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

Optimal. Leaf size=46 \[ \frac {2 \sinh (a+b x) \sqrt {\cosh (a+b x)}}{3 b}-\frac {2 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2635, 2641} \[ \frac {2 \sinh (a+b x) \sqrt {\cosh (a+b x)}}{3 b}-\frac {2 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^(3/2),x]

[Out]

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

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[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 \cosh ^{\frac {3}{2}}(a+b x) \, dx &=\frac {2 \sqrt {\cosh (a+b x)} \sinh (a+b x)}{3 b}+\frac {1}{3} \int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx\\ &=-\frac {2 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{3 b}+\frac {2 \sqrt {\cosh (a+b x)} \sinh (a+b x)}{3 b}\\ \end {align*}

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Mathematica [C]  time = 0.11, size = 81, normalized size = 1.76 \[ \frac {2 \sqrt {\sinh (2 (a+b x))+\cosh (2 (a+b x))+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\cosh (2 (a+b x))-\sinh (2 (a+b x))\right )+\sinh (2 (a+b x))}{3 b \sqrt {\cosh (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]^(3/2),x]

[Out]

(Sinh[2*(a + b*x)] + 2*Hypergeometric2F1[1/4, 1/2, 5/4, -Cosh[2*(a + b*x)] - Sinh[2*(a + b*x)]]*Sqrt[1 + Cosh[
2*(a + b*x)] + Sinh[2*(a + b*x)]])/(3*b*Sqrt[Cosh[a + b*x]])

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fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cosh \left (b x + a\right )^{\frac {3}{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cosh(b*x + a)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cosh(b*x + a)^(3/2), x)

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maple [B]  time = 0.29, size = 174, normalized size = 3.78 \[ \frac {2 \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (4 \left (\cosh ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-6 \left (\cosh ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1}\, \EllipticF \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 \sqrt {2 \left (\sinh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*((2*cosh(1/2*b*x+1/2*a)^2-1)*sinh(1/2*b*x+1/2*a)^2)^(1/2)*(4*cosh(1/2*b*x+1/2*a)^5-6*cosh(1/2*b*x+1/2*a)^3
+(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-2*cosh(1/2*b*x+1/2*a)^2+1)^(1/2)*EllipticF(cosh(1/2*b*x+1/2*a),2^(1/2))+2*co
sh(1/2*b*x+1/2*a))/(2*sinh(1/2*b*x+1/2*a)^4+sinh(1/2*b*x+1/2*a)^2)^(1/2)/sinh(1/2*b*x+1/2*a)/(2*cosh(1/2*b*x+1
/2*a)^2-1)^(1/2)/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cosh(b*x + a)^(3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {cosh}\left (a+b\,x\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)^(3/2),x)

[Out]

int(cosh(a + b*x)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(a + b*x)**(3/2), x)

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