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]

(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

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Rubi [A]  time = 0.161739, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 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 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 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 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 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]

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 (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*}

Mathematica [A]  time = 0.451918, 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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Maple [A]  time = 0.783, size = 192, normalized size = 1.6 \begin{align*} -{\frac{1}{\sin \left ( x \right ) } \left ( -{\frac{{a}^{2}}{3}\sqrt{ \left ( \sin \left ( x \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \cos \left ( x \right ) \right ) ^{2}}{a}}}{\it EllipticF} \left ( \cos \left ( x \right ) ,\sqrt{-{\frac{b}{a}}} \right ) }-{\frac{ab}{3}\sqrt{ \left ( \sin \left ( x \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \cos \left ( x \right ) \right ) ^{2}}{a}}}{\it EllipticF} \left ( \cos \left ( x \right ) ,\sqrt{-{\frac{b}{a}}} \right ) }+{\frac{4\,{a}^{2}}{3}\sqrt{ \left ( \sin \left ( x \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \cos \left ( x \right ) \right ) ^{2}}{a}}}{\it EllipticE} \left ( \cos \left ( x \right ) ,\sqrt{-{\frac{b}{a}}} \right ) }+{\frac{2\,ab}{3}\sqrt{ \left ( \sin \left ( x \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \cos \left ( x \right ) \right ) ^{2}}{a}}}{\it EllipticE} \left ( \cos \left ( x \right ) ,\sqrt{-{\frac{b}{a}}} \right ) }+{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5}{b}^{2}}{3}}+{\frac{ \left ( \cos \left ( x \right ) \right ) ^{3}ba}{3}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{3}{b}^{2}}{3}}-{\frac{ab\cos \left ( x \right ) }{3}} \right ){\frac{1}{\sqrt{a+b \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \end{align*}

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),(-b/a)^(1/2))*a^2-1/3*a*(sin(x)^2)^(1/2)*((a
+b*cos(x)^2)/a)^(1/2)*EllipticF(cos(x),(-b/a)^(1/2))*b+4/3*(sin(x)^2)^(1/2)*((a+b*cos(x)^2)/a)^(1/2)*EllipticE
(cos(x),(-b/a)^(1/2))*a^2+2/3*(sin(x)^2)^(1/2)*((a+b*cos(x)^2)/a)^(1/2)*EllipticE(cos(x),(-b/a)^(1/2))*a*b+1/3
*cos(x)^5*b^2+1/3*cos(x)^3*b*a-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., size = 0, normalized size = 0. \begin{align*} \int{\left (b \cos \left (x\right )^{2} + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \cos \left (x\right )^{2} + a\right )}^{\frac{3}{2}}, x\right ) \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \cos \left (x\right )^{2} + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

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)