\(\int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx\) [542]

Optimal result

Integrand size = 18, antiderivative size = 84 \[ \int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=\frac {2 x}{5 b \text {sech}^{\frac {5}{2}}(a+b x)}+\frac {12 i \sqrt {\cosh (a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right ) \sqrt {\text {sech}(a+b x)}}{25 b^2}-\frac {4 \sinh (a+b x)}{25 b^2 \text {sech}^{\frac {3}{2}}(a+b x)} \]

[Out]

2/5*x/b/sech(b*x+a)^(5/2)-4/25*sinh(b*x+a)/b^2/sech(b*x+a)^(3/2)+12/25*I*(cosh(1/2*a+1/2*b*x)^2)^(1/2)/cosh(1/
2*a+1/2*b*x)*EllipticE(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 = 0.05 (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, 2719} \[ \int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=-\frac {4 \sinh (a+b x)}{25 b^2 \text {sech}^{\frac {3}{2}}(a+b x)}+\frac {12 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{25 b^2}+\frac {2 x}{5 b \text {sech}^{\frac {5}{2}}(a+b x)} \]

[In]

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

[Out]

(2*x)/(5*b*Sech[a + b*x]^(5/2)) + (((12*I)/25)*Sqrt[Cosh[a + b*x]]*EllipticE[(I/2)*(a + b*x), 2]*Sqrt[Sech[a +
b*x]])/b^2 - (4*Sinh[a + b*x])/(25*b^2*Sech[a + b*x]^(3/2))

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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}{5 b \text {sech}^{\frac {5}{2}}(a+b x)}-\frac {2 \int \frac {1}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx}{5 b} \\ & = \frac {2 x}{5 b \text {sech}^{\frac {5}{2}}(a+b x)}-\frac {4 \sinh (a+b x)}{25 b^2 \text {sech}^{\frac {3}{2}}(a+b x)}-\frac {6 \int \frac {1}{\sqrt {\text {sech}(a+b x)}} \, dx}{25 b} \\ & = \frac {2 x}{5 b \text {sech}^{\frac {5}{2}}(a+b x)}-\frac {4 \sinh (a+b x)}{25 b^2 \text {sech}^{\frac {3}{2}}(a+b x)}-\frac {\left (6 \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)}\right ) \int \sqrt {\cosh (a+b x)} \, dx}{25 b} \\ & = \frac {2 x}{5 b \text {sech}^{\frac {5}{2}}(a+b x)}+\frac {12 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)}}{25 b^2}-\frac {4 \sinh (a+b x)}{25 b^2 \text {sech}^{\frac {3}{2}}(a+b x)} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 3.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.49 \[ \int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=\frac {e^{-3 (a+b x)} \left (\left (1+e^{2 (a+b x)}\right ) \left (2+5 b x+2 e^{2 (a+b x)} (-12+5 b x)+e^{4 (a+b x)} (-2+5 b x)\right )+48 e^{2 (a+b x)} \sqrt {1+e^{2 (a+b x)}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-e^{2 (a+b x)}\right )\right ) \sqrt {\text {sech}(a+b x)}}{100 b^2} \]

[In]

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

[Out]

(((1 + E^(2*(a + b*x)))*(2 + 5*b*x + 2*E^(2*(a + b*x))*(-12 + 5*b*x) + E^(4*(a + b*x))*(-2 + 5*b*x)) + 48*E^(2
*(a + b*x))*Sqrt[1 + E^(2*(a + b*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^(2*(a + b*x))])*Sqrt[Sech[a + b*x]]
)/(100*b^2*E^(3*(a + b*x)))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[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)