3.5.59 \(\int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^2} \, dx\) [459]

Optimal. Leaf size=266 \[ -\frac {a \left (4 a^2+b^2\right ) x}{b^5}+\frac {2 a^4 \left (4 a^2-5 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} b^5 (a+b)^{3/2} d}+\frac {\left (12 a^4-7 a^2 b^2-2 b^4\right ) \sin (c+d x)}{3 b^4 \left (a^2-b^2\right ) d}-\frac {a \left (2 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac {\left (4 a^2-b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))} \]

[Out]

-a*(4*a^2+b^2)*x/b^5+2*a^4*(4*a^2-5*b^2)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(3/2)/b^5/(a
+b)^(3/2)/d+1/3*(12*a^4-7*a^2*b^2-2*b^4)*sin(d*x+c)/b^4/(a^2-b^2)/d-a*(2*a^2-b^2)*cos(d*x+c)*sin(d*x+c)/b^3/(a
^2-b^2)/d+1/3*(4*a^2-b^2)*cos(d*x+c)^2*sin(d*x+c)/b^2/(a^2-b^2)/d-a^2*cos(d*x+c)^3*sin(d*x+c)/b/(a^2-b^2)/d/(a
+b*cos(d*x+c))

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Rubi [A]
time = 0.48, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2871, 3128, 3102, 2814, 2738, 211} \begin {gather*} -\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {\left (4 a^2-b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d \left (a^2-b^2\right )}-\frac {a x \left (4 a^2+b^2\right )}{b^5}-\frac {a \left (2 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b^3 d \left (a^2-b^2\right )}+\frac {2 a^4 \left (4 a^2-5 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}+\frac {\left (12 a^4-7 a^2 b^2-2 b^4\right ) \sin (c+d x)}{3 b^4 d \left (a^2-b^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5/(a + b*Cos[c + d*x])^2,x]

[Out]

-((a*(4*a^2 + b^2)*x)/b^5) + (2*a^4*(4*a^2 - 5*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a -
b)^(3/2)*b^5*(a + b)^(3/2)*d) + ((12*a^4 - 7*a^2*b^2 - 2*b^4)*Sin[c + d*x])/(3*b^4*(a^2 - b^2)*d) - (a*(2*a^2
- b^2)*Cos[c + d*x]*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((4*a^2 - b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^
2 - b^2)*d) - (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*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 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 2814

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

Rule 2871

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/
(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e
+ f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 +
c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 +
d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 3128

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

Rubi steps

\begin {align*} \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^2} \, dx &=-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\int \frac {\cos ^2(c+d x) \left (3 a^2-a b \cos (c+d x)-\left (4 a^2-b^2\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{b \left (a^2-b^2\right )}\\ &=\frac {\left (4 a^2-b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (-2 a \left (4 a^2-b^2\right )+b \left (a^2+2 b^2\right ) \cos (c+d x)+6 a \left (2 a^2-b^2\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{3 b^2 \left (a^2-b^2\right )}\\ &=-\frac {a \left (2 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac {\left (4 a^2-b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\int \frac {6 a^2 \left (2 a^2-b^2\right )-2 a b \left (2 a^2+b^2\right ) \cos (c+d x)-2 \left (12 a^4-7 a^2 b^2-2 b^4\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^3 \left (a^2-b^2\right )}\\ &=\frac {\left (12 a^4-7 a^2 b^2-2 b^4\right ) \sin (c+d x)}{3 b^4 \left (a^2-b^2\right ) d}-\frac {a \left (2 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac {\left (4 a^2-b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\int \frac {6 a^2 b \left (2 a^2-b^2\right )+6 a \left (a^2-b^2\right ) \left (4 a^2+b^2\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^4 \left (a^2-b^2\right )}\\ &=-\frac {a \left (4 a^2+b^2\right ) x}{b^5}+\frac {\left (12 a^4-7 a^2 b^2-2 b^4\right ) \sin (c+d x)}{3 b^4 \left (a^2-b^2\right ) d}-\frac {a \left (2 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac {\left (4 a^2-b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\left (a^4 \left (4 a^2-5 b^2\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{b^5 \left (a^2-b^2\right )}\\ &=-\frac {a \left (4 a^2+b^2\right ) x}{b^5}+\frac {\left (12 a^4-7 a^2 b^2-2 b^4\right ) \sin (c+d x)}{3 b^4 \left (a^2-b^2\right ) d}-\frac {a \left (2 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac {\left (4 a^2-b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\left (2 a^4 \left (4 a^2-5 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^5 \left (a^2-b^2\right ) d}\\ &=-\frac {a \left (4 a^2+b^2\right ) x}{b^5}+\frac {2 a^4 \left (4 a^2-5 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} b^5 (a+b)^{3/2} d}+\frac {\left (12 a^4-7 a^2 b^2-2 b^4\right ) \sin (c+d x)}{3 b^4 \left (a^2-b^2\right ) d}-\frac {a \left (2 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac {\left (4 a^2-b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.92, size = 176, normalized size = 0.66 \begin {gather*} \frac {-12 a (2 a-i b) (2 a+i b) (c+d x)+\frac {24 a^4 \left (4 a^2-5 b^2\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{3/2}}+9 b \left (4 a^2+b^2\right ) \sin (c+d x)+\frac {12 a^5 b \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}-6 a b^2 \sin (2 (c+d x))+b^3 \sin (3 (c+d x))}{12 b^5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5/(a + b*Cos[c + d*x])^2,x]

