3.6.52 \(\int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx\) [552]

Optimal. Leaf size=530 \[ \frac {(a-b) \sqrt {a+b} \left (284 a^2+15 b^2\right ) \cot (c+d x) E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{192 a d}+\frac {\sqrt {a+b} \left (72 a^3+284 a^2 b+118 a b^2+15 b^3\right ) \cot (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{192 a d}-\frac {\sqrt {a+b} \left (48 a^4+120 a^2 b^2-5 b^4\right ) \cot (c+d x) \Pi \left (\frac {a+b}{a};\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{64 a^2 d}+\frac {b \left (284 a^2+15 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac {\left (36 a^2+59 b^2\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac {17 a b \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {a^2 \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d} \]

[Out]

1/192*(a-b)*(284*a^2+15*b^2)*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)/a/d+1/192*(72*a^3+284*a^2*b+118*a*b^2+1
5*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)/a/d-1/64*(48*a^4+120*a^2*b^2-5*b^4)*cot(d*x+c)*EllipticPi((a+
b*sec(d*x+c))^(1/2)/(a+b)^(1/2),(a+b)/a,((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)/a^2/d+1/192*b*(284*a^2+15*b^2)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a/d+1/96*(36*a^2+59
*b^2)*cos(d*x+c)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d+17/24*a*b*cos(d*x+c)^2*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/
d+1/4*a^2*cos(d*x+c)^3*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d

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Rubi [A]
time = 0.86, antiderivative size = 530, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3926, 4189, 4143, 4006, 3869, 3917, 4089} \begin {gather*} \frac {(a-b) \sqrt {a+b} \left (284 a^2+15 b^2\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 (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{192 a d}+\frac {b \left (284 a^2+15 b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{192 a d}+\frac {\left (36 a^2+59 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{96 d}+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}-\frac {\sqrt {a+b} \left (48 a^4+120 a^2 b^2-5 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}} \Pi \left (\frac {a+b}{a};\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{64 a^2 d}+\frac {\sqrt {a+b} \left (72 a^3+284 a^2 b+118 a b^2+15 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}} F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{192 a d}+\frac {17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{24 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2),x]

[Out]

((a - b)*Sqrt[a + b]*(284*a^2 + 15*b^2)*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))])/(192*a*d) + (Sqr
t[a + b]*(72*a^3 + 284*a^2*b + 118*a*b^2 + 15*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 + Sec[c + d*x]))/(a - b))])/(192*
a*d) - (Sqrt[a + b]*(48*a^4 + 120*a^2*b^2 - 5*b^4)*Cot[c + d*x]*EllipticPi[(a + b)/a, 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))])/(64*a^2*d) + (b*(284*a^2 + 15*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*a*d) + ((36*a^2 + 59*b^2
)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*d) + (17*a*b*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]
*Sin[c + d*x])/(24*d) + (a^2*Cos[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d)

Rule 3869

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b
*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b)*((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b
*Csc[c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]

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 3926

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

Rule 4006

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[c, In
t[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[d, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 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 4143

