3.1.60 \(\int \cosh (c+d x) (a+b \text {sech}^2(c+d x))^2 \, dx\) [60]

Optimal. Leaf size=56 \[ \frac {b (4 a+b) \text {ArcTan}(\sinh (c+d x))}{2 d}+\frac {a^2 \sinh (c+d x)}{d}+\frac {b^2 \text {sech}(c+d x) \tanh (c+d x)}{2 d} \]

[Out]

1/2*b*(4*a+b)*arctan(sinh(d*x+c))/d+a^2*sinh(d*x+c)/d+1/2*b^2*sech(d*x+c)*tanh(d*x+c)/d

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Rubi [A]
time = 0.05, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4232, 398, 393, 209} \begin {gather*} \frac {a^2 \sinh (c+d x)}{d}+\frac {b (4 a+b) \text {ArcTan}(\sinh (c+d x))}{2 d}+\frac {b^2 \tanh (c+d x) \text {sech}(c+d x)}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(4*a + b)*ArcTan[Sinh[c + d*x]])/(2*d) + (a^2*Sinh[c + d*x])/d + (b^2*Sech[c + d*x]*Tanh[c + d*x])/(2*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 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 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 (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}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (a^2+\frac {b (2 a+b)+2 a b x^2}{\left (1+x^2\right )^2}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {a^2 \sinh (c+d x)}{d}+\frac {\text {Subst}\left (\int \frac {b (2 a+b)+2 a b x^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {a^2 \sinh (c+d x)}{d}+\frac {b^2 \text {sech}(c+d x) \tanh (c+d x)}{2 d}+\frac {(b (4 a+b)) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d}\\ &=\frac {b (4 a+b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {a^2 \sinh (c+d x)}{d}+\frac {b^2 \text {sech}(c+d x) \tanh (c+d x)}{2 d}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 80, normalized size = 1.43 \begin {gather*} \frac {2 a b \text {ArcTan}(\sinh (c+d x))}{d}+\frac {b^2 \text {ArcTan}(\sinh (c+d x))}{2 d}+\frac {a^2 \cosh (d x) \sinh (c)}{d}+\frac {a^2 \cosh (c) \sinh (d x)}{d}+\frac {b^2 \text {sech}(c+d x) \tanh (c+d x)}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains complex when optimal does not.
time = 1.96, size = 144, normalized size = 2.57

method	result	size
risch	\(\frac {a^{2} {\mathrm e}^{d x +c}}{2 d}-\frac {a^{2} {\mathrm e}^{-d x -c}}{2 d}+\frac {b^{2} {\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}+\frac {2 i b a \ln \left ({\mathrm e}^{d x +c}+i\right )}{d}+\frac {i b^{2} \ln \left ({\mathrm e}^{d x +c}+i\right )}{2 d}-\frac {2 i b a \ln \left ({\mathrm e}^{d x +c}-i\right )}{d}-\frac {i b^{2} \ln \left ({\mathrm e}^{d x +c}-i\right )}{2 d}\)	\(144\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2/d*exp(d*x+c)-1/2*a^2/d*exp(-d*x-c)+b^2*exp(d*x+c)*(exp(2*d*x+2*c)-1)/d/(1+exp(2*d*x+2*c))^2+2*I*b*a/d*
ln(exp(d*x+c)+I)+1/2*I*b^2/d*ln(exp(d*x+c)+I)-2*I*b*a/d*ln(exp(d*x+c)-I)-1/2*I*b^2/d*ln(exp(d*x+c)-I)

