3.514 \(\int \sqrt{3+4 \cos (c+d x)} \sec ^2(c+d x) \, dx\)

Optimal. Leaf size=95 \[ \frac{3 F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{\sqrt{7} d}-\frac{\sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{d}+\frac{4 \Pi \left (2;\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{\sqrt{7} d}+\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x)}{d} \]

[Out]

-((Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/d) + (3*EllipticF[(c + d*x)/2, 8/7])/(Sqrt[7]*d) + (4*EllipticPi[2, (c
+ d*x)/2, 8/7])/(Sqrt[7]*d) + (Sqrt[3 + 4*Cos[c + d*x]]*Tan[c + d*x])/d

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Rubi [A]  time = 0.245654, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2796, 3060, 2653, 3002, 2661, 2805} \[ \frac{3 F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{\sqrt{7} d}-\frac{\sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{d}+\frac{4 \Pi \left (2;\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{\sqrt{7} d}+\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[3 + 4*Cos[c + d*x]]*Sec[c + d*x]^2,x]

[Out]

-((Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/d) + (3*EllipticF[(c + d*x)/2, 8/7])/(Sqrt[7]*d) + (4*EllipticPi[2, (c
+ d*x)/2, 8/7])/(Sqrt[7]*d) + (Sqrt[3 + 4*Cos[c + d*x]]*Tan[c + d*x])/d

Rule 2796

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

Rule 3060

Int[((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[
(e_.) + (f_.)*(x_)])), x_Symbol] :> Dist[C/(b*d), Int[Sqrt[a + b*Sin[e + f*x]], x], x] - Dist[1/(b*d), Int[Sim
p[a*c*C - A*b*d + (b*c*C + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /;
FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

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 3002

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

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 2805

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

Rubi steps

\begin{align*} \int \sqrt{3+4 \cos (c+d x)} \sec ^2(c+d x) \, dx &=\frac{\sqrt{3+4 \cos (c+d x)} \tan (c+d x)}{d}+\int \frac{\left (2-2 \cos ^2(c+d x)\right ) \sec (c+d x)}{\sqrt{3+4 \cos (c+d x)}} \, dx\\ &=\frac{\sqrt{3+4 \cos (c+d x)} \tan (c+d x)}{d}-\frac{1}{4} \int \frac{(-8-6 \cos (c+d x)) \sec (c+d x)}{\sqrt{3+4 \cos (c+d x)}} \, dx-\frac{1}{2} \int \sqrt{3+4 \cos (c+d x)} \, dx\\ &=-\frac{\sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{d}+\frac{\sqrt{3+4 \cos (c+d x)} \tan (c+d x)}{d}+\frac{3}{2} \int \frac{1}{\sqrt{3+4 \cos (c+d x)}} \, dx+2 \int \frac{\sec (c+d x)}{\sqrt{3+4 \cos (c+d x)}} \, dx\\ &=-\frac{\sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{d}+\frac{3 F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{\sqrt{7} d}+\frac{4 \Pi \left (2;\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{\sqrt{7} d}+\frac{\sqrt{3+4 \cos (c+d x)} \tan (c+d x)}{d}\\ \end{align*}

Mathematica [C]  time = 1.09799, size = 157, normalized size = 1.65 \[ \frac{6 \sqrt{7} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{8}{7}\right )+21 \sqrt{4 \cos (c+d x)+3} \tan (c+d x)+\frac{i \sqrt{7} \sin (c+d x) \left (-12 F\left (i \sinh ^{-1}\left (\sqrt{4 \cos (c+d x)+3}\right )|-\frac{1}{7}\right )+21 E\left (i \sinh ^{-1}\left (\sqrt{4 \cos (c+d x)+3}\right )|-\frac{1}{7}\right )-8 \Pi \left (-\frac{1}{3};i \sinh ^{-1}\left (\sqrt{4 \cos (c+d x)+3}\right )|-\frac{1}{7}\right )\right )}{\sqrt{\sin ^2(c+d x)}}}{21 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[3 + 4*Cos[c + d*x]]*Sec[c + d*x]^2,x]

[Out]

(6*Sqrt[7]*EllipticPi[2, (c + d*x)/2, 8/7] + (I*Sqrt[7]*(21*EllipticE[I*ArcSinh[Sqrt[3 + 4*Cos[c + d*x]]], -1/
7] - 12*EllipticF[I*ArcSinh[Sqrt[3 + 4*Cos[c + d*x]]], -1/7] - 8*EllipticPi[-1/3, I*ArcSinh[Sqrt[3 + 4*Cos[c +
d*x]]], -1/7])*Sin[c + d*x])/Sqrt[Sin[c + d*x]^2] + 21*Sqrt[3 + 4*Cos[c + d*x]]*Tan[c + d*x])/(21*d)

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Maple [B]  time = 3.312, size = 350, normalized size = 3.7 \begin{align*} -{\frac{1}{d}\sqrt{- \left ( -8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( -4\,{\frac{\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticPi} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,2,2\,\sqrt{2} \right ) }{\sqrt{-8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+7\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}}}-2\,{\frac{\cos \left ( 1/2\,dx+c/2 \right ) \sqrt{-8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+7\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}}{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}+3\,{\frac{\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,2\,\sqrt{2} \right ) }{\sqrt{-8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+7\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}}}+{\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{-8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticE} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,2\,\sqrt{2} \right ){\frac{1}{\sqrt{-8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+7\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}}}} \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^2*(3+4*cos(d*x+c))^(1/2),x)

[Out]

-(-(-8*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-4*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-8*cos(1/2*d*x+1/
2*c)^2+1)^(1/2)/(-8*sin(1/2*d*x+1/2*c)^4+7*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,2*2^(1/
2))-2*cos(1/2*d*x+1/2*c)*(-8*sin(1/2*d*x+1/2*c)^4+7*sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*cos(1/2*d*x+1/2*c)^2-1)+3*(
sin(1/2*d*x+1/2*c)^2)^(1/2)*(-8*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-8*sin(1/2*d*x+1/2*c)^4+7*sin(1/2*d*x+1/2*c)^2)
^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2*2^(1/2))+(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-8*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/
(-8*sin(1/2*d*x+1/2*c)^4+7*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2*2^(1/2)))/sin(1/2*d*x+1/
2*c)/(8*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*(3+4*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(4*cos(d*x + c) + 3)*sec(d*x + c)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*(3+4*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(4*cos(d*x + c) + 3)*sec(d*x + c)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{4 \cos{\left (c + d x \right )} + 3} \sec ^{2}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**2*(3+4*cos(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(4*cos(c + d*x) + 3)*sec(c + d*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*(3+4*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(4*cos(d*x + c) + 3)*sec(d*x + c)^2, x)