\(\int \frac {x \sinh (a+b x)}{\sqrt {\text {sech}(a+b x)}} \, dx\) [541]

Optimal result

Integrand size = 18, antiderivative size = 84 \[ \int \frac {x \sinh (a+b x)}{\sqrt {\text {sech}(a+b x)}} \, dx=\frac {2 x}{3 b \text {sech}^{\frac {3}{2}}(a+b x)}+\frac {4 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right ) \sqrt {\text {sech}(a+b x)}}{9 b^2}-\frac {4 \sinh (a+b x)}{9 b^2 \sqrt {\text {sech}(a+b x)}} \]

[Out]

2/3*x/b/sech(b*x+a)^(3/2)-4/9*sinh(b*x+a)/b^2/sech(b*x+a)^(1/2)+4/9*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 = 0.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5552, 3854, 3856, 2720} \[ \int \frac {x \sinh (a+b x)}{\sqrt {\text {sech}(a+b x)}} \, dx=-\frac {4 \sinh (a+b x)}{9 b^2 \sqrt {\text {sech}(a+b x)}}+\frac {4 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{9 b^2}+\frac {2 x}{3 b \text {sech}^{\frac {3}{2}}(a+b x)} \]

[In]

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

[Out]

(2*x)/(3*b*Sech[a + b*x]^(3/2)) + (((4*I)/9)*Sqrt[Cosh[a + b*x]]*EllipticF[(I/2)*(a + b*x), 2]*Sqrt[Sech[a + b
*x]])/b^2 - (4*Sinh[a + b*x])/(9*b^2*Sqrt[Sech[a + b*x]])

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 3854

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Csc[c + d*x])^(n + 1)/(b*d*n)), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[2*n]

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}{3 b \text {sech}^{\frac {3}{2}}(a+b x)}-\frac {2 \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx}{3 b} \\ & = \frac {2 x}{3 b \text {sech}^{\frac {3}{2}}(a+b x)}-\frac {4 \sinh (a+b x)}{9 b^2 \sqrt {\text {sech}(a+b x)}}-\frac {2 \int \sqrt {\text {sech}(a+b x)} \, dx}{9 b} \\ & = \frac {2 x}{3 b \text {sech}^{\frac {3}{2}}(a+b x)}-\frac {4 \sinh (a+b x)}{9 b^2 \sqrt {\text {sech}(a+b x)}}-\frac {\left (2 \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)}\right ) \int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx}{9 b} \\ & = \frac {2 x}{3 b \text {sech}^{\frac {3}{2}}(a+b x)}+\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)}}{9 b^2}-\frac {4 \sinh (a+b x)}{9 b^2 \sqrt {\text {sech}(a+b x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.85 \[ \int \frac {x \sinh (a+b x)}{\sqrt {\text {sech}(a+b x)}} \, dx=\frac {\sqrt {\text {sech}(a+b x)} \left (3 b x+3 b x \cosh (2 (a+b x))+4 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )-2 \sinh (2 (a+b x))\right )}{9 b^2} \]

[In]

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

[Out]

(Sqrt[Sech[a + b*x]]*(3*b*x + 3*b*x*Cosh[2*(a + b*x)] + (4*I)*Sqrt[Cosh[a + b*x]]*EllipticF[(I/2)*(a + b*x), 2
] - 2*Sinh[2*(a + b*x)]))/(9*b^2)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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