\(\int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx\) [549]

Optimal result

Integrand size = 35, antiderivative size = 333 \[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {(2 A-7 B) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{a^{7/2} d}-\frac {(177 A-637 B) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{64 \sqrt {2} a^{7/2} d}+\frac {(A-B) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)}+\frac {(3 A-7 B) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)}+\frac {(79 A-259 B) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {7 (7 A-27 B) \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \]

[Out]

1/6*(A-B)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(7/2)/sec(d*x+c)^(7/2)+1/16*(3*A-7*B)*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^
(5/2)/sec(d*x+c)^(5/2)+1/192*(79*A-259*B)*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c))^(3/2)/sec(d*x+c)^(3/2)-7/64*(7*A-2
7*B)*sin(d*x+c)/a^3/d/(a+a*cos(d*x+c))^(1/2)/sec(d*x+c)^(1/2)+(2*A-7*B)*arcsin(sin(d*x+c)*a^(1/2)/(a+a*cos(d*x
+c))^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/a^(7/2)/d-1/128*(177*A-637*B)*arctan(1/2*sin(d*x+c)*a^(1/2)*2^(1
/2)/cos(d*x+c)^(1/2)/(a+a*cos(d*x+c))^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/a^(7/2)/d*2^(1/2)

Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3040, 3056, 3062, 3061, 2861, 211, 2853, 222} \[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {(2 A-7 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{a^{7/2} d}-\frac {(177 A-637 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{64 \sqrt {2} a^{7/2} d}-\frac {7 (7 A-27 B) \sin (c+d x)}{64 a^3 d \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+a}}+\frac {(79 A-259 B) \sin (c+d x)}{192 a^2 d \sec ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac {(3 A-7 B) \sin (c+d x)}{16 a d \sec ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}+\frac {(A-B) \sin (c+d x)}{6 d \sec ^{\frac {7}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}} \]

[In]

Int[(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(7/2)),x]

[Out]

((2*A - 7*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a
^(7/2)*d) - ((177*A - 637*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]
])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Sin[c + d*x])/(6*d*(a + a*Cos[c +
d*x])^(7/2)*Sec[c + d*x]^(7/2)) + ((3*A - 7*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^
(5/2)) + ((79*A - 259*B)*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) - (7*(7*A - 2
7*B)*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 2853

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[-2/f, Su
bst[Int[1/Sqrt[1 - x^2/a], x], x, b*(Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, d, e, f}, x]
&& EqQ[a^2 - b^2, 0] && EqQ[d, a/b]

Rule 2861

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> D
ist[-2*(a/f), Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c +
d*Sin[e + f*x]]))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 -
d^2, 0]

Rule 3040

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(g*Csc[e + f*x])^p*(g*Sin[e + f*x])^p, Int[(a + b*Sin[e + f*x])^m*((
c + d*Sin[e + f*x])^n/(g*Sin[e + f*x])^p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d
, 0] &&  !IntegerQ[p] &&  !(IntegerQ[m] && IntegerQ[n])

