3.1.58 \(\int \cosh ^3(c+d x) (a+b \text {sech}^2(c+d x))^2 \, dx\) [58]

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

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4232, 398, 209} \begin {gather*} \frac {a^2 \sinh ^3(c+d x)}{3 d}+\frac {a (a+2 b) \sinh (c+d x)}{d}+\frac {b^2 \text {ArcTan}(\sinh (c+d x))}{d} \end {gather*}

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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 398

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 4232

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]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 72, normalized size = 1.47 \begin {gather*} \frac {b^2 \text {ArcTan}(\sinh (c+d x))}{d}+\frac {2 a b \cosh (d x) \sinh (c)}{d}+\frac {2 a b \cosh (c) \sinh (d x)}{d}+\frac {a^2 \sinh (c+d x)}{d}+\frac {a^2 \sinh ^3(c+d x)}{3 d} \end {gather*}

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 [C] Result contains complex when optimal does not.
time = 1.98, size = 133, normalized size = 2.71

method	result	size
risch	\(\frac {a^{2} {\mathrm e}^{3 d x +3 c}}{24 d}+\frac {3 a^{2} {\mathrm e}^{d x +c}}{8 d}+\frac {a b \,{\mathrm e}^{d x +c}}{d}-\frac {3 a^{2} {\mathrm e}^{-d x -c}}{8 d}-\frac {a \,{\mathrm e}^{-d x -c} b}{d}-\frac {a^{2} {\mathrm e}^{-3 d x -3 c}}{24 d}+\frac {i b^{2} \ln \left ({\mathrm e}^{d x +c}+i\right )}{d}-\frac {i b^{2} \ln \left ({\mathrm e}^{d x +c}-i\right )}{d}\)	\(133\)

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,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)+I*b^2/d*ln(exp(d*x+c)+I)-I*b^2/d*ln(exp(d*x+c)-I)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (47) = 94\).
time = 0.48, size = 105, normalized size = 2.14 \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 {e^{\left (-d x - c\right )}}{d}\right )} - \frac {2 \, b^{2} \arctan \left (e^{\left (-d x - c\right )}\right )}{d} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 414 vs. \(2 (47) = 94\).
time = 0.39, size = 414, normalized size = 8.45 \begin {gather*} \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 {gather*}

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]
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} \cosh ^{3}{\left (c + d x \right )}\, dx \end {gather*}

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]

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

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Giac [A]
time = 0.40, size = 94, normalized size = 1.92 \begin {gather*} \frac {48 \, b^{2} \arctan \left (e^{\left (d x + c\right )}\right ) + a^{2} e^{\left (3 \, d x + 3 \, c\right )} + 9 \, a^{2} e^{\left (d x + c\right )} + 24 \, a b e^{\left (d x + c\right )} - {\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} \end {gather*}

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]

1/24*(48*b^2*arctan(e^(d*x + c)) + a^2*e^(3*d*x + 3*c) + 9*a^2*e^(d*x + c) + 24*a*b*e^(d*x + c) - (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

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Mupad [B]
time = 0.17, size = 114, normalized size = 2.33 \begin {gather*} \frac {2\,\mathrm {atan}\left (\frac {b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {d^2}}{d\,\sqrt {b^4}}\right )\,\sqrt {b^4}}{\sqrt {d^2}}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (3\,a^2+8\,b\,a\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 {a\,{\mathrm {e}}^{c+d\,x}\,\left (3\,a+8\,b\right )}{8\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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