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

Optimal. Leaf size=37 \[ \frac {2 x \sqrt {\cosh (a+b x)}}{b}+\frac {4 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b^2} \]

[Out]

4*I*(cosh(1/2*a+1/2*b*x)^2)^(1/2)/cosh(1/2*a+1/2*b*x)*EllipticE(I*sinh(1/2*a+1/2*b*x),2^(1/2))/b^2+2*x*cosh(b*
x+a)^(1/2)/b

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Rubi [A]  time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5373, 2639} \[ \frac {2 x \sqrt {\cosh (a+b x)}}{b}+\frac {4 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[Cosh[a + b*x]])/b + ((4*I)*EllipticE[(I/2)*(a + b*x), 2])/b^2

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 5373

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp[(x^(m -
n + 1)*Cosh[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Cosh[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 \sinh (a+b x)}{\sqrt {\cosh (a+b x)}} \, dx &=\frac {2 x \sqrt {\cosh (a+b x)}}{b}-\frac {2 \int \sqrt {\cosh (a+b x)} \, dx}{b}\\ &=\frac {2 x \sqrt {\cosh (a+b x)}}{b}+\frac {4 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b^2}\\ \end {align*}

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Mathematica [C]  time = 1.55, size = 190, normalized size = 5.14 \[ \frac {\sqrt {2} e^{-a-b x} \sqrt {e^{2 (a+b x)}+1} \left (18 \, _3F_2\left (-\frac {1}{4},-\frac {1}{4},\frac {1}{2};\frac {3}{4},\frac {3}{4};-e^{2 (a+b x)}\right )-2 e^{2 (a+b x)} \, _3F_2\left (\frac {1}{2},\frac {3}{4},\frac {3}{4};\frac {7}{4},\frac {7}{4};-e^{2 (a+b x)}\right )+3 b x \left (3 \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 (a+b x)}\right )+e^{2 (a+b x)} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 (a+b x)}\right )\right )\right )}{9 b^2 \sqrt {e^{-a-b x}+e^{a+b x}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[2]*E^(-a - b*x)*Sqrt[1 + 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*sinh(b*x+a)/cosh(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 \[ \int \frac {x \sinh \left (b x + a\right )}{\sqrt {\cosh \left (b x + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.15, size = 250, normalized size = 6.76 \[ \frac {\left (b x -2\right ) \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {2}\, {\mathrm e}^{-b x -a}}{b^{2} \sqrt {{\mathrm e}^{-b x -a} \left (1+{\mathrm e}^{2 b x +2 a}\right )}}-\frac {2 \left (-\frac {2 \left (1+{\mathrm e}^{2 b x +2 a}\right )}{\sqrt {\left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{b x +a}}}+\frac {i \sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{b x +a}-i\right )}\, \sqrt {i {\mathrm e}^{b x +a}}\, \left (-2 i \EllipticE \left (\sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \EllipticF \left (\sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 b x +3 a}+{\mathrm e}^{b x +a}}}\right ) \sqrt {2}\, \sqrt {\left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{b x +a}}\, {\mathrm e}^{-b x -a}}{b^{2} \sqrt {{\mathrm e}^{-b x -a} \left (1+{\mathrm e}^{2 b x +2 a}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sinh(b*x+a)/cosh(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)+I*(-I*(exp(b*x+a)+I))^(1/2)*2^(1/2)*(I*(exp(b*x+a)-I))^(1/2)*(I*exp(b*x+a
))^(1/2)/(exp(b*x+a)^3+exp(b*x+a))^(1/2)*(-2*I*EllipticE((-I*(exp(b*x+a)+I))^(1/2),1/2*2^(1/2))+I*EllipticF((-
I*(exp(b*x+a)+I))^(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 \[ \int \frac {x \sinh \left (b x + a\right )}{\sqrt {\cosh \left (b x + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x\,\mathrm {sinh}\left (a+b\,x\right )}{\sqrt {\mathrm {cosh}\left (a+b\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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