3.1.5 \(\int (a+b \sinh ^2(c+d x)) \, dx\) [5]

Optimal. Leaf size=30 \[ a x-\frac {b x}{2}+\frac {b \cosh (c+d x) \sinh (c+d x)}{2 d} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2715, 8} \begin {gather*} a x+\frac {b \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac {b x}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a + b*Sinh[c + d*x]^2,x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 36, normalized size = 1.20 \begin {gather*} a x+\frac {b (-c-d x)}{2 d}+\frac {b \sinh (2 (c+d x))}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a + b*Sinh[c + d*x]^2,x]

[Out]

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

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Maple [A]
time = 0.74, size = 32, normalized size = 1.07

method	result	size
default	\(a x +\frac {b \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d}\)	\(32\)
derivativedivides	\(\frac {\left (d x +c \right ) a +b \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d}\)	\(37\)
risch	\(a x -\frac {b x}{2}+\frac {{\mathrm e}^{2 d x +2 c} b}{8 d}-\frac {{\mathrm e}^{-2 d x -2 c} b}{8 d}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 38, normalized size = 1.27 \begin {gather*} -\frac {1}{8} \, b {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.41, size = 30, normalized size = 1.00 \begin {gather*} \frac {{\left (2 \, a - b\right )} d x + b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.07, size = 51, normalized size = 1.70 \begin {gather*} a x + b \left (\begin {cases} \frac {x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac {x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {\sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \sinh ^{2}{\left (c \right )} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.41, size = 38, normalized size = 1.27 \begin {gather*} -\frac {1}{8} \, b {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.07, size = 23, normalized size = 0.77 \begin {gather*} a\,x-\frac {b\,x}{2}+\frac {b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{4\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*sinh(c + d*x)^2,x)

[Out]

a*x - (b*x)/2 + (b*sinh(2*c + 2*d*x))/(4*d)

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