\(\int \frac {\cos ^2(e+f x)}{(a+b \sec ^2(e+f x))^{5/2}} \, dx\) [293]
Optimal result
Integrand size = 25, antiderivative size = 187 \[
\int \frac {\cos ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\frac {(a-5 b) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{2 a^{7/2} f}+\frac {\cos (e+f x) \sin (e+f x)}{2 a f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b (3 a+5 b) \tan (e+f x)}{6 a^2 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (3 a^2+22 a b+15 b^2\right ) \tan (e+f x)}{6 a^3 (a+b)^2 f \sqrt {a+b+b \tan ^2(e+f x)}}
\]
[Out]
1/2*(a-5*b)*arctan(a^(1/2)*tan(f*x+e)/(a+b+b*tan(f*x+e)^2)^(1/2))/a^(7/2)/f+1/6*b*(3*a^2+22*a*b+15*b^2)*tan(f*
x+e)/a^3/(a+b)^2/f/(a+b+b*tan(f*x+e)^2)^(1/2)+1/2*cos(f*x+e)*sin(f*x+e)/a/f/(a+b+b*tan(f*x+e)^2)^(3/2)+1/6*b*(
3*a+5*b)*tan(f*x+e)/a^2/(a+b)/f/(a+b+b*tan(f*x+e)^2)^(3/2)
Rubi [A] (verified)
Time = 0.28 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of
steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {4231, 425, 541, 12, 385, 209}
\[
\int \frac {\cos ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\frac {(a-5 b) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)+b}}\right )}{2 a^{7/2} f}+\frac {b (3 a+5 b) \tan (e+f x)}{6 a^2 f (a+b) \left (a+b \tan ^2(e+f x)+b\right )^{3/2}}+\frac {b \left (3 a^2+22 a b+15 b^2\right ) \tan (e+f x)}{6 a^3 f (a+b)^2 \sqrt {a+b \tan ^2(e+f x)+b}}+\frac {\sin (e+f x) \cos (e+f x)}{2 a f \left (a+b \tan ^2(e+f x)+b\right )^{3/2}}
\]
[In]
Int[Cos[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(5/2),x]
[Out]
((a - 5*b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*a^(7/2)*f) + (Cos[e + f*x]*Sin[e
+ f*x])/(2*a*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (b*(3*a + 5*b)*Tan[e + f*x])/(6*a^2*(a + b)*f*(a + b + b*Ta
n[e + f*x]^2)^(3/2)) + (b*(3*a^2 + 22*a*b + 15*b^2)*Tan[e + f*x])/(6*a^3*(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 )^2 \left (a+b+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\cos (e+f x) \sin (e+f x)}{2 a f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {-a+b-4 b x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{2 a f} \\ & = \frac {\cos (e+f x) \sin (e+f x)}{2 a f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b (3 a+5 b) \tan (e+f x)}{6 a^2 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {-3 a^2+6 a b+5 b^2-2 b (3 a+5 b) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{6 a^2 (a+b) f} \\ & = \frac {\cos (e+f x) \sin (e+f x)}{2 a f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b (3 a+5 b) \tan (e+f x)}{6 a^2 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (3 a^2+22 a b+15 b^2\right ) \tan (e+f x)}{6 a^3 (a+b)^2 f \sqrt {a+b+b \tan ^2(e+f x)}}-\frac {\text {Subst}\left (\int -\frac {3 (a-5 b) (a+b)^2}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{6 a^3 (a+b)^2 f} \\ & = \frac {\cos (e+f x) \sin (e+f x)}{2 a f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b (3 a+5 b) \tan (e+f x)}{6 a^2 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (3 a^2+22 a b+15 b^2\right ) \tan (e+f x)}{6 a^3 (a+b)^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {(a-5 b) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{2 a^3 f} \\ & = \frac {\cos (e+f x) \sin (e+f x)}{2 a f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b (3 a+5 b) \tan (e+f x)}{6 a^2 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (3 a^2+22 a b+15 b^2\right ) \tan (e+f x)}{6 a^3 (a+b)^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {(a-5 b) \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 )}{2 a^3 f} \\ & = \frac {(a-5 b) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{2 a^{7/2} f}+\frac {\cos (e+f x) \sin (e+f x)}{2 a f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b (3 a+5 b) \tan (e+f x)}{6 a^2 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (3 a^2+22 a b+15 b^2\right ) \tan (e+f x)}{6 a^3 (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 = 16.74 (sec) , antiderivative size = 1775, normalized size of antiderivative = 9.49
\[
\int \frac {\cos ^2(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},-3,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^8(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},-3,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+\left (5 a \operatorname {AppellF1}\left (\frac {3}{2},-3,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-6 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-2,\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},-3,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^7(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},-3,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+\left (5 a \operatorname {AppellF1}\left (\frac {3}{2},-3,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-6 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-2,\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},-3,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^7(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},-3,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+\left (5 a \operatorname {AppellF1}\left (\frac {3}{2},-3,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-6 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-2,\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 {9 