3.9.57 \(\int (a+b \cosh (c+d x) \sinh (c+d x)) \, dx\) [857]

Optimal. Leaf size=20 \[ a x+\frac {b \sinh ^2(c+d x)}{2 d} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2644, 30} \begin {gather*} a x+\frac {b \sinh ^2(c+d x)}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a + b*Cosh[c + d*x]*Sinh[c + d*x],x]

[Out]

a*x + (b*Sinh[c + d*x]^2)/(2*d)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 38, normalized size = 1.90 \begin {gather*} a x+\frac {b \cosh (2 c) \cosh (2 d x)}{4 d}+\frac {b \sinh (2 c) \sinh (2 d x)}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a + b*Cosh[c + d*x]*Sinh[c + d*x],x]

[Out]

a*x + (b*Cosh[2*c]*Cosh[2*d*x])/(4*d) + (b*Sinh[2*c]*Sinh[2*d*x])/(4*d)

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Maple [A]
time = 1.99, size = 19, normalized size = 0.95

method	result	size
default	\(a x +\frac {b \left (\cosh ^{2}\left (d x +c \right )\right )}{2 d}\)	\(19\)
derivativedivides	\(\frac {\left (d x +c \right ) a +\frac {b \left (\cosh ^{2}\left (d x +c \right )\right )}{2}}{d}\)	\(24\)
risch	\(a x +\frac {b \,{\mathrm e}^{2 d x +2 c}}{8 d}+\frac {b \,{\mathrm e}^{-2 d x -2 c}}{8 d}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*cosh(d*x+c)*sinh(d*x+c),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 18, normalized size = 0.90 \begin {gather*} a x + \frac {b \cosh \left (d x + c\right )^{2}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + 1/2*b*cosh(d*x + c)^2/d

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Fricas [A]
time = 0.33, size = 31, normalized size = 1.55 \begin {gather*} \frac {4 \, a d x + b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.07, size = 24, normalized size = 1.20 \begin {gather*} a x + b \left (\begin {cases} \frac {\cosh ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \sinh {\left (c \right )} \cosh {\left (c \right )} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*Piecewise((cosh(c + d*x)**2/(2*d), Ne(d, 0)), (x*sinh(c)*cosh(c), True))

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Giac [A]
time = 0.40, size = 34, normalized size = 1.70 \begin {gather*} a x + \frac {1}{8} \, b {\left (\frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + 1/8*b*(e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d)

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Mupad [B]
time = 1.64, size = 18, normalized size = 0.90 \begin {gather*} a\,x+\frac {b\,{\mathrm {cosh}\left (c+d\,x\right )}^2}{2\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*cosh(c + d*x)*sinh(c + d*x),x)

[Out]

a*x + (b*cosh(c + d*x)^2)/(2*d)

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