\(\int \frac {\cos ^6(e+f x)}{(a+b \sec ^2(e+f x))^{5/2}} \, dx\) [295]
Optimal result
Integrand size = 25, antiderivative size = 332 \[
\int \frac {\cos ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\frac {5 (a-3 b) \left (a^2+7 b^2\right ) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{16 a^{11/2} f}+\frac {\left (5 a^2-10 a b+21 b^2\right ) \cos (e+f x) \sin (e+f x)}{16 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(5 a-9 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (15 a^3-25 a^2 b+49 a b^2+105 b^3\right ) \tan (e+f x)}{48 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (15 a^4-20 a^3 b+38 a^2 b^2+420 a b^3+315 b^4\right ) \tan (e+f x)}{48 a^5 (a+b)^2 f \sqrt {a+b+b \tan ^2(e+f x)}}
\]
[Out]
5/16*(a-3*b)*(a^2+7*b^2)*arctan(a^(1/2)*tan(f*x+e)/(a+b+b*tan(f*x+e)^2)^(1/2))/a^(11/2)/f+1/48*b*(15*a^4-20*a^
3*b+38*a^2*b^2+420*a*b^3+315*b^4)*tan(f*x+e)/a^5/(a+b)^2/f/(a+b+b*tan(f*x+e)^2)^(1/2)+1/16*(5*a^2-10*a*b+21*b^
2)*cos(f*x+e)*sin(f*x+e)/a^3/f/(a+b+b*tan(f*x+e)^2)^(3/2)+1/24*(5*a-9*b)*cos(f*x+e)^3*sin(f*x+e)/a^2/f/(a+b+b*
tan(f*x+e)^2)^(3/2)+1/6*cos(f*x+e)^5*sin(f*x+e)/a/f/(a+b+b*tan(f*x+e)^2)^(3/2)+1/48*b*(15*a^3-25*a^2*b+49*a*b^
2+105*b^3)*tan(f*x+e)/a^4/(a+b)/f/(a+b+b*tan(f*x+e)^2)^(3/2)
Rubi [A] (verified)
Time = 0.49 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.00, number of
steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {4231, 425, 541, 12, 385, 209}
\[
\int \frac {\cos ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\frac {(5 a-9 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \left (a+b \tan ^2(e+f x)+b\right )^{3/2}}+\frac {5 (a-3 b) \left (a^2+7 b^2\right ) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)+b}}\right )}{16 a^{11/2} f}+\frac {\left (5 a^2-10 a b+21 b^2\right ) \sin (e+f x) \cos (e+f x)}{16 a^3 f \left (a+b \tan ^2(e+f x)+b\right )^{3/2}}+\frac {b \left (15 a^3-25 a^2 b+49 a b^2+105 b^3\right ) \tan (e+f x)}{48 a^4 f (a+b) \left (a+b \tan ^2(e+f x)+b\right )^{3/2}}+\frac {b \left (15 a^4-20 a^3 b+38 a^2 b^2+420 a b^3+315 b^4\right ) \tan (e+f x)}{48 a^5 f (a+b)^2 \sqrt {a+b \tan ^2(e+f x)+b}}+\frac {\sin (e+f x) \cos ^5(e+f x)}{6 a f \left (a+b \tan ^2(e+f x)+b\right )^{3/2}}
\]
[In]
Int[Cos[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2),x]
[Out]
(5*(a - 3*b)*(a^2 + 7*b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(11/2)*f) + ((
5*a^2 - 10*a*b + 21*b^2)*Cos[e + f*x]*Sin[e + f*x])/(16*a^3*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + ((5*a - 9*b)
*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^2*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (Cos[e + f*x]^5*Sin[e + f*x])/(6*a
*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (b*(15*a^3 - 25*a^2*b + 49*a*b^2 + 105*b^3)*Tan[e + f*x])/(48*a^4*(a +
b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (b*(15*a^4 - 20*a^3*b + 38*a^2*b^2 + 420*a*b^3 + 315*b^4)*Tan[e + f*x
])/(48*a^5*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 209
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 385
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Rule 425
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]
Rule 541
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
Rule 4231
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fre
eFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 + ff^2*x^2)^(m/2 - 1)*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/
2), x]^p, x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]
Rubi steps \begin{align*}
