3.132 \(\int \frac{1}{(a \cosh ^3(x))^{3/2}} \, dx\)

Optimal. Leaf size=75 \[ -\frac{10 i \cosh ^{\frac{3}{2}}(x) \text{EllipticF}\left (\frac{i x}{2},2\right )}{21 a \sqrt{a \cosh ^3(x)}}+\frac{10 \sinh (x)}{21 a \sqrt{a \cosh ^3(x)}}+\frac{2 \tanh (x) \text{sech}(x)}{7 a \sqrt{a \cosh ^3(x)}} \]

[Out]

(((-10*I)/21)*Cosh[x]^(3/2)*EllipticF[(I/2)*x, 2])/(a*Sqrt[a*Cosh[x]^3]) + (10*Sinh[x])/(21*a*Sqrt[a*Cosh[x]^3
]) + (2*Sech[x]*Tanh[x])/(7*a*Sqrt[a*Cosh[x]^3])

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Rubi [A]  time = 0.0342725, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2636, 2641} \[ \frac{10 \sinh (x)}{21 a \sqrt{a \cosh ^3(x)}}-\frac{10 i \cosh ^{\frac{3}{2}}(x) F\left (\left .\frac{i x}{2}\right |2\right )}{21 a \sqrt{a \cosh ^3(x)}}+\frac{2 \tanh (x) \text{sech}(x)}{7 a \sqrt{a \cosh ^3(x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cosh[x]^3)^(-3/2),x]

[Out]

(((-10*I)/21)*Cosh[x]^(3/2)*EllipticF[(I/2)*x, 2])/(a*Sqrt[a*Cosh[x]^3]) + (10*Sinh[x])/(21*a*Sqrt[a*Cosh[x]^3
]) + (2*Sech[x]*Tanh[x])/(7*a*Sqrt[a*Cosh[x]^3])

Rule 3207

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 2636

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

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 \frac{1}{\left (a \cosh ^3(x)\right )^{3/2}} \, dx &=\frac{\cosh ^{\frac{3}{2}}(x) \int \frac{1}{\cosh ^{\frac{9}{2}}(x)} \, dx}{a \sqrt{a \cosh ^3(x)}}\\ &=\frac{2 \text{sech}(x) \tanh (x)}{7 a \sqrt{a \cosh ^3(x)}}+\frac{\left (5 \cosh ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\cosh ^{\frac{5}{2}}(x)} \, dx}{7 a \sqrt{a \cosh ^3(x)}}\\ &=\frac{10 \sinh (x)}{21 a \sqrt{a \cosh ^3(x)}}+\frac{2 \text{sech}(x) \tanh (x)}{7 a \sqrt{a \cosh ^3(x)}}+\frac{\left (5 \cosh ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\sqrt{\cosh (x)}} \, dx}{21 a \sqrt{a \cosh ^3(x)}}\\ &=-\frac{10 i \cosh ^{\frac{3}{2}}(x) F\left (\left .\frac{i x}{2}\right |2\right )}{21 a \sqrt{a \cosh ^3(x)}}+\frac{10 \sinh (x)}{21 a \sqrt{a \cosh ^3(x)}}+\frac{2 \text{sech}(x) \tanh (x)}{7 a \sqrt{a \cosh ^3(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0588545, size = 48, normalized size = 0.64 \[ \frac{2 \cosh ^2(x) \left (-5 i \cosh ^{\frac{5}{2}}(x) \text{EllipticF}\left (\frac{i x}{2},2\right )+3 \tanh (x)+5 \sinh (x) \cosh (x)\right )}{21 \left (a \cosh ^3(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cosh[x]^3)^(-3/2),x]

[Out]

(2*Cosh[x]^2*((-5*I)*Cosh[x]^(5/2)*EllipticF[(I/2)*x, 2] + 5*Cosh[x]*Sinh[x] + 3*Tanh[x]))/(21*(a*Cosh[x]^3)^(
3/2))

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int \left ( a \left ( \cosh \left ( x \right ) \right ) ^{3} \right ) ^{-{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cosh(x)^3)^(3/2),x)

[Out]

int(1/(a*cosh(x)^3)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \cosh \left (x\right )^{3}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)^3)^(3/2),x, algorithm="maxima")

[Out]

integrate((a*cosh(x)^3)^(-3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \cosh \left (x\right )^{3}}}{a^{2} \cosh \left (x\right )^{6}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)^3)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*cosh(x)^3)/(a^2*cosh(x)^6), 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(1/(a*cosh(x)**3)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \cosh \left (x\right )^{3}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)^3)^(3/2),x, algorithm="giac")

[Out]

integrate((a*cosh(x)^3)^(-3/2), x)