3.47 \(\int (1+\cosh ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=55 \[ \frac {2}{3} i F\left (\left .i x+\frac {\pi }{2}\right |-1\right )-2 i E\left (\left .i x+\frac {\pi }{2}\right |-1\right )+\frac {1}{3} \sinh (x) \cosh (x) \sqrt {\cosh ^2(x)+1} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3180, 3172, 3177, 3182} \[ \frac {2}{3} i F\left (\left .i x+\frac {\pi }{2}\right |-1\right )-2 i E\left (\left .i x+\frac {\pi }{2}\right |-1\right )+\frac {1}{3} \sinh (x) \cosh (x) \sqrt {\cosh ^2(x)+1} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cosh[x]^2)^(3/2),x]

[Out]

(-2*I)*EllipticE[Pi/2 + I*x, -1] + ((2*I)/3)*EllipticF[Pi/2 + I*x, -1] + (Cosh[x]*Sqrt[1 + Cosh[x]^2]*Sinh[x])
/3

Rule 3172

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Dist[
B/b, Int[Sqrt[a + b*Sin[e + f*x]^2], x], x] + Dist[(A*b - a*B)/b, Int[1/Sqrt[a + b*Sin[e + f*x]^2], x], x] /;
FreeQ[{a, b, e, f, A, B}, x]

Rule 3177

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[e + f*x, -(b/a)])/f, x]
/; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]

Rule 3180

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

Rule 3182

Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(1*EllipticF[e + f*x, -(b/a)])/(Sqrt[a]*
f), x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]

Rubi steps

\begin {align*} \int \left (1+\cosh ^2(x)\right )^{3/2} \, dx &=\frac {1}{3} \cosh (x) \sqrt {1+\cosh ^2(x)} \sinh (x)+\frac {1}{3} \int \frac {4+6 \cosh ^2(x)}{\sqrt {1+\cosh ^2(x)}} \, dx\\ &=\frac {1}{3} \cosh (x) \sqrt {1+\cosh ^2(x)} \sinh (x)-\frac {2}{3} \int \frac {1}{\sqrt {1+\cosh ^2(x)}} \, dx+2 \int \sqrt {1+\cosh ^2(x)} \, dx\\ &=-2 i E\left (\left .\frac {\pi }{2}+i x\right |-1\right )+\frac {2}{3} i F\left (\left .\frac {\pi }{2}+i x\right |-1\right )+\frac {1}{3} \cosh (x) \sqrt {1+\cosh ^2(x)} \sinh (x)\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 51, normalized size = 0.93 \[ \frac {4 i F\left (i x\left |\frac {1}{2}\right .\right )-24 i E\left (i x\left |\frac {1}{2}\right .\right )+\sinh (2 x) \sqrt {\cosh (2 x)+3}}{6 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cosh[x]^2)^(3/2),x]

[Out]

((-24*I)*EllipticE[I*x, 1/2] + (4*I)*EllipticF[I*x, 1/2] + Sqrt[3 + Cosh[2*x]]*Sinh[2*x])/(6*Sqrt[2])

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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\cosh \relax (x)^{2} + 1\right )}^{\frac {3}{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((cosh(x)^2 + 1)^(3/2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\cosh \relax (x)^{2} + 1\right )}^{\frac {3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((cosh(x)^2 + 1)^(3/2), x)

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maple [A]  time = 0.38, size = 99, normalized size = 1.80 \[ -\frac {\sqrt {\left (1+\cosh ^{2}\relax (x )\right ) \left (\sinh ^{2}\relax (x )\right )}\, \left (-\left (\cosh ^{5}\relax (x )\right )+10 i \sqrt {1+\cosh ^{2}\relax (x )}\, \sqrt {-\left (\sinh ^{2}\relax (x )\right )}\, \EllipticF \left (i \cosh \relax (x ), i\right )-6 i \sqrt {1+\cosh ^{2}\relax (x )}\, \sqrt {-\left (\sinh ^{2}\relax (x )\right )}\, \EllipticE \left (i \cosh \relax (x ), i\right )+\cosh \relax (x )\right )}{3 \sqrt {\cosh ^{4}\relax (x )-1}\, \sinh \relax (x ) \sqrt {1+\cosh ^{2}\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+cosh(x)^2)^(3/2),x)

[Out]

-1/3*((1+cosh(x)^2)*sinh(x)^2)^(1/2)*(-cosh(x)^5+10*I*(1+cosh(x)^2)^(1/2)*(-sinh(x)^2)^(1/2)*EllipticF(I*cosh(
x),I)-6*I*(1+cosh(x)^2)^(1/2)*(-sinh(x)^2)^(1/2)*EllipticE(I*cosh(x),I)+cosh(x))/(cosh(x)^4-1)^(1/2)/sinh(x)/(
1+cosh(x)^2)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\cosh \relax (x)^{2} + 1\right )}^{\frac {3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((cosh(x)^2 + 1)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(x)^2 + 1)^(3/2),x)

[Out]

int((cosh(x)^2 + 1)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\cosh ^{2}{\relax (x )} + 1\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((cosh(x)**2 + 1)**(3/2), x)

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