\text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^4 \left (a+b+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {-5 a+b-8 b x^2}{\left (1+x^2\right )^3 \left (a+b+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{6 a f} \\ & = \frac {(5 a-9 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {3 \left (5 a^2+3 b^2\right )+6 (5 a-9 b) b x^2}{\left (1+x^2\right )^2 \left (a+b+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{24 a^2 f} \\ & = \frac {\left (5 a^2-10 a b+21 b^2\right ) \cos (e+f x) \sin (e+f x)}{16 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(5 a-9 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {-3 \left (5 a^3+5 a^2 b-5 a b^2-21 b^3\right )-12 b \left (5 a^2-10 a b+21 b^2\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{48 a^3 f} \\ & = \frac {\left (5 a^2-10 a b+21 b^2\right ) \cos (e+f x) \sin (e+f x)}{16 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(5 a-9 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (15 a^3-25 a^2 b+49 a b^2+105 b^3\right ) \tan (e+f x)}{48 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {-3 \left (15 a^4+10 a^2 b^2-112 a b^3-105 b^4\right )-6 b \left (15 a^3-25 a^2 b+49 a b^2+105 b^3\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{144 a^4 (a+b) f} \\ & = \frac {\left (5 a^2-10 a b+21 b^2\right ) \cos (e+f x) \sin (e+f x)}{16 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(5 a-9 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (15 a^3-25 a^2 b+49 a b^2+105 b^3\right ) \tan (e+f x)}{48 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (15 a^4-20 a^3 b+38 a^2 b^2+420 a b^3+315 b^4\right ) \tan (e+f x)}{48 a^5 (a+b)^2 f \sqrt {a+b+b \tan ^2(e+f x)}}-\frac {\text {Subst}\left (\int -\frac {45 (a-3 b) (a+b)^2 \left (a^2+7 b^2\right )}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{144 a^5 (a+b)^2 f} \\ & = \frac {\left (5 a^2-10 a b+21 b^2\right ) \cos (e+f x) \sin (e+f x)}{16 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(5 a-9 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (15 a^3-25 a^2 b+49 a b^2+105 b^3\right ) \tan (e+f x)}{48 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (15 a^4-20 a^3 b+38 a^2 b^2+420 a b^3+315 b^4\right ) \tan (e+f x)}{48 a^5 (a+b)^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\left (5 (a-3 b) \left (a^2+7 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{16 a^5 f} \\ & = \frac {\left (5 a^2-10 a b+21 b^2\right ) \cos (e+f x) \sin (e+f x)}{16 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(5 a-9 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (15 a^3-25 a^2 b+49 a b^2+105 b^3\right ) \tan (e+f x)}{48 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (15 a^4-20 a^3 b+38 a^2 b^2+420 a b^3+315 b^4\right ) \tan (e+f x)}{48 a^5 (a+b)^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\left (5 (a-3 b) \left (a^2+7 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,\frac {\tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{16 a^5 f} \\ & = \frac {5 (a-3 b) \left (a^2+7 b^2\right ) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{16 a^{11/2} f}+\frac {\left (5 a^2-10 a b+21 b^2\right ) \cos (e+f x) \sin (e+f x)}{16 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(5 a-9 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (15 a^3-25 a^2 b+49 a b^2+105 b^3\right ) \tan (e+f x)}{48 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (15 a^4-20 a^3 b+38 a^2 b^2+420 a b^3+315 b^4\right ) \tan (e+f x)}{48 a^5 (a+b)^2 f \sqrt {a+b+b \tan ^2(e+f x)}} \\
\end{align*}
Mathematica [C] (warning: unable to verify)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 21.31 (sec) , antiderivative size = 1776, normalized size of antiderivative = 5.35
\[
\int \frac {\cos ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\frac {3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-5,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^{16}(e+f x) \sin (e+f x)}{4 \sqrt {2} f \left (a+b \sec ^2(e+f x)\right )^{5/2} \left (a+b-a \sin ^2(e+f x)\right )^{5/2} \left (3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-5,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+5 \left (a \operatorname {AppellF1}\left (\frac {3}{2},-5,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-2 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-4,\frac {5}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )\right ) \sin ^2(e+f x)\right ) \left (\frac {15 a (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-5,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^{11}(e+f x) \sin ^2(e+f x)}{4 \sqrt {2} \left (a+b-a \sin ^2(e+f x)\right )^{7/2} \left (3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-5,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+5 \left (a \operatorname {AppellF1}\left (\frac {3}{2},-5,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-2 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-4,\frac {5}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )\right ) \sin ^2(e+f x)\right )}+\frac {3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-5,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^{11}(e+f x)}{4 \sqrt {2} \left (a+b-a \sin ^2(e+f x)\right )^{5/2} \left (3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-5,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+5 \left (a \operatorname {AppellF1}\left (\frac {3}{2},-5,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-2 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-4,\frac {5}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )\right ) \sin ^2(e+f x)\right )}-\frac {15 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-5,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^9(e+f x) \sin ^2(e+f x)}{2 \sqrt {2} \left (a+b-a \sin ^2(e+f x)\right )^{5/2} \left (3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-5,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+5 \left (a \operatorname {AppellF1}\left (\frac {3}{2},-5,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-2 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-4,\frac {5}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )\right ) \sin ^2(e+f x)\right )}+\frac {3 (a+b) \cos ^{10}(e+f x) \sin (e+f x) \left (\frac {5 a f \operatorname {AppellF1}\left (\frac {3}{2},-5,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-\frac {10}{3} f \operatorname {AppellF1}\left (\frac {3}{2},-4,\frac {5}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos (e+f x) \sin (e+f x)\right )}{4 \sqrt {2} f \left (a+b-a \sin ^2(e+f x)\right )^{5/2} \left (3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-5,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+5 \left (a \operatorname {AppellF1}\left (\frac {3}{2},-5,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-2 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-4,\frac {5}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )\right ) \sin ^2(e+f x)\right )}-\frac {3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-5,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^{10}(e+f x) \sin (e+f x) \left (10 f \left (a \operatorname {AppellF1}\left (\frac {3}{2},-5,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-2 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-4,\frac {5}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )\right ) \cos (e+f x) \sin (e+f x)+3 (a+b) \left (\frac {5 a f \operatorname {AppellF1}\left (\frac {3}{2},-5,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-\frac {10}{3} f \operatorname {AppellF1}\left (\frac {3}{2},-4,\frac {5}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos (e+f x) \sin (e+f x)\right )+5 \sin ^2(e+f x) \left (a \left (\frac {21 a f \operatorname {AppellF1}\left (\frac {5}{2},-5,\frac {9}{2},\frac {7}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos (e+f x) \sin (e+f x)}{5 (a+b)}-6 f \operatorname {AppellF1}\left (\frac {5}{2},-4,\frac {7}{2},\frac {7}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos (e+f x) \sin (e+f x)\right )-2 (a+b) \left (\frac {3 a f \operatorname {AppellF1}\left (\frac {5}{2},-4,\frac {7}{2},\frac {7}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos (e+f x) \sin (e+f x)}{a+b}-\frac {24}{5} f \operatorname {AppellF1}\left (\frac {5}{2},-3,\frac {5}{2},\frac {7}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos (e+f x) \sin (e+f x)\right )\right )\right )}{4 \sqrt {2} f \left (a+b-a \sin ^2(e+f x)\right )^{5/2} \left (3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-5,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+5 \left (a \operatorname {AppellF1}\left (\frac {3}{2},-5,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-2 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-4,\frac {5}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )\right ) \sin ^2(e+f x)\right )^2}\right )}
