3.1.52 \(\int \cosh (c+d x) (a+b \text {sech}^2(c+d x)) \, dx\) [52]

Optimal. Leaf size=24 \[ \frac {b \text {ArcTan}(\sinh (c+d x))}{d}+\frac {a \sinh (c+d x)}{d} \]

[Out]

b*arctan(sinh(d*x+c))/d+a*sinh(d*x+c)/d

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Rubi [A]
time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {4130, 3855} \begin {gather*} \frac {a \sinh (c+d x)}{d}+\frac {b \text {ArcTan}(\sinh (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4130

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[A*Cot[e
+ f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + Dist[(C*m + A*(m + 1))/(b^2*m), Int[(b*Csc[e + f*x])^(m + 2), x], x] /
; FreeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 35, normalized size = 1.46 \begin {gather*} \frac {b \text {ArcTan}(\sinh (c+d x))}{d}+\frac {a \cosh (d x) \sinh (c)}{d}+\frac {a \cosh (c) \sinh (d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 1.50, size = 24, normalized size = 1.00

method	result	size
derivativedivides	\(\frac {a \sinh \left (d x +c \right )+2 b \arctan \left ({\mathrm e}^{d x +c}\right )}{d}\)	\(24\)
default	\(\frac {a \sinh \left (d x +c \right )+2 b \arctan \left ({\mathrm e}^{d x +c}\right )}{d}\)	\(24\)
risch	\(\frac {a \,{\mathrm e}^{d x +c}}{2 d}-\frac {{\mathrm e}^{-d x -c} a}{2 d}+\frac {i b \ln \left ({\mathrm e}^{d x +c}+i\right )}{d}-\frac {i b \ln \left ({\mathrm e}^{d x +c}-i\right )}{d}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.52, size = 28, normalized size = 1.17 \begin {gather*} -\frac {2 \, b \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {a \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),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (24) = 48\).
time = 0.37, size = 93, normalized size = 3.88 \begin {gather*} \frac {a \cosh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - a}{2 \, {\left (d \cosh \left (d x + c\right ) + d \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),x, algorithm="fricas")

[Out]

1/2*(a*cosh(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c) + b*sinh(d*x
+ c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - a)/(d*cosh(d*x + c) + 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 ) \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),x)

[Out]

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

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Giac [A]
time = 0.40, size = 36, normalized size = 1.50 \begin {gather*} \frac {4 \, b \arctan \left (e^{\left (d x + c\right )}\right ) + a e^{\left (d x + c\right )} - a e^{\left (-d x - c\right )}}{2 \, 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),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.44, size = 62, normalized size = 2.58 \begin {gather*} \frac {2\,\mathrm {atan}\left (\frac {b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {d^2}}{d\,\sqrt {b^2}}\right )\,\sqrt {b^2}}{\sqrt {d^2}}-\frac {a\,{\mathrm {e}}^{-c-d\,x}}{2\,d}+\frac {a\,{\mathrm {e}}^{c+d\,x}}{2\,d} \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),x)

[Out]

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

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