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

Optimal. Leaf size=182 \[ \frac{2 \left (5 a^2 A+7 b (2 a B+A b)\right ) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}+\frac{2 \left (3 a^2 B+6 a A b+5 b^2 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 \left (5 a^2 A+7 b (2 a B+A b)\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a (7 a B+9 A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)}{7 d} \]

[Out]

(2*(6*a*A*b + 3*a^2*B + 5*b^2*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(5*a^2*A + 7*b*(A*b + 2*a*B))*EllipticF
[(c + d*x)/2, 2])/(21*d) + (2*(5*a^2*A + 7*b*(A*b + 2*a*B))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(9*
A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*a*A*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])*Sin[c +
d*x])/(7*d)

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Rubi [A]  time = 0.368102, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {2954, 2990, 3023, 2748, 2639, 2635, 2641} \[ \frac{2 \left (3 a^2 B+6 a A b+5 b^2 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 \left (5 a^2 A+7 b (2 a B+A b)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (5 a^2 A+7 b (2 a B+A b)\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a (7 a B+9 A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(6*a*A*b + 3*a^2*B + 5*b^2*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(5*a^2*A + 7*b*(A*b + 2*a*B))*EllipticF
[(c + d*x)/2, 2])/(21*d) + (2*(5*a^2*A + 7*b*(A*b + 2*a*B))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(9*
A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*a*A*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])*Sin[c +
d*x])/(7*d)

Rule 2954

Int[((a_.) + csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.)*((g_.)*sin[(e_.
) + (f_.)*(x_)])^(p_.), x_Symbol] :> Dist[g^(m + n), Int[(g*Sin[e + f*x])^(p - m - n)*(b + a*Sin[e + f*x])^m*(
d + c*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[p] && I
ntegerQ[m] && IntegerQ[n]

Rule 2990

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x]
)^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x
])^n*Simp[a^2*A*d*(m + n + 1) + b*B*(b*c*(m - 1) + a*d*(n + 1)) + (a*d*(2*A*b + a*B)*(m + n + 1) - b*B*(a*c -
b*d*(m + n)))*Sin[e + f*x] + b*(A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x]^2, x], x], x] /; F
reeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m,
1] &&  !(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

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 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 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]

Rubi steps

\begin{align*} \int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx &=\int \sqrt{\cos (c+d x)} (b+a \cos (c+d x))^2 (B+A \cos (c+d x)) \, dx\\ &=\frac{2 a A \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x)) \sin (c+d x)}{7 d}+\frac{2}{7} \int \sqrt{\cos (c+d x)} \left (\frac{1}{2} b (3 a A+7 b B)+\frac{1}{2} \left (5 a^2 A+7 b (A b+2 a B)\right ) \cos (c+d x)+\frac{1}{2} a (9 A b+7 a B) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 a (9 A b+7 a B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 a A \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x)) \sin (c+d x)}{7 d}+\frac{4}{35} \int \sqrt{\cos (c+d x)} \left (\frac{7}{4} \left (6 a A b+3 a^2 B+5 b^2 B\right )+\frac{5}{4} \left (5 a^2 A+7 b (A b+2 a B)\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{2 a (9 A b+7 a B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 a A \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x)) \sin (c+d x)}{7 d}+\frac{1}{5} \left (6 a A b+3 a^2 B+5 b^2 B\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{7} \left (5 a^2 A+7 b (A b+2 a B)\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 \left (6 a A b+3 a^2 B+5 b^2 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 \left (5 a^2 A+7 b (A b+2 a B)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a (9 A b+7 a B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 a A \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x)) \sin (c+d x)}{7 d}+\frac{1}{21} \left (5 a^2 A+7 b (A b+2 a B)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (6 a A b+3 a^2 B+5 b^2 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 \left (5 a^2 A+7 b (A b+2 a B)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (5 a^2 A+7 b (A b+2 a B)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a (9 A b+7 a B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 a A \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x)) \sin (c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 1.21557, size = 139, normalized size = 0.76 \[ \frac{10 \left (5 a^2 A+14 a b B+7 A b^2\right ) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+42 \left (3 a^2 B+6 a A b+5 b^2 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\sin (c+d x) \sqrt{\cos (c+d x)} \left (5 \left (3 a^2 A \cos (2 (c+d x))+13 a^2 A+28 a b B+14 A b^2\right )+42 a (a B+2 A b) \cos (c+d x)\right )}{105 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(42*(6*a*A*b + 3*a^2*B + 5*b^2*B)*EllipticE[(c + d*x)/2, 2] + 10*(5*a^2*A + 7*A*b^2 + 14*a*b*B)*EllipticF[(c +
d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(42*a*(2*A*b + a*B)*Cos[c + d*x] + 5*(13*a^2*A + 14*A*b^2 + 28*a*b*B + 3*a^2*
A*Cos[2*(c + d*x)]))*Sin[c + d*x])/(105*d)

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Maple [B]  time = 1.966, size = 548, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)),x)

[Out]

-2/105*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(240*a^2*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c
)^8+(-360*A*a^2-336*A*a*b-168*B*a^2)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(280*A*a^2+336*A*a*b+140*A*b^2+16
8*B*a^2+280*B*a*b)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-80*A*a^2-84*A*a*b-70*A*b^2-42*B*a^2-140*B*a*b)*si
n(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-126*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)*Elli
pticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b+25*a^2*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)*E
llipticF(cos(1/2*d*x+1/2*c),2^(1/2))+35*A*b^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*El
lipticF(cos(1/2*d*x+1/2*c),2^(1/2))-63*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Ellipti
cE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2-105*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Ellipti
cE(cos(1/2*d*x+1/2*c),2^(1/2))*b^2+70*B*a*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Elli
pticF(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)^(7/2)*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)),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 (B b^{2} \cos \left (d x + c\right )^{3} \sec \left (d x + c\right )^{3} + A a^{2} \cos \left (d x + c\right )^{3} +{\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{3} \sec \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{3} \sec \left (d x + c\right )\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)^(7/2)*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)),x, algorithm="fricas")

[Out]

integral((B*b^2*cos(d*x + c)^3*sec(d*x + c)^3 + A*a^2*cos(d*x + c)^3 + (2*B*a*b + A*b^2)*cos(d*x + c)^3*sec(d*
x + c)^2 + (B*a^2 + 2*A*a*b)*cos(d*x + c)^3*sec(d*x + c))*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)**(7/2)*(a+b*sec(d*x+c))**2*(A+B*sec(d*x+c)),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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