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

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

[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 = 0.26, 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 = {3791, 3769, 3771, 2641, 2639} \[ \frac {128 a^4 \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {150 a^4 \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {8 a^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a^4 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {904 a^4 \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {904 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {128 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d} \]

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 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]

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 3791

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]  time = 3.38, size = 296, normalized size = 1.39 \[ \frac {a^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) (\sec (c+d x)+1)^4 \left (\frac {137055 \sin (2 (c+d x))+48664 \sin (3 (c+d x))+14760 \sin (4 (c+d x))+3080 \sin (5 (c+d x))+315 \sin (6 (c+d x))-213752 \csc (c) \cos (d x)-259336 \csc (c) \cos (2 c+d x)}{384 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {i \sqrt {2} \left (1232 \left (-1+e^{2 i c}\right ) \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 i (c+d x)}\right )-565 \left (-1+e^{2 i c}\right ) e^{i (c+d x)} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-e^{2 i (c+d x)}\right )+1232 \sqrt {1+e^{2 i (c+d x)}}\right ) \cos ^4(c+d x)}{\left (-1+e^{2 i c}\right ) d \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}}}\right )}{2310} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*Sec[(c + d*x)/2]^8*(1 + Sec[c + d*x])^4*((I*Sqrt[2]*Cos[c + d*x]^4*(1232*Sqrt[1 + E^((2*I)*(c + d*x))] +
1232*(-1 + E^((2*I)*c))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 565*E^(I*(c + d*x))*(-1 + E^
((2*I)*c))*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*(-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))
/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]) + (-213752*Cos[d*x]*Csc[c] - 259336*Cos[2*c + d*x]*
Csc[c] + 137055*Sin[2*(c + d*x)] + 48664*Sin[3*(c + d*x)] + 14760*Sin[4*(c + d*x)] + 3080*Sin[5*(c + d*x)] + 3
15*Sin[6*(c + d*x)])/(384*d*Sec[c + d*x]^(7/2))))/2310

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fricas [F]  time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{4} \sec \left (d x + c\right )^{4} + 4 \, a^{4} \sec \left (d x + c\right )^{3} + 6 \, a^{4} \sec \left (d x + c\right )^{2} + 4 \, a^{4} \sec \left (d x + c\right ) + a^{4}}{\sec \left (d x + c\right )^{\frac {11}{2}}}, x\right ) \]

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]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{4}}{\sec \left (d x + c\right )^{\frac {11}{2}}}\,{d x} \]

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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maple [A]  time = 3.74, size = 273, normalized size = 1.28 \[ -\frac {8 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, a^{4} \left (5040 \left (\cos ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5320 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1740 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+326 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+678 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4465 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1695 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3696 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+2001 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3465 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]

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)

[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(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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

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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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