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

Rule 4189

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

Rubi steps

\begin {align*} \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx &=\frac {a^2 \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {1}{4} \int \frac {\cos ^3(c+d x) \left (\frac {17 a^2 b}{2}+3 a \left (a^2+4 b^2\right ) \sec (c+d x)+\frac {1}{2} b \left (5 a^2+8 b^2\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx\\ &=\frac {17 a b \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {a^2 \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}-\frac {\int \frac {\cos ^2(c+d x) \left (-\frac {1}{4} a^2 \left (36 a^2+59 b^2\right )-\frac {1}{2} a b \left (49 a^2+24 b^2\right ) \sec (c+d x)-\frac {51}{4} a^2 b^2 \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{12 a}\\ &=\frac {\left (36 a^2+59 b^2\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac {17 a b \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {a^2 \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {\int \frac {\cos (c+d x) \left (\frac {1}{8} a^2 b \left (284 a^2+15 b^2\right )+\frac {1}{4} a^3 \left (36 a^2+161 b^2\right ) \sec (c+d x)+\frac {1}{8} a^2 b \left (36 a^2+59 b^2\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{24 a^2}\\ &=\frac {b \left (284 a^2+15 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac {\left (36 a^2+59 b^2\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac {17 a b \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {a^2 \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}-\frac {\int \frac {-\frac {3}{16} a^2 \left (48 a^4+120 a^2 b^2-5 b^4\right )-\frac {1}{8} a^3 b \left (36 a^2+59 b^2\right ) \sec (c+d x)+\frac {1}{16} a^2 b^2 \left (284 a^2+15 b^2\right ) \sec ^2(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{24 a^3}\\ &=\frac {b \left (284 a^2+15 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac {\left (36 a^2+59 b^2\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac {17 a b \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {a^2 \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}-\frac {\int \frac {-\frac {3}{16} a^2 \left (48 a^4+120 a^2 b^2-5 b^4\right )+\left (-\frac {1}{16} a^2 b^2 \left (284 a^2+15 b^2\right )-\frac {1}{8} a^3 b \left (36 a^2+59 b^2\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{24 a^3}-\frac {\left (b^2 \left (284 a^2+15 b^2\right )\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx}{384 a}\\ &=\frac {(a-b) \sqrt {a+b} \left (284 a^2+15 b^2\right ) \cot (c+d x) E\left (\sin ^{-1}\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}}}{192 a d}+\frac {b \left (284 a^2+15 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac {\left (36 a^2+59 b^2\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac {17 a b \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {a^2 \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {\left (b \left (72 a^3+284 a^2 b+118 a b^2+15 b^3\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{384 a}+\frac {\left (48 a^4+120 a^2 b^2-5 b^4\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}} \, dx}{128 a}\\ &=\frac {(a-b) \sqrt {a+b} \left (284 a^2+15 b^2\right ) \cot (c+d x) E\left (\sin ^{-1}\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}}}{192 a d}+\frac {\sqrt {a+b} \left (72 a^3+284 a^2 b+118 a b^2+15 b^3\right ) \cot (c+d x) F\left (\sin ^{-1}\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}}}{192 a d}-\frac {\sqrt {a+b} \left (48 a^4+120 a^2 b^2-5 b^4\right ) \cot (c+d x) \Pi \left (\frac {a+b}{a};\sin ^{-1}\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}}}{64 a^2 d}+\frac {b \left (284 a^2+15 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac {\left (36 a^2+59 b^2\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac {17 a b \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {a^2 \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 14.07, size = 1688, normalized size = 3.18 \begin {gather*} \frac {\cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left (\frac {17}{96} a b \sin (c+d x)+\frac {1}{192} \left (48 a^2+59 b^2\right ) \sin (2 (c+d x))+\frac {17}{96} a b \sin (3 (c+d x))+\frac {1}{32} a^2 \sin (4 (c+d x))\right )}{d (b+a \cos (c+d x))^2}+\frac {(a+b \sec (c+d x))^{5/2} \left (-284 a^3 b \sqrt {\frac {-a+b}{a+b}} \tan \left (\frac {1}{2} (c+d x)\right )-284 a^2 b^2 \sqrt {\frac {-a+b}{a+b}} \tan \left (\frac {1}{2} (c+d x)\right )-15 a b^3 \sqrt {\frac {-a+b}{a+b}} \tan \left (\frac {1}{2} (c+d x)\right )-15 b^4 \sqrt {\frac {-a+b}{a+b}} \tan \left (\frac {1}{2} (c+d x)\right )+568 a^3 b \sqrt {\frac {-a+b}{a+b}} \tan ^3\left (\frac {1}{2} (c+d x)\right )+30 a b^3 \sqrt {\frac {-a+b}{a+b}} \tan ^3\left (\frac {1}{2} (c+d x)\right )-284 a^3 b \sqrt {\frac {-a+b}{a+b}} \tan ^5\left (\frac {1}{2} (c+d x)\right )+284 a^2 b^2 \sqrt {\frac {-a+b}{a+b}} \tan ^5\left (\frac {1}{2} (c+d x)\right )-15 a b^3 \sqrt {\frac {-a+b}{a+b}} \tan ^5\left (\frac {1}{2} (c+d x)\right )+15 b^4 \sqrt {\frac {-a+b}{a+b}} \tan ^5\left (\frac {1}{2} (c+d x)\right )+288 i a^4 \Pi \left (-\frac {a+b}{a-b};i \sinh ^{-1}\left (\sqrt {\frac {-a+b}{a+b}} \tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a+b}{a-b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+720 i a^2 b^2 \Pi \left (-\frac {a+b}{a-b};i \sinh ^{-1}\left (\sqrt {\frac {-a+b}{a+b}} \tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a+b}{a-b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}-30 i b^4 \Pi \left (-\frac {a+b}{a-b};i \sinh ^{-1}\left (\sqrt {\frac {-a+b}{a+b}} \tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a+b}{a-b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+288 i a^4 \Pi \left (-\frac {a+b}{a-b};i \sinh ^{-1}\left (\sqrt {\frac {-a+b}{a+b}} \tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a+b}{a-b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+720 i a^2 b^2 \Pi \left (-\frac {a+b}{a-b};i \sinh ^{-1}\left (\sqrt {\frac {-a+b}{a+b}} \tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a+b}{a-b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}-30 i b^4 \Pi \left (-\frac {a+b}{a-b};i \sinh ^{-1}\left (\sqrt {\frac {-a+b}{a+b}} \tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a+b}{a-b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+i b \left (284 a^3-284 a^2 b+15 a b^2-15 b^3\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {-a+b}{a+b}} \tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a+b}{a-b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}-2 i \left (72 a^4-36 a^3 b+38 a^2 b^2-59 a b^3-15 b^4\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {-a+b}{a+b}} \tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a+b}{a-b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}\right )}{192 a \sqrt {\frac {-a+b}{a+b}} d (b+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x) \sqrt {\frac {1}{1-\tan ^2\left (\frac {1}{2} (c+d x)\right )}} \left (-1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2),x]

