3.11.28 \(\int \cosh (x) \sinh (x) \sqrt {a+b \sinh ^2(x)} \, dx\) [1028]

Optimal. Leaf size=19 \[ \frac {\left (a+b \sinh ^2(x)\right )^{3/2}}{3 b} \]

[Out]

1/3*(a+b*sinh(x)^2)^(3/2)/b

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Rubi [A]
time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3277, 267} \begin {gather*} \frac {\left (a+b \sinh ^2(x)\right )^{3/2}}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[x]*Sinh[x]*Sqrt[a + b*Sinh[x]^2],x]

[Out]

(a + b*Sinh[x]^2)^(3/2)/(3*b)

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 3277

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^
2)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(d*ff*x)^n*(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, d, e, f, n, p}, x] && IntegerQ[
(m - 1)/2]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} \frac {\left (a+b \sinh ^2(x)\right )^{3/2}}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]*Sinh[x]*Sqrt[a + b*Sinh[x]^2],x]

[Out]

(a + b*Sinh[x]^2)^(3/2)/(3*b)

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Maple [A]
time = 0.22, size = 16, normalized size = 0.84

method	result	size
derivativedivides	\(\frac {\left (a +b \left (\sinh ^{2}\left (x \right )\right )\right )^{\frac {3}{2}}}{3 b}\)	\(16\)
default	\(\frac {\left (a +b \left (\sinh ^{2}\left (x \right )\right )\right )^{\frac {3}{2}}}{3 b}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*sinh(x)*(a+b*sinh(x)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(a+b*sinh(x)^2)^(3/2)/b

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Maxima [A]
time = 0.26, size = 15, normalized size = 0.79 \begin {gather*} \frac {{\left (b \sinh \left (x\right )^{2} + a\right )}^{\frac {3}{2}}}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sinh(x)*(a+b*sinh(x)^2)^(1/2),x, algorithm="maxima")

[Out]

1/3*(b*sinh(x)^2 + a)^(3/2)/b

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (15) = 30\).
time = 0.34, size = 154, normalized size = 8.11 \begin {gather*} \frac {\sqrt {2} {\left (b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} + 2 \, {\left (2 \, a - b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b \cosh \left (x\right )^{2} + 2 \, a - b\right )} \sinh \left (x\right )^{2} + 4 \, {\left (b \cosh \left (x\right )^{3} + {\left (2 \, a - b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + b\right )} \sqrt {\frac {b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a - b}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{24 \, {\left (b \cosh \left (x\right )^{3} + 3 \, b \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, b \cosh \left (x\right ) \sinh \left (x\right )^{2} + b \sinh \left (x\right )^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sinh(x)*(a+b*sinh(x)^2)^(1/2),x, algorithm="fricas")

[Out]

1/24*sqrt(2)*(b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 + 2*(2*a - b)*cosh(x)^2 + 2*(3*b*cosh(x)^2 + 2
*a - b)*sinh(x)^2 + 4*(b*cosh(x)^3 + (2*a - b)*cosh(x))*sinh(x) + b)*sqrt((b*cosh(x)^2 + b*sinh(x)^2 + 2*a - b
)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(b*cosh(x)^3 + 3*b*cosh(x)^2*sinh(x) + 3*b*cosh(x)*sinh(x)^2 +
b*sinh(x)^3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (14) = 28\).
time = 0.27, size = 46, normalized size = 2.42 \begin {gather*} \begin {cases} \frac {a \sqrt {a + b \sinh ^{2}{\left (x \right )}}}{3 b} + \frac {\sqrt {a + b \sinh ^{2}{\left (x \right )}} \sinh ^{2}{\left (x \right )}}{3} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} \cosh ^{2}{\left (x \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sinh(x)*(a+b*sinh(x)**2)**(1/2),x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sinh(x)*(a+b*sinh(x)^2)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*sinh(x)^2 + a)*cosh(x)*sinh(x), x)

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Mupad [B]
time = 1.87, size = 15, normalized size = 0.79 \begin {gather*} \frac {{\left (b\,{\mathrm {sinh}\left (x\right )}^2+a\right )}^{3/2}}{3\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*sinh(x)*(a + b*sinh(x)^2)^(1/2),x)

[Out]

(a + b*sinh(x)^2)^(3/2)/(3*b)

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