### 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]

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

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Rubi [A]  time = 0.0295638, antiderivative size = 37, normalized size of antiderivative = 1., 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 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]

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 \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*}

Mathematica [C]  time = 1.46792, size = 190, normalized size = 5.14 $\frac{\sqrt{2} e^{-a-b x} \sqrt{e^{2 (a+b x)}+1} \left (18 \text{HypergeometricPFQ}\left (\left \{-\frac{1}{4},-\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{3}{4},\frac{3}{4}\right \},-e^{2 (a+b x)}\right )-2 e^{2 (a+b x)} \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{3}{4},\frac{3}{4}\right \},\left \{\frac{7}{4},\frac{7}{4}\right \},-e^{2 (a+b x)}\right )+3 b x \left (3 \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 (a+b x)}\right )+e^{2 (a+b x)} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 (a+b x)}\right )\right )\right )}{9 b^2 \sqrt{e^{-a-b x}+e^{a+b x}}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[(x*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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Maple [B]  time = 0.066, size = 250, normalized size = 6.8 \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{i\sqrt{-i \left ({{\rm e}^{bx+a}}+i \right ) }\sqrt{2}\sqrt{i \left ({{\rm e}^{bx+a}}-i \right ) }\sqrt{i{{\rm e}^{bx+a}}} \left ( -2\,i{\it EllipticE} \left ( \sqrt{-i \left ({{\rm e}^{bx+a}}+i \right ) },1/2\,\sqrt{2} \right ) +i{\it EllipticF} \left ( \sqrt{-i \left ({{\rm e}^{bx+a}}+i \right ) },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*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., size = 0, normalized size = 0. \begin{align*} \int \frac{x \sinh \left (b x + a\right )}{\sqrt{\cosh \left (b x + a\right )}}\,{d x} \end{align*}

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \sinh \left (b x + a\right )}{\sqrt{\cosh \left (b x + a\right )}}\,{d x} \end{align*}

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)