\]
[In]
Integrate[Cos[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2),x]
[Out]
(3*(a + b)*AppellF1[1/2, -5, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^16*Sin[e + f*x
])/(4*Sqrt[2]*f*(a + b*Sec[e + f*x]^2)^(5/2)*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -5, 5/2
, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + 5*(a*AppellF1[3/2, -5, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e
+ f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e
+ f*x]^2)*((15*a*(a + b)*AppellF1[1/2, -5, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]
^11*Sin[e + f*x]^2)/(4*Sqrt[2]*(a + b - a*Sin[e + f*x]^2)^(7/2)*(3*(a + b)*AppellF1[1/2, -5, 5/2, 3/2, Sin[e +
f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + 5*(a*AppellF1[3/2, -5, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a
+ b)] - 2*(a + b)*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) +
(3*(a + b)*AppellF1[1/2, -5, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^11)/(4*Sqrt[2]
*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -5, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a
+ b)] + 5*(a*AppellF1[3/2, -5, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2
, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (15*(a + b)*AppellF1[1/2, -5,
5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^9*Sin[e + f*x]^2)/(2*Sqrt[2]*(a + b - a*Sin
[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -5, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + 5*(a*A
ppellF1[3/2, -5, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2, -4, 5/2, 5/2,
Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + (3*(a + b)*Cos[e + f*x]^10*Sin[e + f*x]*((5*a
*f*AppellF1[3/2, -5, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(3*(a +
b)) - (10*f*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])
/3))/(4*Sqrt[2]*f*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -5, 5/2, 3/2, Sin[e + f*x]^2, (a*S
in[e + f*x]^2)/(a + b)] + 5*(a*AppellF1[3/2, -5, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 2*(a
+ b)*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (3*(a + b)*Ap
pellF1[1/2, -5, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^10*Sin[e + f*x]*(10*f*(a*Ap
pellF1[3/2, -5, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2, -4, 5/2, 5/2,
Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Cos[e + f*x]*Sin[e + f*x] + 3*(a + b)*((5*a*f*AppellF1[3/2, -5, 7
/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)) - (10*f*AppellF1[3
/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/3) + 5*Sin[e + f*x]^2
*(a*((21*a*f*AppellF1[5/2, -5, 9/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x]
)/(5*(a + b)) - 6*f*AppellF1[5/2, -4, 7/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e
+ f*x]) - 2*(a + b)*((3*a*f*AppellF1[5/2, -4, 7/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f
*x]*Sin[e + f*x])/(a + b) - (24*f*AppellF1[5/2, -3, 5/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[
e + f*x]*Sin[e + f*x])/5))))/(4*Sqrt[2]*f*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -5, 5/2, 3
/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + 5*(a*AppellF1[3/2, -5, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e +
f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e +
f*x]^2)^2)))
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(3398\) vs. \(2(304)=608\).
Time = 10.81 (sec) , antiderivative size = 3399, normalized size of antiderivative =
10.24
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\(\text {Expression too large to display}\) |
\(3399\) |
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[In]
