### 3.540 $$\int x \sqrt{\text{sech}(a+b x)} \sinh (a+b x) \, dx$$

Optimal. Leaf size=57 $\frac{2 x}{b \sqrt{\text{sech}(a+b x)}}+\frac{4 i \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{b^2}$

[Out]

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

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Rubi [A]  time = 0.039545, antiderivative size = 57, 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 = {5444, 3771, 2639} $\frac{2 x}{b \sqrt{\text{sech}(a+b x)}}+\frac{4 i \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{b^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[x*Sqrt[Sech[a + b*x]]*Sinh[a + b*x],x]

[Out]

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

Rule 5444

Int[(x_)^(m_.)*Sech[(a_.) + (b_.)*(x_)^(n_.)]^(p_)*Sinh[(a_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> -Simp[(x^(m -
n + 1)*Sech[a + b*x^n]^(p - 1))/(b*n*(p - 1)), x] + Dist[(m - n + 1)/(b*n*(p - 1)), Int[x^(m - n)*Sech[a + b*x
^n]^(p - 1), x], x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

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 x \sqrt{\text{sech}(a+b x)} \sinh (a+b x) \, dx &=\frac{2 x}{b \sqrt{\text{sech}(a+b x)}}-\frac{2 \int \frac{1}{\sqrt{\text{sech}(a+b x)}} \, dx}{b}\\ &=\frac{2 x}{b \sqrt{\text{sech}(a+b x)}}-\frac{\left (2 \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)}\right ) \int \sqrt{\cosh (a+b x)} \, dx}{b}\\ &=\frac{2 x}{b \sqrt{\text{sech}(a+b x)}}+\frac{4 i \sqrt{\cosh (a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right ) \sqrt{\text{sech}(a+b x)}}{b^2}\\ \end{align*}

Mathematica [C]  time = 1.07534, size = 100, normalized size = 1.75 $\frac{\sqrt{2} e^{-a-b x} \sqrt{\frac{e^{a+b x}}{e^{2 (a+b x)}+1}} \left (4 \sqrt{e^{2 (a+b x)}+1} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 (a+b x)}\right )+(b x-2) \left (e^{2 (a+b x)}+1\right )\right )}{b^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*Sqrt[Sech[a + b*x]]*Sinh[a + b*x],x]

[Out]

(Sqrt[2]*E^(-a - b*x)*Sqrt[E^(a + b*x)/(1 + E^(2*(a + b*x)))]*((1 + E^(2*(a + b*x)))*(-2 + b*x) + 4*Sqrt[1 + E
^(2*(a + b*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^(2*(a + b*x))]))/b^2

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Maple [B]  time = 0.053, size = 250, normalized size = 4.4 \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}}}\sqrt{{\frac{{{\rm e}^{bx+a}}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}}}}-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 ) \sqrt{{\frac{{{\rm e}^{bx+a}}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x-2)*(exp(b*x+a)^2+1)/b^2*2^(1/2)*(exp(b*x+a)/(exp(b*x+a)^2+1))^(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)/(exp(b*x+a)^2+1))^(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 x \sqrt{\operatorname{sech}\left (b x + a\right )} \sinh \left (b x + a\right )\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sinh{\left (a + b x \right )} \sqrt{\operatorname{sech}{\left (a + b x \right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sinh(a + b*x)*sqrt(sech(a + b*x)), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*sqrt(sech(b*x + a))*sinh(b*x + a), x)