = 111, normalized size = 2.52

method	result	size
risch	\(\frac {a \,{\mathrm e}^{3 d x +3 c}}{24 d}-\frac {3 a \,{\mathrm e}^{d x +c}}{8 d}+\frac {b \,{\mathrm e}^{d x +c}}{2 d}-\frac {3 \,{\mathrm e}^{-d x -c} a}{8 d}+\frac {{\mathrm e}^{-d x -c} b}{2 d}+\frac {a \,{\mathrm e}^{-3 d x -3 c}}{24 d}+\frac {2 b \,{\mathrm e}^{d x +c}}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )}\)	\(111\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (42) = 84\).
time = 0.29, size = 111, normalized size = 2.52 \begin {gather*} \frac {1}{24} \, a {\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 )} + \frac {1}{2} \, 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*a*(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) + 1/2*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))))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (42) = 84\).
time = 0.45, size = 85, normalized size = 1.93 \begin {gather*} \frac {a \cosh \left (d x + c\right )^{4} + a \sinh \left (d x + c\right )^{4} - 4 \, {\left (2 \, a - 3 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} - 4 \, a + 6 \, b\right )} \sinh \left (d x + c\right )^{2} - 9 \, a + 36 \, b}{24 \, d \cosh \left (d x + c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(a*cosh(d*x + c)^4 + a*sinh(d*x + c)^4 - 4*(2*a - 3*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - 4*a + 6
*b)*sinh(d*x + c)^2 - 9*a + 36*b)/(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 ) \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)*sinh(d*x+c)**3,x)

[Out]

Integral((a + b*sech(c + d*x)**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. 85 vs. \(2 (42) = 84\).
time = 0.40, size = 85, normalized size = 1.93 \begin {gather*} \frac {a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 12 \, a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 12 \, b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + \frac {48 \, b}{e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(a*(e^(d*x + c) + e^(-d*x - c))^3 - 12*a*(e^(d*x + c) + e^(-d*x - c)) + 12*b*(e^(d*x + c) + e^(-d*x - c))
+ 48*b/(e^(d*x + c) + e^(-d*x - c)))/d

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Mupad [B]
time = 0.18, size = 44, normalized size = 1.00 \begin {gather*} \frac {a\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3\,d}-\frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (a-b\right )}{d}+\frac {b}{d\,\mathrm {cosh}\left (c+d\,x\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),x)

[Out]

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

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