3.1.23 \(\int \frac {A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx\) [23]

Optimal. Leaf size=112 \[ \frac {2 (7 A+9 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b^4 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 (7 A+9 C) \sin (c+d x)}{45 b^3 d (b \sec (c+d x))^{3/2}}+\frac {2 A \tan (c+d x)}{9 d (b \sec (c+d x))^{9/2}} \]

[Out]

2/45*(7*A+9*C)*sin(d*x+c)/b^3/d/(b*sec(d*x+c))^(3/2)+2/15*(7*A+9*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1
/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/b^4/d/cos(d*x+c)^(1/2)/(b*sec(d*x+c))^(1/2)+2/9*A*tan(d*x+c)/d/(b*
sec(d*x+c))^(9/2)

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Rubi [A]
time = 0.06, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {4130, 3854, 3856, 2719} \begin {gather*} \frac {2 (7 A+9 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b^4 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 (7 A+9 C) \sin (c+d x)}{45 b^3 d (b \sec (c+d x))^{3/2}}+\frac {2 A \tan (c+d x)}{9 d (b \sec (c+d x))^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(7*A + 9*C)*EllipticE[(c + d*x)/2, 2])/(15*b^4*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*(7*A + 9*C)*
Sin[c + d*x])/(45*b^3*d*(b*Sec[c + d*x])^(3/2)) + (2*A*Tan[c + d*x])/(9*d*(b*Sec[c + d*x])^(9/2))

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 3854

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 3856

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 4130

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[A*Cot[e
+ f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + Dist[(C*m + A*(m + 1))/(b^2*m), Int[(b*Csc[e + f*x])^(m + 2), x], x] /
; FreeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.95, size = 143, normalized size = 1.28 \begin {gather*} \frac {e^{-i d x} (\cos (d x)+i \sin (d x)) \left (336 i A+432 i C-\frac {32 i (7 A+9 C) e^{2 i (c+d x)} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )}{\sqrt {1+e^{2 i (c+d x)}}}+(76 A+72 C) \sin (2 (c+d x))+10 A \sin (4 (c+d x))\right )}{360 b^4 d \sqrt {b \sec (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Cos[d*x] + I*Sin[d*x])*((336*I)*A + (432*I)*C - ((32*I)*(7*A + 9*C)*E^((2*I)*(c + d*x))*Hypergeometric2F1[1/
2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] + (76*A + 72*C)*Sin[2*(c + d*x)] + 10*A*Sin[
4*(c + d*x)]))/(360*b^4*d*E^(I*d*x)*Sqrt[b*Sec[c + d*x]])

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Maple [C] Result contains complex when optimal does not.
time = 23.05, size = 636, normalized size = 5.68

method	result	size
default	\(\frac {\frac {14 i A \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )}{15}-\frac {14 i A \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )}{15}-\frac {2 A \left (\cos ^{6}\left (d x +c \right )\right )}{9}+\frac {6 i C \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )}{5}-\frac {6 i C \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )}{5}+\frac {14 i A \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{15}-\frac {14 i A \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{15}+\frac {6 i C \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{5}-\frac {6 i C \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{5}-\frac {4 A \left (\cos ^{4}\left (d x +c \right )\right )}{45}-\frac {2 C \left (\cos ^{4}\left (d x +c \right )\right )}{5}-\frac {28 A \left (\cos ^{2}\left (d x +c \right )\right )}{45}-\frac {4 C \left (\cos ^{2}\left (d x +c \right )\right )}{5}+\frac {14 A \cos \left (d x +c \right )}{15}+\frac {6 C \cos \left (d x +c \right )}{5}}{d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{5} \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {9}{2}}}\)	\(636\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(9/2),x,method=_RETURNVERBOSE)

[Out]

2/45/d*(21*I*A*sin(d*x+c)*cos(d*x+c)*EllipticF(I*(-1+cos(d*x+c))/sin(d*x+c),I)*(1/(1+cos(d*x+c)))^(1/2)*(cos(d
*x+c)/(1+cos(d*x+c)))^(1/2)-21*I*A*sin(d*x+c)*cos(d*x+c)*EllipticE(I*(-1+cos(d*x+c))/sin(d*x+c),I)*(1/(1+cos(d
*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)-5*A*cos(d*x+c)^6+27*I*C*sin(d*x+c)*cos(d*x+c)*EllipticF(I*(-1+
cos(d*x+c))/sin(d*x+c),I)*(1/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)-27*I*C*sin(d*x+c)*cos(d*x
+c)*EllipticE(I*(-1+cos(d*x+c))/sin(d*x+c),I)*(1/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+21*I*
A*sin(d*x+c)*EllipticF(I*(-1+cos(d*x+c))/sin(d*x+c),I)*(1/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1
/2)-21*I*A*sin(d*x+c)*EllipticE(I*(-1+cos(d*x+c))/sin(d*x+c),I)*(1/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*
x+c)))^(1/2)+27*I*C*sin(d*x+c)*EllipticF(I*(-1+cos(d*x+c))/sin(d*x+c),I)*(1/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/
(1+cos(d*x+c)))^(1/2)-27*I*C*sin(d*x+c)*EllipticE(I*(-1+cos(d*x+c))/sin(d*x+c),I)*(1/(1+cos(d*x+c)))^(1/2)*(co
s(d*x+c)/(1+cos(d*x+c)))^(1/2)-2*A*cos(d*x+c)^4-9*C*cos(d*x+c)^4-14*A*cos(d*x+c)^2-18*C*cos(d*x+c)^2+21*A*cos(
d*x+c)+27*C*cos(d*x+c))/sin(d*x+c)/cos(d*x+c)^5/(b/cos(d*x+c))^(9/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(9/2),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.20, size = 129, normalized size = 1.15 \begin {gather*} -\frac {3 \, \sqrt {2} {\left (-7 i \, A - 9 i \, C\right )} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, \sqrt {2} {\left (7 i \, A + 9 i \, C\right )} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (5 \, A \cos \left (d x + c\right )^{4} + {\left (7 \, A + 9 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{45 \, b^{5} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(9/2),x, algorithm="fricas")

[Out]

-1/45*(3*sqrt(2)*(-7*I*A - 9*I*C)*sqrt(b)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*s
in(d*x + c))) + 3*sqrt(2)*(7*I*A + 9*I*C)*sqrt(b)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x +
c) - I*sin(d*x + c))) - 2*(5*A*cos(d*x + c)^4 + (7*A + 9*C)*cos(d*x + c)^2)*sqrt(b/cos(d*x + c))*sin(d*x + c))
/(b^5*d)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+C*sec(d*x+c)**2)/(b*sec(d*x+c))**(9/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3879 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(9/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {A+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + C/cos(c + d*x)^2)/(b/cos(c + d*x))^(9/2),x)

[Out]

int((A + C/cos(c + d*x)^2)/(b/cos(c + d*x))^(9/2), x)

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