3.16 \(\int (b \text{sech}(c+d x))^{5/2} \, dx\)

Optimal. Leaf size=74 \[ \frac{2 b \sinh (c+d x) (b \text{sech}(c+d x))^{3/2}}{3 d}-\frac{2 i b^2 \sqrt{\cosh (c+d x)} \text{EllipticF}\left (\frac{1}{2} i (c+d x),2\right ) \sqrt{b \text{sech}(c+d x)}}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.0381697, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3768, 3771, 2641} \[ \frac{2 b \sinh (c+d x) (b \text{sech}(c+d x))^{3/2}}{3 d}-\frac{2 i b^2 \sqrt{\cosh (c+d x)} F\left (\left .\frac{1}{2} i (c+d x)\right |2\right ) \sqrt{b \text{sech}(c+d x)}}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(b*Sech[c + d*x])^(5/2),x]

[Out]

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

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 2641

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

Rubi steps

\begin{align*} \int (b \text{sech}(c+d x))^{5/2} \, dx &=\frac{2 b (b \text{sech}(c+d x))^{3/2} \sinh (c+d x)}{3 d}+\frac{1}{3} b^2 \int \sqrt{b \text{sech}(c+d x)} \, dx\\ &=\frac{2 b (b \text{sech}(c+d x))^{3/2} \sinh (c+d x)}{3 d}+\frac{1}{3} \left (b^2 \sqrt{\cosh (c+d x)} \sqrt{b \text{sech}(c+d x)}\right ) \int \frac{1}{\sqrt{\cosh (c+d x)}} \, dx\\ &=-\frac{2 i b^2 \sqrt{\cosh (c+d x)} F\left (\left .\frac{1}{2} i (c+d x)\right |2\right ) \sqrt{b \text{sech}(c+d x)}}{3 d}+\frac{2 b (b \text{sech}(c+d x))^{3/2} \sinh (c+d x)}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.0695537, size = 56, normalized size = 0.76 \[ \frac{2 b^2 \sqrt{b \text{sech}(c+d x)} \left (\tanh (c+d x)-i \sqrt{\cosh (c+d x)} \text{EllipticF}\left (\frac{1}{2} i (c+d x),2\right )\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Sech[c + d*x])^(5/2),x]

[Out]

(2*b^2*Sqrt[b*Sech[c + d*x]]*((-I)*Sqrt[Cosh[c + d*x]]*EllipticF[(I/2)*(c + d*x), 2] + Tanh[c + d*x]))/(3*d)

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \begin{align*} \int \left ( b{\rm sech} \left (dx+c\right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*sech(d*x+c))^(5/2),x)

[Out]

int((b*sech(d*x+c))^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \operatorname{sech}\left (d x + c\right )\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*sech(d*x + c))^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b \operatorname{sech}\left (d x + c\right )} b^{2} \operatorname{sech}\left (d x + c\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*sech(d*x + c))*b^2*sech(d*x + c)^2, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sech(d*x+c))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \operatorname{sech}\left (d x + c\right )\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*sech(d*x + c))^(5/2), x)