### 3.546 $$\int x \cosh (a+b x) \sqrt{\sinh (a+b x)} \, dx$$

Optimal. Leaf size=98 $-\frac{4 i \sqrt{i \sinh (a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right ),2\right )}{9 b^2 \sqrt{\sinh (a+b x)}}-\frac{4 \sqrt{\sinh (a+b x)} \cosh (a+b x)}{9 b^2}+\frac{2 x \sinh ^{\frac{3}{2}}(a+b x)}{3 b}$

[Out]

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

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Rubi [A]  time = 0.0530854, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.222, Rules used = {5372, 2635, 2642, 2641} $-\frac{4 \sqrt{\sinh (a+b x)} \cosh (a+b x)}{9 b^2}-\frac{4 i \sqrt{i \sinh (a+b x)} F\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{9 b^2 \sqrt{\sinh (a+b x)}}+\frac{2 x \sinh ^{\frac{3}{2}}(a+b x)}{3 b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 5372

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[(x^(m -
n + 1)*Sinh[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sinh[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 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

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

Mathematica [A]  time = 0.199701, size = 77, normalized size = 0.79 $\frac{2 \left (2 i \sqrt{i \sinh (a+b x)} \text{EllipticF}\left (\frac{1}{4} (-2 i a-2 i b x+\pi ),2\right )+3 b x \sinh ^2(a+b x)-\sinh (2 (a+b x))\right )}{9 b^2 \sqrt{\sinh (a+b x)}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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