\(\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \, dx\) [545]

Optimal result

Integrand size = 23, antiderivative size = 399 \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\frac {2 (a-b) \sqrt {a+b} \left (10 a^4-279 a^2 b^2-147 b^4\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{315 b^3 d}+\frac {2 (a-b) \sqrt {a+b} \left (10 a^3+165 a^2 b-114 a b^2+147 b^3\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{315 b^2 d}-\frac {4 a \left (5 a^2-57 b^2\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b d}-\frac {2 \left (10 a^2-49 b^2\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b d}-\frac {4 a (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b d}+\frac {2 (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d} \]

[Out]

2/315*(a-b)*(10*a^4-279*a^2*b^2-147*b^4)*cot(d*x+c)*EllipticE((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))
^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/b^3/d+2/315*(a-b)*(10*a^3+1
65*a^2*b-114*a*b^2+147*b^3)*cot(d*x+c)*EllipticF((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(a+b)
^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/b^2/d-2/315*(10*a^2-49*b^2)*(a+b*sec(d*x
+c))^(3/2)*tan(d*x+c)/b/d-4/63*a*(a+b*sec(d*x+c))^(5/2)*tan(d*x+c)/b/d+2/9*(a+b*sec(d*x+c))^(7/2)*tan(d*x+c)/b
/d-4/315*a*(5*a^2-57*b^2)*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b/d

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3925, 4087, 4090, 3917, 4089} \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=-\frac {2 \left (10 a^2-49 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{315 b d}-\frac {4 a \left (5 a^2-57 b^2\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{315 b d}+\frac {2 (a-b) \sqrt {a+b} \left (10 a^4-279 a^2 b^2-147 b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{315 b^3 d}+\frac {2 (a-b) \sqrt {a+b} \left (10 a^3+165 a^2 b-114 a b^2+147 b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{315 b^2 d}+\frac {2 \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{9 b d}-\frac {4 a \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b d} \]

[In]

Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2),x]

[Out]

(2*(a - b)*Sqrt[a + b]*(10*a^4 - 279*a^2*b^2 - 147*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]
/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/
(315*b^3*d) + (2*(a - b)*Sqrt[a + b]*(10*a^3 + 165*a^2*b - 114*a*b^2 + 147*b^3)*Cot[c + d*x]*EllipticF[ArcSin[
Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Se
c[c + d*x]))/(a - b))])/(315*b^2*d) - (4*a*(5*a^2 - 57*b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b*d) -
(2*(10*a^2 - 49*b^2)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b*d) - (4*a*(a + b*Sec[c + d*x])^(5/2)*Tan
[c + d*x])/(63*b*d) + (2*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*b*d)

Rule 3917

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(Rt[a + b, 2]/(b*
f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin
[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3925

Int[csc[(e_.) + (f_.)*(x_)]^3*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-Cot[e + f*x])*(
(a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(b*(m + 2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*
(b*(m + 1) - a*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1]

Rule 4087

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

Rule 4089

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)
], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + C
sc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a
*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]

