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

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

[Out]

2/3*b*sinh(x)*(a+b*cosh(x))^(1/2)-8/3*I*a*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticE(I*sinh(1/2*x),2^(1/2)*(b
/(a+b))^(1/2))*(a+b*cosh(x))^(1/2)/((a+b*cosh(x))/(a+b))^(1/2)+2/3*I*(a^2-b^2)*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*
x)*EllipticF(I*sinh(1/2*x),2^(1/2)*(b/(a+b))^(1/2))*((a+b*cosh(x))/(a+b))^(1/2)/(a+b*cosh(x))^(1/2)

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Rubi [A]  time = 0.16, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, 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 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]

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 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 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 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 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]

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*}

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Mathematica [A]  time = 0.24, size = 111, normalized size = 0.90 \[ \frac {2 i \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )+2 b \sinh (x) (a+b \cosh (x))-8 i a (a+b) \sqrt {\frac {a+b \cosh (x)}{a+b}} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 \sqrt {a+b \cosh (x)}} \]

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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fricas [F]  time = 1.23, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \cosh \relax (x) + a\right )}^{\frac {3}{2}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \cosh \relax (x) + a\right )}^{\frac {3}{2}}\,{d x} \]

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)

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maple [B]  time = 0.53, size = 458, normalized size = 3.69 \[ \frac {2 \left (4 \sqrt {-\frac {2 b}{a -b}}\, b^{2} \cosh \left (\frac {x}{2}\right ) \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\left (2 \sqrt {-\frac {2 b}{a -b}}\, a b +2 \sqrt {-\frac {2 b}{a -b}}\, b^{2}\right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )+3 a^{2} \sqrt {\frac {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticF \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )+4 a b \sqrt {\frac {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticF \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )+b^{2} \sqrt {\frac {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticF \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )-8 \sqrt {\frac {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticE \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right ) a b \right ) \sqrt {\left (2 b \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+a -b \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}}{3 \sqrt {-\frac {2 b}{a -b}}\, \sqrt {2 b \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\left (a +b \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )+a +b}} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \cosh \relax (x) + a\right )}^{\frac {3}{2}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {cosh}\relax (x)\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*cosh(x))^(3/2),x)

[Out]

int((a + b*cosh(x))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \cosh {\relax (x )}\right )^{\frac {3}{2}}\, dx \]

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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