\(\int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx\) [552]
Optimal result
Integrand size = 23, antiderivative size = 530 \[
\int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\frac {(a-b) \sqrt {a+b} \left (284 a^2+15 b^2\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{192 a d}+\frac {\sqrt {a+b} \left (72 a^3+284 a^2 b+118 a b^2+15 b^3\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{192 a d}-\frac {\sqrt {a+b} \left (48 a^4+120 a^2 b^2-5 b^4\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{64 a^2 d}+\frac {b \left (284 a^2+15 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac {\left (36 a^2+59 b^2\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac {17 a b \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {a^2 \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}
\]
[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
Rubi [A] (verified)
Time = 1.50 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.00, number of
steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3926, 4189, 4143, 4006, 3869,
3917, 4089} \[
\int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\frac {(a-b) \sqrt {a+b} \left (284 a^2+15 b^2\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{192 a d}+\frac {b \left (284 a^2+15 b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{192 a d}+\frac {\left (36 a^2+59 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{96 d}+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}-\frac {\sqrt {a+b} \left (48 a^4+120 a^2 b^2-5 b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{64 a^2 d}+\frac {\sqrt {a+b} \left (72 a^3+284 a^2 b+118 a b^2+15 b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{192 a d}+\frac {17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{24 d}
\]
[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*}
\text {integral}& = \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 (\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 {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 (\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) \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}}}{192 a d}-\frac {\sqrt {a+b} \left (48 a^4+120 a^2 b^2-5 b^4\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\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} \\
\end{align*}
Mathematica [B] (warning: unable to verify)
Leaf count is larger than twice the leaf count of optimal. \(1274\) vs. \(2(530)=1060\).
Time = 14.49 (sec) , antiderivative size = 1274, normalized size of antiderivative = 2.40
\[
\int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\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 \tan \left (\frac {1}{2} (c+d x)\right )-284 a^2 b^2 \tan \left (\frac {1}{2} (c+d x)\right )-15 a b^3 \tan \left (\frac {1}{2} (c+d x)\right )-15 b^4 \tan \left (\frac {1}{2} (c+d x)\right )+568 a^3 b \tan ^3\left (\frac {1}{2} (c+d x)\right )+30 a b^3 \tan ^3\left (\frac {1}{2} (c+d x)\right )-284 a^3 b \tan ^5\left (\frac {1}{2} (c+d x)\right )+284 a^2 b^2 \tan ^5\left (\frac {1}{2} (c+d x)\right )-15 a b^3 \tan ^5\left (\frac {1}{2} (c+d x)\right )+15 b^4 \tan ^5\left (\frac {1}{2} (c+d x)\right )-288 a^4 \operatorname {EllipticPi}\left (-1,\arcsin \left (\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 a^2 b^2 \operatorname {EllipticPi}\left (-1,\arcsin \left (\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 b^4 \operatorname {EllipticPi}\left (-1,\arcsin \left (\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 a^4 \operatorname {EllipticPi}\left (-1,\arcsin \left (\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 a^2 b^2 \operatorname {EllipticPi}\left (-1,\arcsin \left (\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 b^4 \operatorname {EllipticPi}\left (-1,\arcsin \left (\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}}-b \left (284 a^3+284 a^2 b+15 a b^2+15 b^3\right ) E\left (\arcsin \left (\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 a \left (72 a^3-36 a^2 b+322 a b^2-59 b^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\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 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 )}}}
\]
[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*Tan[(c + d*x)/2] - 284*a^2*b^2*Tan[(c + d*x)/2] - 15*a*b^3*Tan[(c + d*x)/2] - 15*b^4*Tan
[(c + d*x)/2] + 568*a^3*b*Tan[(c + d*x)/2]^3 + 30*a*b^3*Tan[(c + d*x)/2]^3 - 284*a^3*b*Tan[(c + d*x)/2]^5 + 28
4*a^2*b^2*Tan[(c + d*x)/2]^5 - 15*a*b^3*Tan[(c + d*x)/2]^5 + 15*b^4*Tan[(c + d*x)/2]^5 - 288*a^4*EllipticPi[-1
, ArcSin[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*a^2*b^2*EllipticPi[-1, ArcSin[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*b^4*EllipticPi
[-1, ArcSin[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*a^4*EllipticPi[-1, ArcSin[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*a^2*b^2*EllipticPi[-1, ArcSin[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*b^4*EllipticPi[-1, ArcSin[Ta
n[(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)] - b*(284*a^3 + 284*a^2*b + 15*a*b^2 + 15*b^3)*EllipticE[ArcSin[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*a*(72*a^3 - 36*a^2*b + 322*a*b^2 - 59*b^3)*EllipticF[ArcSin[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)]))/(192*a*d*(b + a*Cos[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)])
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(3191\) vs. \(2(481)=962\).
