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

Optimal. Leaf size=181 \[ \frac{11 \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{2 a^3 d}-\frac{119 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{11 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{119 \sin (c+d x) \sqrt{\cos (c+d x)}}{30 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \sec (c+d x)+a)^3} \]

[Out]

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

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Rubi [A]  time = 0.396354, antiderivative size = 181, normalized size of antiderivative = 1., 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 = {4264, 3817, 4020, 3787, 3769, 3771, 2641, 2639} \[ \frac{11 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{2 a^3 d}-\frac{119 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{11 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{119 \sin (c+d x) \sqrt{\cos (c+d x)}}{30 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \sec (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^(3/2)/(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*Sqrt[Cos[c + d*x]
]*Sin[c + d*x])/(2*a^3*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (2*Sqrt[Cos[c + d
*x]]*Sin[c + d*x])/(3*a*d*(a + a*Sec[c + d*x])^2) - (119*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(30*d*(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 3817

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

Rule 4020

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> -Simp[((A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(b*f*(2
*m + 1)), x] - Dist[1/(a^2*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[b*B*n - a*A*(2
*m + n + 1) + (A*b - a*B)*(m + n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, 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 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{\cos ^{\frac{3}{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{1}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx\\ &=-\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{13 a}{2}+\frac{7}{2} a \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{2 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{69 a^2}{2}+25 a^2 \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx}{15 a^4}\\ &=-\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{2 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac{119 \sqrt{\cos (c+d x)} \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{495 a^3}{4}+\frac{357}{4} a^3 \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{15 a^6}\\ &=-\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{2 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac{119 \sqrt{\cos (c+d x)} \sin (c+d x)}{30 d \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 \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{20 a^3}+\frac{\left (33 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{4 a^3}\\ &=\frac{11 \sqrt{\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{2 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac{119 \sqrt{\cos (c+d x)} \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{119 \int \sqrt{\cos (c+d x)} \, dx}{20 a^3}+\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{119 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{11 \sqrt{\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{2 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac{119 \sqrt{\cos (c+d x)} \sin (c+d x)}{30 d \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 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 \sqrt{\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{2 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac{119 \sqrt{\cos (c+d x)} \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 2.48146, size = 375, normalized size = 2.07 \[ \frac{\cos ^6\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\frac{1}{4} \sec \left (\frac{c}{2}\right ) \left (-709 \sin \left (c+\frac{d x}{2}\right )+715 \sin \left (c+\frac{3 d x}{2}\right )-170 \sin \left (2 c+\frac{3 d x}{2}\right )+202 \sin \left (2 c+\frac{5 d x}{2}\right )+25 \sin \left (3 c+\frac{5 d x}{2}\right )+5 \sin \left (3 c+\frac{7 d x}{2}\right )+5 \sin \left (4 c+\frac{7 d x}{2}\right )+1061 \sin \left (\frac{d x}{2}\right )\right ) \sec ^5\left (\frac{1}{2} (c+d x)\right )+720 \cot (c)+708 \csc (c)}{3 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{4 i \sqrt{2} e^{-i (c+d x)} \sec ^3(c+d x) \left (119 \left (-1+e^{2 i c}\right ) \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},-e^{2 i (c+d x)}\right )+55 \left (-1+e^{2 i c}\right ) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},-e^{2 i (c+d x)}\right )+119 \left (1+e^{2 i (c+d x)}\right )\right )}{\left (-1+e^{2 i c}\right ) d \sqrt{e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{5 a^3 (\sec (c+d x)+1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[(c + d*x)/2]^6*(((-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))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 55*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*S
qrt[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))]) + (720*Cot[c] + 708*Csc[c] + (Se
c[c/2]*Sec[(c + d*x)/2]^5*(1061*Sin[(d*x)/2] - 709*Sin[c + (d*x)/2] + 715*Sin[c + (3*d*x)/2] - 170*Sin[2*c + (
3*d*x)/2] + 202*Sin[2*c + (5*d*x)/2] + 25*Sin[3*c + (5*d*x)/2] + 5*Sin[3*c + (7*d*x)/2] + 5*Sin[4*c + (7*d*x)/
2]))/4)/(3*d*Cos[c + d*x]^(5/2))))/(5*a^3*(1 + Sec[c + d*x])^3)

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Maple [A]  time = 1.446, size = 283, normalized size = 1.6 \begin{align*} -{\frac{1}{60\,{a}^{3}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 ( 160\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}+468\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+330\,\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 ) \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+714\,\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} \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -1058\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+474\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-47\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+3 \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 ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5} \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(cos(d*x+c)^(3/2)/(a+a*sec(d*x+c))^3,x)

[Out]

-1/60*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(160*cos(1/2*d*x+1/2*c)^10+468*cos(1/2*d*x+1/2*c
)^8+330*(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))*c
os(1/2*d*x+1/2*c)^5+714*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*cos(1/2*d*x+1/2*c)^5*El
lipticE(cos(1/2*d*x+1/2*c),2^(1/2))-1058*cos(1/2*d*x+1/2*c)^6+474*cos(1/2*d*x+1/2*c)^4-47*cos(1/2*d*x+1/2*c)^2
+3)/a^3/(-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/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., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{\frac{3}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cos(d*x + c)^(3/2)/(a^3*sec(d*x + c)^3 + 3*a^3*sec(d*x + c)^2 + 3*a^3*sec(d*x + c) + a^3), 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(cos(d*x+c)**(3/2)/(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{\cos \left (d x + c\right )^{\frac{3}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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