3.7.25 \(\int \frac {(a+b \cos (c+d x))^{5/2}}{\cos ^{\frac {13}{2}}(c+d x)} \, dx\) [625]

Optimal. Leaf size=522 \[ \frac {2 (a-b) b \sqrt {a+b} \left (741 a^4+51 a^2 b^2+8 b^4\right ) \cot (c+d x) E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{693 a^4 d}+\frac {2 (a-b) \sqrt {a+b} \left (135 a^4-606 a^3 b+57 a^2 b^2+6 a b^3+8 b^4\right ) \cot (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{693 a^3 d}+\frac {2 a^2 \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {46 a b \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2+113 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b \left (229 a^2+3 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (135 a^4+205 a^2 b^2-4 b^4\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a^2 d \cos ^{\frac {3}{2}}(c+d x)} \]

[Out]

2/11*a^2*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/d/cos(d*x+c)^(11/2)+46/99*a*b*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/d/c
os(d*x+c)^(9/2)+2/693*(81*a^2+113*b^2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/d/cos(d*x+c)^(7/2)+2/693*b*(229*a^2+3
*b^2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/a/d/cos(d*x+c)^(5/2)+2/693*(135*a^4+205*a^2*b^2-4*b^4)*sin(d*x+c)*(a+b
*cos(d*x+c))^(1/2)/a^2/d/cos(d*x+c)^(3/2)+2/693*(a-b)*b*(741*a^4+51*a^2*b^2+8*b^4)*cot(d*x+c)*EllipticE((a+b*c
os(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*(a*(1-sec(d*x+c))/(a+b))^(1/2)
*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a^4/d+2/693*(a-b)*(135*a^4-606*a^3*b+57*a^2*b^2+6*a*b^3+8*b^4)*cot(d*x+c)*Elli
pticF((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*(a*(1-sec(d*x+c))/
(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a^3/d

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Rubi [A]
time = 1.14, antiderivative size = 522, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2871, 3134, 3077, 2895, 3073} \begin {gather*} \frac {2 \left (81 a^2+113 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b \left (229 a^2+3 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{693 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 a^2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 b (a-b) \sqrt {a+b} \left (741 a^4+51 a^2 b^2+8 b^4\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{693 a^4 d}+\frac {2 \left (135 a^4+205 a^2 b^2-4 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{693 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (a-b) \sqrt {a+b} \left (135 a^4-606 a^3 b+57 a^2 b^2+6 a b^3+8 b^4\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{693 a^3 d}+\frac {46 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{99 d \cos ^{\frac {9}{2}}(c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(13/2),x]

[Out]

(2*(a - b)*b*Sqrt[a + b]*(741*a^4 + 51*a^2*b^2 + 8*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]
/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[
c + d*x]))/(a - b)])/(693*a^4*d) + (2*(a - b)*Sqrt[a + b]*(135*a^4 - 606*a^3*b + 57*a^2*b^2 + 6*a*b^3 + 8*b^4)
*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]
*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(693*a^3*d) + (2*a^2*Sqrt[a + b*Co
s[c + d*x]]*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + (46*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(99*d*Co
s[c + d*x]^(9/2)) + (2*(81*a^2 + 113*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*d*Cos[c + d*x]^(7/2)) +
(2*b*(229*a^2 + 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*a*d*Cos[c + d*x]^(5/2)) + (2*(135*a^4 + 205
*a^2*b^2 - 4*b^4)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*a^2*d*Cos[c + d*x]^(3/2))

Rule 2871

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/
(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e
+ f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 +
c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 +
d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 2895

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(
Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqrt[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]
*EllipticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2]], -(a + b)/(a - b)], x] /; Fr
eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]

Rule 3073

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A*(c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e +
f*x])/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e +
f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ
[A, B] && PosQ[(c + d)/b]

Rule 3077

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*s
in[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[(A - B)/(a - b), Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e
+ f*x]]), x], x] - Dist[(A*b - a*B)/(a - b), Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin
[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2
- d^2, 0] && NeQ[A, B]

