3.397 \(\int \frac{1}{\cos ^{\frac{11}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=207 \[ \frac{11 \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{2 a^3 d}+\frac{119 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{11 \sin (c+d x)}{2 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{119 \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{119 \sin (c+d x)}{30 d \cos ^{\frac{5}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}-\frac{2 \sin (c+d x)}{3 a d \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^3} \]

[Out]

(119*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + (11*EllipticF[(c + d*x)/2, 2])/(2*a^3*d) + (11*Sin[c + d*x])/(2*a
^3*d*Cos[c + d*x]^(3/2)) - (119*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(5*d*Cos[c + d*x]^(
9/2)*(a + a*Sec[c + d*x])^3) - (2*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2) - (119*Sin[c
+ d*x])/(30*d*Cos[c + d*x]^(5/2)*(a^3 + a^3*Sec[c + d*x]))

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Rubi [A]  time = 0.434142, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {4264, 3816, 4019, 3787, 3768, 3771, 2639, 2641} \[ \frac{11 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{2 a^3 d}+\frac{119 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{11 \sin (c+d x)}{2 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{119 \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{119 \sin (c+d x)}{30 d \cos ^{\frac{5}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}-\frac{2 \sin (c+d x)}{3 a d \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^3),x]

[Out]

(119*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + (11*EllipticF[(c + d*x)/2, 2])/(2*a^3*d) + (11*Sin[c + d*x])/(2*a
^3*d*Cos[c + d*x]^(3/2)) - (119*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(5*d*Cos[c + d*x]^(
9/2)*(a + a*Sec[c + d*x])^3) - (2*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2) - (119*Sin[c
+ d*x])/(30*d*Cos[c + d*x]^(5/2)*(a^3 + a^3*Sec[c + d*x]))

Rule 4264

Int[(u_)*((c_.)*sin[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Dist[(c*Csc[a + b*x])^m*(c*Sin[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Csc[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSecantIntegrandQ[
u, x]

Rule 3816

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Simp[(d^2*
Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 2))/(f*(2*m + 1)), x] + Dist[d^2/(a*b*(2*m + 1)), In
t[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)*(b*(n - 2) + a*(m - n + 2)*Csc[e + f*x]), x], x] /; Fr
eeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 2] && (IntegersQ[2*m, 2*n] || IntegerQ[m]
)

Rule 4019

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[(d*(A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1))/
(a*f*(2*m + 1)), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)*Simp[A
*(a*d*(n - 1)) - B*(b*d*(n - 1)) - d*(a*B*(m - n + 1) + A*b*(m + n))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b,
d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)] && GtQ[n, 0]

