\(\int (b \sec (c+d x))^{5/2} (A+C \sec ^2(c+d x)) \, dx\) [16]

Optimal result

Integrand size = 25, antiderivative size = 110 \[ \int (b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 b^2 (7 A+5 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {b \sec (c+d x)}}{21 d}+\frac {2 b (7 A+5 C) (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 d}+\frac {2 C (b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d} \]

[Out]

2/21*b*(7*A+5*C)*(b*sec(d*x+c))^(3/2)*sin(d*x+c)/d+2/21*b^2*(7*A+5*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x
+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*(b*sec(d*x+c))^(1/2)/d+2/7*C*(b*sec(d*x+c))^(5/
2)*tan(d*x+c)/d

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {4131, 3853, 3856, 2720} \[ \int (b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 b^2 (7 A+5 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {b \sec (c+d x)}}{21 d}+\frac {2 b (7 A+5 C) \sin (c+d x) (b \sec (c+d x))^{3/2}}{21 d}+\frac {2 C \tan (c+d x) (b \sec (c+d x))^{5/2}}{7 d} \]

[In]

Int[(b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]

[Out]

(2*b^2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[b*Sec[c + d*x]])/(21*d) + (2*b*(7*A + 5*C
)*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(21*d) + (2*C*(b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)

Rule 2720

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

Rule 3853

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 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 4131

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 C (b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac {1}{7} (7 A+5 C) \int (b \sec (c+d x))^{5/2} \, dx \\ & = \frac {2 b (7 A+5 C) (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 d}+\frac {2 C (b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac {1}{21} \left (b^2 (7 A+5 C)\right ) \int \sqrt {b \sec (c+d x)} \, dx \\ & = \frac {2 b (7 A+5 C) (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 d}+\frac {2 C (b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac {1}{21} \left (b^2 (7 A+5 C) \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 b^2 (7 A+5 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {b \sec (c+d x)}}{21 d}+\frac {2 b (7 A+5 C) (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 d}+\frac {2 C (b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.76 \[ \int (b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {(b \sec (c+d x))^{7/2} \left (4 (7 A+5 C) \cos ^{\frac {7}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 (7 A+11 C+(7 A+5 C) \cos (2 (c+d x))) \sin (c+d x)\right )}{42 b d} \]

[In]

Integrate[(b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]

[Out]

((b*Sec[c + d*x])^(7/2)*(4*(7*A + 5*C)*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 2*(7*A + 11*C + (7*A + 5
*C)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(42*b*d)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 22.69 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.65

method	result	size
default	\(\frac {2 b^{2} \sqrt {b \sec \left (d x +c \right )}\, \left (7 i A \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \cos \left (d x +c \right )+5 i C \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \cos \left (d x +c \right )+7 i A \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right )+5 i C \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right )+7 A \tan \left (d x +c \right )+5 C \tan \left (d x +c \right )+3 C \tan \left (d x +c \right ) \sec \left (d x +c \right )^{2}\right )}{21 d}\)	\(291\)
parts	\(\frac {2 A \sqrt {b \sec \left (d x +c \right )}\, b^{2} \left (i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \cos \left (d x +c \right )+i \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\tan \left (d x +c \right )\right )}{3 d}+\frac {2 i C \sqrt {b \sec \left (d x +c \right )}\, b^{2} \left (5 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \cos \left (d x +c \right )+5 \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-5 i \tan \left (d x +c \right )-3 i \tan \left (d x +c \right ) \sec \left (d x +c \right )^{2}\right )}{21 d}\)	\(305\)

[In]

int((b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

2/21*b^2/d*(b*sec(d*x+c))^(1/2)*(7*I*A*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*EllipticF(I*
(cot(d*x+c)-csc(d*x+c)),I)*cos(d*x+c)+5*I*C*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*Ellipti
cF(I*(cot(d*x+c)-csc(d*x+c)),I)*cos(d*x+c)+7*I*A*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*El
lipticF(I*(cot(d*x+c)-csc(d*x+c)),I)+5*I*C*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*Elliptic
F(I*(cot(d*x+c)-csc(d*x+c)),I)+7*A*tan(d*x+c)+5*C*tan(d*x+c)+3*C*tan(d*x+c)*sec(d*x+c)^2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.29 \[ \int (b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {-i \, \sqrt {2} {\left (7 \, A + 5 \, C\right )} b^{\frac {5}{2}} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + i \, \sqrt {2} {\left (7 \, A + 5 \, C\right )} b^{\frac {5}{2}} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left ({\left (7 \, A + 5 \, C\right )} b^{2} \cos \left (d x + c\right )^{2} + 3 \, C b^{2}\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{21 \, d \cos \left (d x + c\right )^{3}} \]

[In]

integrate((b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/21*(-I*sqrt(2)*(7*A + 5*C)*b^(5/2)*cos(d*x + c)^3*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))
+ I*sqrt(2)*(7*A + 5*C)*b^(5/2)*cos(d*x + c)^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2*(
(7*A + 5*C)*b^2*cos(d*x + c)^2 + 3*C*b^2)*sqrt(b/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^3)

Sympy [F]

\[ \int (b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (b \sec {\left (c + d x \right )}\right )^{\frac {5}{2}} \left (A + C \sec ^{2}{\left (c + d x \right )}\right )\, dx \]

[In]

integrate((b*sec(d*x+c))**(5/2)*(A+C*sec(d*x+c)**2),x)

[Out]

Integral((b*sec(c + d*x))**(5/2)*(A + C*sec(c + d*x)**2), x)

Maxima [F]

\[ \int (b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}} \,d x } \]

[In]

integrate((b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

integrate((C*sec(d*x + c)^2 + A)*(b*sec(d*x + c))^(5/2), x)

Giac [F]

\[ \int (b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}} \,d x } \]

[In]

integrate((b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + A)*(b*sec(d*x + c))^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int (b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]

[In]

int((A + C/cos(c + d*x)^2)*(b/cos(c + d*x))^(5/2),x)

[Out]

int((A + C/cos(c + d*x)^2)*(b/cos(c + d*x))^(5/2), x)