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

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

[Out]

(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]])

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Rubi [A]  time = 0.258238, antiderivative size = 213, normalized size of antiderivative = 1., 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 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]

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 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 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]

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*}

Mathematica [C]  time = 3.2189, size = 306, normalized size = 1.44 \[ -\frac{i a^4 e^{-6 i (c+d x)} \sec ^8\left (\frac{1}{2} (c+d x)\right ) (\sec (c+d x)+1)^4 \left (-946176 e^{5 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},-e^{2 i (c+d x)}\right )+433920 e^{6 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},-e^{2 i (c+d x)}\right )-3080 e^{i (c+d x)}-14760 e^{2 i (c+d x)}-48664 e^{3 i (c+d x)}-137055 e^{4 i (c+d x)}+427504 e^{5 i (c+d x)}+518672 e^{7 i (c+d x)}+137055 e^{8 i (c+d x)}+48664 e^{9 i (c+d x)}+14760 e^{10 i (c+d x)}+3080 e^{11 i (c+d x)}+315 e^{12 i (c+d x)}-315\right )}{1774080 d \sec ^{\frac{7}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/1774080)*a^4*(-315 - 3080*E^(I*(c + d*x)) - 14760*E^((2*I)*(c + d*x)) - 48664*E^((3*I)*(c + d*x)) - 13705
5*E^((4*I)*(c + d*x)) + 427504*E^((5*I)*(c + d*x)) + 518672*E^((7*I)*(c + d*x)) + 137055*E^((8*I)*(c + d*x)) +
48664*E^((9*I)*(c + d*x)) + 14760*E^((10*I)*(c + d*x)) + 3080*E^((11*I)*(c + d*x)) + 315*E^((12*I)*(c + d*x))
- 946176*E^((5*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d
*x))] + 433920*E^((6*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c
+ d*x))])*Sec[(c + d*x)/2]^8*(1 + Sec[c + d*x])^4)/(d*E^((6*I)*(c + d*x))*Sec[c + d*x]^(7/2))

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Maple [A]  time = 1.388, size = 273, normalized size = 1.3 \begin{align*} -{\frac{8\,{a}^{4}}{3465\,d}\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 5040\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{13}-5320\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{11}+1740\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{9}+326\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}+678\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}-4465\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+1695\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -3696\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +2001\,\cos \left ( 1/2\,dx+c/2 \right ) \right ){\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}}} \end{align*}

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., 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))^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{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 ) \end{align*}

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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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))**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 (a \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+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)