3.538 \(\int x \text {sech}^{\frac {5}{2}}(a+b x) \sinh (a+b x) \, dx\)

Optimal. Leaf size=84 \[ \frac {4 \sinh (a+b x) \sqrt {\text {sech}(a+b x)}}{3 b^2}+\frac {4 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{3 b^2}-\frac {2 x \text {sech}^{\frac {3}{2}}(a+b x)}{3 b} \]

[Out]

-2/3*x*sech(b*x+a)^(3/2)/b+4/3*sinh(b*x+a)*sech(b*x+a)^(1/2)/b^2+4/3*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

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Rubi [A]  time = 0.05, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5444, 3768, 3771, 2639} \[ \frac {4 \sinh (a+b x) \sqrt {\text {sech}(a+b x)}}{3 b^2}+\frac {4 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{3 b^2}-\frac {2 x \text {sech}^{\frac {3}{2}}(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[x*Sech[a + b*x]^(5/2)*Sinh[a + b*x],x]

[Out]

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

Rule 2639

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

Rule 3768

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

Rule 3771

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 5444

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*} \int x \text {sech}^{\frac {5}{2}}(a+b x) \sinh (a+b x) \, dx &=-\frac {2 x \text {sech}^{\frac {3}{2}}(a+b x)}{3 b}+\frac {2 \int \text {sech}^{\frac {3}{2}}(a+b x) \, dx}{3 b}\\ &=-\frac {2 x \text {sech}^{\frac {3}{2}}(a+b x)}{3 b}+\frac {4 \sqrt {\text {sech}(a+b x)} \sinh (a+b x)}{3 b^2}-\frac {2 \int \frac {1}{\sqrt {\text {sech}(a+b x)}} \, dx}{3 b}\\ &=-\frac {2 x \text {sech}^{\frac {3}{2}}(a+b x)}{3 b}+\frac {4 \sqrt {\text {sech}(a+b x)} \sinh (a+b x)}{3 b^2}-\frac {\left (2 \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)}\right ) \int \sqrt {\cosh (a+b x)} \, dx}{3 b}\\ &=\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)}}{3 b^2}-\frac {2 x \text {sech}^{\frac {3}{2}}(a+b x)}{3 b}+\frac {4 \sqrt {\text {sech}(a+b x)} \sinh (a+b x)}{3 b^2}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 57, normalized size = 0.68 \[ \frac {2 \text {sech}^{\frac {3}{2}}(a+b x) \left (\sinh (2 (a+b x))+2 i \cosh ^{\frac {3}{2}}(a+b x) E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )-b x\right )}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sech[a + b*x]^(5/2)*Sinh[a + b*x],x]

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sech(b*x+a)^(5/2)*sinh(b*x+a),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {sech}\left (b x + a\right )^{\frac {5}{2}} \sinh \left (b x + a\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sech(b*x+a)^(5/2)*sinh(b*x+a),x, algorithm="giac")

[Out]

integrate(x*sech(b*x + a)^(5/2)*sinh(b*x + a), x)

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maple [F]  time = 0.15, size = 0, normalized size = 0.00 \[ \int x \mathrm {sech}\left (b x +a \right )^{\frac {5}{2}} \sinh \left (b x +a \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sech(b*x+a)^(5/2)*sinh(b*x+a),x)

[Out]

int(x*sech(b*x+a)^(5/2)*sinh(b*x+a),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {sech}\left (b x + a\right )^{\frac {5}{2}} \sinh \left (b x + a\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sech(b*x+a)^(5/2)*sinh(b*x+a),x, algorithm="maxima")

[Out]

integrate(x*sech(b*x + a)^(5/2)*sinh(b*x + a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\mathrm {sinh}\left (a+b\,x\right )\,{\left (\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sinh(a + b*x)*(1/cosh(a + b*x))^(5/2),x)

[Out]

int(x*sinh(a + b*x)*(1/cosh(a + b*x))^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sech(b*x+a)**(5/2)*sinh(b*x+a),x)

[Out]

Timed out

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