\(\int \frac {x \cosh (a+b x)}{\sqrt {\sinh (a+b x)}} \, dx\) [547]

Optimal result

Integrand size = 18, antiderivative size = 71 \[ \int \frac {x \cosh (a+b x)}{\sqrt {\sinh (a+b x)}} \, dx=\frac {2 x \sqrt {\sinh (a+b x)}}{b}+\frac {4 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {\sinh (a+b x)}}{b^2 \sqrt {i \sinh (a+b x)}} \]

[Out]

2*x*sinh(b*x+a)^(1/2)/b-4*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)*EllipticE(co
s(1/2*I*a+1/4*Pi+1/2*I*b*x),2^(1/2))*sinh(b*x+a)^(1/2)/b^2/(I*sinh(b*x+a))^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5480, 2721, 2719} \[ \int \frac {x \cosh (a+b x)}{\sqrt {\sinh (a+b x)}} \, dx=\frac {2 x \sqrt {\sinh (a+b x)}}{b}+\frac {4 i \sqrt {\sinh (a+b x)} E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b^2 \sqrt {i \sinh (a+b x)}} \]

[In]

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

[Out]

(2*x*Sqrt[Sinh[a + b*x]])/b + ((4*I)*EllipticE[(I*a - Pi/2 + I*b*x)/2, 2]*Sqrt[Sinh[a + b*x]])/(b^2*Sqrt[I*Sin
h[a + b*x]])

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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 \sqrt {\sinh (a+b x)}}{b}-\frac {2 \int \sqrt {\sinh (a+b x)} \, dx}{b} \\ & = \frac {2 x \sqrt {\sinh (a+b x)}}{b}-\frac {\left (2 \sqrt {\sinh (a+b x)}\right ) \int \sqrt {i \sinh (a+b x)} \, dx}{b \sqrt {i \sinh (a+b x)}} \\ & = \frac {2 x \sqrt {\sinh (a+b x)}}{b}+\frac {4 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {\sinh (a+b x)}}{b^2 \sqrt {i \sinh (a+b x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 1.67 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.62 \[ \int \frac {x \cosh (a+b x)}{\sqrt {\sinh (a+b x)}} \, dx=\frac {(-\cosh (a+b x)+\sinh (a+b x)) \left (-2 (-2+b x) \sinh (a+b x) (\cosh (a+b x)+\sinh (a+b x))+4 \sqrt {2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},\cosh (2 (a+b x))+\sinh (2 (a+b x))\right ) \sqrt {-\sinh (a+b x) (\cosh (a+b x)+\sinh (a+b x))}\right )}{b^2 \sqrt {\sinh (a+b x)}} \]

[In]

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

[Out]

((-Cosh[a + b*x] + Sinh[a + b*x])*(-2*(-2 + b*x)*Sinh[a + b*x]*(Cosh[a + b*x] + Sinh[a + b*x]) + 4*Sqrt[2]*Hyp
ergeometric2F1[-1/4, 1/2, 3/4, Cosh[2*(a + b*x)] + Sinh[2*(a + b*x)]]*Sqrt[-(Sinh[a + b*x]*(Cosh[a + b*x] + Si
nh[a + b*x]))]))/(b^2*Sqrt[Sinh[a + b*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(228\) vs. \(2(92)=184\).

Time = 0.23 (sec) , antiderivative size = 229, normalized size of antiderivative = 3.23

method	result	size
risch	\(\frac {\left (b x -2\right ) \left ({\mathrm e}^{2 b x +2 a}-1\right ) \sqrt {2}\, {\mathrm e}^{-b x -a}}{b^{2} \sqrt {\left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{-b x -a}}}+\frac {2 \left (\frac {2 \,{\mathrm e}^{2 b x +2 a}-2}{\sqrt {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a}-1\right )}}-\frac {\sqrt {{\mathrm e}^{b x +a}+1}\, \sqrt {-2 \,{\mathrm e}^{b x +a}+2}\, \sqrt {-{\mathrm e}^{b x +a}}\, \left (-2 \operatorname {EllipticE}\left (\sqrt {{\mathrm e}^{b x +a}+1}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticF}\left (\sqrt {{\mathrm e}^{b x +a}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 b x +3 a}-{\mathrm e}^{b x +a}}}\right ) \sqrt {2}\, \sqrt {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a}-1\right )}\, {\mathrm e}^{-b x -a}}{b^{2} \sqrt {\left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{-b x -a}}}\)	\(229\)

[In]

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

[Out]

(b*x-2)*(exp(b*x+a)^2-1)/b^2*2^(1/2)/((exp(b*x+a)^2-1)/exp(b*x+a))^(1/2)/exp(b*x+a)+2/b^2*(2*(exp(b*x+a)^2-1)/
(exp(b*x+a)*(exp(b*x+a)^2-1))^(1/2)-(exp(b*x+a)+1)^(1/2)*(-2*exp(b*x+a)+2)^(1/2)*(-exp(b*x+a))^(1/2)/(exp(b*x+
a)^3-exp(b*x+a))^(1/2)*(-2*EllipticE((exp(b*x+a)+1)^(1/2),1/2*2^(1/2))+EllipticF((exp(b*x+a)+1)^(1/2),1/2*2^(1
/2))))*2^(1/2)/((exp(b*x+a)^2-1)/exp(b*x+a))^(1/2)*(exp(b*x+a)*(exp(b*x+a)^2-1))^(1/2)/exp(b*x+a)

Fricas [F(-2)]

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

[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]

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

[In]

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

[Out]

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

Maxima [F]

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

[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]

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

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

[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)