[Out]

(-12*a*(2*a - I*b)*(2*a + I*b)*(c + d*x) + (24*a^4*(4*a^2 - 5*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^
2 + b^2]])/(-a^2 + b^2)^(3/2) + 9*b*(4*a^2 + b^2)*Sin[c + d*x] + (12*a^5*b*Sin[c + d*x])/((a - b)*(a + b)*(a +
b*Cos[c + d*x])) - 6*a*b^2*Sin[2*(c + d*x)] + b^3*Sin[3*(c + d*x)])/(12*b^5*d)

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Maple [A]
time = 0.40, size = 257, normalized size = 0.97

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {\left (-3 a^{2} b -b^{2} a -b^{3}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-6 a^{2} b -\frac {2}{3} b^{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 a^{2} b +b^{2} a -b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+a \left (4 a^{2}+b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{b^{5}}+\frac {2 a^{4} \left (\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}+\frac {\left (4 a^{2}-5 b^{2}\right ) \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}}{d}\)	\(257\)
default	\(\frac {-\frac {2 \left (\frac {\left (-3 a^{2} b -b^{2} a -b^{3}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-6 a^{2} b -\frac {2}{3} b^{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 a^{2} b +b^{2} a -b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+a \left (4 a^{2}+b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{b^{5}}+\frac {2 a^{4} \left (\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}+\frac {\left (4 a^{2}-5 b^{2}\right ) \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}}{d}\)	\(257\)
risch	\(-\frac {4 a^{3} x}{b^{5}}-\frac {a x}{b^{3}}+\frac {i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{4 b^{3} d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a^{2}}{2 b^{4} d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )}}{8 b^{2} d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a^{2}}{2 b^{4} d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )}}{8 b^{2} d}-\frac {i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{4 b^{3} d}+\frac {2 i a^{5} \left (a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}{b^{5} \left (a^{2}-b^{2}\right ) d \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}-\frac {4 a^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{5}}+\frac {5 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{3}}+\frac {4 a^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{5}}-\frac {5 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{3}}+\frac {\sin \left (3 d x +3 c \right )}{12 b^{2} d}\)	\(558\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5/(a+b*cos(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(-2/b^5*(((-3*a^2*b-a*b^2-b^3)*tan(1/2*d*x+1/2*c)^5+(-6*a^2*b-2/3*b^3)*tan(1/2*d*x+1/2*c)^3+(-3*a^2*b+a*b^
2-b^3)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^3+a*(4*a^2+b^2)*arctan(tan(1/2*d*x+1/2*c)))+2*a^4/b^5*(a*b
/(a^2-b^2)*tan(1/2*d*x+1/2*c)/(a*tan(1/2*d*x+1/2*c)^2-b*tan(1/2*d*x+1/2*c)^2+a+b)+(4*a^2-5*b^2)/(a-b)/(a+b)/((
a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+b*cos(d*x+c))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

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Fricas [A]
time = 0.45, size = 747, normalized size = 2.81 \begin {gather*} \left [-\frac {6 \, {\left (4 \, a^{7} b - 7 \, a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\right )} d x \cos \left (d x + c\right ) + 6 \, {\left (4 \, a^{8} - 7 \, a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}\right )} d x + 3 \, {\left (4 \, a^{7} - 5 \, a^{5} b^{2} + {\left (4 \, a^{6} b - 5 \, a^{4} b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - 2 \, {\left (12 \, a^{7} b - 19 \, a^{5} b^{3} + 5 \, a^{3} b^{5} + 2 \, a b^{7} + {\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{6} b^{2} - 5 \, a^{4} b^{4} + a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left ({\left (a^{4} b^{6} - 2 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right ) + {\left (a^{5} b^{5} - 2 \, a^{3} b^{7} + a b^{9}\right )} d\right )}}, -\frac {3 \, {\left (4 \, a^{7} b - 7 \, a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (4 \, a^{8} - 7 \, a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}\right )} d x - 3 \, {\left (4 \, a^{7} - 5 \, a^{5} b^{2} + {\left (4 \, a^{6} b - 5 \, a^{4} b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) - {\left (12 \, a^{7} b - 19 \, a^{5} b^{3} + 5 \, a^{3} b^{5} + 2 \, a b^{7} + {\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{6} b^{2} - 5 \, a^{4} b^{4} + a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3 \, {\left ({\left (a^{4} b^{6} - 2 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right ) + {\left (a^{5} b^{5} - 2 \, a^{3} b^{7} + a b^{9}\right )} d\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+b*cos(d*x+c))^2,x, algorithm="fricas")

