3.1107 \(\int \frac{\cos ^{\frac{7}{2}}(c+d x) (A+C \sec ^2(c+d x))}{a+a \sec (c+d x)} \, dx\)

Optimal. Leaf size=192 \[ \frac{5 (9 A+7 C) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 a d}-\frac{3 (7 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(9 A+7 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 a d}-\frac{(7 A+5 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}+\frac{5 (9 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.287431, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4114, 3042, 2748, 2635, 2639, 2641} \[ \frac{5 (9 A+7 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a d}-\frac{3 (7 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(9 A+7 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 a d}-\frac{(7 A+5 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}+\frac{5 (9 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 a d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^(7/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]

[Out]

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

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 3042

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(a*(A + C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x
])^(n + 1))/(f*(b*c - a*d)*(2*m + 1)), x] + Dist[1/(b*(b*c - a*d)*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)
*(c + d*Sin[e + f*x])^n*Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) - C*(a*c*m + b*d*(n + 1)) + (a*A*d*(m + n + 2
) + C*(b*c*(2*m + 1) - a*d*(m - n - 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] &&
NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-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 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{\cos ^{\frac{7}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx &=\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left (C+A \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx\\ &=-\frac{(A+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac{\int \cos ^{\frac{5}{2}}(c+d x) \left (-\frac{1}{2} a (7 A+5 C)+\frac{1}{2} a (9 A+7 C) \cos (c+d x)\right ) \, dx}{a^2}\\ &=-\frac{(A+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{(7 A+5 C) \int \cos ^{\frac{5}{2}}(c+d x) \, dx}{2 a}+\frac{(9 A+7 C) \int \cos ^{\frac{7}{2}}(c+d x) \, dx}{2 a}\\ &=-\frac{(7 A+5 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 a d}+\frac{(9 A+7 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{7 a d}-\frac{(A+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{(3 (7 A+5 C)) \int \sqrt{\cos (c+d x)} \, dx}{10 a}+\frac{(5 (9 A+7 C)) \int \cos ^{\frac{3}{2}}(c+d x) \, dx}{14 a}\\ &=-\frac{3 (7 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d}+\frac{5 (9 A+7 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 a d}-\frac{(7 A+5 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 a d}+\frac{(9 A+7 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{7 a d}-\frac{(A+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac{(5 (9 A+7 C)) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{42 a}\\ &=-\frac{3 (7 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d}+\frac{5 (9 A+7 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a d}+\frac{5 (9 A+7 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 a d}-\frac{(7 A+5 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 a d}+\frac{(9 A+7 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{7 a d}-\frac{(A+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}\\ \end{align*}

Mathematica [C]  time = 6.72502, size = 1393, normalized size = 7.26 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Cos[c + d*x]^(7/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]

[Out]

(((-21*I)/10)*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*H
ypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (
2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*
I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^
((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^
(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(
-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (((3*I)/2)*C*Cos[c/2 + (
d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4,
7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*S
in[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Co
s[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin
[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*
d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])
))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]^(3/2)*(A + C*Sec
[c + d*x]^2)*((4*(5*A + 5*C + 16*A*Cos[c] + 10*C*Cos[c])*Csc[c])/(5*d) + (2*(51*A + 28*C)*Cos[d*x]*Sin[c])/(21
*d) - (4*A*Cos[2*d*x]*Sin[2*c])/(5*d) + (2*A*Cos[3*d*x]*Sin[3*c])/(7*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Si
n[(d*x)/2] + C*Sin[(d*x)/2]))/d + (2*(51*A + 28*C)*Cos[c]*Sin[d*x])/(21*d) - (4*A*Cos[2*c]*Sin[2*d*x])/(5*d) +
(2*A*Cos[3*c]*Sin[3*d*x])/(7*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (30*A*Cos[c/2 + (d*
x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A +
C*Sec[c + d*x]^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]]]])/(7*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1
+ Cot[c]^2]*(a + a*Sec[c + d*x])) - (10*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4,
1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - S
in[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcT
an[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))

________________________________________________________________________________________

Maple [A]  time = 2.47, size = 295, normalized size = 1.5 \begin{align*} -{\frac{1}{105\,ad}\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 ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1} \left ( 225\,A{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +441\,A{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +175\,C{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +315\,C{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \right ) -480\,A \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}+864\,A \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+ \left ( -888\,A-280\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}+ \left ( 930\,A+630\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+ \left ( -321\,A-245\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\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)^(7/2)*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c)),x)

[Out]

-1/105*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(cos(1/2*d*x+1/2*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)*(225*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+441*A*EllipticE(cos(1/2*d*x+1/
2*c),2^(1/2))+175*C*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+315*C*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))-480*A*s
in(1/2*d*x+1/2*c)^10+864*A*sin(1/2*d*x+1/2*c)^8+(-888*A-280*C)*sin(1/2*d*x+1/2*c)^6+(930*A+630*C)*sin(1/2*d*x+
1/2*c)^4+(-321*A-245*C)*sin(1/2*d*x+1/2*c)^2)/a/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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \sec \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac{7}{2}}}{a \sec \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(7/2)*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

integrate((C*sec(d*x + c)^2 + A)*cos(d*x + c)^(7/2)/(a*sec(d*x + c) + a), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(7/2)*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

integral((C*cos(d*x + c)^3*sec(d*x + c)^2 + A*cos(d*x + c)^3)*sqrt(cos(d*x + c))/(a*sec(d*x + c) + a), x)

________________________________________________________________________________________

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)**(7/2)*(A+C*sec(d*x+c)**2)/(a+a*sec(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \sec \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac{7}{2}}}{a \sec \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(7/2)*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c)),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + A)*cos(d*x + c)^(7/2)/(a*sec(d*x + c) + a), x)