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

Optimal. Leaf size=121 \[ \frac {1}{3} b \sin (x) \cos (x) \sqrt {a+b \cos ^2(x)}-\frac {a (a+b) \sqrt {\frac {b \cos ^2(x)}{a}+1} F\left (x+\frac {\pi }{2}|-\frac {b}{a}\right )}{3 \sqrt {a+b \cos ^2(x)}}+\frac {2 (2 a+b) \sqrt {a+b \cos ^2(x)} E\left (x+\frac {\pi }{2}|-\frac {b}{a}\right )}{3 \sqrt {\frac {b \cos ^2(x)}{a}+1}} \]

[Out]

1/3*b*cos(x)*sin(x)*(a+b*cos(x)^2)^(1/2)-2/3*(2*a+b)*(sin(x)^2)^(1/2)/sin(x)*EllipticE(cos(x),(-b/a)^(1/2))*(a
+b*cos(x)^2)^(1/2)/(1+b*cos(x)^2/a)^(1/2)+1/3*a*(a+b)*(sin(x)^2)^(1/2)/sin(x)*EllipticF(cos(x),(-b/a)^(1/2))*(
1+b*cos(x)^2/a)^(1/2)/(a+b*cos(x)^2)^(1/2)

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Rubi [A]  time = 0.16, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3180, 3172, 3178, 3177, 3183, 3182} \[ \frac {1}{3} b \sin (x) \cos (x) \sqrt {a+b \cos ^2(x)}-\frac {a (a+b) \sqrt {\frac {b \cos ^2(x)}{a}+1} F\left (x+\frac {\pi }{2}|-\frac {b}{a}\right )}{3 \sqrt {a+b \cos ^2(x)}}+\frac {2 (2 a+b) \sqrt {a+b \cos ^2(x)} E\left (x+\frac {\pi }{2}|-\frac {b}{a}\right )}{3 \sqrt {\frac {b \cos ^2(x)}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cos[x]^2)^(3/2),x]

[Out]

(2*(2*a + b)*Sqrt[a + b*Cos[x]^2]*EllipticE[Pi/2 + x, -(b/a)])/(3*Sqrt[1 + (b*Cos[x]^2)/a]) - (a*(a + b)*Sqrt[
1 + (b*Cos[x]^2)/a]*EllipticF[Pi/2 + x, -(b/a)])/(3*Sqrt[a + b*Cos[x]^2]) + (b*Cos[x]*Sqrt[a + b*Cos[x]^2]*Sin
[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 3178

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Dist[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[1 + (b*Sin
[e + f*x]^2)/a], Int[Sqrt[1 + (b*Sin[e + f*x]^2)/a], x], 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]

Rule 3183

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

Rubi steps

\begin {align*} \int \left (a+b \cos ^2(x)\right )^{3/2} \, dx &=\frac {1}{3} b \cos (x) \sqrt {a+b \cos ^2(x)} \sin (x)+\frac {1}{3} \int \frac {a (3 a+b)+2 b (2 a+b) \cos ^2(x)}{\sqrt {a+b \cos ^2(x)}} \, dx\\ &=\frac {1}{3} b \cos (x) \sqrt {a+b \cos ^2(x)} \sin (x)-\frac {1}{3} (a (a+b)) \int \frac {1}{\sqrt {a+b \cos ^2(x)}} \, dx+\frac {1}{3} (2 (2 a+b)) \int \sqrt {a+b \cos ^2(x)} \, dx\\ &=\frac {1}{3} b \cos (x) \sqrt {a+b \cos ^2(x)} \sin (x)+\frac {\left (2 (2 a+b) \sqrt {a+b \cos ^2(x)}\right ) \int \sqrt {1+\frac {b \cos ^2(x)}{a}} \, dx}{3 \sqrt {1+\frac {b \cos ^2(x)}{a}}}-\frac {\left (a (a+b) \sqrt {1+\frac {b \cos ^2(x)}{a}}\right ) \int \frac {1}{\sqrt {1+\frac {b \cos ^2(x)}{a}}} \, dx}{3 \sqrt {a+b \cos ^2(x)}}\\ &=\frac {2 (2 a+b) \sqrt {a+b \cos ^2(x)} E\left (\frac {\pi }{2}+x|-\frac {b}{a}\right )}{3 \sqrt {1+\frac {b \cos ^2(x)}{a}}}-\frac {a (a+b) \sqrt {1+\frac {b \cos ^2(x)}{a}} F\left (\frac {\pi }{2}+x|-\frac {b}{a}\right )}{3 \sqrt {a+b \cos ^2(x)}}+\frac {1}{3} b \cos (x) \sqrt {a+b \cos ^2(x)} \sin (x)\\ \end {align*}