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[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 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{1}{\cos ^{\frac{11}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec ^{\frac{11}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx\\ &=-\frac{\sin (c+d x)}{5 d \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec ^{\frac{7}{2}}(c+d x) \left (\frac{7 a}{2}-\frac{13}{2} a \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{\sin (c+d x)}{5 d \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac{2 \sin (c+d x)}{3 a d \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec ^{\frac{5}{2}}(c+d x) \left (25 a^2-\frac{69}{2} a^2 \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4}\\ &=-\frac{\sin (c+d x)}{5 d \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac{2 \sin (c+d x)}{3 a d \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac{119 \sin (c+d x)}{30 d \cos ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{3}{2}}(c+d x) \left (\frac{357 a^3}{4}-\frac{495}{4} a^3 \sec (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac{\sin (c+d x)}{5 d \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac{2 \sin (c+d x)}{3 a d \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac{119 \sin (c+d x)}{30 d \cos ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}-\frac{\left (119 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx}{20 a^3}+\frac{\left (33 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{5}{2}}(c+d x) \, dx}{4 a^3}\\ &=\frac{11 \sin (c+d x)}{2 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{119 \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac{2 \sin (c+d x)}{3 a d \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac{119 \sin (c+d x)}{30 d \cos ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}+\frac{\left (11 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \, dx}{4 a^3}+\frac{\left (119 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{20 a^3}\\ &=\frac{11 \sin (c+d x)}{2 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{119 \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac{2 \sin (c+d x)}{3 a d \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac{119 \sin (c+d x)}{30 d \cos ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}+\frac{11 \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{4 a^3}+\frac{119 \int \sqrt{\cos (c+d x)} \, dx}{20 a^3}\\ &=\frac{119 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{11 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{2 a^3 d}+\frac{11 \sin (c+d x)}{2 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{119 \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac{2 \sin (c+d x)}{3 a d \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac{119 \sin (c+d x)}{30 d \cos ^{\frac{5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 3.55563, size = 402, normalized size = 1.94 \[ \frac{\cos ^6\left (\frac{1}{2} (c+d x)\right ) \left (-\frac{\csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \left (5134 \cos \left (\frac{1}{2} (c-d x)\right )+4148 \cos \left (\frac{1}{2} (3 c+d x)\right )+4664 \cos \left (\frac{1}{2} (c+3 d x)\right )+2476 \cos \left (\frac{1}{2} (5 c+3 d x)\right )+3340 \cos \left (\frac{1}{2} (3 c+5 d x)\right )+944 \cos \left (\frac{1}{2} (7 c+5 d x)\right )+1620 \cos \left (\frac{1}{2} (5 c+7 d x)\right )+165 \cos \left (\frac{1}{2} (9 c+7 d x)\right )+357 \cos \left (\frac{1}{2} (7 c+9 d x)\right )\right ) \sec ^5\left (\frac{1}{2} (c+d x)\right )}{96 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{4 i \sqrt{2} e^{-i (c+d x)} \sec ^3(c+d x) \left (119 \left (-1+e^{2 i c}\right ) \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},-e^{2 i (c+d x)}\right )-55 \left (-1+e^{2 i c}\right ) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},-e^{2 i (c+d x)}\right )+119 \left (1+e^{2 i (c+d x)}\right )\right )}{\left (-1+e^{2 i c}\right ) d \sqrt{e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{5 a^3 (\sec (c+d x)+1)^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^3),x]

[Out]

(Cos[(c + d*x)/2]^6*(-((5134*Cos[(c - d*x)/2] + 4148*Cos[(3*c + d*x)/2] + 4664*Cos[(c + 3*d*x)/2] + 2476*Cos[(
5*c + 3*d*x)/2] + 3340*Cos[(3*c + 5*d*x)/2] + 944*Cos[(7*c + 5*d*x)/2] + 1620*Cos[(5*c + 7*d*x)/2] + 165*Cos[(
9*c + 7*d*x)/2] + 357*Cos[(7*c + 9*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5)/(96*d*Cos[c + d*x]^(9/2)) +
((4*I)*Sqrt[2]*(119*(1 + E^((2*I)*(c + d*x))) + 119*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeom
etric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 55*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*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]^3)/(d*E^(I*(c + d*x))*(-1 + E^((2*
I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))])))/(5*a^3*(1 + Sec[c + d*x])^3)

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Maple [A]  time = 2.977, size = 453, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(d*x+c)^(11/2)/(a+a*sec(d*x+c))^3,x)

[Out]

-1/4*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)/a^3*(32/15*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x
+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)^3+118/5*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1
/2*c)-128/5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*
x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+238/5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2
*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-Ell
ipticE(cos(1/2*d*x+1/2*c),2^(1/2)))-4/3*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
)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+1/5*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)^5+4
8*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2+1)*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(1/cos(d*x+c)^(11/2)/(a+a*sec(d*x+c))^3,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{\sqrt{\cos \left (d x + c\right )}}{a^{3} \cos \left (d x + c\right )^{6} \sec \left (d x + c\right )^{3} + 3 \, a^{3} \cos \left (d x + c\right )^{6} \sec \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right )^{6} \sec \left (d x + c\right ) + a^{3} \cos \left (d x + c\right )^{6}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(d*x+c)^(11/2)/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

integral(sqrt(cos(d*x + c))/(a^3*cos(d*x + c)^6*sec(d*x + c)^3 + 3*a^3*cos(d*x + c)^6*sec(d*x + c)^2 + 3*a^3*c
os(d*x + c)^6*sec(d*x + c) + a^3*cos(d*x + c)^6), 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(1/cos(d*x+c)**(11/2)/(a+a*sec(d*x+c))**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac{11}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(d*x+c)^(11/2)/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

integrate(1/((a*sec(d*x + c) + a)^3*cos(d*x + c)^(11/2)), x)