### 3.530 $$\int x \sqrt{\cosh (a+b x)} \sinh (a+b x) \, dx$$

Optimal. Leaf size=64 $\frac{4 i \text{EllipticF}\left (\frac{1}{2} i (a+b x),2\right )}{9 b^2}-\frac{4 \sinh (a+b x) \sqrt{\cosh (a+b x)}}{9 b^2}+\frac{2 x \cosh ^{\frac{3}{2}}(a+b x)}{3 b}$

[Out]

(2*x*Cosh[a + b*x]^(3/2))/(3*b) + (((4*I)/9)*EllipticF[(I/2)*(a + b*x), 2])/b^2 - (4*Sqrt[Cosh[a + b*x]]*Sinh[
a + b*x])/(9*b^2)

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Rubi [A]  time = 0.0422691, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {5373, 2635, 2641} $\frac{4 i F\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{9 b^2}-\frac{4 \sinh (a+b x) \sqrt{\cosh (a+b x)}}{9 b^2}+\frac{2 x \cosh ^{\frac{3}{2}}(a+b x)}{3 b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*x*Cosh[a + b*x]^(3/2))/(3*b) + (((4*I)/9)*EllipticF[(I/2)*(a + b*x), 2])/b^2 - (4*Sqrt[Cosh[a + b*x]]*Sinh[
a + b*x])/(9*b^2)

Rule 5373

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp[(x^(m -
n + 1)*Cosh[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Cosh[a + b*x
^n]^(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

Rule 2635

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] && Integer
Q[2*n]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int x \sqrt{\cosh (a+b x)} \sinh (a+b x) \, dx &=\frac{2 x \cosh ^{\frac{3}{2}}(a+b x)}{3 b}-\frac{2 \int \cosh ^{\frac{3}{2}}(a+b x) \, dx}{3 b}\\ &=\frac{2 x \cosh ^{\frac{3}{2}}(a+b x)}{3 b}-\frac{4 \sqrt{\cosh (a+b x)} \sinh (a+b x)}{9 b^2}-\frac{2 \int \frac{1}{\sqrt{\cosh (a+b x)}} \, dx}{9 b}\\ &=\frac{2 x \cosh ^{\frac{3}{2}}(a+b x)}{3 b}+\frac{4 i F\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{9 b^2}-\frac{4 \sqrt{\cosh (a+b x)} \sinh (a+b x)}{9 b^2}\\ \end{align*}

Mathematica [A]  time = 0.159411, size = 56, normalized size = 0.88 $\frac{2 \sqrt{\cosh (a+b x)} (3 b x \cosh (a+b x)-2 \sinh (a+b x))+4 i \text{EllipticF}\left (\frac{1}{2} i (a+b x),2\right )}{9 b^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((4*I)*EllipticF[(I/2)*(a + b*x), 2] + 2*Sqrt[Cosh[a + b*x]]*(3*b*x*Cosh[a + b*x] - 2*Sinh[a + b*x]))/(9*b^2)

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \begin{align*} \int x\sinh \left ( bx+a \right ) \sqrt{\cosh \left ( bx+a \right ) }\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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