Rule 3134

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + D
ist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*
(b*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(
b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x]
/; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rubi steps

\begin {align*} \int \frac {(a+b \cos (c+d x))^{5/2}}{\cos ^{\frac {13}{2}}(c+d x)} \, dx &=\frac {2 a^2 \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2}{11} \int \frac {\frac {23 a^2 b}{2}+\frac {3}{2} a \left (3 a^2+11 b^2\right ) \cos (c+d x)+\frac {1}{2} b \left (8 a^2+11 b^2\right ) \cos ^2(c+d x)}{\cos ^{\frac {11}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx\\ &=\frac {2 a^2 \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {46 a b \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {4 \int \frac {\frac {1}{4} a^2 \left (81 a^2+113 b^2\right )+\frac {1}{4} a b \left (233 a^2+99 b^2\right ) \cos (c+d x)+\frac {69}{2} a^2 b^2 \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{99 a}\\ &=\frac {2 a^2 \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {46 a b \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2+113 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {8 \int \frac {\frac {5}{8} a^2 b \left (229 a^2+3 b^2\right )+\frac {1}{8} a^3 \left (405 a^2+1531 b^2\right ) \cos (c+d x)+\frac {1}{2} a^2 b \left (81 a^2+113 b^2\right ) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{693 a^2}\\ &=\frac {2 a^2 \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {46 a b \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2+113 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b \left (229 a^2+3 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {16 \int \frac {\frac {15}{16} a^2 \left (135 a^4+205 a^2 b^2-4 b^4\right )+\frac {5}{16} a^3 b \left (1011 a^2+461 b^2\right ) \cos (c+d x)+\frac {5}{8} a^2 b^2 \left (229 a^2+3 b^2\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{3465 a^3}\\ &=\frac {2 a^2 \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {46 a b \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2+113 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b \left (229 a^2+3 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (135 a^4+205 a^2 b^2-4 b^4\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {32 \int \frac {\frac {15}{32} a^2 b \left (741 a^4+51 a^2 b^2+8 b^4\right )+\frac {15}{32} a^3 \left (135 a^4+663 a^2 b^2+2 b^4\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{10395 a^4}\\ &=\frac {2 a^2 \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {46 a b \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2+113 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b \left (229 a^2+3 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (135 a^4+205 a^2 b^2-4 b^4\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (b \left (741 a^4+51 a^2 b^2+8 b^4\right )\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{693 a^2}+\frac {\left ((a-b) \left (135 a^4-606 a^3 b+57 a^2 b^2+6 a b^3+8 b^4\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{693 a^2}\\ &=\frac {2 (a-b) b \sqrt {a+b} \left (741 a^4+51 a^2 b^2+8 b^4\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{693 a^4 d}+\frac {2 (a-b) \sqrt {a+b} \left (135 a^4-606 a^3 b+57 a^2 b^2+6 a b^3+8 b^4\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{693 a^3 d}+\frac {2 a^2 \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {46 a b \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2+113 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b \left (229 a^2+3 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (135 a^4+205 a^2 b^2-4 b^4\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a^2 d \cos ^{\frac {3}{2}}(c+d x)}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 6.34, size = 1431, normalized size = 2.74 \begin {gather*} \frac {-\frac {4 a \left (135 a^6-78 a^4 b^2-49 a^2 b^4-8 b^6\right ) \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right )|-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-4 a \left (-741 a^5 b-51 a^3 b^3-8 a b^5\right ) \left (\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right )|-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \Pi \left (-\frac {a}{b};\text {ArcSin}\left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right )|-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}\right )+2 \left (-741 a^4 b^2-51 a^2 b^4-8 b^6\right ) \left (\frac {i \cos \left (\frac {1}{2} (c+d x)\right ) \sqrt {a+b \cos (c+d x)} E\left (i \sinh ^{-1}\left (\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\cos (c+d x)}}\right )|-\frac {2 a}{-a-b}\right ) \sec (c+d x)}{b \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)} \sqrt {\frac {(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac {2 a \left (\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right )|-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \Pi \left (-\frac {a}{b};\text {ArcSin}\left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right )|-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}\right )}{b}+\frac {\sqrt {a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt {\cos (c+d x)}}\right )}{693 a^3 d}+\frac {\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (\frac {2}{693} \sec ^4(c+d x) \left (81 a^2 \sin (c+d x)+113 b^2 \sin (c+d x)\right )+\frac {2 \sec ^3(c+d x) \left (229 a^2 b \sin (c+d x)+3 b^3 \sin (c+d x)\right )}{693 a}+\frac {2 \sec ^2(c+d x) \left (135 a^4 \sin (c+d x)+205 a^2 b^2 \sin (c+d x)-4 b^4 \sin (c+d x)\right )}{693 a^2}+\frac {2 \sec (c+d x) \left (741 a^4 b \sin (c+d x)+51 a^2 b^3 \sin (c+d x)+8 b^5 \sin (c+d x)\right )}{693 a^3}+\frac {46}{99} a b \sec ^4(c+d x) \tan (c+d x)+\frac {2}{11} a^2 \sec ^5(c+d x) \tan (c+d x)\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(13/2),x]

