3.186 \(\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx\)

Optimal. Leaf size=234 \[ \frac{4 a^2 (7 A+6 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}+\frac{2 a^2 (7 A+9 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{4 a^2 (7 A+6 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{4 a^2 (4 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{4 a^2 (4 A+3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{7 d} \]

[Out]

(-4*a^2*(4*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(7*A + 6*B
)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(4*A + 3*B)*Sqrt[Sec[c + d*
x]]*Sin[c + d*x])/(5*d) + (4*a^2*(7*A + 6*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a^2*(7*A + 9*B)*Sec[
c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*B*Sec[c + d*x]^(5/2)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(7*d)

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Rubi [A]  time = 0.342083, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {4018, 3997, 3787, 3768, 3771, 2639, 2641} \[ \frac{2 a^2 (7 A+9 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{4 a^2 (7 A+6 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{4 a^2 (4 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^2 (7 A+6 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{4 a^2 (4 A+3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]

[Out]

(-4*a^2*(4*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(7*A + 6*B
)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(4*A + 3*B)*Sqrt[Sec[c + d*
x]]*Sin[c + d*x])/(5*d) + (4*a^2*(7*A + 6*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a^2*(7*A + 9*B)*Sec[
c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*B*Sec[c + d*x]^(5/2)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(7*d)

Rule 4018

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

Rule 3997

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.
) + (A_)), x_Symbol] :> -Simp[(b*B*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*(n + 1)), x] + Dist[1/(n + 1), Int[(d*C
sc[e + f*x])^n*Simp[A*a*(n + 1) + B*b*n + (A*b + B*a)*(n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f
, A, B}, x] && NeQ[A*b - a*B, 0] &&  !LeQ[n, -1]

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 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[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 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 \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx &=\frac{2 B \sec ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{7 d}+\frac{2}{7} \int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x)) \left (\frac{1}{2} a (7 A+3 B)+\frac{1}{2} a (7 A+9 B) \sec (c+d x)\right ) \, dx\\ &=\frac{2 a^2 (7 A+9 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 B \sec ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{7 d}+\frac{4}{35} \int \sec ^{\frac{3}{2}}(c+d x) \left (\frac{7}{2} a^2 (4 A+3 B)+\frac{5}{2} a^2 (7 A+6 B) \sec (c+d x)\right ) \, dx\\ &=\frac{2 a^2 (7 A+9 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 B \sec ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{7 d}+\frac{1}{5} \left (2 a^2 (4 A+3 B)\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{7} \left (2 a^2 (7 A+6 B)\right ) \int \sec ^{\frac{5}{2}}(c+d x) \, dx\\ &=\frac{4 a^2 (4 A+3 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{4 a^2 (7 A+6 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 a^2 (7 A+9 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 B \sec ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{7 d}-\frac{1}{5} \left (2 a^2 (4 A+3 B)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{21} \left (2 a^2 (7 A+6 B)\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{4 a^2 (4 A+3 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{4 a^2 (7 A+6 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 a^2 (7 A+9 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 B \sec ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{7 d}-\frac{1}{5} \left (2 a^2 (4 A+3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{21} \left (2 a^2 (7 A+6 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{4 a^2 (4 A+3 B) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^2 (7 A+6 B) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{21 d}+\frac{4 a^2 (4 A+3 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{4 a^2 (7 A+6 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 a^2 (7 A+9 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 B \sec ^{\frac{5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{7 d}\\ \end{align*}

