3.80 \(\int (a+b \cosh (x))^{3/2} \, dx\)

Optimal. Leaf size=124 \[ \frac{2 i \left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}} \text{EllipticF}\left (\frac{i x}{2},\frac{2 b}{a+b}\right )}{3 \sqrt{a+b \cosh (x)}}+\frac{2}{3} b \sinh (x) \sqrt{a+b \cosh (x)}-\frac{8 i a \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{3 \sqrt{\frac{a+b \cosh (x)}{a+b}}} \]

[Out]

(((-8*I)/3)*a*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/Sqrt[(a + b*Cosh[x])/(a + b)] + (((2*I)/3
)*(a^2 - b^2)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)])/Sqrt[a + b*Cosh[x]] + (2*b*Sqrt
[a + b*Cosh[x]]*Sinh[x])/3

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Rubi [A]  time = 0.156512, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {2656, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 i \left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}} F\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{3 \sqrt{a+b \cosh (x)}}+\frac{2}{3} b \sinh (x) \sqrt{a+b \cosh (x)}-\frac{8 i a \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{3 \sqrt{\frac{a+b \cosh (x)}{a+b}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[x])^(3/2),x]

[Out]

(((-8*I)/3)*a*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/Sqrt[(a + b*Cosh[x])/(a + b)] + (((2*I)/3
)*(a^2 - b^2)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)])/Sqrt[a + b*Cosh[x]] + (2*b*Sqrt
[a + b*Cosh[x]]*Sinh[x])/3

Rule 2656

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

Rule 2752

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]

Rule 2663

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

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

\begin{align*} \int (a+b \cosh (x))^{3/2} \, dx &=\frac{2}{3} b \sqrt{a+b \cosh (x)} \sinh (x)+\frac{2}{3} \int \frac{\frac{1}{2} \left (3 a^2+b^2\right )+2 a b \cosh (x)}{\sqrt{a+b \cosh (x)}} \, dx\\ &=\frac{2}{3} b \sqrt{a+b \cosh (x)} \sinh (x)+\frac{1}{3} (4 a) \int \sqrt{a+b \cosh (x)} \, dx+\frac{1}{3} \left (-a^2+b^2\right ) \int \frac{1}{\sqrt{a+b \cosh (x)}} \, dx\\ &=\frac{2}{3} b \sqrt{a+b \cosh (x)} \sinh (x)+\frac{\left (4 a \sqrt{a+b \cosh (x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \cosh (x)}{a+b}} \, dx}{3 \sqrt{\frac{a+b \cosh (x)}{a+b}}}+\frac{\left (\left (-a^2+b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \cosh (x)}{a+b}}} \, dx}{3 \sqrt{a+b \cosh (x)}}\\ &=-\frac{8 i a \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{3 \sqrt{\frac{a+b \cosh (x)}{a+b}}}+\frac{2 i \left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}} F\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{3 \sqrt{a+b \cosh (x)}}+\frac{2}{3} b \sqrt{a+b \cosh (x)} \sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.227168, size = 111, normalized size = 0.9 \[ \frac{2 i \left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}} \text{EllipticF}\left (\frac{i x}{2},\frac{2 b}{a+b}\right )+2 b \sinh (x) (a+b \cosh (x))-8 i a (a+b) \sqrt{\frac{a+b \cosh (x)}{a+b}} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{3 \sqrt{a+b \cosh (x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cosh[x])^(3/2),x]

[Out]

((-8*I)*a*(a + b)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticE[(I/2)*x, (2*b)/(a + b)] + (2*I)*(a^2 - b^2)*Sqrt[(a
+ b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)] + 2*b*(a + b*Cosh[x])*Sinh[x])/(3*Sqrt[a + b*Cosh[x]])

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Maple [B]  time = 0.063, size = 458, normalized size = 3.7 \begin{align*}{\frac{2}{3} \left ( 4\,\sqrt{-2\,{\frac{b}{a-b}}}{b}^{2}\cosh \left ( x/2 \right ) \left ( \sinh \left ( x/2 \right ) \right ) ^{4}+ \left ( 2\,\sqrt{-2\,{\frac{b}{a-b}}}ab+2\,\sqrt{-2\,{\frac{b}{a-b}}}{b}^{2} \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}\cosh \left ({\frac{x}{2}} \right ) +3\,{a}^{2}\sqrt{2\,{\frac{ \left ( \sinh \left ( x/2 \right ) \right ) ^{2}b}{a-b}}+{\frac{a+b}{a-b}}}\sqrt{- \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}{\it EllipticF} \left ( \cosh \left ( x/2 \right ) \sqrt{-2\,{\frac{b}{a-b}}},1/2\,\sqrt{-2\,{\frac{a-b}{b}}} \right ) +4\,ab\sqrt{2\,{\frac{ \left ( \sinh \left ( x/2 \right ) \right ) ^{2}b}{a-b}}+{\frac{a+b}{a-b}}}\sqrt{- \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}{\it EllipticF} \left ( \cosh \left ( x/2 \right ) \sqrt{-2\,{\frac{b}{a-b}}},1/2\,\sqrt{-2\,{\frac{a-b}{b}}} \right ) +{b}^{2}\sqrt{2\,{\frac{ \left ( \sinh \left ( x/2 \right ) \right ) ^{2}b}{a-b}}+{\frac{a+b}{a-b}}}\sqrt{- \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}}{\it EllipticF} \left ( \cosh \left ({\frac{x}{2}} \right ) \sqrt{-2\,{\frac{b}{a-b}}},{\frac{1}{2}\sqrt{-2\,{\frac{a-b}{b}}}} \right ) -8\,\sqrt{2\,{\frac{ \left ( \sinh \left ( x/2 \right ) \right ) ^{2}b}{a-b}}+{\frac{a+b}{a-b}}}\sqrt{- \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cosh \left ( x/2 \right ) \sqrt{-2\,{\frac{b}{a-b}}},1/2\,\sqrt{-2\,{\frac{a-b}{b}}} \right ) ab \right ) \sqrt{ \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}b+a-b \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}}{\frac{1}{\sqrt{-2\,{\frac{b}{a-b}}}}}{\frac{1}{\sqrt{2\,b \left ( \sinh \left ( x/2 \right ) \right ) ^{4}+ \left ( a+b \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}}}} \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{2}b+a+b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(4*(-2*b/(a-b))^(1/2)*b^2*cosh(1/2*x)*sinh(1/2*x)^4+(2*(-2*b/(a-b))^(1/2)*a*b+2*(-2*b/(a-b))^(1/2)*b^2)*si
nh(1/2*x)^2*cosh(1/2*x)+3*a^2*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(cos
h(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))+4*a*b*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1
/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))+b^2*(2*b/(a-b)*sinh(1/2*x)^2+(
a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))-8*(2
*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*
(-2*(a-b)/b)^(1/2))*a*b)*((2*cosh(1/2*x)^2*b+a-b)*sinh(1/2*x)^2)^(1/2)/(-2*b/(a-b))^(1/2)/(2*b*sinh(1/2*x)^4+(
a+b)*sinh(1/2*x)^2)^(1/2)/sinh(1/2*x)/(2*sinh(1/2*x)^2*b+a+b)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*cosh(x) + a)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*cosh(x) + a)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*cosh(x))**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*cosh(x) + a)^(3/2), x)