3.7.7 \(\int \frac {(a+b \sec (c+d x))^4}{\sec ^{\frac {11}{2}}(c+d x)} \, dx\) [607]

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

[Out]

52/99*a^3*b*sin(d*x+c)/d/sec(d*x+c)^(7/2)+2/77*a^2*(9*a^2+59*b^2)*sin(d*x+c)/d/sec(d*x+c)^(5/2)+8/45*a*b*(7*a^
2+9*b^2)*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/11*a^2*(a+b*sec(d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(9/2)+2/231*(45*a^4
+330*a^2*b^2+77*b^4)*sin(d*x+c)/d/sec(d*x+c)^(1/2)+8/15*a*b*(7*a^2+9*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2
*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+2/231*(45*a^4+330*a^2*b^
2+77*b^4)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/
2)*sec(d*x+c)^(1/2)/d

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Rubi [A]
time = 0.30, antiderivative size = 289, normalized size of antiderivative = 1.00, 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 = {3926, 4159, 4132, 3854, 3856, 2719, 4130, 2720} \begin {gather*} \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 {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 {2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{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 \left (45 a^4+330 a^2 b^2+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} \end {gather*}

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 2719

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

Rule 2720

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

Rule 3854

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 3856

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 3926

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[a^2*Co
t[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 4130

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 4132

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 4159

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]

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*}

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Mathematica [A]
time = 1.23, size = 199, normalized size = 0.69 \begin {gather*} \frac {\sqrt {\sec (c+d x)} \left (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 )+240 \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 )+\left (616 a b \left (43 a^2+36 b^2\right ) \cos (c+d x)+5 \left (1593 a^4+10296 a^2 b^2+1848 b^4+72 \left (8 a^4+33 a^2 b^2\right ) \cos (2 (c+d x))+616 a^3 b \cos (3 (c+d x))+63 a^4 \cos (4 (c+d x))\right )\right ) \sin (2 (c+d x))\right )}{27720 d} \end {gather*}

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 = 0.14, size = 586, normalized size = 2.03

method	result	size
default	\(-\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (20160 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}+\left (-50400 a^{4}-49280 b \,a^{3}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (56880 a^{4}+98560 b \,a^{3}+47520 b^{2} a^{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-34920 a^{4}-91168 b \,a^{3}-71280 b^{2} a^{2}-22176 b^{3} a \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (13860 a^{4}+41888 b \,a^{3}+55440 b^{2} a^{2}+22176 b^{3} a +4620 b^{4}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-2790 a^{4}-7392 b \,a^{3}-15840 b^{2} a^{2}-5544 b^{3} a -2310 b^{4}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+675 a^{4} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+4950 b^{2} a^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+1155 b^{4} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-6468 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{3} b -8316 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \,b^{3}\right )}{3465 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\)	\(586\)

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,method=_RETURNVERBOSE)

[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*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^1
2*a^4+(-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*
b^2*a^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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.90, size = 286, normalized size = 0.99 \begin {gather*} -\frac {15 \, \sqrt {2} {\left (45 i \, a^{4} + 330 i \, a^{2} b^{2} + 77 i \, b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-45 i \, a^{4} - 330 i \, a^{2} b^{2} - 77 i \, b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 924 \, \sqrt {2} {\left (-7 i \, a^{3} b - 9 i \, a b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 924 \, \sqrt {2} {\left (7 i \, a^{3} b + 9 i \, a b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (315 \, a^{4} \cos \left (d x + c\right )^{5} + 1540 \, a^{3} b \cos \left (d x + c\right )^{4} + 135 \, {\left (3 \, a^{4} + 22 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} + 308 \, {\left (7 \, a^{3} b + 9 \, a b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (45 \, a^{4} + 330 \, a^{2} b^{2} + 77 \, b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{3465 \, d} \end {gather*}

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]

-1/3465*(15*sqrt(2)*(45*I*a^4 + 330*I*a^2*b^2 + 77*I*b^4)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x
+ c)) + 15*sqrt(2)*(-45*I*a^4 - 330*I*a^2*b^2 - 77*I*b^4)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x
+ c)) + 924*sqrt(2)*(-7*I*a^3*b - 9*I*a*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) +
I*sin(d*x + c))) + 924*sqrt(2)*(7*I*a^3*b + 9*I*a*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d
*x + c) - I*sin(d*x + c))) - 2*(315*a^4*cos(d*x + c)^5 + 1540*a^3*b*cos(d*x + c)^4 + 135*(3*a^4 + 22*a^2*b^2)*
cos(d*x + c)^3 + 308*(7*a^3*b + 9*a*b^3)*cos(d*x + c)^2 + 15*(45*a^4 + 330*a^2*b^2 + 77*b^4)*cos(d*x + c))*sin
(d*x + c)/sqrt(cos(d*x + c)))/d

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/cos(c + d*x))^4/(1/cos(c + d*x))^(11/2),x)

[Out]

int((a + b/cos(c + d*x))^4/(1/cos(c + d*x))^(11/2), x)

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