Mathematica [C]  time = 4.45944, size = 463, normalized size = 1.98 \[ \frac{a^2 \csc (c) e^{-i d x} \cos ^3(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (\sec (c+d x)+1)^2 (A+B \sec (c+d x)) \left (7 \sqrt{2} \left (-1+e^{2 i c}\right ) (4 A+3 B) e^{2 i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-\frac{\left (-1+e^{2 i c}\right ) e^{-i (c-d x)} \sqrt{\sec (c+d x)} \left (5 i (7 A+6 B) \left (1+e^{2 i (c+d x)}\right )^3 \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+7 A \left (9 e^{i (c+d x)}-5 e^{2 i (c+d x)}+36 e^{3 i (c+d x)}+5 e^{4 i (c+d x)}+39 e^{5 i (c+d x)}+5 e^{6 i (c+d x)}+12 e^{7 i (c+d x)}-5\right )+3 B \left (7 e^{i (c+d x)}-20 e^{2 i (c+d x)}+63 e^{3 i (c+d x)}+20 e^{4 i (c+d x)}+77 e^{5 i (c+d x)}+10 e^{6 i (c+d x)}+21 e^{7 i (c+d x)}-10\right )\right )}{\left (1+e^{2 i (c+d x)}\right )^3}\right )}{210 d (A \cos (c+d x)+B)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*Cos[c + d*x]^3*Csc[c]*Sec[(c + d*x)/2]^4*(7*Sqrt[2]*(4*A + 3*B)*E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Sqrt[E^(
I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*
I)*(c + d*x))] - ((-1 + E^((2*I)*c))*(7*A*(-5 + 9*E^(I*(c + d*x)) - 5*E^((2*I)*(c + d*x)) + 36*E^((3*I)*(c + d
*x)) + 5*E^((4*I)*(c + d*x)) + 39*E^((5*I)*(c + d*x)) + 5*E^((6*I)*(c + d*x)) + 12*E^((7*I)*(c + d*x))) + 3*B*
(-10 + 7*E^(I*(c + d*x)) - 20*E^((2*I)*(c + d*x)) + 63*E^((3*I)*(c + d*x)) + 20*E^((4*I)*(c + d*x)) + 77*E^((5
*I)*(c + d*x)) + 10*E^((6*I)*(c + d*x)) + 21*E^((7*I)*(c + d*x))) + (5*I)*(7*A + 6*B)*(1 + E^((2*I)*(c + d*x))
)^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])*Sqrt[Sec[c + d*x]])/(E^(I*(c - d*x))*(1 + E^((2*I)*(c + d*x)
))^3))*(1 + Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/(210*d*E^(I*d*x)*(B + A*Cos[c + d*x]))

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Maple [B]  time = 6.524, size = 852, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^(3/2)*(a+a*sec(d*x+c))^2*(A+B*sec(d*x+c)),x)

[Out]

-a^2*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(8*(1/2*A+1/4*B)*(-1/6*cos(1/2*d*x+1/2*c)*(-2*s
in(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)^2-1/2)^2+1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*
(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1
/2*c),2^(1/2)))-8/5*(1/4*A+1/2*B)/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/si
n(1/2*d*x+1/2*c)^2*(12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2
*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4-24*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-12*EllipticE(cos(1/2*d*x+1/2*c),2
^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^2+24*sin(1/2*d*x+1/2*
c)^4*cos(1/2*d*x+1/2*c)+3*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+
1/2*c)^2)^(1/2)-8*sin(1/2*d*x+1/2*c)^2*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
)+2*B*(-1/56*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)/(cos(1/2*d*x+1/2*c)^2-1/2
)^4-5/42*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)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+
5/21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c
)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+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)*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)+2*(-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)*sin(1/2*d*x+1/2*c)^2)/sin(1/2*d*x+1/2*c)^2/(2*s
in(1/2*d*x+1/2*c)^2-1))/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(sec(d*x+c)^(3/2)*(a+a*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 a^{2} \sec \left (d x + c\right )^{4} +{\left (A + 2 \, B\right )} a^{2} \sec \left (d x + c\right )^{3} +{\left (2 \, A + B\right )} a^{2} \sec \left (d x + c\right )^{2} + A a^{2} \sec \left (d x + c\right )\right )} \sqrt{\sec \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*a^2*sec(d*x + c)^4 + (A + 2*B)*a^2*sec(d*x + c)^3 + (2*A + B)*a^2*sec(d*x + c)^2 + A*a^2*sec(d*x +
c))*sqrt(sec(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(sec(d*x+c)**(3/2)*(a+a*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 (a \sec \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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