\(\int x \text {sech}^{\frac {3}{2}}(a+b x) \sinh (a+b x) \, dx\) [539]

Optimal result

Integrand size = 18, antiderivative size = 57 \[ \int x \text {sech}^{\frac {3}{2}}(a+b x) \sinh (a+b x) \, dx=-\frac {2 x \sqrt {\text {sech}(a+b x)}}{b}-\frac {4 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right ) \sqrt {\text {sech}(a+b x)}}{b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5552, 3856, 2720} \[ \int x \text {sech}^{\frac {3}{2}}(a+b x) \sinh (a+b x) \, dx=-\frac {2 x \sqrt {\text {sech}(a+b x)}}{b}-\frac {4 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{b^2} \]

[In]

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

[Out]

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

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 3856

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 5552

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 x \sqrt {\text {sech}(a+b x)}}{b}+\frac {2 \int \sqrt {\text {sech}(a+b x)} \, dx}{b} \\ & = -\frac {2 x \sqrt {\text {sech}(a+b x)}}{b}+\frac {\left (2 \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)}\right ) \int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx}{b} \\ & = -\frac {2 x \sqrt {\text {sech}(a+b x)}}{b}-\frac {4 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right ) \sqrt {\text {sech}(a+b x)}}{b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81 \[ \int x \text {sech}^{\frac {3}{2}}(a+b x) \sinh (a+b x) \, dx=-\frac {2 \left (b x+2 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )\right ) \sqrt {\text {sech}(a+b x)}}{b^2} \]

[In]

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

[Out]

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

Maple [F]

\[\int x \operatorname {sech}\left (b x +a \right )^{\frac {3}{2}} \sinh \left (b x +a \right )d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int x \text {sech}^{\frac {3}{2}}(a+b x) \sinh (a+b x) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x*sech(b*x+a)^(3/2)*sinh(b*x+a),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F(-1)]

Timed out. \[ \int x \text {sech}^{\frac {3}{2}}(a+b x) \sinh (a+b x) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int x \text {sech}^{\frac {3}{2}}(a+b x) \sinh (a+b x) \, dx=\int { x \operatorname {sech}\left (b x + a\right )^{\frac {3}{2}} \sinh \left (b x + a\right ) \,d x } \]

[In]

integrate(x*sech(b*x+a)^(3/2)*sinh(b*x+a),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int x \text {sech}^{\frac {3}{2}}(a+b x) \sinh (a+b x) \, dx=\int { x \operatorname {sech}\left (b x + a\right )^{\frac {3}{2}} \sinh \left (b x + a\right ) \,d x } \]

[In]

integrate(x*sech(b*x+a)^(3/2)*sinh(b*x+a),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x \text {sech}^{\frac {3}{2}}(a+b x) \sinh (a+b x) \, dx=\int x\,\mathrm {sinh}\left (a+b\,x\right )\,{\left (\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}\right )}^{3/2} \,d x \]

[In]

int(x*sinh(a + b*x)*(1/cosh(a + b*x))^(3/2),x)

[Out]

int(x*sinh(a + b*x)*(1/cosh(a + b*x))^(3/2), x)