[Out]

((-4*a*(135*a^6 - 78*a^4*b^2 - 49*a^2*b^4 - 8*b^6)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)
*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF
[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a +
b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-741*a^5*b - 51*a^3*b^3 - 8*a*b^5)*((Sqrt[((a + b)*Cot
[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc
[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]],
(-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Co
t[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Cs
c[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/
Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-741*a^4*
b^2 - 51*a^2*b^4 - 8*b^6)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/S
qrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c
+ d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c
+ d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin
[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqr
t[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos
[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(
a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/
(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c +
d*x]])))/(693*a^3*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^4*(81*a^2*Sin[c + d*x] +
113*b^2*Sin[c + d*x]))/693 + (2*Sec[c + d*x]^3*(229*a^2*b*Sin[c + d*x] + 3*b^3*Sin[c + d*x]))/(693*a) + (2*Sec
[c + d*x]^2*(135*a^4*Sin[c + d*x] + 205*a^2*b^2*Sin[c + d*x] - 4*b^4*Sin[c + d*x]))/(693*a^2) + (2*Sec[c + d*x
]*(741*a^4*b*Sin[c + d*x] + 51*a^2*b^3*Sin[c + d*x] + 8*b^5*Sin[c + d*x]))/(693*a^3) + (46*a*b*Sec[c + d*x]^4*
Tan[c + d*x])/99 + (2*a^2*Sec[c + d*x]^5*Tan[c + d*x])/11))/d

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2788\) vs. \(2(472)=944\).
time = 0.57, size = 2789, normalized size = 5.34

method	result	size
default	\(\text {Expression too large to display}\)	\(2789\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cos(d*x+c))^(5/2)/cos(d*x+c)^(13/2),x,method=_RETURNVERBOSE)

[Out]

