3.1075 \(\int \cos ^{\frac{9}{2}}(c+d x) (A+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=80 \[ \frac{2 (7 A+9 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 (7 A+9 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d} \]

[Out]

(2*(7*A + 9*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(7*A + 9*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2
*A*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)

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Rubi [A]  time = 0.0790408, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {4066, 3014, 2635, 2639} \[ \frac{2 (7 A+9 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 (7 A+9 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^(9/2)*(A + C*Sec[c + d*x]^2),x]

[Out]

(2*(7*A + 9*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(7*A + 9*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2
*A*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)

Rule 4066

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(m_)*((A_.) + (C_.)*sec[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[b^2, Int
[(b*Cos[e + f*x])^(m - 2)*(C + A*Cos[e + f*x]^2), x], x] /; FreeQ[{b, e, f, A, C, m}, x] &&  !IntegerQ[m]

Rule 3014

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[
e + f*x]*(b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[(A*(m + 2) + C*(m + 1))/(m + 2), Int[(b*Sin[e + f*
x])^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] &&  !LtQ[m, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

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 \cos ^{\frac{9}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\int \cos ^{\frac{5}{2}}(c+d x) \left (C+A \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 A \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{9} (7 A+9 C) \int \cos ^{\frac{5}{2}}(c+d x) \, dx\\ &=\frac{2 (7 A+9 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 A \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{15} (7 A+9 C) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 (7 A+9 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 (7 A+9 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 A \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}\\ \end{align*}

Mathematica [A]  time = 0.329501, size = 65, normalized size = 0.81 \[ \frac{12 (7 A+9 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\sin (2 (c+d x)) \sqrt{\cos (c+d x)} (5 A \cos (2 (c+d x))+19 A+18 C)}{90 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^(9/2)*(A + C*Sec[c + d*x]^2),x]

[Out]

(12*(7*A + 9*C)*EllipticE[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(19*A + 18*C + 5*A*Cos[2*(c + d*x)])*Sin[2*(c +
d*x)])/(90*d)

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Maple [B]  time = 1.864, size = 313, normalized size = 3.9 \begin{align*} -{\frac{2}{45\,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\,A\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}+320\,A \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}\cos \left ( 1/2\,dx+c/2 \right ) + \left ( -296\,A-72\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( 136\,A+72\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( -24\,A-18\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -21\,A\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -27\,C\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{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(cos(d*x+c)^(9/2)*(A+C*sec(d*x+c)^2),x)

[Out]

-2/45*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-160*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10
+320*A*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-296*A-72*C)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(136*A+72
*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-24*A-18*C)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-21*A*(sin(1/2
*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-27*C*(sin(1/2*d*x+
1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))/(-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., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac{9}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(9/2)*(A+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

integrate((C*sec(d*x + c)^2 + A)*cos(d*x + c)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{2} + A \cos \left (d x + c\right )^{4}\right )} \sqrt{\cos \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(9/2)*(A+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

integral((C*cos(d*x + c)^4*sec(d*x + c)^2 + A*cos(d*x + c)^4)*sqrt(cos(d*x + c)), 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)**(9/2)*(A+C*sec(d*x+c)**2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac{9}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(9/2)*(A+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + A)*cos(d*x + c)^(9/2), x)