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

Optimal. Leaf size=230 \[ \frac{5 (9 A-7 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 a d}-\frac{(A-B) \sin (c+d x)}{d \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)}-\frac{7 (A-B) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{(9 A-7 B) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{5 (9 A-7 B) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}-\frac{21 (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d} \]

[Out]

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

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Rubi [A]  time = 0.230189, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4020, 3787, 3769, 3771, 2641, 2639} \[ -\frac{(A-B) \sin (c+d x)}{d \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)}-\frac{7 (A-B) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{(9 A-7 B) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{5 (9 A-7 B) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}+\frac{5 (9 A-7 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a d}-\frac{21 (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4020

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

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 3769

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Csc[c + d*x])^(n + 1))/(b*d*n), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[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 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 \frac{A+B \sec (c+d x)}{\sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx &=-\frac{(A-B) \sin (c+d x)}{d \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))}+\frac{\int \frac{\frac{1}{2} a (9 A-7 B)-\frac{7}{2} a (A-B) \sec (c+d x)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx}{a^2}\\ &=-\frac{(A-B) \sin (c+d x)}{d \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))}+\frac{(9 A-7 B) \int \frac{1}{\sec ^{\frac{7}{2}}(c+d x)} \, dx}{2 a}-\frac{(7 (A-B)) \int \frac{1}{\sec ^{\frac{5}{2}}(c+d x)} \, dx}{2 a}\\ &=\frac{(9 A-7 B) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{7 (A-B) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(A-B) \sin (c+d x)}{d \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))}+\frac{(5 (9 A-7 B)) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{14 a}-\frac{(21 (A-B)) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{10 a}\\ &=\frac{(9 A-7 B) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{7 (A-B) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{5 (9 A-7 B) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}-\frac{(A-B) \sin (c+d x)}{d \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))}+\frac{(5 (9 A-7 B)) \int \sqrt{\sec (c+d x)} \, dx}{42 a}-\frac{\left (21 (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{10 a}\\ &=-\frac{21 (A-B) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 a d}+\frac{(9 A-7 B) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{7 (A-B) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{5 (9 A-7 B) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}-\frac{(A-B) \sin (c+d x)}{d \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))}+\frac{\left (5 (9 A-7 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{42 a}\\ &=-\frac{21 (A-B) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 a d}+\frac{5 (9 A-7 B) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{21 a d}+\frac{(9 A-7 B) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{7 (A-B) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{5 (9 A-7 B) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}-\frac{(A-B) \sin (c+d x)}{d \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))}\\ \end{align*}

Mathematica [C]  time = 3.93781, size = 568, normalized size = 2.47 \[ \frac{\cos ^2\left (\frac{1}{2} (c+d x)\right ) (A+B \sec (c+d x)) \left (588 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right )+1800 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )-588 \sqrt{2} B \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right )-1400 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+\sqrt{\sec (c+d x)} \left (20 (27 A-14 B) \sin (2 c) \cos (2 d x)-84 (A-B) \sin (3 c) \cos (3 d x)-2772 (A-B) \cos (c) \sin (d x)+20 (27 A-14 B) \cos (2 c) \sin (2 d x)-84 (A-B) \cos (3 c) \sin (3 d x)-840 (A-B) \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \sec \left (\frac{1}{2} (c+d x)\right )+63 (A-B) (11 \cos (2 c)+17) \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \cos (d x)-840 (A-B) \tan \left (\frac{c}{2}\right )+30 A \sin (4 c) \cos (4 d x)+30 A \cos (4 c) \sin (4 d x)\right )\right )}{420 a d (\sec (c+d x)+1) (A \cos (c+d x)+B)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[(c + d*x)/2]^2*(A + B*Sec[c + d*x])*((588*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[
1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeom
etric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) - (588*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*
(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((
2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + 1800*A*Sqrt[Cos[c + d*x]]*Ellipt
icF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] - 1400*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]
] + Sqrt[Sec[c + d*x]]*(63*(A - B)*(17 + 11*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2] + 20*(27*A - 14*B)*Cos[2*d*x]
*Sin[2*c] - 84*(A - B)*Cos[3*d*x]*Sin[3*c] + 30*A*Cos[4*d*x]*Sin[4*c] - 840*(A - B)*Sec[c/2]*Sec[(c + d*x)/2]*
Sin[(d*x)/2] - 2772*(A - B)*Cos[c]*Sin[d*x] + 20*(27*A - 14*B)*Cos[2*c]*Sin[2*d*x] - 84*(A - B)*Cos[3*c]*Sin[3
*d*x] + 30*A*Cos[4*c]*Sin[4*d*x] - 840*(A - B)*Tan[c/2])))/(420*a*d*(B + A*Cos[c + d*x])*(1 + Sec[c + d*x]))

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Maple [A]  time = 1.687, size = 300, normalized size = 1.3 \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{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \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\,B{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -441\,B{\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}+ \left ( 864\,A+336\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{8}+ \left ( -888\,A-392\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}+ \left ( 930\,A-210\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+ \left ( -321\,A+161\,B \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((A+B*sec(d*x+c))/sec(d*x+c)^(7/2)/(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)*(2*sin(1/2*d*x+1/2*c)^2-1)^
(1/2)*(sin(1/2*d*x+1/2*c)^2)^(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*B*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-441*B*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+336*B)*sin(1/2*d*x+1/2*c)^8+(-888*A-392*B)*sin(1/2*d*x+1/2*c)^6+(930*A-210*B)*sin(
1/2*d*x+1/2*c)^4+(-321*A+161*B)*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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B \sec \left (d x + c\right ) + A\right )} \sqrt{\sec \left (d x + c\right )}}{a \sec \left (d x + c\right )^{5} + a \sec \left (d x + c\right )^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*sec(d*x + c) + A)*sqrt(sec(d*x + c))/(a*sec(d*x + c)^5 + a*sec(d*x + c)^4), 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((A+B*sec(d*x+c))/sec(d*x+c)**(7/2)/(a+a*sec(d*x+c)),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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