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Mathematica [A]  time = 0.48, size = 123, normalized size = 1.02 \[ \frac {8 \left (2 a^2+3 a b+b^2\right ) \sqrt {\frac {2 a+b \cos (2 x)+b}{a+b}} E\left (x\left |\frac {b}{a+b}\right .\right )+\sqrt {2} b \sin (2 x) (2 a+b \cos (2 x)+b)-4 a (a+b) \sqrt {\frac {2 a+b \cos (2 x)+b}{a+b}} F\left (x\left |\frac {b}{a+b}\right .\right )}{12 \sqrt {2 a+b \cos (2 x)+b}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cos[x]^2)^(3/2),x]

[Out]

(8*(2*a^2 + 3*a*b + b^2)*Sqrt[(2*a + b + b*Cos[2*x])/(a + b)]*EllipticE[x, b/(a + b)] - 4*a*(a + b)*Sqrt[(2*a
+ b + b*Cos[2*x])/(a + b)]*EllipticF[x, b/(a + b)] + Sqrt[2]*b*(2*a + b + b*Cos[2*x])*Sin[2*x])/(12*Sqrt[2*a +
b + b*Cos[2*x]])

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fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \cos \relax (x)^{2} + a\right )}^{\frac {3}{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*cos(x)^2 + a)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*cos(x)^2 + a)^(3/2), x)

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maple [A]  time = 1.79, size = 192, normalized size = 1.59 \[ -\frac {-\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \left (\cos ^{2}\relax (x )\right )}{a}}\, \EllipticF \left (\cos \relax (x ), \sqrt {-\frac {b}{a}}\right ) a^{2}}{3}-\frac {a \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \left (\cos ^{2}\relax (x )\right )}{a}}\, \EllipticF \left (\cos \relax (x ), \sqrt {-\frac {b}{a}}\right ) b}{3}+\frac {4 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \left (\cos ^{2}\relax (x )\right )}{a}}\, \EllipticE \left (\cos \relax (x ), \sqrt {-\frac {b}{a}}\right ) a^{2}}{3}+\frac {2 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \left (\cos ^{2}\relax (x )\right )}{a}}\, \EllipticE \left (\cos \relax (x ), \sqrt {-\frac {b}{a}}\right ) a b}{3}+\frac {\left (\cos ^{5}\relax (x )\right ) b^{2}}{3}+\frac {\left (\cos ^{3}\relax (x )\right ) a b}{3}-\frac {\left (\cos ^{3}\relax (x )\right ) b^{2}}{3}-\frac {a b \cos \relax (x )}{3}}{\sin \relax (x ) \sqrt {a +b \left (\cos ^{2}\relax (x )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-1/3*(sin(x)^2)^(1/2)*((a+b*cos(x)^2)/a)^(1/2)*EllipticF(cos(x),(-1/a*b)^(1/2))*a^2-1/3*a*(sin(x)^2)^(1/2)*(
(a+b*cos(x)^2)/a)^(1/2)*EllipticF(cos(x),(-1/a*b)^(1/2))*b+4/3*(sin(x)^2)^(1/2)*((a+b*cos(x)^2)/a)^(1/2)*Ellip
ticE(cos(x),(-1/a*b)^(1/2))*a^2+2/3*(sin(x)^2)^(1/2)*((a+b*cos(x)^2)/a)^(1/2)*EllipticE(cos(x),(-1/a*b)^(1/2))
*a*b+1/3*cos(x)^5*b^2+1/3*cos(x)^3*a*b-1/3*cos(x)^3*b^2-1/3*a*b*cos(x))/sin(x)/(a+b*cos(x)^2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*cos(x)^2 + a)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*cos(x)^2)^(3/2),x)

[Out]

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

[Out]

Timed out

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