Rule 3056

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

Rule 3061

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

Rule 3062

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

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx \\ & = \frac {(A-B) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (\frac {7}{2} a (A-B)-a (A-7 B) \cos (c+d x)\right )}{(a+a \cos (c+d x))^{5/2}} \, dx}{6 a^2} \\ & = \frac {(A-B) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)}+\frac {(3 A-7 B) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (\frac {15}{4} a^2 (3 A-7 B)-\frac {1}{2} a^2 (17 A-77 B) \cos (c+d x)\right )}{(a+a \cos (c+d x))^{3/2}} \, dx}{24 a^4} \\ & = \frac {(A-B) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)}+\frac {(3 A-7 B) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)}+\frac {(79 A-259 B) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\cos (c+d x)} \left (\frac {3}{8} a^3 (79 A-259 B)-\frac {21}{4} a^3 (7 A-27 B) \cos (c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx}{48 a^6} \\ & = \frac {(A-B) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)}+\frac {(3 A-7 B) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)}+\frac {(79 A-259 B) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {7 (7 A-27 B) \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {21}{8} a^4 (7 A-27 B)+24 a^4 (2 A-7 B) \cos (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{48 a^7} \\ & = \frac {(A-B) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)}+\frac {(3 A-7 B) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)}+\frac {(79 A-259 B) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {7 (7 A-27 B) \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}}-\frac {\left ((177 A-637 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{128 a^3}+\frac {\left ((2 A-7 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx}{2 a^4} \\ & = \frac {(A-B) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)}+\frac {(3 A-7 B) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)}+\frac {(79 A-259 B) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {7 (7 A-27 B) \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}}+\frac {\left ((177 A-637 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{2 a^2+a x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{64 a^2 d}-\frac {\left ((2 A-7 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}}} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{a^4 d} \\ & = \frac {(2 A-7 B) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{a^{7/2} d}-\frac {(177 A-637 B) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{64 \sqrt {2} a^{7/2} d}+\frac {(A-B) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)}+\frac {(3 A-7 B) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)}+\frac {(79 A-259 B) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {7 (7 A-27 B) \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.35 (sec) , antiderivative size = 1017, normalized size of antiderivative = 3.05 \[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {49 i A e^{-\frac {1}{2} i (c+d x)} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \text {arctanh}\left (\frac {1-e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right ) \cos ^7\left (\frac {c}{2}+\frac {d x}{2}\right )}{8 d (a (1+\cos (c+d x)))^{7/2}}+\frac {189 i B e^{-\frac {1}{2} i (c+d x)} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \text {arctanh}\left (\frac {1-e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right ) \cos ^7\left (\frac {c}{2}+\frac {d x}{2}\right )}{8 d (a (1+\cos (c+d x)))^{7/2}}+\frac {8 i \sqrt {2} A e^{-\frac {1}{2} i (c+d x)} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \left (-\text {arcsinh}\left (e^{i (c+d x)}\right )+\sqrt {2} \text {arctanh}\left (\frac {-1+e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right )+\text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )\right ) \cos ^7\left (\frac {c}{2}+\frac {d x}{2}\right )}{d (a (1+\cos (c+d x)))^{7/2}}-\frac {28 i \sqrt {2} B e^{-\frac {1}{2} i (c+d x)} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \left (-\text {arcsinh}\left (e^{i (c+d x)}\right )+\sqrt {2} \text {arctanh}\left (\frac {-1+e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right )+\text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )\right ) \cos ^7\left (\frac {c}{2}+\frac {d x}{2}\right )}{d (a (1+\cos (c+d x)))^{7/2}}+\frac {\cos ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\sec (c+d x)} \left (\frac {(-247 A+427 B) \cos \left (\frac {d x}{2}\right ) \sin \left (\frac {c}{2}\right )}{12 d}+\frac {8 B \cos \left (\frac {3 d x}{2}\right ) \sin \left (\frac {3 c}{2}\right )}{d}-\frac {(247 A-427 B) \cos \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )}{12 d}+\frac {\sec \left (\frac {c}{2}\right ) \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (379 A \sin \left (\frac {d x}{2}\right )-703 B \sin \left (\frac {d x}{2}\right )\right )}{24 d}+\frac {\sec \left (\frac {c}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A \sin \left (\frac {d x}{2}\right )-B \sin \left (\frac {d x}{2}\right )\right )}{3 d}+\frac {\sec \left (\frac {c}{2}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-41 A \sin \left (\frac {d x}{2}\right )+53 B \sin \left (\frac {d x}{2}\right )\right )}{12 d}+\frac {8 B \cos \left (\frac {3 c}{2}\right ) \sin \left (\frac {3 d x}{2}\right )}{d}+\frac {(379 A-703 B) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{24 d}-\frac {(41 A-53 B) \sec ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{12 d}+\frac {(A-B) \sec ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{3 d}\right )}{(a (1+\cos (c+d x)))^{7/2}} \]

[In]

Integrate[(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(7/2)),x]

[Out]

