3.607 \(\int \frac{(a+b \sec (c+d x))^4}{\sec ^{\frac{11}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=289 \[ \frac{2 \left (330 a^2 b^2+45 a^4+77 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{231 d}+\frac{2 a^2 \left (9 a^2+59 b^2\right ) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a b \left (7 a^2+9 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (330 a^2 b^2+45 a^4+77 b^4\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{8 a b \left (7 a^2+9 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{52 a^3 b \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac{9}{2}}(c+d x)} \]

[Out]

(8*a*b*(7*a^2 + 9*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(45*a^4 +
330*a^2*b^2 + 77*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (52*a^3*b*Sin
[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*a^2*(9*a^2 + 59*b^2)*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (8*a*
b*(7*a^2 + 9*b^2)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(45*a^4 + 330*a^2*b^2 + 77*b^4)*Sin[c + d*x])/(
231*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))

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Rubi [A]  time = 0.459503, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {3841, 4074, 4047, 3769, 3771, 2639, 4045, 2641} \[ \frac{2 a^2 \left (9 a^2+59 b^2\right ) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a b \left (7 a^2+9 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (330 a^2 b^2+45 a^4+77 b^4\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left (330 a^2 b^2+45 a^4+77 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{8 a b \left (7 a^2+9 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{52 a^3 b \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac{9}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sec[c + d*x])^4/Sec[c + d*x]^(11/2),x]

[Out]

(8*a*b*(7*a^2 + 9*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(45*a^4 +
330*a^2*b^2 + 77*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (52*a^3*b*Sin
[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*a^2*(9*a^2 + 59*b^2)*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (8*a*
b*(7*a^2 + 9*b^2)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(45*a^4 + 330*a^2*b^2 + 77*b^4)*Sin[c + d*x])/(
231*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))

Rule 3841

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(a^2*C
ot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n)/(f*n), x] - Dist[1/(d*n), Int[(a + b*Csc[e + f*x]
)^(m - 3)*(d*Csc[e + f*x])^(n + 1)*Simp[a^2*b*(m - 2*n - 2) - a*(3*b^2*n + a^2*(n + 1))*Csc[e + f*x] - b*(b^2*
n + a^2*(m + n - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 2]
&& ((IntegerQ[m] && LtQ[n, -1]) || (IntegersQ[m + 1/2, 2*n] && LeQ[n, -1]))

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+b \sec (c+d x))^4}{\sec ^{\frac{11}{2}}(c+d x)} \, dx &=\frac{2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2}{11} \int \frac{(a+b \sec (c+d x)) \left (13 a^2 b+\frac{3}{2} a \left (3 a^2+11 b^2\right ) \sec (c+d x)+\frac{1}{2} b \left (5 a^2+11 b^2\right ) \sec ^2(c+d x)\right )}{\sec ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{52 a^3 b \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}-\frac{4}{99} \int \frac{-\frac{9}{4} a^2 \left (9 a^2+59 b^2\right )-11 a b \left (7 a^2+9 b^2\right ) \sec (c+d x)-\frac{9}{4} b^2 \left (5 a^2+11 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{52 a^3 b \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}-\frac{4}{99} \int \frac{-\frac{9}{4} a^2 \left (9 a^2+59 b^2\right )-\frac{9}{4} b^2 \left (5 a^2+11 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx+\frac{1}{9} \left (4 a b \left (7 a^2+9 b^2\right )\right ) \int \frac{1}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{52 a^3 b \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a^2 \left (9 a^2+59 b^2\right ) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a b \left (7 a^2+9 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{1}{15} \left (4 a b \left (7 a^2+9 b^2\right )\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx-\frac{1}{77} \left (-45 a^4-330 a^2 b^2-77 b^4\right ) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{52 a^3 b \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a^2 \left (9 a^2+59 b^2\right ) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a b \left (7 a^2+9 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (45 a^4+330 a^2 b^2+77 b^4\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}-\frac{1}{231} \left (-45 a^4-330 a^2 b^2-77 b^4\right ) \int \sqrt{\sec (c+d x)} \, dx+\frac{1}{15} \left (4 a b \left (7 a^2+9 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{8 a b \left (7 a^2+9 b^2\right ) \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{52 a^3 b \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a^2 \left (9 a^2+59 b^2\right ) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a b \left (7 a^2+9 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (45 a^4+330 a^2 b^2+77 b^4\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}-\frac{1}{231} \left (\left (-45 a^4-330 a^2 b^2-77 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{8 a b \left (7 a^2+9 b^2\right ) \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 \left (45 a^4+330 a^2 b^2+77 b^4\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{231 d}+\frac{52 a^3 b \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a^2 \left (9 a^2+59 b^2\right ) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a b \left (7 a^2+9 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (45 a^4+330 a^2 b^2+77 b^4\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}\\ \end{align*}

