\(\int x \sqrt {\text {sech}(a+b x)} \sinh (a+b x) \, dx\) [540]

Optimal result

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

[Out]

2*x/b/sech(b*x+a)^(1/2)+4*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 = 0.03 (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, 2719} \[ \int x \sqrt {\text {sech}(a+b x)} \sinh (a+b x) \, dx=\frac {2 x}{b \sqrt {\text {sech}(a+b x)}}+\frac {4 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b^2} \]

[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 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 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}{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] (verified)

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

Time = 1.40 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.75 \[ \int x \sqrt {\text {sech}(a+b x)} \sinh (a+b x) \, dx=\frac {\sqrt {2} e^{-a-b x} \sqrt {\frac {e^{a+b x}}{1+e^{2 (a+b x)}}} \left (\left (1+e^{2 (a+b x)}\right ) (-2+b x)+4 \sqrt {1+e^{2 (a+b x)}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-e^{2 (a+b x)}\right )\right )}{b^2} \]

[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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (77 ) = 154\).

Time = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 4.39

method	result	size
risch	\(\frac {\left (b x -2\right ) \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {2}\, \sqrt {\frac {{\mathrm e}^{b x +a}}{1+{\mathrm e}^{2 b x +2 a}}}\, {\mathrm e}^{-b x -a}}{b^{2}}-\frac {2 \left (-\frac {2 \left (1+{\mathrm e}^{2 b x +2 a}\right )}{\sqrt {\left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{b x +a}}}+\frac {i \sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{b x +a}-i\right )}\, \sqrt {i {\mathrm e}^{b x +a}}\, \left (-2 i \operatorname {EllipticE}\left (\sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \operatorname {EllipticF}\left (\sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 b x +3 a}+{\mathrm e}^{b x +a}}}\right ) \sqrt {2}\, \sqrt {\frac {{\mathrm e}^{b x +a}}{1+{\mathrm e}^{2 b x +2 a}}}\, \sqrt {\left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{b x +a}}\, {\mathrm e}^{-b x -a}}{b^{2}}\)	\(250\)

[In]

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

[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)

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[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)