(((-49*I)/8)*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(1 - E^(I
*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])]*Cos[c/2 + (d*x)/2]^7)/(d*E^((I/2)*(c + d*x))*(a*(1 + Cos
[c + d*x]))^(7/2)) + (((189*I)/8)*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x
))]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])]*Cos[c/2 + (d*x)/2]^7)/(d*E^((I/2)*(
c + d*x))*(a*(1 + Cos[c + d*x]))^(7/2)) + ((8*I)*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqr
t[1 + E^((2*I)*(c + d*x))]*(-ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*ArcTanh[(-1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1
+ E^((2*I)*(c + d*x))])] + ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Cos[c/2 + (d*x)/2]^7)/(d*E^((I/2)*(c + d*x
))*(a*(1 + Cos[c + d*x]))^(7/2)) - ((28*I)*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 +
E^((2*I)*(c + d*x))]*(-ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*ArcTanh[(-1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^(
(2*I)*(c + d*x))])] + ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Cos[c/2 + (d*x)/2]^7)/(d*E^((I/2)*(c + d*x))*(a*
(1 + Cos[c + d*x]))^(7/2)) + (Cos[c/2 + (d*x)/2]^7*Sqrt[Sec[c + d*x]]*(((-247*A + 427*B)*Cos[(d*x)/2]*Sin[c/2]
)/(12*d) + (8*B*Cos[(3*d*x)/2]*Sin[(3*c)/2])/d - ((247*A - 427*B)*Cos[c/2]*Sin[(d*x)/2])/(12*d) + (Sec[c/2]*Se
c[c/2 + (d*x)/2]^2*(379*A*Sin[(d*x)/2] - 703*B*Sin[(d*x)/2]))/(24*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^6*(A*Sin[(
d*x)/2] - B*Sin[(d*x)/2]))/(3*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^4*(-41*A*Sin[(d*x)/2] + 53*B*Sin[(d*x)/2]))/(1
2*d) + (8*B*Cos[(3*c)/2]*Sin[(3*d*x)/2])/d + ((379*A - 703*B)*Sec[c/2 + (d*x)/2]*Tan[c/2])/(24*d) - ((41*A - 5
3*B)*Sec[c/2 + (d*x)/2]^3*Tan[c/2])/(12*d) + ((A - B)*Sec[c/2 + (d*x)/2]^5*Tan[c/2])/(3*d)))/(a*(1 + Cos[c + d
*x]))^(7/2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(810\) vs. \(2(282)=564\).

Time = 20.98 (sec) , antiderivative size = 811, normalized size of antiderivative = 2.44

method	result	size
default	\(\text {Expression too large to display}\)	\(811\)
parts	\(\text {Expression too large to display}\)	\(843\)

[In]

int((A+B*cos(d*x+c))/(a+cos(d*x+c)*a)^(7/2)/sec(d*x+c)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/384/a^4/d*(a*(1+cos(d*x+c)))^(1/2)/(1+cos(d*x+c))^4/sec(d*x+c)^(7/2)/(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(-19
2*B*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+247*tan(d*x+c)*A*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^
(1/2)-384*A*2^(1/2)*arctan(tan(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-1099*tan(d*x+c)*B*2^(1/2)*(cos(d*x+c)
/(1+cos(d*x+c)))^(1/2)+1344*B*2^(1/2)*arctan(tan(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+362*tan(d*x+c)*sec(
d*x+c)*A*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)-531*A*arcsin(cot(d*x+c)-csc(d*x+c))-1152*sec(d*x+c)*A*2^(1/
2)*arctan(tan(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-1442*tan(d*x+c)*sec(d*x+c)*B*2^(1/2)*(cos(d*x+c)/(1+co
s(d*x+c)))^(1/2)+1911*B*arcsin(cot(d*x+c)-csc(d*x+c))+4032*sec(d*x+c)*B*2^(1/2)*arctan(tan(d*x+c)*(cos(d*x+c)/
(1+cos(d*x+c)))^(1/2))+147*tan(d*x+c)*sec(d*x+c)^2*A*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)-1593*sec(d*x+c)
*A*arcsin(cot(d*x+c)-csc(d*x+c))-1152*sec(d*x+c)^2*A*2^(1/2)*arctan(tan(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/
2))-567*tan(d*x+c)*sec(d*x+c)^2*B*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+5733*sec(d*x+c)*B*arcsin(cot(d*x+c
)-csc(d*x+c))+4032*sec(d*x+c)^2*B*2^(1/2)*arctan(tan(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-1593*sec(d*x+c)
^2*A*arcsin(cot(d*x+c)-csc(d*x+c))-384*sec(d*x+c)^3*A*2^(1/2)*arctan(tan(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1
/2))+5733*sec(d*x+c)^2*B*arcsin(cot(d*x+c)-csc(d*x+c))+1344*sec(d*x+c)^3*B*2^(1/2)*arctan(tan(d*x+c)*(cos(d*x+
c)/(1+cos(d*x+c)))^(1/2))-531*sec(d*x+c)^3*A*arcsin(cot(d*x+c)-csc(d*x+c))+1911*sec(d*x+c)^3*B*arcsin(cot(d*x+
c)-csc(d*x+c)))*2^(1/2)

