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*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]])

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Rubi [A]  time = 0.03856, antiderivative size = 71, 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 = {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 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 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 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]

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*}

Mathematica [C]  time = 1.69049, size = 182, normalized size = 2.56 \[ \frac{e^{-a-b x} \sqrt{2-2 e^{2 (a+b x)}} \left (-18 \text{HypergeometricPFQ}\left (\left \{-\frac{1}{4},-\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{3}{4},\frac{3}{4}\right \},e^{2 (a+b x)}\right )-2 e^{2 (a+b x)} \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{3}{4},\frac{3}{4}\right \},\left \{\frac{7}{4},\frac{7}{4}\right \},e^{2 (a+b x)}\right )-3 b x \left (3 \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};e^{2 (a+b x)}\right )-e^{2 (a+b x)} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};e^{2 (a+b x)}\right )\right )\right )}{9 b^2 \sqrt{e^{a+b x}-e^{-a-b x}}} \]

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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Maple [B]  time = 0.062, size = 229, normalized size = 3.2 \begin{align*}{\frac{ \left ( bx-2 \right ) \left ( \left ({{\rm e}^{bx+a}} \right ) ^{2}-1 \right ) \sqrt{2}}{{b}^{2}{{\rm e}^{bx+a}}}{\frac{1}{\sqrt{{\frac{ \left ({{\rm e}^{bx+a}} \right ) ^{2}-1}{{{\rm e}^{bx+a}}}}}}}}+2\,{\frac{\sqrt{2}\sqrt{ \left ( \left ({{\rm e}^{bx+a}} \right ) ^{2}-1 \right ){{\rm e}^{bx+a}}}}{{b}^{2}{{\rm e}^{bx+a}}} \left ( 2\,{\frac{ \left ({{\rm e}^{bx+a}} \right ) ^{2}-1}{\sqrt{ \left ( \left ({{\rm e}^{bx+a}} \right ) ^{2}-1 \right ){{\rm e}^{bx+a}}}}}-{\frac{\sqrt{1+{{\rm e}^{bx+a}}}\sqrt{2-2\,{{\rm e}^{bx+a}}}\sqrt{-{{\rm e}^{bx+a}}} \left ( -2\,{\it EllipticE} \left ( \sqrt{1+{{\rm e}^{bx+a}}},1/2\,\sqrt{2} \right ) +{\it EllipticF} \left ( \sqrt{1+{{\rm e}^{bx+a}}},1/2\,\sqrt{2} \right ) \right ) }{\sqrt{ \left ({{\rm e}^{bx+a}} \right ) ^{3}-{{\rm e}^{bx+a}}}}} \right ){\frac{1}{\sqrt{{\frac{ \left ({{\rm e}^{bx+a}} \right ) ^{2}-1}{{{\rm e}^{bx+a}}}}}}}} \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]

(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., size = 0, normalized size = 0. \begin{align*} \int \frac{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 \frac{x \cosh{\left (a + b x \right )}}{\sqrt{\sinh{\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*cosh(a + b*x)/sqrt(sinh(a + b*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{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)