\(\int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx\) [208]

Optimal result

Integrand size = 35, antiderivative size = 293 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx=\frac {(2 A-7 B) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{a^{7/2} d}-\frac {(177 A-637 B) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{64 \sqrt {2} a^{7/2} d}+\frac {(A-B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(3 A-7 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}+\frac {(79 A-259 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}-\frac {7 (7 A-27 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}} \]

[Out]

(2*A-7*B)*arcsin(sin(d*x+c)*a^(1/2)/(a+a*cos(d*x+c))^(1/2))/a^(7/2)/d+1/6*(A-B)*cos(d*x+c)^(7/2)*sin(d*x+c)/d/
(a+a*cos(d*x+c))^(7/2)+1/16*(3*A-7*B)*cos(d*x+c)^(5/2)*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^(5/2)+1/192*(79*A-259*B
)*cos(d*x+c)^(3/2)*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c))^(3/2)-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))/a^(7/2)/d*2^(1/2)-7/64*(7*A-27*B)*sin(d*x+c)*cos(d*x+c)^(1/2)/
a^3/d/(a+a*cos(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 1.66 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3056, 3062, 3061, 2861, 211, 2853, 222} \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx=\frac {(2 A-7 B) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{a^{7/2} d}-\frac {(177 A-637 B) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{64 \sqrt {2} a^{7/2} d}-\frac {7 (7 A-27 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{64 a^3 d \sqrt {a \cos (c+d x)+a}}+\frac {(79 A-259 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}+\frac {(3 A-7 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{5/2}} \]

[In]

Int[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[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]]])/(a^(7/2)*d) - ((177*A - 637*B)*ArcTan[(S
qrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)
*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + ((3*A - 7*B)*Cos[c + d*x]^(5/2)*Sin[c + d
*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) + ((79*A - 259*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(192*a^2*d*(a + a*
Cos[c + d*x])^(3/2)) - (7*(7*A - 27*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[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 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}& = \frac {(A-B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {\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) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(3 A-7 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}+\frac {\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) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(3 A-7 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}+\frac {(79 A-259 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {\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) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(3 A-7 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}+\frac {(79 A-259 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}-\frac {7 (7 A-27 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}+\frac {\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) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(3 A-7 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}+\frac {(79 A-259 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}-\frac {7 (7 A-27 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}-\frac {(177 A-637 B) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{128 a^3}+\frac {(2 A-7 B) \int \frac {\sqrt {a+a \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx}{2 a^4} \\ & = \frac {(A-B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(3 A-7 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}+\frac {(79 A-259 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}-\frac {7 (7 A-27 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}+\frac {(177 A-637 B) \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 {(2 A-7 B) \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 )}{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 )}{64 \sqrt {2} a^{7/2} d}+\frac {(A-B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(3 A-7 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}+\frac {(79 A-259 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}-\frac {7 (7 A-27 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 2.94 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.29 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx=\frac {\sqrt {a (1+\cos (c+d x))} \left (-336 (7 A-27 B) \arcsin \left (\sqrt {1-\cos (c+d x)}\right ) \cos ^6\left (\frac {1}{2} (c+d x)\right )-8496 A \arcsin \left (\sqrt {\cos (c+d x)}\right ) \cos ^6\left (\frac {1}{2} (c+d x)\right )+30576 B \arcsin \left (\sqrt {\cos (c+d x)}\right ) \cos ^6\left (\frac {1}{2} (c+d x)\right )+24 \sqrt {2} (177 A-637 B) \arctan \left (\frac {\sqrt {\cos (c+d x)}}{\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}}\right ) \cos ^6\left (\frac {1}{2} (c+d x)\right )-724 A \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)+2884 B \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)-494 A \sqrt {1-\cos (c+d x)} \cos ^{\frac {5}{2}}(c+d x)+2198 B \sqrt {1-\cos (c+d x)} \cos ^{\frac {5}{2}}(c+d x)+384 B \sqrt {1-\cos (c+d x)} \cos ^{\frac {7}{2}}(c+d x)-294 A \sqrt {-((-1+\cos (c+d x)) \cos (c+d x))}+1134 B \sqrt {-((-1+\cos (c+d x)) \cos (c+d x))}\right ) \sin (c+d x)}{384 a^4 d \sqrt {1-\cos (c+d x)} (1+\cos (c+d x))^4} \]

[In]

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

[Out]