Rule 4090

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)
], x_Symbol] :> Dist[A - B, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[B, Int[Csc[e + f*x]*((1 +
Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && NeQ[A
^2 - B^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d}+\frac {2 \int \sec (c+d x) \left (\frac {7 b}{2}-a \sec (c+d x)\right ) (a+b \sec (c+d x))^{5/2} \, dx}{9 b} \\ & = -\frac {4 a (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b d}+\frac {2 (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d}+\frac {4 \int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac {39 a b}{4}-\frac {1}{4} \left (10 a^2-49 b^2\right ) \sec (c+d x)\right ) \, dx}{63 b} \\ & = -\frac {2 \left (10 a^2-49 b^2\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b d}-\frac {4 a (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b d}+\frac {2 (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d}+\frac {8 \int \sec (c+d x) \sqrt {a+b \sec (c+d x)} \left (\frac {3}{8} b \left (55 a^2+49 b^2\right )-\frac {3}{4} a \left (5 a^2-57 b^2\right ) \sec (c+d x)\right ) \, dx}{315 b} \\ & = -\frac {4 a \left (5 a^2-57 b^2\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b d}-\frac {2 \left (10 a^2-49 b^2\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b d}-\frac {4 a (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b d}+\frac {2 (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d}+\frac {16 \int \frac {\sec (c+d x) \left (\frac {3}{16} a b \left (155 a^2+261 b^2\right )-\frac {3}{16} \left (10 a^4-279 a^2 b^2-147 b^4\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{945 b} \\ & = -\frac {4 a \left (5 a^2-57 b^2\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b d}-\frac {2 \left (10 a^2-49 b^2\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b d}-\frac {4 a (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b d}+\frac {2 (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d}+\frac {\left ((a-b) \left (10 a^3+165 a^2 b-114 a b^2+147 b^3\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{315 b}-\frac {\left (10 a^4-279 a^2 b^2-147 b^4\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx}{315 b} \\ & = \frac {2 (a-b) \sqrt {a+b} \left (10 a^4-279 a^2 b^2-147 b^4\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{315 b^3 d}+\frac {2 (a-b) \sqrt {a+b} \left (10 a^3+165 a^2 b-114 a b^2+147 b^3\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{315 b^2 d}-\frac {4 a \left (5 a^2-57 b^2\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b d}-\frac {2 \left (10 a^2-49 b^2\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b d}-\frac {4 a (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b d}+\frac {2 (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 13.33 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.38 \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\frac {2 \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (2 \left (10 a^5+10 a^4 b-279 a^3 b^2-279 a^2 b^3-147 a b^4-147 b^5\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {b+a \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right )+2 b \left (-10 a^4+155 a^3 b+279 a^2 b^2+261 a b^3+147 b^4\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {b+a \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right )+\left (10 a^4-279 a^2 b^2-147 b^4\right ) \cos (c+d x) (b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{315 b^2 d (b+a \cos (c+d x))^3 \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sec ^{\frac {5}{2}}(c+d x)}+\frac {\cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left (\frac {2 \left (-10 a^4+279 a^2 b^2+147 b^4\right ) \sin (c+d x)}{315 b^2}+\frac {2}{315} \sec ^2(c+d x) \left (75 a^2 \sin (c+d x)+49 b^2 \sin (c+d x)\right )+\frac {2 \sec (c+d x) \left (5 a^3 \sin (c+d x)+163 a b^2 \sin (c+d x)\right )}{315 b}+\frac {38}{63} a b \sec ^2(c+d x) \tan (c+d x)+\frac {2}{9} b^2 \sec ^3(c+d x) \tan (c+d x)\right )}{d (b+a \cos (c+d x))^2} \]

[In]

Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2),x]

[Out]

(2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(2*(10*a^5 + 10*a^4*b - 279*a^3*b^2 - 279*
a^2*b^3 - 147*a*b^4 - 147*b^5)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + C
os[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(-10*a^4 + 155*a^3*b + 279*a^2*b^2 +
261*a*b^3 + 147*b^4)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*
x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (10*a^4 - 279*a^2*b^2 - 147*b^4)*Cos[c + d*x]*(b
+ a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^2*d*(b + a*Cos[c + d*x])^3*Sqrt[Sec[(c + d*x)/2
]^2]*Sec[c + d*x]^(5/2)) + (Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((2*(-10*a^4 + 279*a^2*b^2 + 147*b^4)*Si
n[c + d*x])/(315*b^2) + (2*Sec[c + d*x]^2*(75*a^2*Sin[c + d*x] + 49*b^2*Sin[c + d*x]))/315 + (2*Sec[c + d*x]*(
5*a^3*Sin[c + d*x] + 163*a*b^2*Sin[c + d*x]))/(315*b) + (38*a*b*Sec[c + d*x]^2*Tan[c + d*x])/63 + (2*b^2*Sec[c
+ d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x])^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3238\) vs. \(2(361)=722\).

Time = 65.48 (sec) , antiderivative size = 3239, normalized size of antiderivative = 8.12

method	result	size
default	\(\text {Expression too large to display}\)	\(3239\)

