3.540 \(\int \frac{(a+a \sec (c+d x)) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=215 \[ \frac{2 a (5 (A+B)+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}+\frac{2 a (7 A+9 (B+C)) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (5 (A+B)+7 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+9 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a (A+B) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]

[Out]

(2*a*(7*A + 9*(B + C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*a*(5*(A +
B) + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*A*Sin[c + d*x])/(9*d*
Sec[c + d*x]^(7/2)) + (2*a*(A + B)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(7*A + 9*(B + C))*Sin[c + d*x
])/(45*d*Sec[c + d*x]^(3/2)) + (2*a*(5*(A + B) + 7*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])

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Rubi [A]  time = 0.282104, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4074, 4047, 3769, 3771, 2639, 4045, 2641} \[ \frac{2 a (7 A+9 (B+C)) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (5 (A+B)+7 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (5 (A+B)+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 a (7 A+9 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a (A+B) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]

[Out]

(2*a*(7*A + 9*(B + C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*a*(5*(A +
B) + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*A*Sin[c + d*x])/(9*d*
Sec[c + d*x]^(7/2)) + (2*a*(A + B)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(7*A + 9*(B + C))*Sin[c + d*x
])/(45*d*Sec[c + d*x]^(3/2)) + (2*a*(5*(A + B) + 7*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])

Rule 4074

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(A*a*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*n), x]
+ Dist[1/(d*n), Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (n*(a*C + B*b) + A*a*(n + 1))*Csc[e + f*x]
+ b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]

Rule 4047

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(
C_.)), x_Symbol] :> Dist[B/b, Int[(b*Csc[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x
]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, x]

Rule 3769

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Csc[c + d*x])^(n + 1))/(b*d*n), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[2*n]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

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 4045

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(A*Cot[e
+ f*x]*(b*Csc[e + f*x])^m)/(f*m), x] + Dist[(C*m + A*(m + 1))/(b^2*m), Int[(b*Csc[e + f*x])^(m + 2), x], x] /
; FreeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]

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+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac{9}{2}}(c+d x)} \, dx &=\frac{2 a A \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}-\frac{2}{9} \int \frac{-\frac{9}{2} a (A+B)-\frac{1}{2} a (7 A+9 (B+C)) \sec (c+d x)-\frac{9}{2} a C \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a A \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}-\frac{2}{9} \int \frac{-\frac{9}{2} a (A+B)-\frac{9}{2} a C \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx+\frac{1}{9} (a (7 A+9 (B+C))) \int \frac{1}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a A \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a (A+B) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a (7 A+9 (B+C)) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{1}{7} (a (5 (A+B)+7 C)) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx+\frac{1}{15} (a (7 A+9 (B+C))) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a A \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a (A+B) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a (7 A+9 (B+C)) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (5 (A+B)+7 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{1}{21} (a (5 (A+B)+7 C)) \int \sqrt{\sec (c+d x)} \, dx+\frac{1}{15} \left (a (7 A+9 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 a (7 A+9 (B+C)) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 a A \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a (A+B) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a (7 A+9 (B+C)) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (5 (A+B)+7 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{1}{21} \left (a (5 (A+B)+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 a (7 A+9 (B+C)) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 a (5 (A+B)+7 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{21 d}+\frac{2 a A \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a (A+B) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a (7 A+9 (B+C)) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (5 (A+B)+7 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 2.51103, size = 231, normalized size = 1.07 \[ \frac{a e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (-112 i (7 A+9 (B+C)) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )+240 (5 A+5 B+7 C) \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+2 \cos (c+d x) (30 (23 A+23 B+28 C) \sin (c+d x)+14 (19 A+18 (B+C)) \sin (2 (c+d x))+90 A \sin (3 (c+d x))+35 A \sin (4 (c+d x))+1176 i A+90 B \sin (3 (c+d x))+1512 i B+1512 i C)\right )}{2520 d} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]

[Out]

(a*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(240*(5*A + 5*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2,
2] - (112*I)*(7*A + 9*(B + C))*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4,
-E^((2*I)*(c + d*x))] + 2*Cos[c + d*x]*((1176*I)*A + (1512*I)*B + (1512*I)*C + 30*(23*A + 23*B + 28*C)*Sin[c
+ d*x] + 14*(19*A + 18*(B + C))*Sin[2*(c + d*x)] + 90*A*Sin[3*(c + d*x)] + 90*B*Sin[3*(c + d*x)] + 35*A*Sin[4*
(c + d*x)])))/(2520*d*E^(I*d*x))

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Maple [B]  time = 2.228, size = 512, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x)

[Out]

-2/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a*(-1120*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c
)^10+(2960*A+720*B)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-3152*A-1584*B-504*C)*sin(1/2*d*x+1/2*c)^6*cos(1/
2*d*x+1/2*c)+(1792*A+1344*B+924*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-408*A-366*B-336*C)*sin(1/2*d*x+1/
2*c)^2*cos(1/2*d*x+1/2*c)+75*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2
*d*x+1/2*c),2^(1/2))-147*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x
+1/2*c),2^(1/2))+75*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*
c),2^(1/2))-189*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2
^(1/2))+105*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/
2))-189*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))
/(-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(-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+a*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{C a \sec \left (d x + c\right )^{3} +{\left (B + C\right )} a \sec \left (d x + c\right )^{2} +{\left (A + B\right )} a \sec \left (d x + c\right ) + A a}{\sec \left (d x + c\right )^{\frac{9}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x, algorithm="fricas")

[Out]

integral((C*a*sec(d*x + c)^3 + (B + C)*a*sec(d*x + c)^2 + (A + B)*a*sec(d*x + c) + A*a)/sec(d*x + c)^(9/2), 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+a*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/sec(d*x+c)**(9/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}}{\sec \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)/sec(d*x + c)^(9/2), x)