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-3,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^5(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},-3,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+\left (5 a \operatorname {AppellF1}\left (\frac {3}{2},-3,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-6 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-2,\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 ^6(e+f x) \sin (e+f x) \left (\frac {5 a f \operatorname {AppellF1}\left (\frac {3}{2},-3,\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)}-2 f \operatorname {AppellF1}\left (\frac {3}{2},-2,\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},-3,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+\left (5 a \operatorname {AppellF1}\left (\frac {3}{2},-3,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-6 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-2,\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},-3,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^6(e+f x) \sin (e+f x) \left (2 f \left (5 a \operatorname {AppellF1}\left (\frac {3}{2},-3,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-6 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-2,\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},-3,\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)}-2 f \operatorname {AppellF1}\left (\frac {3}{2},-2,\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 )+\sin ^2(e+f x) \left (5 a \left (\frac {21 a f \operatorname {AppellF1}\left (\frac {5}{2},-3,\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)}-\frac {18}{5} f \operatorname {AppellF1}\left (\frac {5}{2},-2,\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 )-6 (a+b) \left (\frac {3 a f \operatorname {AppellF1}\left (\frac {5}{2},-2,\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 {12}{5} f \operatorname {AppellF1}\left (\frac {5}{2},-1,\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},-3,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+\left (5 a \operatorname {AppellF1}\left (\frac {3}{2},-3,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-6 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-2,\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]^2/(a + b*Sec[e + f*x]^2)^(5/2),x]
[Out]
(3*(a + b)*AppellF1[1/2, -3, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^8*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, -3, 5/2,
3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -3, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e
+ f*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 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, -3, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^
7*Sin[e + f*x]^2)/(4*Sqrt[2]*(a + b - a*Sin[e + f*x]^2)^(7/2)*(3*(a + b)*AppellF1[1/2, -3, 5/2, 3/2, Sin[e + f
*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -3, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a +
b)] - 6*(a + b)*AppellF1[3/2, -2, 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, -3, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^7)/(4*Sqrt[2]*(a
+ b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -3, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a +
b)] + (5*a*AppellF1[3/2, -3, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -
2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (9*(a + b)*AppellF1[1/2, -3, 5/2,
3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^5*Sin[e + f*x]^2)/(2*Sqrt[2]*(a + b - a*Sin[e +
f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -3, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*Appel
lF1[3/2, -3, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 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]^6*Sin[e + f*x]*((5*a*f*Ap
pellF1[3/2, -3, 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)) -
2*f*AppellF1[3/2, -2, 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]))/(4*Sq
rt[2]*f*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -3, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x
]^2)/(a + b)] + (5*a*AppellF1[3/2, -3, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 6*(a + b)*Appel
lF1[3/2, -2, 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
, -3, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^6*Sin[e + f*x]*(2*f*(5*a*AppellF1[3/2
, -3, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 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, -3, 7/2, 5/2, S
in[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)) - 2*f*AppellF1[3/2, -2, 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]) + Sin[e + f*x]^2*(5*a*((21*a*f*Ap
pellF1[5/2, -3, 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)) -
(18*f*AppellF1[5/2, -2, 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])/5) -
6*(a + b)*((3*a*f*AppellF1[5/2, -2, 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) - (12*f*AppellF1[5/2, -1, 5/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*S
in[e + f*x])/5))))/(4*Sqrt[2]*f*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -3, 5/2, 3/2, Sin[e
+ f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -3, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a
+ b)] - 6*(a + b)*AppellF1[3/2, -2, 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. \(2142\) vs. \(2(167)=334\).