[In]

int(sec(d*x+c)^3*(a+b*sec(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/315/d/b^2*(a+b*sec(d*x+c))^(1/2)/(b+a*cos(d*x+c))/(cos(d*x+c)+1)*(147*b^5*sin(d*x+c)+442*a^2*b^3*sin(d*x+c)+
49*b^5*tan(d*x+c)+80*a^3*b^2*sin(d*x+c)-279*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d
*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^3*cos(d*x+c)^2-261*(1/(a+b)*(b+a*co
s(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b)
)^(1/2))*a*b^4*cos(d*x+c)^2-20*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/
2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^4*b*cos(d*x+c)+558*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*
x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b^2*
cos(d*x+c)+558*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(cot
(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^3*cos(d*x+c)-10*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*
(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^4*b*cos(d*x+c)^2+294*
(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x
+c),((a-b)/(a+b))^(1/2))*a*b^4*cos(d*x+c)+20*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(
d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^4*b*cos(d*x+c)-310*(1/(a+b)*(b+a*cos(d
*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(
1/2))*a^3*b^2*cos(d*x+c)-558*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)
*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^3*cos(d*x+c)-522*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*
x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a*b^4*co
s(d*x+c)+147*a*b^4*cos(d*x+c)*sin(d*x+c)-10*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+
c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^5+147*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*
x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*b^5-147*
(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+
c)/(cos(d*x+c)+1))^(1/2)*b^5+5*a^4*b*cos(d*x+c)*sin(d*x+c)+279*a^3*b^2*cos(d*x+c)*sin(d*x+c)+163*a^2*b^3*cos(d
*x+c)*sin(d*x+c)+170*a^2*b^3*tan(d*x+c)*sec(d*x+c)+130*a*b^4*tan(d*x+c)*sec(d*x+c)+130*a*b^4*tan(d*x+c)*sec(d*
x+c)^2-10*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))
*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^4*b+279*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x
+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^3*b^2+279*(1/(a+b)*(b+a*cos(d*x+c))/(c
os(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^2
*b^3+147*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*
(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a*b^4-147*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d
*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*b^5*cos(d*x+c)^2-20*(1/(a+b)*(b+a*cos(d*x
+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/
2))*a^5*cos(d*x+c)+294*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*Ellip
ticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*b^5*cos(d*x+c)-294*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(
1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*b^5*cos(d*x+c)+10*
(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+
c)/(cos(d*x+c)+1))^(1/2)*a^4*b-155*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*
x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^3*b^2-279*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+
1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^2*b^3-261*(
1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c
)/(cos(d*x+c)+1))^(1/2)*a*b^4-10*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(
1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^5*cos(d*x+c)^2+147*(1/(a+b)*(b+a*cos(d*x+c))/(cos(
d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*b^5*co
s(d*x+c)^2+279*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(cot
(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b^2*cos(d*x+c)^2+279*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/
2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^3*cos(d*x+c)^2
+147*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-cs
c(d*x+c),((a-b)/(a+b))^(1/2))*a*b^4*cos(d*x+c)^2+10*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c
)/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^4*b*cos(d*x+c)^2-155*(1/(a+b)*(
b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)
/(a+b))^(1/2))*a^3*b^2*cos(d*x+c)^2+212*a*b^4*sin(d*x+c)+80*a^3*b^2*tan(d*x+c)+170*a^2*b^3*tan(d*x+c)+212*a*b^
4*tan(d*x+c)+49*b^5*tan(d*x+c)*sec(d*x+c)+35*b^5*tan(d*x+c)*sec(d*x+c)^2-10*a^5*cos(d*x+c)*sin(d*x+c)-5*a^4*b*
sin(d*x+c)+35*b^5*tan(d*x+c)*sec(d*x+c)^3)

Fricas [F]

\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{3} \,d x } \]

[In]

integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

integral((b^2*sec(d*x + c)^5 + 2*a*b*sec(d*x + c)^4 + a^2*sec(d*x + c)^3)*sqrt(b*sec(d*x + c) + a), x)

Sympy [F]

\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{\frac {5}{2}} \sec ^{3}{\left (c + d x \right )}\, dx \]

[In]

integrate(sec(d*x+c)**3*(a+b*sec(d*x+c))**(5/2),x)

[Out]

Integral((a + b*sec(c + d*x))**(5/2)*sec(c + d*x)**3, x)

Maxima [F]

\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{3} \,d x } \]

[In]

integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*sec(d*x + c) + a)^(5/2)*sec(d*x + c)^3, x)

Giac [F]

\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{3} \,d x } \]

[In]

integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate((b*sec(d*x + c) + a)^(5/2)*sec(d*x + c)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^3} \,d x \]

[In]

int((a + b/cos(c + d*x))^(5/2)/cos(c + d*x)^3,x)

[Out]

int((a + b/cos(c + d*x))^(5/2)/cos(c + d*x)^3, x)