3.79 \(\int (a+b \cosh (x))^{5/2} \, dx\)

Optimal. Leaf size=153 \[ \frac {16 i a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 \sqrt {a+b \cosh (x)}}-\frac {2 i \left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2}{5} b \sinh (x) (a+b \cosh (x))^{3/2}+\frac {16}{15} a b \sinh (x) \sqrt {a+b \cosh (x)} \]

[Out]

2/5*b*(a+b*cosh(x))^(3/2)*sinh(x)+16/15*a*b*sinh(x)*(a+b*cosh(x))^(1/2)-2/15*I*(23*a^2+9*b^2)*(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)+16/15*I*a*(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 = 0.24, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {2656, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac {16 i a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 \sqrt {a+b \cosh (x)}}-\frac {2 i \left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2}{5} b \sinh (x) (a+b \cosh (x))^{3/2}+\frac {16}{15} a b \sinh (x) \sqrt {a+b \cosh (x)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[x])^(5/2),x]

[Out]

(((-2*I)/15)*(23*a^2 + 9*b^2)*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/Sqrt[(a + b*Cosh[x])/(a +
b)] + (((16*I)/15)*a*(a^2 - b^2)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)])/Sqrt[a + b*
Cosh[x]] + (16*a*b*Sqrt[a + b*Cosh[x]]*Sinh[x])/15 + (2*b*(a + b*Cosh[x])^(3/2)*Sinh[x])/5

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]

Rule 2753

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

Rubi steps

\begin {align*} \int (a+b \cosh (x))^{5/2} \, dx &=\frac {2}{5} b (a+b \cosh (x))^{3/2} \sinh (x)+\frac {2}{5} \int \sqrt {a+b \cosh (x)} \left (\frac {1}{2} \left (5 a^2+3 b^2\right )+4 a b \cosh (x)\right ) \, dx\\ &=\frac {16}{15} a b \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{5} b (a+b \cosh (x))^{3/2} \sinh (x)+\frac {4}{15} \int \frac {\frac {1}{4} a \left (15 a^2+17 b^2\right )+\frac {1}{4} b \left (23 a^2+9 b^2\right ) \cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx\\ &=\frac {16}{15} a b \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{5} b (a+b \cosh (x))^{3/2} \sinh (x)-\frac {1}{15} \left (8 a \left (a^2-b^2\right )\right ) \int \frac {1}{\sqrt {a+b \cosh (x)}} \, dx+\frac {1}{15} \left (23 a^2+9 b^2\right ) \int \sqrt {a+b \cosh (x)} \, dx\\ &=\frac {16}{15} a b \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{5} b (a+b \cosh (x))^{3/2} \sinh (x)+\frac {\left (\left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}} \, dx}{15 \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {\left (8 a \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}{15 \sqrt {a+b \cosh (x)}}\\ &=-\frac {2 i \left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {16 i a \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 )}{15 \sqrt {a+b \cosh (x)}}+\frac {16}{15} a b \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{5} b (a+b \cosh (x))^{3/2} \sinh (x)\\ \end {align*}

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Mathematica [A]  time = 0.55, size = 150, normalized size = 0.98 \[ \frac {b \sinh (x) \left (22 a^2+28 a b \cosh (x)+3 b^2 \cosh (2 x)+3 b^2\right )+16 i a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )-2 i \left (23 a^3+23 a^2 b+9 a b^2+9 b^3\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 \sqrt {a+b \cosh (x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cosh[x])^(5/2),x]

[Out]

((-2*I)*(23*a^3 + 23*a^2*b + 9*a*b^2 + 9*b^3)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticE[(I/2)*x, (2*b)/(a + b)]
+ (16*I)*a*(a^2 - b^2)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)] + b*(22*a^2 + 3*b^2 + 2
8*a*b*Cosh[x] + 3*b^2*Cosh[2*x])*Sinh[x])/(15*Sqrt[a + b*Cosh[x]])

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fricas [F]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \cosh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + a^{2}\right )} \sqrt {b \cosh \relax (x) + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(x))^(5/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \cosh \relax (x) + a\right )}^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(x))^(5/2),x, algorithm="giac")

[Out]

integrate((b*cosh(x) + a)^(5/2), x)

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maple [B]  time = 0.49, size = 685, normalized size = 4.48 \[ \frac {2 \left (24 \sqrt {-\frac {2 b}{a -b}}\, b^{3} \cosh \left (\frac {x}{2}\right ) \left (\sinh ^{6}\left (\frac {x}{2}\right )\right )+\left (56 \sqrt {-\frac {2 b}{a -b}}\, a \,b^{2}+24 \sqrt {-\frac {2 b}{a -b}}\, b^{3}\right ) \left (\sinh ^{4}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )+\left (22 \sqrt {-\frac {2 b}{a -b}}\, a^{2} b +28 \sqrt {-\frac {2 b}{a -b}}\, a \,b^{2}+6 \sqrt {-\frac {2 b}{a -b}}\, b^{3}\right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )+15 a^{3} \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 )+23 a^{2} 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 )+17 a \,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 )+9 b^{3} \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 )-46 \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^{2} b -18 \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 ) b^{3}\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 )}}{15 \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))^(5/2),x)

[Out]

2/15*(24*(-2*b/(a-b))^(1/2)*b^3*cosh(1/2*x)*sinh(1/2*x)^6+(56*(-2*b/(a-b))^(1/2)*a*b^2+24*(-2*b/(a-b))^(1/2)*b
^3)*sinh(1/2*x)^4*cosh(1/2*x)+(22*(-2*b/(a-b))^(1/2)*a^2*b+28*(-2*b/(a-b))^(1/2)*a*b^2+6*(-2*b/(a-b))^(1/2)*b^
3)*sinh(1/2*x)^2*cosh(1/2*x)+15*a^3*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*Ellipti
cF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))+23*a^2*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))+17*a*b^2*(2*b/(a-b)*s
inh(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))+9*b^3*(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))-46*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*El
lipticE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))*a^2*b-18*(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))*b^3)*((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 \[ \int {\left (b \cosh \relax (x) + a\right )}^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(x))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*cosh(x) + a)^(5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {cosh}\relax (x)\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*cosh(x))^(5/2),x)

[Out]

int((a + b*cosh(x))^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(x))**(5/2),x)

[Out]

Timed out

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