int(cos(f*x+e)^6/(a+b*sec(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)
[Out]
1/48/f/a^5/(-a)^(1/2)/(a+b)^2*(b+a*cos(f*x+e)^2)*(315*(-a)^(1/2)*b^6*sin(f*x+e)-315*((b+a*cos(f*x+e)^2)/(1+cos
(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a*co
s(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*b^6+420*(-a)^(1/2)*a*b^5*cos(f*x+e)^2*sin(f*x+e)-15*((b+a*
cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+4
*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a^5*b*cos(f*x+e)^2+30*((b+a*cos(f*x+e)
^2)/(1+cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+4*(-a)^(1/2
)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a^4*b^2*cos(f*x+e)^2-150*((b+a*cos(f*x+e)^2)/(1+
cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a
*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a^3*b^3*cos(f*x+e)^2-525*((b+a*cos(f*x+e)^2)/(1+cos(f*x
+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a*cos(f*
x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a^2*b^4*cos(f*x+e)^2-315*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)
^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)
/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a*b^5*cos(f*x+e)^2+15*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*ln(
4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*
x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a^5*b*cos(f*x+e)+16*(-a)^(1/2)*a^5*b*cos(f*x+e)^8*sin(f*x+e)+8*(-a)^(1/2)*a^4*b
^2*cos(f*x+e)^8*sin(f*x+e)+2*(-a)^(1/2)*a^5*b*cos(f*x+e)^6*sin(f*x+e)-26*(-a)^(1/2)*a^4*b^2*cos(f*x+e)^6*sin(f
*x+e)-18*(-a)^(1/2)*a^3*b^3*cos(f*x+e)^6*sin(f*x+e)+18*(-a)^(1/2)*a^4*b^2*cos(f*x+e)^4*sin(f*x+e)+96*(-a)^(1/2
)*a^3*b^3*cos(f*x+e)^4*sin(f*x+e)-15*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x
+e)^2)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e
)*a)*a^4*b^2*cos(f*x+e)+30*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+
cos(f*x+e))^2)^(1/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a^3*b
^3*cos(f*x+e)-150*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e
))^2)^(1/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a^2*b^4*cos(f*
x+e)-525*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/
2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a*b^5*cos(f*x+e)+63*(-a
)^(1/2)*a^2*b^4*cos(f*x+e)^4*sin(f*x+e)-15*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*c
os(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin
(f*x+e)*a)*a^5*b*cos(f*x+e)^3+30*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^
2)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)
*a^4*b^2*cos(f*x+e)^3-150*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+c
os(f*x+e))^2)^(1/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a^3*b^
3*cos(f*x+e)^3-525*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+
e))^2)^(1/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a^2*b^4*cos(f
*x+e)^3-315*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^
(1/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a*b^5*cos(f*x+e)^3+3
0*(-a)^(1/2)*a^5*b*cos(f*x+e)^2*sin(f*x+e)-30*(-a)^(1/2)*a^4*b^2*cos(f*x+e)^2*sin(f*x+e)+62*(-a)^(1/2)*a^3*b^3
*cos(f*x+e)^2*sin(f*x+e)+574*(-a)^(1/2)*a^2*b^4*cos(f*x+e)^2*sin(f*x+e)+15*(-a)^(1/2)*a^4*b^2*sin(f*x+e)-20*(-
a)^(1/2)*a^3*b^3*sin(f*x+e)+38*(-a)^(1/2)*a^2*b^4*sin(f*x+e)+420*(-a)^(1/2)*a*b^5*sin(f*x+e)+8*(-a)^(1/2)*a^6*
cos(f*x+e)^8*sin(f*x+e)+10*(-a)^(1/2)*a^6*cos(f*x+e)^6*sin(f*x+e)+15*(-a)^(1/2)*a^6*cos(f*x+e)^4*sin(f*x+e)+15
*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*cos(f
*x+e)+4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a^6*cos(f*x+e)^3+15*((b+a*cos(f
*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+4*(-a)
^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a^6*cos(f*x+e)^2-315*((b+a*cos(f*x+e)^2)/(1
+cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+
a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*b^6*cos(f*x+e)+15*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2
