### 3.594 $$\int \frac{1}{(a \cosh (x)+b \sinh (x))^{3/2}} \, dx$$

Optimal. Leaf size=112 $\frac{2 (a \sinh (x)+b \cosh (x))}{\left (a^2-b^2\right ) \sqrt{a \cosh (x)+b \sinh (x)}}+\frac{2 i \sqrt{a \cosh (x)+b \sinh (x)} E\left (\left .\frac{1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{\left (a^2-b^2\right ) \sqrt{\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}}}$

[Out]

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

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Rubi [A]  time = 0.0505367, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.231, Rules used = {3076, 3078, 2639} $\frac{2 (a \sinh (x)+b \cosh (x))}{\left (a^2-b^2\right ) \sqrt{a \cosh (x)+b \sinh (x)}}+\frac{2 i \sqrt{a \cosh (x)+b \sinh (x)} E\left (\left .\frac{1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{\left (a^2-b^2\right ) \sqrt{\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*Cosh[x] + b*Sinh[x])^(-3/2),x]

[Out]

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

Rule 3076

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[((b*Cos[c + d*x] -
a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1))/(d*(n + 1)*(a^2 + b^2)), x] + Dist[(n + 2)/((n + 1
)*(a^2 + b^2)), Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b
^2, 0] && LtQ[n, -1] && NeQ[n, -2]

Rule 3078

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(a*Cos[c + d*x] +
b*Sin[c + d*x])^n/((a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2])^n, Int[Cos[c + d*x - ArcTan[a, b]]^n, x]
, x] /; FreeQ[{a, b, c, d, n}, x] &&  !(GeQ[n, 1] || LeQ[n, -1]) &&  !(GtQ[a^2 + b^2, 0] || EqQ[a^2 + b^2, 0])

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]

Rubi steps

\begin{align*} \int \frac{1}{(a \cosh (x)+b \sinh (x))^{3/2}} \, dx &=\frac{2 (b \cosh (x)+a \sinh (x))}{\left (a^2-b^2\right ) \sqrt{a \cosh (x)+b \sinh (x)}}+\frac{\int \sqrt{a \cosh (x)+b \sinh (x)} \, dx}{-a^2+b^2}\\ &=\frac{2 (b \cosh (x)+a \sinh (x))}{\left (a^2-b^2\right ) \sqrt{a \cosh (x)+b \sinh (x)}}+\frac{\sqrt{a \cosh (x)+b \sinh (x)} \int \sqrt{\cosh \left (x+i \tan ^{-1}(a,-i b)\right )} \, dx}{\left (-a^2+b^2\right ) \sqrt{\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}}}\\ &=\frac{2 (b \cosh (x)+a \sinh (x))}{\left (a^2-b^2\right ) \sqrt{a \cosh (x)+b \sinh (x)}}+\frac{2 i E\left (\left .\frac{1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right ) \sqrt{a \cosh (x)+b \sinh (x)}}{\left (a^2-b^2\right ) \sqrt{\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}}}\\ \end{align*}

Mathematica [C]  time = 0.446826, size = 148, normalized size = 1.32 $\frac{b \sinh \left (\tanh ^{-1}\left (\frac{b}{a}\right )+x\right ) \text{HypergeometricPFQ}\left (\left \{-\frac{1}{2},-\frac{1}{4}\right \},\left \{\frac{3}{4}\right \},\cosh ^2\left (\tanh ^{-1}\left (\frac{b}{a}\right )+x\right )\right )-\sqrt{-\sinh ^2\left (\tanh ^{-1}\left (\frac{b}{a}\right )+x\right )} \left (2 a \sqrt{1-\frac{b^2}{a^2}} \cosh (x)+b \sinh \left (\tanh ^{-1}\left (\frac{b}{a}\right )+x\right )-2 a \cosh \left (\tanh ^{-1}\left (\frac{b}{a}\right )+x\right )\right )}{a b \sqrt{1-\frac{b^2}{a^2}} \sqrt{-\sinh ^2\left (\tanh ^{-1}\left (\frac{b}{a}\right )+x\right )} \sqrt{a \cosh (x)+b \sinh (x)}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(b*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cosh[x + ArcTanh[b/a]]^2]*Sinh[x + ArcTanh[b/a]] - Sqrt[-Sinh[x + Ar
cTanh[b/a]]^2]*(2*a*Sqrt[1 - b^2/a^2]*Cosh[x] - 2*a*Cosh[x + ArcTanh[b/a]] + b*Sinh[x + ArcTanh[b/a]]))/(a*b*S
qrt[1 - b^2/a^2]*Sqrt[a*Cosh[x] + b*Sinh[x]]*Sqrt[-Sinh[x + ArcTanh[b/a]]^2])

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Maple [A]  time = 0.12, size = 33, normalized size = 0.3 \begin{align*}{{\it Artanh} \left ( \cosh \left ( x \right ) \right ){\frac{1}{\sqrt{{a}^{2}-{b}^{2}}}}{\frac{1}{\sqrt{-\sinh \left ( x \right ) \sqrt{{a}^{2}-{b}^{2}}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a^2-b^2)^(1/2)/(-sinh(x)*(a^2-b^2)^(1/2))^(1/2)*arctanh(cosh(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 ) + b \sinh \left (x\right )\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*cosh(x) + b*sinh(x))^(-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 ) + b \sinh \left (x\right )}}{a^{2} \cosh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) \sinh \left (x\right ) + b^{2} \sinh \left (x\right )^{2}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a*cosh(x) + b*sinh(x))/(a^2*cosh(x)^2 + 2*a*b*cosh(x)*sinh(x) + b^2*sinh(x)^2), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)+b*sinh(x))**(3/2),x)

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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