Mathematica [A]  time = 1.83525, size = 199, normalized size = 0.69 \[ \frac{\sqrt{\sec (c+d x)} \left (240 \left (330 a^2 b^2+45 a^4+77 b^4\right ) \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+\sin (2 (c+d x)) \left (616 a b \left (43 a^2+36 b^2\right ) \cos (c+d x)+5 \left (72 \left (33 a^2 b^2+8 a^4\right ) \cos (2 (c+d x))+10296 a^2 b^2+616 a^3 b \cos (3 (c+d x))+63 a^4 \cos (4 (c+d x))+1593 a^4+1848 b^4\right )\right )+14784 a b \left (7 a^2+9 b^2\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{27720 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sec[c + d*x])^4/Sec[c + d*x]^(11/2),x]

[Out]

(Sqrt[Sec[c + d*x]]*(14784*a*b*(7*a^2 + 9*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 240*(45*a^4 + 33
0*a^2*b^2 + 77*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (616*a*b*(43*a^2 + 36*b^2)*Cos[c + d*x] + 5
*(1593*a^4 + 10296*a^2*b^2 + 1848*b^4 + 72*(8*a^4 + 33*a^2*b^2)*Cos[2*(c + d*x)] + 616*a^3*b*Cos[3*(c + d*x)]
+ 63*a^4*Cos[4*(c + d*x)]))*Sin[2*(c + d*x)]))/(27720*d)

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Maple [A]  time = 1.592, size = 586, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sec(d*x+c))^4/sec(d*x+c)^(11/2),x)

[Out]

-2/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(20160*a^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*
c)^12+(-50400*a^4-49280*a^3*b)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(56880*a^4+98560*a^3*b+47520*a^2*b^2)*
sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-34920*a^4-91168*a^3*b-71280*a^2*b^2-22176*a*b^3)*sin(1/2*d*x+1/2*c)^
6*cos(1/2*d*x+1/2*c)+(13860*a^4+41888*a^3*b+55440*a^2*b^2+22176*a*b^3+4620*b^4)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d
*x+1/2*c)+(-2790*a^4-7392*a^3*b-15840*a^2*b^2-5544*a*b^3-2310*b^4)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+675
*a^4*(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))+4950*
a^2*b^2*(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))+11
55*b^4*(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))-646
8*(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))*a^3*b-83
16*(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))*a*b^3)/
(-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+b*sec(d*x+c))^4/sec(d*x+c)^(11/2),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{b^{4} \sec \left (d x + c\right )^{4} + 4 \, a b^{3} \sec \left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \sec \left (d x + c\right )^{2} + 4 \, a^{3} b \sec \left (d x + c\right ) + a^{4}}{\sec \left (d x + c\right )^{\frac{11}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^4/sec(d*x+c)^(11/2),x, algorithm="fricas")

[Out]

integral((b^4*sec(d*x + c)^4 + 4*a*b^3*sec(d*x + c)^3 + 6*a^2*b^2*sec(d*x + c)^2 + 4*a^3*b*sec(d*x + c) + a^4)
/sec(d*x + c)^(11/2), 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*sec(d*x+c))**4/sec(d*x+c)**(11/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sec \left (d x + c\right ) + a\right )}^{4}}{\sec \left (d x + c\right )^{\frac{11}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^4/sec(d*x+c)^(11/2),x, algorithm="giac")

[Out]

integrate((b*sec(d*x + c) + a)^4/sec(d*x + c)^(11/2), x)