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

Optimal. Leaf size=71 \[ \frac {2 x \sqrt {\sinh (a+b x)}}{b}+\frac {4 i \sqrt {\sinh (a+b x)} E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b^2 \sqrt {i \sinh (a+b x)}} \]

[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)

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Rubi [A]  time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, 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 = {5372, 2640, 2639} \[ \frac {2 x \sqrt {\sinh (a+b x)}}{b}+\frac {4 i \sqrt {\sinh (a+b x)} E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b^2 \sqrt {i \sinh (a+b x)}} \]

Antiderivative was successfully verified.

[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 2639

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

Rule 2640

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

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]

Rubi steps

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

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Mathematica [C]  time = 1.77, size = 182, normalized size = 2.56 \[ \frac {e^{-a-b x} \sqrt {2-2 e^{2 (a+b x)}} \left (-18 \, _3F_2\left (-\frac {1}{4},-\frac {1}{4},\frac {1}{2};\frac {3}{4},\frac {3}{4};e^{2 (a+b x)}\right )-2 e^{2 (a+b x)} \, _3F_2\left (\frac {1}{2},\frac {3}{4},\frac {3}{4};\frac {7}{4},\frac {7}{4};e^{2 (a+b x)}\right )-3 b x \left (3 \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};e^{2 (a+b x)}\right )-e^{2 (a+b x)} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};e^{2 (a+b x)}\right )\right )\right )}{9 b^2 \sqrt {e^{a+b x}-e^{-a-b x}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(E^(-a - b*x)*Sqrt[2 - 2*E^(2*(a + b*x))]*(-3*b*x*(3*Hypergeometric2F1[-1/4, 1/2, 3/4, E^(2*(a + b*x))] - E^(2
*(a + b*x))*Hypergeometric2F1[1/2, 3/4, 7/4, E^(2*(a + b*x))]) - 18*HypergeometricPFQ[{-1/4, -1/4, 1/2}, {3/4,
3/4}, E^(2*(a + b*x))] - 2*E^(2*(a + b*x))*HypergeometricPFQ[{1/2, 3/4, 3/4}, {7/4, 7/4}, E^(2*(a + b*x))]))/
(9*b^2*Sqrt[-E^(-a - b*x) + E^(a + b*x)])

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \cosh \left (b x + a\right )}{\sqrt {\sinh \left (b x + a\right )}}\,{d x} \]

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)

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maple [B]  time = 0.16, size = 229, normalized size = 3.23 \[ \frac {\left (b x -2\right ) \left ({\mathrm e}^{2 b x +2 a}-1\right ) \sqrt {2}\, {\mathrm e}^{-b x -a}}{b^{2} \sqrt {\left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{-b x -a}}}+\frac {2 \left (\frac {2 \,{\mathrm e}^{2 b x +2 a}-2}{\sqrt {\left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{b x +a}}}-\frac {\sqrt {1+{\mathrm e}^{b x +a}}\, \sqrt {2-2 \,{\mathrm e}^{b x +a}}\, \sqrt {-{\mathrm e}^{b x +a}}\, \left (-2 \EllipticE \left (\sqrt {1+{\mathrm e}^{b x +a}}, \frac {\sqrt {2}}{2}\right )+\EllipticF \left (\sqrt {1+{\mathrm e}^{b x +a}}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 b x +3 a}-{\mathrm e}^{b x +a}}}\right ) \sqrt {2}\, \sqrt {\left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{b x +a}}\, {\mathrm e}^{-b x -a}}{b^{2} \sqrt {\left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{-b x -a}}} \]

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]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \cosh \left (b x + a\right )}{\sqrt {\sinh \left (b x + a\right )}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\mathrm {cosh}\left (a+b\,x\right )}{\sqrt {\mathrm {sinh}\left (a+b\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \cosh {\left (a + b x \right )}}{\sqrt {\sinh {\left (a + b x \right )}}}\, dx \]

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

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