)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a*cos(f*x+e)^2
)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a^5*b-15*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)
*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/
2)-4*sin(f*x+e)*a)*a^4*b^2+30*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/
(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a^
3*b^3-150*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1
/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a^2*b^4-525*((b+a*cos(
f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+4*(-a
)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a*b^5)/(a+b*sec(f*x+e)^2)^(5/2)*sec(f*x+e)
^5
Fricas [A] (verification not implemented)
none
Time = 16.21 (sec) , antiderivative size = 1337, normalized size of antiderivative = 4.03
\[
\int \frac {\cos ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display}
\]
[In]
integrate(cos(f*x+e)^6/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm="fricas")
[Out]
[1/384*(15*(a^5*b^2 - a^4*b^3 + 2*a^3*b^4 - 10*a^2*b^5 - 35*a*b^6 - 21*b^7 + (a^7 - a^6*b + 2*a^5*b^2 - 10*a^4
*b^3 - 35*a^3*b^4 - 21*a^2*b^5)*cos(f*x + e)^4 + 2*(a^6*b - a^5*b^2 + 2*a^4*b^3 - 10*a^3*b^4 - 35*a^2*b^5 - 21
*a*b^6)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 1
4*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2
*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*
b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 +
b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(8*(a^7 + 2*a^6*b + a^5*b^2)*cos(f*x + e)^9 + 2*(5*a^7 + a^6*b - 13*a^5*b
^2 - 9*a^4*b^3)*cos(f*x + e)^7 + 3*(5*a^7 + 6*a^5*b^2 + 32*a^4*b^3 + 21*a^3*b^4)*cos(f*x + e)^5 + 2*(15*a^6*b
- 15*a^5*b^2 + 31*a^4*b^3 + 287*a^3*b^4 + 210*a^2*b^5)*cos(f*x + e)^3 + (15*a^5*b^2 - 20*a^4*b^3 + 38*a^3*b^4
+ 420*a^2*b^5 + 315*a*b^6)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^10 + 2*
a^9*b + a^8*b^2)*f*cos(f*x + e)^4 + 2*(a^9*b + 2*a^8*b^2 + a^7*b^3)*f*cos(f*x + e)^2 + (a^8*b^2 + 2*a^7*b^3 +
a^6*b^4)*f), -1/192*(15*(a^5*b^2 - a^4*b^3 + 2*a^3*b^4 - 10*a^2*b^5 - 35*a*b^6 - 21*b^7 + (a^7 - a^6*b + 2*a^5
*b^2 - 10*a^4*b^3 - 35*a^3*b^4 - 21*a^2*b^5)*cos(f*x + e)^4 + 2*(a^6*b - a^5*b^2 + 2*a^4*b^3 - 10*a^3*b^4 - 35
*a^2*b^5 - 21*a*b^6)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 +
(a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4
- a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 4*(8*(a^7 + 2*a^6*b + a^5*b^2)*cos(f*x + e)
^9 + 2*(5*a^7 + a^6*b - 13*a^5*b^2 - 9*a^4*b^3)*cos(f*x + e)^7 + 3*(5*a^7 + 6*a^5*b^2 + 32*a^4*b^3 + 21*a^3*b^
4)*cos(f*x + e)^5 + 2*(15*a^6*b - 15*a^5*b^2 + 31*a^4*b^3 + 287*a^3*b^4 + 210*a^2*b^5)*cos(f*x + e)^3 + (15*a^
5*b^2 - 20*a^4*b^3 + 38*a^3*b^4 + 420*a^2*b^5 + 315*a*b^6)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x +
e)^2)*sin(f*x + e))/((a^10 + 2*a^9*b + a^8*b^2)*f*cos(f*x + e)^4 + 2*(a^9*b + 2*a^8*b^2 + a^7*b^3)*f*cos(f*x
+ e)^2 + (a^8*b^2 + 2*a^7*b^3 + a^6*b^4)*f)]
Sympy [F(-1)]
Timed out. \[
\int \frac {\cos ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out}
\]
[In]
integrate(cos(f*x+e)**6/(a+b*sec(f*x+e)**2)**(5/2),x)
[Out]
Timed out
Maxima [F]
\[
\int \frac {\cos ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{6}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x }
\]
[In]
integrate(cos(f*x+e)^6/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm="maxima")
[Out]
integrate(cos(f*x + e)^6/(b*sec(f*x + e)^2 + a)^(5/2), x)
Giac [F]
\[
\int \frac {\cos ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{6}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x }
\]
[In]
integrate(cos(f*x+e)^6/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm="giac")
[Out]
sage0*x
Mupad [F(-1)]
Timed out. \[
\int \frac {\cos ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^6}{{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{5/2}} \,d x
\]
[In]
int(cos(e + f*x)^6/(a + b/cos(e + f*x)^2)^(5/2),x)
[Out]
int(cos(e + f*x)^6/(a + b/cos(e + f*x)^2)^(5/2), x)