Time = 410.77 (sec) , antiderivative size = 3192, normalized size of antiderivative =
6.02
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\(\text {Expression too large to display}\) |
\(3192\) |
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[In]
int(cos(d*x+c)^4*(a+b*sec(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
[Out]
1/192/d/a*(-1440*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticPi(
cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))*a^2*b^2*cos(d*x+c)-720*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b
)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))*a^2*b^2*cos(
d*x+c)^2-720*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+
b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b^2+60*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c
))/(cos(d*x+c)+1))^(1/2)*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))*b^4*cos(d*x+c)-30*(cos(d*x+c
)/(cos(d*x+c)+1))^(1/2)*(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))*b^4*cos(d*x+c)+144*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(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))*a^4*cos(d*x+c)^2-288*(cos(d*x+c)/(cos(d*x+c)+1))^(1
/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))*a
^4*cos(d*x+c)^2+30*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticP
i(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))*b^4*cos(d*x+c)^2-15*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(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))*b^4*cos(d*x+c)^2+
288*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(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))*a^4*cos(d*x+c)-576*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(c
os(d*x+c)+1))^(1/2)*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))*a^4*cos(d*x+c)+184*a^3*b*cos(d*x+
c)^4*sin(d*x+c)+184*a^3*b*cos(d*x+c)^3*sin(d*x+c)+254*a^2*b^2*cos(d*x+c)^3*sin(d*x+c)+356*a^3*b*cos(d*x+c)^2*s
in(d*x+c)+254*a^2*b^2*cos(d*x+c)^2*sin(d*x+c)+133*a*b^3*cos(d*x+c)^2*sin(d*x+c)+144*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)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a
^4-288*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+
a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^4+30*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))*(cos(d*x+c
)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*b^4-15*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)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*b^4-28
4*(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-284*(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^2*b^2-15*(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^3-72*(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+644*(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^2-118*(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^3+48*a^4*cos
(d*x+c)^5*sin(d*x+c)+48*a^4*cos(d*x+c)^4*sin(d*x+c)+72*a^4*cos(d*x+c)^3*sin(d*x+c)+72*a^4*cos(d*x+c)^2*sin(d*x
+c)+15*b^4*cos(d*x+c)*sin(d*x+c)+118*a*b^3*cos(d*x+c)*sin(d*x+c)+72*a^3*b*cos(d*x+c)*sin(d*x+c)+284*a^2*b^2*co
s(d*x+c)*sin(d*x+c)-284*(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)*Elli
pticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^2*cos(d*x+c)^2-15*(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^3*cos(d*
x+c)^2-144*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(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)*a^3*b*cos(d*x+c)+1288*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))
*(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)*a^2*b^2*cos(d*x+c)-236*Elli
pticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(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)*a*b^3*cos(d*x+c)-568*(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*cos(d*x+c)-568*(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^2*cos(d*x+c)-30*(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)*Ell
ipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a*b^3*cos(d*x+c)-72*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(
a+b))^(1/2))*(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)*a^3*b*cos(d*x+c
)^2+644*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(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)*a^2*b^2*cos(d*x+c)^2-118*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))
*(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)*a*b^3*cos(d*x+c)^2-284*(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*cos(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2)/(b+a*cos(d*x+c))/(cos(d*x+c)+1)
Fricas [F]
\[
\int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x }
\]
[In]
integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^(5/2),x, algorithm="fricas")
[Out]
integral((b^2*cos(d*x + c)^4*sec(d*x + c)^2 + 2*a*b*cos(d*x + c)^4*sec(d*x + c) + a^2*cos(d*x + c)^4)*sqrt(b*s
ec(d*x + c) + a), x)
Sympy [F(-1)]
Timed out. \[
\int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\text {Timed out}
\]
[In]
integrate(cos(d*x+c)**4*(a+b*sec(d*x+c))**(5/2),x)
[Out]
Timed out
Maxima [F]
\[
\int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x }
\]
[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)
Giac [F]
\[
\int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x }
\]
[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)
Mupad [F(-1)]
Timed out. \[
\int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\int {\cos \left (c+d\,x\right )}^4\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x
\]
[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)