\(\int x \cosh (a+b x) \sqrt {\sinh (a+b x)} \, dx\) [546]

Optimal result

Integrand size = 18, antiderivative size = 98 \[ \int x \cosh (a+b x) \sqrt {\sinh (a+b x)} \, dx=-\frac {4 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right ),2\right ) \sqrt {i \sinh (a+b x)}}{9 b^2 \sqrt {\sinh (a+b x)}}-\frac {4 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{9 b^2}+\frac {2 x \sinh ^{\frac {3}{2}}(a+b x)}{3 b} \]

[Out]

2/3*x*sinh(b*x+a)^(3/2)/b+4/9*I*(sin(1/2*I*a+1/4*Pi+1/2*I*b*x)^2)^(1/2)/sin(1/2*I*a+1/4*Pi+1/2*I*b*x)*Elliptic
F(cos(1/2*I*a+1/4*Pi+1/2*I*b*x),2^(1/2))*(I*sinh(b*x+a))^(1/2)/b^2/sinh(b*x+a)^(1/2)-4/9*cosh(b*x+a)*sinh(b*x+
a)^(1/2)/b^2

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5480, 2715, 2721, 2720} \[ \int x \cosh (a+b x) \sqrt {\sinh (a+b x)} \, dx=-\frac {4 \sqrt {\sinh (a+b x)} \cosh (a+b x)}{9 b^2}-\frac {4 i \sqrt {i \sinh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right ),2\right )}{9 b^2 \sqrt {\sinh (a+b x)}}+\frac {2 x \sinh ^{\frac {3}{2}}(a+b x)}{3 b} \]

[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 2715

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] && Integ
erQ[2*n]

Rule 2720

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

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rule 5480

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]

Rubi steps \begin{align*} \text {integral}& = \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 \operatorname {EllipticF}\left (\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\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] (verified)

Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int x \cosh (a+b x) \sqrt {\sinh (a+b x)} \, dx=\frac {2 \left (2 i \operatorname {EllipticF}\left (\frac {1}{4} (-2 i a+\pi -2 i b x),2\right ) \sqrt {i \sinh (a+b x)}+3 b x \sinh ^2(a+b x)-\sinh (2 (a+b x))\right )}{9 b^2 \sqrt {\sinh (a+b x)}} \]

[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]])

Maple [F]

\[\int x \cosh \left (b x +a \right ) \sqrt {\sinh \left (b x +a \right )}d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int x \cosh (a+b x) \sqrt {\sinh (a+b x)} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

\[ \int x \cosh (a+b x) \sqrt {\sinh (a+b x)} \, dx=\int x \sqrt {\sinh {\left (a + b x \right )}} \cosh {\left (a + b x \right )}\, dx \]

[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)

Maxima [F]

\[ \int x \cosh (a+b x) \sqrt {\sinh (a+b x)} \, dx=\int { x \cosh \left (b x + a\right ) \sqrt {\sinh \left (b x + a\right )} \,d x } \]

[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)

Giac [F]

\[ \int x \cosh (a+b x) \sqrt {\sinh (a+b x)} \, dx=\int { x \cosh \left (b x + a\right ) \sqrt {\sinh \left (b x + a\right )} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int x \cosh (a+b x) \sqrt {\sinh (a+b x)} \, dx=\int x\,\mathrm {cosh}\left (a+b\,x\right )\,\sqrt {\mathrm {sinh}\left (a+b\,x\right )} \,d x \]

[In]

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

[Out]

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