Fricas [A] (verification not implemented)

none

Time = 13.73 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.14 \[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {3 \, \sqrt {2} {\left ({\left (177 \, A - 637 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (177 \, A - 637 \, B\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (177 \, A - 637 \, B\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (177 \, A - 637 \, B\right )} \cos \left (d x + c\right ) + 177 \, A - 637 \, B\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) - 384 \, {\left ({\left (2 \, A - 7 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (2 \, A - 7 \, B\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (2 \, A - 7 \, B\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, A - 7 \, B\right )} \cos \left (d x + c\right ) + 2 \, A - 7 \, B\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) + \frac {2 \, {\left (192 \, B \cos \left (d x + c\right )^{4} - {\left (247 \, A - 1099 \, B\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (181 \, A - 721 \, B\right )} \cos \left (d x + c\right )^{2} - 21 \, {\left (7 \, A - 27 \, B\right )} \cos \left (d x + c\right )\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{384 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]

[In]

integrate((A+B*cos(d*x+c))/(a+a*cos(d*x+c))^(7/2)/sec(d*x+c)^(7/2),x, algorithm="fricas")

[Out]

1/384*(3*sqrt(2)*((177*A - 637*B)*cos(d*x + c)^4 + 4*(177*A - 637*B)*cos(d*x + c)^3 + 6*(177*A - 637*B)*cos(d*
x + c)^2 + 4*(177*A - 637*B)*cos(d*x + c) + 177*A - 637*B)*sqrt(a)*arctan(sqrt(2)*sqrt(a*cos(d*x + c) + a)*sqr
t(cos(d*x + c))/(sqrt(a)*sin(d*x + c))) - 384*((2*A - 7*B)*cos(d*x + c)^4 + 4*(2*A - 7*B)*cos(d*x + c)^3 + 6*(
2*A - 7*B)*cos(d*x + c)^2 + 4*(2*A - 7*B)*cos(d*x + c) + 2*A - 7*B)*sqrt(a)*arctan(sqrt(a*cos(d*x + c) + a)*sq
rt(cos(d*x + c))/(sqrt(a)*sin(d*x + c))) + 2*(192*B*cos(d*x + c)^4 - (247*A - 1099*B)*cos(d*x + c)^3 - 2*(181*
A - 721*B)*cos(d*x + c)^2 - 21*(7*A - 27*B)*cos(d*x + c))*sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/sqrt(cos(d*x +
c)))/(a^4*d*cos(d*x + c)^4 + 4*a^4*d*cos(d*x + c)^3 + 6*a^4*d*cos(d*x + c)^2 + 4*a^4*d*cos(d*x + c) + a^4*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((A+B*cos(d*x+c))/(a+a*cos(d*x+c))**(7/2)/sec(d*x+c)**(7/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {B \cos \left (d x + c\right ) + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]

[In]

integrate((A+B*cos(d*x+c))/(a+a*cos(d*x+c))^(7/2)/sec(d*x+c)^(7/2),x, algorithm="maxima")

[Out]

integrate((B*cos(d*x + c) + A)/((a*cos(d*x + c) + a)^(7/2)*sec(d*x + c)^(7/2)), x)

Giac [F(-1)]

Timed out. \[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((A+B*cos(d*x+c))/(a+a*cos(d*x+c))^(7/2)/sec(d*x+c)^(7/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {A+B\,\cos \left (c+d\,x\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]

[In]

int((A + B*cos(c + d*x))/((1/cos(c + d*x))^(7/2)*(a + a*cos(c + d*x))^(7/2)),x)

[Out]

int((A + B*cos(c + d*x))/((1/cos(c + d*x))^(7/2)*(a + a*cos(c + d*x))^(7/2)), x)