-2/693/d*(-116*cos(d*x+c)^3*a^3*b^3-274*cos(d*x+c)^2*a^4*b^2-224*cos(d*x+c)*a^5*b+135*cos(d*x+c)^7*a^5*b+741*c
os(d*x+c)^7*a^4*b^2+205*cos(d*x+c)^7*a^3*b^3+51*cos(d*x+c)^7*a^2*b^4-4*cos(d*x+c)^7*a*b^5+741*cos(d*x+c)^6*a^5
*b-307*cos(d*x+c)^6*a^4*b^2+741*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2
)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*cos(d*x+c)^6*sin(d*x+c)*a^5*b+663*(cos(d*x+c)/(1+
cos(d*x+c)))^(1/2)*((a+b*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/
(a+b))^(1/2))*cos(d*x+c)^6*sin(d*x+c)*a^4*b^2+51*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos(d*x+c))/(1+cos(d*
x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*cos(d*x+c)^6*sin(d*x+c)*a^3*b^3+
2*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))/si
n(d*x+c),(-(a-b)/(a+b))^(1/2))*cos(d*x+c)^6*sin(d*x+c)*a^2*b^4+8*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos(d
*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*cos(d*x+c)^6*sin
(d*x+c)*a*b^5-741*cos(d*x+c)^6*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos(d*x+c))/(1+cos(d*x+c))/(
a+b))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^5*b-741*cos(d*x+c)^6*sin(d*x+c)*(cos(
d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c
),(-(a-b)/(a+b))^(1/2))*a^4*b^2-51*cos(d*x+c)^6*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos(d*x+c))
/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^3*b^3-51*cos(d*x+c)^
6*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos
(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^2*b^4-8*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos(d*x+c))/(1+cos
(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*cos(d*x+c)^6*sin(d*x+c)*a*b^5
+741*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*E
llipticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^5*b+663*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(1+cos
(d*x+c)))^(1/2)*((a+b*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+
b))^(1/2))*a^4*b^2+51*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos(d*x+c))/(1+cos(d*x+c
))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^3*b^3+2*cos(d*x+c)^5*sin(d*x+c)*(
cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d
*x+c),(-(a-b)/(a+b))^(1/2))*a^2*b^4+8*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos(d*x+
c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a*b^5+135*(cos(d*x+
c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),(-
(a-b)/(a+b))^(1/2))*cos(d*x+c)^6*sin(d*x+c)*a^6-8*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos(d*x+c))/(1+cos(d
*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*cos(d*x+c)^6*sin(d*x+c)*b^6+135
*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*Ellip
ticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^6-8*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c))
)^(1/2)*((a+b*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2
))*b^6+51*cos(d*x+c)^6*a^3*b^3-52*cos(d*x+c)^6*a^2*b^4+8*cos(d*x+c)^6*a*b^5-566*cos(d*x+c)^5*a^5*b-140*cos(d*x
+c)^5*a^3*b^3-4*cos(d*x+c)^5*a*b^5-160*cos(d*x+c)^4*a^4*b^2+cos(d*x+c)^4*a^2*b^4-86*cos(d*x+c)^3*a^5*b+8*cos(d
*x+c)^7*b^6-8*cos(d*x+c)^6*b^6+135*cos(d*x+c)^6*a^6-54*cos(d*x+c)^4*a^6-18*cos(d*x+c)^2*a^6-741*cos(d*x+c)^5*s
in(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*
x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^5*b-741*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a
+b*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^4*b^2-
51*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*x+c))/s
in(d*x+c),(-(a-b)/(a+b))^(1/2))*cos(d*x+c)^5*sin(d*x+c)*a^3*b^3-51*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos
(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*cos(d*x+c)^5*s
in(d*x+c)*a^2*b^4-8*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+b*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE(
(-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*cos(d*x+c)^5*sin(d*x+c)*a*b^5-63*a^6)/(a+b*cos(d*x+c))^(1/2)/
sin(d*x+c)/cos(d*x+c)^(11/2)/a^3

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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+b*cos(d*x+c))^(5/2)/cos(d*x+c)^(13/2),x, algorithm="maxima")

[Out]

integrate((b*cos(d*x + c) + a)^(5/2)/cos(d*x + c)^(13/2), x)

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Fricas [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+b*cos(d*x+c))^(5/2)/cos(d*x+c)^(13/2),x, algorithm="fricas")

[Out]

integral((b^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + a^2)*sqrt(b*cos(d*x + c) + a)/cos(d*x + c)^(13/2), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))**(5/2)/cos(d*x+c)**(13/2),x)

[Out]

Timed out

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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+b*cos(d*x+c))^(5/2)/cos(d*x+c)^(13/2),x, algorithm="giac")

[Out]

integrate((b*cos(d*x + c) + a)^(5/2)/cos(d*x + c)^(13/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^{13/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*cos(c + d*x))^(5/2)/cos(c + d*x)^(13/2),x)

[Out]

int((a + b*cos(c + d*x))^(5/2)/cos(c + d*x)^(13/2), x)

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