(Sqrt[a*(1 + Cos[c + d*x])]*(-336*(7*A - 27*B)*ArcSin[Sqrt[1 - Cos[c + d*x]]]*Cos[(c + d*x)/2]^6 - 8496*A*ArcS
in[Sqrt[Cos[c + d*x]]]*Cos[(c + d*x)/2]^6 + 30576*B*ArcSin[Sqrt[Cos[c + d*x]]]*Cos[(c + d*x)/2]^6 + 24*Sqrt[2]
*(177*A - 637*B)*ArcTan[Sqrt[Cos[c + d*x]]/Sqrt[Sin[(c + d*x)/2]^2]]*Cos[(c + d*x)/2]^6 - 724*A*Sqrt[1 - Cos[c
+ d*x]]*Cos[c + d*x]^(3/2) + 2884*B*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2) - 494*A*Sqrt[1 - Cos[c + d*x]]*
Cos[c + d*x]^(5/2) + 2198*B*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(5/2) + 384*B*Sqrt[1 - Cos[c + d*x]]*Cos[c + d
*x]^(7/2) - 294*A*Sqrt[-((-1 + Cos[c + d*x])*Cos[c + d*x])] + 1134*B*Sqrt[-((-1 + Cos[c + d*x])*Cos[c + d*x])]
)*Sin[c + d*x])/(384*a^4*d*Sqrt[1 - Cos[c + d*x]]*(1 + Cos[c + d*x])^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(794\) vs. \(2(250)=500\).

Time = 17.28 (sec) , antiderivative size = 795, normalized size of antiderivative = 2.71

method	result	size
default	\(\text {Expression too large to display}\)	\(795\)
parts	\(\text {Expression too large to display}\)	\(828\)

[In]

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

[Out]

1/384/a^4/d*(531*A*2^(1/2)*arcsin(cot(d*x+c)-csc(d*x+c))*cos(d*x+c)^3-1911*B*2^(1/2)*arcsin(cot(d*x+c)-csc(d*x
+c))*cos(d*x+c)^3+384*B*cos(d*x+c)^3*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+1593*A*2^(1/2)*arcsin(cot(d*
x+c)-csc(d*x+c))*cos(d*x+c)^2+768*A*arctan(tan(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2))*cos(d*x+c)^3-494*A*co
s(d*x+c)^2*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)-5733*B*2^(1/2)*arcsin(cot(d*x+c)-csc(d*x+c))*cos(d*x+c
)^2-2688*B*arctan(tan(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2))*cos(d*x+c)^3+2198*B*cos(d*x+c)^2*sin(d*x+c)*(c
os(d*x+c)/(1+cos(d*x+c)))^(1/2)+1593*A*2^(1/2)*arcsin(cot(d*x+c)-csc(d*x+c))*cos(d*x+c)+2304*A*cos(d*x+c)^2*ar
ctan(tan(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-724*A*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/
2)-5733*B*2^(1/2)*arcsin(cot(d*x+c)-csc(d*x+c))*cos(d*x+c)-8064*B*cos(d*x+c)^2*arctan(tan(d*x+c)*(cos(d*x+c)/(
1+cos(d*x+c)))^(1/2))+2884*B*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+531*A*2^(1/2)*arcsin(cot(
d*x+c)-csc(d*x+c))+2304*A*cos(d*x+c)*arctan(tan(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-294*A*(cos(d*x+c)/(1
+cos(d*x+c)))^(1/2)*sin(d*x+c)-1911*B*2^(1/2)*arcsin(cot(d*x+c)-csc(d*x+c))-8064*B*cos(d*x+c)*arctan(tan(d*x+c
)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+1134*B*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+768*A*arctan(tan(d*x+
c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-2688*B*arctan(tan(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)))*(a*(1+cos(d
*x+c)))^(1/2)*cos(d*x+c)^(1/2)/(1+cos(d*x+c))^4/(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 14.79 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.26 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx=\frac {3 \, \sqrt {2} {\left ({\left (177 \, A - 637 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (177 \, A - 637 \, B\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (177 \, A - 637 \, B\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (177 \, A - 637 \, B\right )} \cos \left (d x + c\right ) + 177 \, A - 637 \, B\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) + 2 \, {\left (192 \, B \cos \left (d x + c\right )^{3} - {\left (247 \, A - 1099 \, B\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (181 \, A - 721 \, B\right )} \cos \left (d x + c\right ) - 147 \, A + 567 \, B\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 384 \, {\left ({\left (2 \, A - 7 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (2 \, A - 7 \, B\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (2 \, A - 7 \, B\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, A - 7 \, B\right )} \cos \left (d x + c\right ) + 2 \, A - 7 \, B\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right )}{384 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]

[In]

integrate(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c))/(a+a*cos(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))) + 2*(192*B*cos(d*x + c)^3 - (247*A - 1099*B)*cos(d*x + c)^2 - 2*(181*A
- 721*B)*cos(d*x + c) - 147*A + 567*B)*sqrt(a*cos(d*x + c) + a)*sqrt(cos(d*x + c))*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)*sqrt(cos(d*x + c))/(sqrt(a)*sin(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. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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