3.3.86 \(\int \cosh (c+d x) (a+b \sinh ^2(c+d x)) \, dx\) [286]

Optimal. Leaf size=28 \[ \frac {a \sinh (c+d x)}{d}+\frac {b \sinh ^3(c+d x)}{3 d} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {3269} \begin {gather*} \frac {a \sinh (c+d x)}{d}+\frac {b \sinh ^3(c+d x)}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sinh[c + d*x])/d + (b*Sinh[c + d*x]^3)/(3*d)

Rule 3269

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 39, normalized size = 1.39 \begin {gather*} \frac {a \cosh (d x) \sinh (c)}{d}+\frac {a \cosh (c) \sinh (d x)}{d}+\frac {b \sinh ^3(c+d x)}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Cosh[d*x]*Sinh[c])/d + (a*Cosh[c]*Sinh[d*x])/d + (b*Sinh[c + d*x]^3)/(3*d)

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Maple [A]
time = 0.57, size = 25, normalized size = 0.89

method	result	size
derivativedivides	\(\frac {\frac {b \left (\sinh ^{3}\left (d x +c \right )\right )}{3}+a \sinh \left (d x +c \right )}{d}\)	\(25\)
default	\(\frac {\frac {b \left (\sinh ^{3}\left (d x +c \right )\right )}{3}+a \sinh \left (d x +c \right )}{d}\)	\(25\)
risch	\(\frac {{\mathrm e}^{3 d x +3 c} b}{24 d}+\frac {a \,{\mathrm e}^{d x +c}}{2 d}-\frac {b \,{\mathrm e}^{d x +c}}{8 d}-\frac {{\mathrm e}^{-d x -c} a}{2 d}+\frac {{\mathrm e}^{-d x -c} b}{8 d}-\frac {{\mathrm e}^{-3 d x -3 c} b}{24 d}\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 26, normalized size = 0.93 \begin {gather*} \frac {b \sinh \left (d x + c\right )^{3}}{3 \, 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*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.58, size = 41, normalized size = 1.46 \begin {gather*} \frac {b \sinh \left (d x + c\right )^{3} + 3 \, {\left (b \cosh \left (d x + c\right )^{2} + 4 \, a - b\right )} \sinh \left (d x + c\right )}{12 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(b*sinh(d*x + c)^3 + 3*(b*cosh(d*x + c)^2 + 4*a - b)*sinh(d*x + c))/d

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Sympy [A]
time = 0.11, size = 36, normalized size = 1.29 \begin {gather*} \begin {cases} \frac {a \sinh {\left (c + d x \right )}}{d} + \frac {b \sinh ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right ) \cosh {\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*sinh(c + d*x)/d + b*sinh(c + d*x)**3/(3*d), Ne(d, 0)), (x*(a + b*sinh(c)**2)*cosh(c), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (26) = 52\).
time = 0.41, size = 70, normalized size = 2.50 \begin {gather*} \frac {b e^{\left (3 \, d x + 3 \, c\right )}}{24 \, d} + \frac {{\left (4 \, a - b\right )} e^{\left (d x + c\right )}}{8 \, d} - \frac {{\left (4 \, a - b\right )} e^{\left (-d x - c\right )}}{8 \, d} - \frac {b 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)*(a+b*sinh(d*x+c)^2),x, algorithm="giac")

[Out]

1/24*b*e^(3*d*x + 3*c)/d + 1/8*(4*a - b)*e^(d*x + c)/d - 1/8*(4*a - b)*e^(-d*x - c)/d - 1/24*b*e^(-3*d*x - 3*c
)/d

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Mupad [B]
time = 0.09, size = 25, normalized size = 0.89 \begin {gather*} \frac {\mathrm {sinh}\left (c+d\,x\right )\,\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+3\,a\right )}{3\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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