[Out]

(Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((17*a*b*Sin[c + d*x])/96 + ((48*a^2 + 59*b^2)*Sin[2*(c + d*x)])/19
2 + (17*a*b*Sin[3*(c + d*x)])/96 + (a^2*Sin[4*(c + d*x)])/32))/(d*(b + a*Cos[c + d*x])^2) + ((a + b*Sec[c + d*
x])^(5/2)*(-284*a^3*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 284*a^2*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*
x)/2] - 15*a*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 15*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 56
8*a^3*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + 30*a*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 - 284*a
^3*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 284*a^2*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 15*a*
b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 15*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + (288*I)*a^4
*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 -
Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (720*I)*a^2*b^2*Elli
pticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(
c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (30*I)*b^4*EllipticPi[-((
a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2
]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (288*I)*a^4*EllipticPi[-((a + b)/(a
- b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(
c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (720*I)*a^2*b^2*EllipticP
i[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*
Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (30*I)*b^4*
EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d
*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*
b*(284*a^3 - 284*a^2*b + 15*a*b^2 - 15*b^3)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a +
b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[
(c + d*x)/2]^2)/(a + b)] - (2*I)*(72*a^4 - 36*a^3*b + 38*a^2*b^2 - 59*a*b^3 - 15*b^4)*EllipticF[I*ArcSinh[Sqrt
[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*S
qrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(192*a*Sqrt[(-a + b)/(a + b)]*d*(b + a*Co
s[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c
+ d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2329\) vs. \(2(481)=962\).
time = 0.26, size = 2330, normalized size = 4.40

method	result	size
default	\(\text {Expression too large to display}\)	\(2330\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*(a+b*sec(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/192/d*(-1+cos(d*x+c))^2*(-72*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+co
s(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+644*cos(d*x+c)*sin(d*x+
c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))/s
in(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^2-118*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*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^3-284*cos(d*x
+c)*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+c
os(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b-284*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(
(b+a*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^2
-15*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*Elli
pticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^3-720*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c
)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))
^(1/2))*a^2*b^2-48*cos(d*x+c)^6*a^4+72*cos(d*x+c)^2*a^4+15*cos(d*x+c)*b^4-288*a^4*(cos(d*x+c)/(1+cos(d*x+c)))^
(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/
2))*sin(d*x+c)+30*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticPi((
-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*b^4*sin(d*x+c)-15*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(1+cos(d
*x+c)))^(1/2)*((b+a*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^4-15*cos(d*x+c)^2*b^4-24*cos(d*x+c)^4*a^4-184*cos(d*x+c)^5*a^3*b-254*cos(d*x+c)^4*a^2*b^2-172*cos(d*
x+c)^3*a^3*b-133*cos(d*x+c)^3*a*b^3-30*cos(d*x+c)^2*a^2*b^2+118*cos(d*x+c)*a*b^3+144*a^4*(cos(d*x+c)/(1+cos(d*
x+c)))^(1/2)*((b+a*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))*sin(d*x+c)-15*b^4*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*Ellip
ticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)-288*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(1+cos(d
*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a
+b))^(1/2))*a^4+30*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a
+b))^(1/2)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*b^4-72*(cos(d*x+c)/(1+cos(d*x+c)))^(1
/2)*((b+a*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*sin(d*x+c)+644*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*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^2*sin(d*x+c)-118*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*
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^3*sin(d*
x+c)-284*a^3*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*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))*sin(d*x+c)*b-284*a^2*b^2*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*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))*sin(d*x+c)-15*b^
3*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*x+c))/si
n(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*a-720*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+
c))/(a+b))^(1/2)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*a^2*b^2*sin(d*x+c)+144*cos(d*x+
c)*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+co
s(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^4+284*cos(d*x+c)^2*a^3*b+15*cos(d*x+c)^2*a*b^3+72*cos(d*x+c)*a^3*b
+284*cos(d*x+c)*a^2*b^2)*(1+cos(d*x+c))^2*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)/(b+a*cos(d*x+c))/sin(d*x+c)^5/a

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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(cos(d*x+c)^4*(a+b*sec(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*sec(d*x + c) + a)^(5/2)*cos(d*x + c)^4, x)

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Fricas [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(cos(d*x+c)^4*(a+b*sec(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

Timed out

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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(cos(d*x+c)**4*(a+b*sec(d*x+c))**(5/2),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(cos(d*x+c)^4*(a+b*sec(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate((b*sec(d*x + c) + a)^(5/2)*cos(d*x + c)^4, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\cos \left (c+d\,x\right )}^4\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^4*(a + b/cos(c + d*x))^(5/2),x)

[Out]

int(cos(c + d*x)^4*(a + b/cos(c + d*x))^(5/2), x)

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