Time = 5.35 (sec) , antiderivative size = 2143, normalized size of antiderivative =
11.46
| | |
| method | result | size |
| | |
| default |
\(\text {Expression too large to display}\) |
\(2143\) |
| | |
|
|
|
|
[In]
int(cos(f*x+e)^2/(a+b*sec(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)
[Out]
1/6/f/(a+b)^2/a^3/(-a)^(1/2)*(b+a*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*b^3*cos(f*x+e)^2+6*(-a)^(1/2)*a^3*b*cos(f*x+e)^2*sin(f*x+e)+30*(-a)^(1/2)*a^2*b^2*cos(f
*x+e)^2*sin(f*x+e)+20*(-a)^(1/2)*a*b^3*cos(f*x+e)^2*sin(f*x+e)+3*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*l
n(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*cos(f*x+e)-9*((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^2*cos(f*x+e)-27*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*ln(4*(-a)^(1/2)*((b+a*co
s(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^3*cos(f*x+e)+6*(-a)^(1/2)*a^3*b*cos(f*x+e)^4*sin(f*x+e)+3*(-a)^(1/2)*a^2*b^2*cos(f*x+e)^4*sin(f*
x+e)-9*((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*cos(f*x+e)^3-27*((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^2*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*b^3*cos(f*x+e)^3-9*((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*cos(f*x+e)^2+15*(-a)^(1/2)*b^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)*b^4+3*(-a)^(1/2)*a^2*b^2*sin(f*x+e)
+22*(-a)^(1/2)*a*b^3*sin(f*x+e)+3*((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*cos(f*x+e)^3+3*((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*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)*b^4*cos(f*x+e)+3*((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-9*((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^2-27*((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^3+3*(-a)^(1/2)*a^4*cos(f*x+e)^4*sin(f*x+e)-27*((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^2*cos(f*x+e)^2)/(a+b*sec(f*x+e)^2)^(5/2)*sec(f*x+e)^5
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (167) = 334\).
Time = 1.78 (sec) , antiderivative size = 1023, normalized size of antiderivative = 5.47
\[
\int \frac {\cos ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display}
\]
[In]
integrate(cos(f*x+e)^2/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm="fricas")
[Out]
[1/48*(3*(a^3*b^2 - 3*a^2*b^3 - 9*a*b^4 - 5*b^5 + (a^5 - 3*a^4*b - 9*a^3*b^2 - 5*a^2*b^3)*cos(f*x + e)^4 + 2*(
a^4*b - 3*a^3*b^2 - 9*a^2*b^3 - 5*a*b^4)*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 - 14*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)*co
s(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))*s
qrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(3*(a^5 + 2*a^4*b + a^3*b^2)*cos(f*x + e
)^5 + 2*(3*a^4*b + 15*a^3*b^2 + 10*a^2*b^3)*cos(f*x + e)^3 + (3*a^3*b^2 + 22*a^2*b^3 + 15*a*b^4)*cos(f*x + e))
*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^8 + 2*a^7*b + a^6*b^2)*f*cos(f*x + e)^4 + 2*(a^
7*b + 2*a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^2 + (a^6*b^2 + 2*a^5*b^3 + a^4*b^4)*f), -1/24*(3*(a^3*b^2 - 3*a^2*b^
3 - 9*a*b^4 - 5*b^5 + (a^5 - 3*a^4*b - 9*a^3*b^2 - 5*a^2*b^3)*cos(f*x + e)^4 + 2*(a^4*b - 3*a^3*b^2 - 9*a^2*b^
3 - 5*a*b^4)*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*(3*(a^5 + 2*a^4*b + a^3*b^2)*cos(f*x + e)^5 + 2*(
3*a^4*b + 15*a^3*b^2 + 10*a^2*b^3)*cos(f*x + e)^3 + (3*a^3*b^2 + 22*a^2*b^3 + 15*a*b^4)*cos(f*x + e))*sqrt((a*
cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^8 + 2*a^7*b + a^6*b^2)*f*cos(f*x + e)^4 + 2*(a^7*b + 2*a
^6*b^2 + a^5*b^3)*f*cos(f*x + e)^2 + (a^6*b^2 + 2*a^5*b^3 + a^4*b^4)*f)]
Sympy [F]
\[
\int \frac {\cos ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cos ^{2}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx
\]
[In]
integrate(cos(f*x+e)**2/(a+b*sec(f*x+e)**2)**(5/2),x)
[Out]
Integral(cos(e + f*x)**2/(a + b*sec(e + f*x)**2)**(5/2), x)
Maxima [F]
\[
\int \frac {\cos ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{2}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x }
\]
[In]
integrate(cos(f*x+e)^2/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm="maxima")
[Out]
integrate(cos(f*x + e)^2/(b*sec(f*x + e)^2 + a)^(5/2), x)
Giac [F]
\[
\int \frac {\cos ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{2}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x }
\]
[In]
integrate(cos(f*x+e)^2/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm="giac")
[Out]
sage0*x
Mupad [F(-1)]
Timed out. \[
\int \frac {\cos ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^2}{{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{5/2}} \,d x
\]
[In]
int(cos(e + f*x)^2/(a + b/cos(e + f*x)^2)^(5/2),x)
[Out]
int(cos(e + f*x)^2/(a + b/cos(e + f*x)^2)^(5/2), x)