### 3.532 $$\int \frac{x \sinh (a+b x)}{\cosh ^{\frac{3}{2}}(a+b x)} \, dx$$

Optimal. Leaf size=37 $-\frac{2 x}{b \sqrt{\cosh (a+b x)}}-\frac{4 i \text{EllipticF}\left (\frac{1}{2} i (a+b x),2\right )}{b^2}$

[Out]

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

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Rubi [A]  time = 0.0293948, 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, 2641} $-\frac{2 x}{b \sqrt{\cosh (a+b x)}}-\frac{4 i F\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{b^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(x*Sinh[a + b*x])/Cosh[a + b*x]^(3/2),x]

[Out]

(-2*x)/(b*Sqrt[Cosh[a + b*x]]) - ((4*I)*EllipticF[(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 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(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)}{\cosh ^{\frac{3}{2}}(a+b x)} \, dx &=-\frac{2 x}{b \sqrt{\cosh (a+b x)}}+\frac{2 \int \frac{1}{\sqrt{\cosh (a+b x)}} \, dx}{b}\\ &=-\frac{2 x}{b \sqrt{\cosh (a+b x)}}-\frac{4 i F\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{b^2}\\ \end{align*}

Mathematica [A]  time = 0.162898, size = 37, normalized size = 1. $-\frac{2 x}{b \sqrt{\cosh (a+b x)}}-\frac{4 i \text{EllipticF}\left (\frac{1}{2} i (a+b x),2\right )}{b^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(x*Sinh[a + b*x])/Cosh[a + b*x]^(3/2),x]

[Out]

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

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{x\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*sinh(b*x+a)/cosh(b*x+a)^(3/2),x)

[Out]

int(x*sinh(b*x+a)/cosh(b*x+a)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{\frac{3}{2}}}\,{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)^(3/2),x, algorithm="maxima")

[Out]

integrate(x*sinh(b*x + a)/cosh(b*x + a)^(3/2), 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)^(3/2),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)**(3/2),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 )}{\cosh \left (b x + a\right )^{\frac{3}{2}}}\,{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)^(3/2),x, algorithm="giac")

[Out]

integrate(x*sinh(b*x + a)/cosh(b*x + a)^(3/2), x)