3.1083 \(\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x)) (A+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=165 \[ \frac{2 a (5 A+7 C) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}+\frac{2 a (7 A+9 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a (7 A+9 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a (5 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d} \]

[Out]

(2*a*(7*A + 9*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*a*(5*A + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a
*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(7*A + 9*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d
) + (2*a*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a*A*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)

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Rubi [A]  time = 0.241436, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {4114, 3034, 3023, 2748, 2635, 2641, 2639} \[ \frac{2 a (5 A+7 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 a (7 A+9 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a (7 A+9 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a (5 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*(7*A + 9*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*a*(5*A + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a
*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(7*A + 9*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d
) + (2*a*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a*A*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)

Rule 4114

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

Rule 3034

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (C_.)*sin[(e
_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*d*Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m
+ 3)), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*d*(C*(m + 2) + A*(m
+ 3))*Sin[e + f*x] - (2*a*C*d - b*c*C*(m + 3))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}
, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1]

Rule 3023

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

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

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 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 \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x)) \left (C+A \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 a A \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{2}{9} \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{9 a C}{2}+\frac{1}{2} a (7 A+9 C) \cos (c+d x)+\frac{9}{2} a A \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 a A \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{2 a A \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{4}{63} \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{9}{4} a (5 A+7 C)+\frac{7}{4} a (7 A+9 C) \cos (c+d x)\right ) \, dx\\ &=\frac{2 a A \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{2 a A \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{7} (a (5 A+7 C)) \int \cos ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{9} (a (7 A+9 C)) \int \cos ^{\frac{5}{2}}(c+d x) \, dx\\ &=\frac{2 a (5 A+7 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a (7 A+9 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 a A \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{2 a A \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{21} (a (5 A+7 C)) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\frac{1}{15} (a (7 A+9 C)) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 a (7 A+9 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a (5 A+7 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 a (5 A+7 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a (7 A+9 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 a A \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{2 a A \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}\\ \end{align*}

Mathematica [C]  time = 6.29184, size = 918, normalized size = 5.56 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-((7*A + 9*C)*Cot[c])/(15*d) + ((23*A + 28*C)*C
os[d*x]*Sin[c])/(84*d) + ((19*A + 18*C)*Cos[2*d*x]*Sin[2*c])/(180*d) + (A*Cos[3*d*x]*Sin[3*c])/(28*d) + (A*Cos
[4*d*x]*Sin[4*c])/(72*d) + ((23*A + 28*C)*Cos[c]*Sin[d*x])/(84*d) + ((19*A + 18*C)*Cos[2*c]*Sin[2*d*x])/(180*d
) + (A*Cos[3*c]*Sin[3*d*x])/(28*d) + (A*Cos[4*c]*Sin[4*d*x])/(72*d)) - (5*A*(1 + Cos[c + d*x])*Csc[c]*Hypergeo
metricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[
1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x -
ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4
}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c
]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sq
rt[1 + Cot[c]^2]) - (7*A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4
}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1
+ Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - (
(Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^
2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d) - (3*C*(1 + Cos[
c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Si
n[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[
Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])
/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Co
s[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d))

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Maple [B]  time = 3.306, size = 406, normalized size = 2.5 \begin{align*} -{\frac{2\,a}{315\,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 ( -1120\,A\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}+2960\,A \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}\cos \left ( 1/2\,dx+c/2 \right ) + \left ( -3152\,A-504\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( 1792\,A+924\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( -408\,A-336\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +75\,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 EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -147\,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 ) +105\,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 EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -189\,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+a*sec(d*x+c))*(A+C*sec(d*x+c)^2),x)

[Out]

-2/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a*(-1120*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c
)^10+2960*A*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-3152*A-504*C)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(1
792*A+924*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-408*A-336*C)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+75
*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-147*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))+105*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-189*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(-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+a*sec(d*x+c))*(A+C*sec(d*x+c)^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 ({\left (C a \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{3} + C a \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{2} + A a \cos \left (d x + c\right )^{4} \sec \left (d x + c\right ) + A 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+a*sec(d*x+c))*(A+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

integral((C*a*cos(d*x + c)^4*sec(d*x + c)^3 + C*a*cos(d*x + c)^4*sec(d*x + c)^2 + A*a*cos(d*x + c)^4*sec(d*x +
c) + A*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+a*sec(d*x+c))*(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 )}{\left (a \sec \left (d x + c\right ) + 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+a*sec(d*x+c))*(A+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

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