3.1296 \(\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=250 \[ -\frac{(5 A-7 B+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{(7 A-7 B+9 C) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{5 (7 A-7 B+9 C) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{5 (7 A-7 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a d}-\frac{3 (5 A-7 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d} \]

[Out]

(-3*(5*A - 7*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) + (5*(7*A - 7*B
+ 9*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*a*d) - ((A - B + C)*Sin[c + d*x])
/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)) + ((7*A - 7*B + 9*C)*Sin[c + d*x])/(7*a*d*Sec[c + d*x]^(5/2)) - (
(5*A - 7*B + 7*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) + (5*(7*A - 7*B + 9*C)*Sin[c + d*x])/(21*a*d*Sqrt[S
ec[c + d*x]])

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Rubi [A]  time = 0.365452, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used = {4221, 3041, 2748, 2635, 2639, 2641} \[ -\frac{(5 A-7 B+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{(7 A-7 B+9 C) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{5 (7 A-7 B+9 C) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{5 (7 A-7 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a d}-\frac{3 (5 A-7 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)),x]

[Out]

(-3*(5*A - 7*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) + (5*(7*A - 7*B
+ 9*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*a*d) - ((A - B + C)*Sin[c + d*x])
/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)) + ((7*A - 7*B + 9*C)*Sin[c + d*x])/(7*a*d*Sec[c + d*x]^(5/2)) - (
(5*A - 7*B + 7*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) + (5*(7*A - 7*B + 9*C)*Sin[c + d*x])/(21*a*d*Sqrt[S
ec[c + d*x]])

Rule 4221

Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Dist[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u,
x]

Rule 3041

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((a*A - b*B + a*C)*Cos[e + f*x]*(
a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(f*(b*c - a*d)*(2*m + 1)), x] + Dist[1/(b*(b*c - a*d)*(2*m
+ 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) + B*(
b*c*m + a*d*(n + 1)) - C*(a*c*m + b*d*(n + 1)) + (d*(a*A - b*B)*(m + n + 2) + C*(b*c*(2*m + 1) - a*d*(m - n -
1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^
2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)]

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

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 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

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 \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx\\ &=-\frac{(A-B+C) \sin (c+d x)}{d (a+a \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{5}{2}}(c+d x) \left (-\frac{1}{2} a (5 A-7 B+7 C)+\frac{1}{2} a (7 A-7 B+9 C) \cos (c+d x)\right ) \, dx}{a^2}\\ &=-\frac{(A-B+C) \sin (c+d x)}{d (a+a \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}-\frac{\left ((5 A-7 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{5}{2}}(c+d x) \, dx}{2 a}+\frac{\left ((7 A-7 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{7}{2}}(c+d x) \, dx}{2 a}\\ &=-\frac{(A-B+C) \sin (c+d x)}{d (a+a \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}+\frac{(7 A-7 B+9 C) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{(5 A-7 B+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{\left (3 (5 A-7 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{10 a}+\frac{\left (5 (7 A-7 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx}{14 a}\\ &=-\frac{3 (5 A-7 B+7 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 a d}-\frac{(A-B+C) \sin (c+d x)}{d (a+a \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}+\frac{(7 A-7 B+9 C) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{(5 A-7 B+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{5 (7 A-7 B+9 C) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}+\frac{\left (5 (7 A-7 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{42 a}\\ &=-\frac{3 (5 A-7 B+7 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 a d}+\frac{5 (7 A-7 B+9 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{21 a d}-\frac{(A-B+C) \sin (c+d x)}{d (a+a \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}+\frac{(7 A-7 B+9 C) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{(5 A-7 B+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{5 (7 A-7 B+9 C) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 1.80131, size = 198, normalized size = 0.79 \[ -\frac{\cos ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{\sec (c+d x)} \left (-100 (7 A-7 B+9 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+252 (5 A-7 B+7 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\left (\sin \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{3}{2} (c+d x)\right )\right ) \sec \left (\frac{1}{2} (c+d x)\right ) ((140 A-56 B+201 C) \cos (c+d x)+350 A+6 (7 B-2 C) \cos (2 (c+d x))-308 B+15 C \cos (3 (c+d x))+438 C)\right )}{210 a d (\cos (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)),x]

[Out]

-(Cos[(c + d*x)/2]^2*Sqrt[Sec[c + d*x]]*(252*(5*A - 7*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] -
100*(7*A - 7*B + 9*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (350*A - 308*B + 438*C + (140*A - 56*B +
201*C)*Cos[c + d*x] + 6*(7*B - 2*C)*Cos[2*(c + d*x)] + 15*C*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]*(Sin[(c + d*x)/
2] - Sin[(3*(c + d*x))/2])))/(210*a*d*(1 + Cos[c + d*x]))

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Maple [A]  time = 1.234, size = 341, normalized size = 1.4 \begin{align*} -{\frac{1}{105\,da}\sqrt{ \left ( 2\, \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 ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 175\,A{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +315\,A{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -175\,B{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -441\,B{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +225\,C{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +441\,C{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \right ) -480\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}+ \left ( 336\,B+864\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{8}+ \left ( -280\,A-392\,B-888\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}+ \left ( 630\,A-210\,B+930\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+ \left ( -245\,A+161\,B-321\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \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((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))/sec(d*x+c)^(5/2),x)

[Out]

-1/105*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(cos(1/2*d*x+1/2*c)*(2*sin(1/2*d*x+1/2*c)^2-1)^
(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(175*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+315*A*EllipticE(cos(1/2*d*x+1/
2*c),2^(1/2))-175*B*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-441*B*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+225*C*El
lipticF(cos(1/2*d*x+1/2*c),2^(1/2))+441*C*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))-480*C*sin(1/2*d*x+1/2*c)^10+(
336*B+864*C)*sin(1/2*d*x+1/2*c)^8+(-280*A-392*B-888*C)*sin(1/2*d*x+1/2*c)^6+(630*A-210*B+930*C)*sin(1/2*d*x+1/
2*c)^4+(-245*A+161*B-321*C)*sin(1/2*d*x+1/2*c)^2)/a/cos(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/
2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2*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 \frac{C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (a \cos \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))/sec(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)/((a*cos(d*x + c) + a)*sec(d*x + c)^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (a \cos \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac{5}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))/sec(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

integral((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)/((a*cos(d*x + c) + a)*sec(d*x + c)^(5/2)), 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(d*x+c)+C*cos(d*x+c)**2)/(a+a*cos(d*x+c))/sec(d*x+c)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (a \cos \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))/sec(d*x+c)^(5/2),x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)/((a*cos(d*x + c) + a)*sec(d*x + c)^(5/2)), x)