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Maxima [A]
time = 0.50, size = 101, normalized size = 1.80 \begin {gather*} -b^{2} {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} - \frac {4 \, a b \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {a^{2} \sinh \left (d x + c\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 653 vs. \(2 (52) = 104\).
time = 0.43, size = 653, normalized size = 11.66 \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} + {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + {\left (15 \, a^{2} \cosh \left (d x + c\right )^{2} + a^{2} + 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, a^{2} \cosh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + {\left (15 \, a^{2} \cosh \left (d x + c\right )^{4} + 6 \, {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} - a^{2} - 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} - a^{2} + 2 \, {\left ({\left (4 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{5} + 5 \, {\left (4 \, a b + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + {\left (4 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{5} + 2 \, {\left (4 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{3} + 2 \, {\left (5 \, {\left (4 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 4 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (5 \, {\left (4 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (4 \, a b + b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + {\left (4 \, a b + b^{2}\right )} \cosh \left (d x + c\right ) + {\left (5 \, {\left (4 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{4} + 6 \, {\left (4 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 4 \, a b + b^{2}\right )} \sinh \left (d x + c\right )\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + 2 \, {\left (3 \, a^{2} \cosh \left (d x + c\right )^{5} + 2 \, {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + d \sinh \left (d x + c\right )^{5} + 2 \, d \cosh \left (d x + c\right )^{3} + 2 \, {\left (5 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (5 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + d \cosh \left (d x + c\right ) + {\left (5 \, d \cosh \left (d x + c\right )^{4} + 6 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(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 + (a^2 + 2*b^2)*cosh(d*x
+ c)^4 + (15*a^2*cosh(d*x + c)^2 + a^2 + 2*b^2)*sinh(d*x + c)^4 + 4*(5*a^2*cosh(d*x + c)^3 + (a^2 + 2*b^2)*cos
h(d*x + c))*sinh(d*x + c)^3 - (a^2 + 2*b^2)*cosh(d*x + c)^2 + (15*a^2*cosh(d*x + c)^4 + 6*(a^2 + 2*b^2)*cosh(d
*x + c)^2 - a^2 - 2*b^2)*sinh(d*x + c)^2 - a^2 + 2*((4*a*b + b^2)*cosh(d*x + c)^5 + 5*(4*a*b + b^2)*cosh(d*x +
c)*sinh(d*x + c)^4 + (4*a*b + b^2)*sinh(d*x + c)^5 + 2*(4*a*b + b^2)*cosh(d*x + c)^3 + 2*(5*(4*a*b + b^2)*cos
h(d*x + c)^2 + 4*a*b + b^2)*sinh(d*x + c)^3 + 2*(5*(4*a*b + b^2)*cosh(d*x + c)^3 + 3*(4*a*b + b^2)*cosh(d*x +
c))*sinh(d*x + c)^2 + (4*a*b + b^2)*cosh(d*x + c) + (5*(4*a*b + b^2)*cosh(d*x + c)^4 + 6*(4*a*b + b^2)*cosh(d*
x + c)^2 + 4*a*b + b^2)*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) + 2*(3*a^2*cosh(d*x + c)^5 + 2*(a
^2 + 2*b^2)*cosh(d*x + c)^3 - (a^2 + 2*b^2)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^5 + 5*d*cosh(d*x +
c)*sinh(d*x + c)^4 + d*sinh(d*x + c)^5 + 2*d*cosh(d*x + c)^3 + 2*(5*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^3 + 2
*(5*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^2 + d*cosh(d*x + c) + (5*d*cosh(d*x + c)^4 + 6*d*cosh
(d*x + c)^2 + d)*sinh(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} \cosh {\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (52) = 104\).
time = 0.39, size = 112, normalized size = 2.00 \begin {gather*} \frac {2 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (4 \, a b + b^{2}\right )} + \frac {4 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*a^2*(e^(d*x + c) - e^(-d*x - c)) + (pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(4*a*b + b^2
) + 4*b^2*(e^(d*x + c) - e^(-d*x - c))/((e^(d*x + c) - e^(-d*x - c))^2 + 4))/d

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Mupad [B]
time = 1.44, size = 172, normalized size = 3.07 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (b^2\,\sqrt {d^2}+4\,a\,b\,\sqrt {d^2}\right )}{d\,\sqrt {16\,a^2\,b^2+8\,a\,b^3+b^4}}\right )\,\sqrt {16\,a^2\,b^2+8\,a\,b^3+b^4}}{\sqrt {d^2}}+\frac {a^2\,{\mathrm {e}}^{c+d\,x}}{2\,d}-\frac {a^2\,{\mathrm {e}}^{-c-d\,x}}{2\,d}+\frac {b^2\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,b^2\,{\mathrm {e}}^{c+d\,x}}{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(cosh(c + d*x)*(a + b/cosh(c + d*x)^2)^2,x)

[Out]

(atan((exp(d*x)*exp(c)*(b^2*(d^2)^(1/2) + 4*a*b*(d^2)^(1/2)))/(d*(8*a*b^3 + b^4 + 16*a^2*b^2)^(1/2)))*(8*a*b^3
+ b^4 + 16*a^2*b^2)^(1/2))/(d^2)^(1/2) + (a^2*exp(c + d*x))/(2*d) - (a^2*exp(- c - d*x))/(2*d) + (b^2*exp(c +
d*x))/(d*(exp(2*c + 2*d*x) + 1)) - (2*b^2*exp(c + d*x))/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1))

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