3.2.93 \(\int \frac {(a+a \sec (c+d x))^4}{\sec ^{\frac {11}{2}}(c+d x)} \, dx\) [193]

Optimal. Leaf size=213 \[ \frac {128 a^4 \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {904 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 a^4 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {8 a^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {150 a^4 \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {128 a^4 \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {904 a^4 \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}} \]

[Out]

2/11*a^4*sin(d*x+c)/d/sec(d*x+c)^(9/2)+8/9*a^4*sin(d*x+c)/d/sec(d*x+c)^(7/2)+150/77*a^4*sin(d*x+c)/d/sec(d*x+c
)^(5/2)+128/45*a^4*sin(d*x+c)/d/sec(d*x+c)^(3/2)+904/231*a^4*sin(d*x+c)/d/sec(d*x+c)^(1/2)+128/15*a^4*(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+904/231*a^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 = 0.21, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3876, 3854, 3856, 2720, 2719} \begin {gather*} \frac {128 a^4 \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {150 a^4 \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {8 a^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a^4 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {904 a^4 \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {904 a^4 \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 {128 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(128*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (904*a^4*Sqrt[Cos[c + d*x]]
*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a^4*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + (8*a
^4*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (150*a^4*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (128*a^4*Sin[c
+ d*x])/(45*d*Sec[c + d*x]^(3/2)) + (904*a^4*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])

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 3876

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Int[Expand
Trig[(a + b*csc[e + f*x])^m*(d*csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0]
&& IGtQ[m, 0] && RationalQ[n]

Rubi steps

\begin {align*} \int \frac {(a+a \sec (c+d x))^4}{\sec ^{\frac {11}{2}}(c+d x)} \, dx &=\int \left (\frac {a^4}{\sec ^{\frac {11}{2}}(c+d x)}+\frac {4 a^4}{\sec ^{\frac {9}{2}}(c+d x)}+\frac {6 a^4}{\sec ^{\frac {7}{2}}(c+d x)}+\frac {4 a^4}{\sec ^{\frac {5}{2}}(c+d x)}+\frac {a^4}{\sec ^{\frac {3}{2}}(c+d x)}\right ) \, dx\\ &=a^4 \int \frac {1}{\sec ^{\frac {11}{2}}(c+d x)} \, dx+a^4 \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx+\left (4 a^4\right ) \int \frac {1}{\sec ^{\frac {9}{2}}(c+d x)} \, dx+\left (4 a^4\right ) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx+\left (6 a^4\right ) \int \frac {1}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 a^4 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {8 a^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {12 a^4 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {8 a^4 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 a^4 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {1}{3} a^4 \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{11} \left (9 a^4\right ) \int \frac {1}{\sec ^{\frac {7}{2}}(c+d x)} \, dx+\frac {1}{5} \left (12 a^4\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{9} \left (28 a^4\right ) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx+\frac {1}{7} \left (30 a^4\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a^4 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {8 a^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {150 a^4 \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {128 a^4 \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {74 a^4 \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {1}{77} \left (45 a^4\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx+\frac {1}{7} \left (10 a^4\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{15} \left (28 a^4\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{3} \left (a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{5} \left (12 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {24 a^4 \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{3 d}+\frac {2 a^4 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {8 a^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {150 a^4 \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {128 a^4 \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {904 a^4 \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {1}{77} \left (15 a^4\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{7} \left (10 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{15} \left (28 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {128 a^4 \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 {74 a^4 \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^4 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {8 a^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {150 a^4 \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {128 a^4 \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {904 a^4 \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {1}{77} \left (15 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {128 a^4 \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 {904 a^4 \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 {2 a^4 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {8 a^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {150 a^4 \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {128 a^4 \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {904 a^4 \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 4.70, size = 293, normalized size = 1.38 \begin {gather*} -\frac {a^4 \csc (c) \sqrt {\sec (c+d x)} \left (427504 \cos (d x)+518672 \cos (2 c+d x)-137055 \cos (c+2 d x)+137055 \cos (3 c+2 d x)-48664 \cos (2 c+3 d x)+48664 \cos (4 c+3 d x)-14760 \cos (3 c+4 d x)+14760 \cos (5 c+4 d x)-3080 \cos (4 c+5 d x)+3080 \cos (6 c+5 d x)-315 \cos (5 c+6 d x)+315 \cos (7 c+6 d x)-473088 e^{-i d x} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 i (c+d x)}\right )-157696 e^{i d x} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )+433920 i \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-e^{2 i (c+d x)}\right ) \sin (c)\right )}{110880 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/110880*(a^4*Csc[c]*Sqrt[Sec[c + d*x]]*(427504*Cos[d*x] + 518672*Cos[2*c + d*x] - 137055*Cos[c + 2*d*x] + 13
7055*Cos[3*c + 2*d*x] - 48664*Cos[2*c + 3*d*x] + 48664*Cos[4*c + 3*d*x] - 14760*Cos[3*c + 4*d*x] + 14760*Cos[5
*c + 4*d*x] - 3080*Cos[4*c + 5*d*x] + 3080*Cos[6*c + 5*d*x] - 315*Cos[5*c + 6*d*x] + 315*Cos[7*c + 6*d*x] - (4
73088*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/E^(I*d*x) - 15769
6*E^(I*d*x)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + (433920*I)*
Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sin[c]))/d

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Maple [A]
time = 0.07, size = 273, normalized size = 1.28

method	result	size
default	\(-\frac {8 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, a^{4} \left (5040 \left (\cos ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5320 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1740 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+326 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+678 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4465 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1695 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3696 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+2001 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3465 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\)	\(273\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^4*(5040*cos(1/2*d*x+1/2*c)^13-5320*cos(1/2*d
*x+1/2*c)^11+1740*cos(1/2*d*x+1/2*c)^9+326*cos(1/2*d*x+1/2*c)^7+678*cos(1/2*d*x+1/2*c)^5-4465*cos(1/2*d*x+1/2*
c)^3+1695*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))
-3696*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+200
1*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 = 0.00, 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+a*sec(d*x+c))^4/sec(d*x+c)^(11/2),x, algorithm="maxima")

[Out]

integrate((a*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 = 0.89, size = 196, normalized size = 0.92 \begin {gather*} -\frac {2 \, {\left (3390 i \, \sqrt {2} a^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 3390 i \, \sqrt {2} a^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 7392 i \, \sqrt {2} a^{4} {\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 ) + 7392 i \, \sqrt {2} a^{4} {\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 {{\left (315 \, a^{4} \cos \left (d x + c\right )^{5} + 1540 \, a^{4} \cos \left (d x + c\right )^{4} + 3375 \, a^{4} \cos \left (d x + c\right )^{3} + 4928 \, a^{4} \cos \left (d x + c\right )^{2} + 6780 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{3465 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3465*(3390*I*sqrt(2)*a^4*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 3390*I*sqrt(2)*a^4*wei
erstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 7392*I*sqrt(2)*a^4*weierstrassZeta(-4, 0, weierstrass
PInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 7392*I*sqrt(2)*a^4*weierstrassZeta(-4, 0, weierstrassPInvers
e(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (315*a^4*cos(d*x + c)^5 + 1540*a^4*cos(d*x + c)^4 + 3375*a^4*cos(d*
x + c)^3 + 4928*a^4*cos(d*x + c)^2 + 6780*a^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+a*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+a*sec(d*x+c))^4/sec(d*x+c)^(11/2),x, algorithm="giac")

[Out]

integrate((a*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 {a}{\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 + a/cos(c + d*x))^4/(1/cos(c + d*x))^(11/2),x)

[Out]

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

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