### 3.1028 $$\int \cosh (x) \sinh (x) \sqrt{a+b \sinh ^2(x)} \, dx$$

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

[Out]

(a + b*Sinh[x]^2)^(3/2)/(3*b)

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Rubi [A]  time = 0.0622082, antiderivative size = 19, normalized size of antiderivative = 1., 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 = {3198, 261} $\frac{\left (a+b \sinh ^2(x)\right )^{3/2}}{3 b}$

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 3198

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]

Rule 261

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]

Rubi steps

\begin{align*} \int \cosh (x) \sinh (x) \sqrt{a+b \sinh ^2(x)} \, dx &=\operatorname{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*}

Mathematica [A]  time = 0.0120327, size = 19, normalized size = 1. $\frac{\left (a+b \sinh ^2(x)\right )^{3/2}}{3 b}$

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 = 0.009, size = 16, normalized size = 0.8 \begin{align*}{\frac{1}{3\,b} \left ( a+b \left ( \sinh \left ( x \right ) \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

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]

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

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Maxima [A]  time = 1.06767, size = 20, normalized size = 1.05 \begin{align*} \frac{{\left (b \sinh \left (x\right )^{2} + a\right )}^{\frac{3}{2}}}{3 \, b} \end{align*}

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]  time = 2.19973, size = 464, normalized size = 24.42 \begin{align*} \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{align*}

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 [A]  time = 1.90936, size = 46, normalized size = 2.42 \begin{align*} \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} \sinh ^{2}{\left (x \right )}}{2} & \text{otherwise} \end{cases} \end{align*}

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)*sinh(x)**
2/2, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sinh \left (x\right )^{2} + a} \cosh \left (x\right ) \sinh \left (x\right )\,{d x} \end{align*}

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)