[Out]

[-1/6*(6*(4*a^7*b - 7*a^5*b^3 + 2*a^3*b^5 + a*b^7)*d*x*cos(d*x + c) + 6*(4*a^8 - 7*a^6*b^2 + 2*a^4*b^4 + a^2*b
^6)*d*x + 3*(4*a^7 - 5*a^5*b^2 + (4*a^6*b - 5*a^4*b^3)*cos(d*x + c))*sqrt(-a^2 + b^2)*log((2*a*b*cos(d*x + c)
+ (2*a^2 - b^2)*cos(d*x + c)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(d*x + c) + b)*sin(d*x + c) - a^2 + 2*b^2)/(b^2*cos(
d*x + c)^2 + 2*a*b*cos(d*x + c) + a^2)) - 2*(12*a^7*b - 19*a^5*b^3 + 5*a^3*b^5 + 2*a*b^7 + (a^4*b^4 - 2*a^2*b^
6 + b^8)*cos(d*x + c)^3 - 2*(a^5*b^3 - 2*a^3*b^5 + a*b^7)*cos(d*x + c)^2 + 2*(3*a^6*b^2 - 5*a^4*b^4 + a^2*b^6
+ b^8)*cos(d*x + c))*sin(d*x + c))/((a^4*b^6 - 2*a^2*b^8 + b^10)*d*cos(d*x + c) + (a^5*b^5 - 2*a^3*b^7 + a*b^9
)*d), -1/3*(3*(4*a^7*b - 7*a^5*b^3 + 2*a^3*b^5 + a*b^7)*d*x*cos(d*x + c) + 3*(4*a^8 - 7*a^6*b^2 + 2*a^4*b^4 +
a^2*b^6)*d*x - 3*(4*a^7 - 5*a^5*b^2 + (4*a^6*b - 5*a^4*b^3)*cos(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*cos(d*x +
c) + b)/(sqrt(a^2 - b^2)*sin(d*x + c))) - (12*a^7*b - 19*a^5*b^3 + 5*a^3*b^5 + 2*a*b^7 + (a^4*b^4 - 2*a^2*b^6
+ b^8)*cos(d*x + c)^3 - 2*(a^5*b^3 - 2*a^3*b^5 + a*b^7)*cos(d*x + c)^2 + 2*(3*a^6*b^2 - 5*a^4*b^4 + a^2*b^6 +
b^8)*cos(d*x + c))*sin(d*x + c))/((a^4*b^6 - 2*a^2*b^8 + b^10)*d*cos(d*x + c) + (a^5*b^5 - 2*a^3*b^7 + a*b^9)
*d)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5/(a+b*cos(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]
time = 0.48, size = 333, normalized size = 1.25 \begin {gather*} \frac {\frac {6 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{2} b^{4} - b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )}} - \frac {6 \, {\left (4 \, a^{6} - 5 \, a^{4} b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{2} b^{5} - b^{7}\right )} \sqrt {a^{2} - b^{2}}} - \frac {3 \, {\left (4 \, a^{3} + a b^{2}\right )} {\left (d x + c\right )}}{b^{5}} + \frac {2 \, {\left (9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} b^{4}}}{3 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+b*cos(d*x+c))^2,x, algorithm="giac")

[Out]

1/3*(6*a^5*tan(1/2*d*x + 1/2*c)/((a^2*b^4 - b^6)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 + a + b)
) - 6*(4*a^6 - 5*a^4*b^2)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c)
- b*tan(1/2*d*x + 1/2*c))/sqrt(a^2 - b^2)))/((a^2*b^5 - b^7)*sqrt(a^2 - b^2)) - 3*(4*a^3 + a*b^2)*(d*x + c)/b^
5 + 2*(9*a^2*tan(1/2*d*x + 1/2*c)^5 + 3*a*b*tan(1/2*d*x + 1/2*c)^5 + 3*b^2*tan(1/2*d*x + 1/2*c)^5 + 18*a^2*tan
(1/2*d*x + 1/2*c)^3 + 2*b^2*tan(1/2*d*x + 1/2*c)^3 + 9*a^2*tan(1/2*d*x + 1/2*c) - 3*a*b*tan(1/2*d*x + 1/2*c) +
3*b^2*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*b^4))/d

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Mupad [B]
time = 7.21, size = 2500, normalized size = 9.40 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^5/(a + b*cos(c + d*x))^2,x)

[Out]

- ((2*tan(c/2 + (d*x)/2)^3*(8*a*b^4 - 6*a^4*b - 36*a^5 - b^5 + 7*a^2*b^3 + 19*a^3*b^2))/(3*b^4*(a + b)*(a - b)
) - (2*tan(c/2 + (d*x)/2)^7*(4*a^5 - 2*a^4*b + b^5 + a^2*b^3 - 3*a^3*b^2))/(b^4*(a + b)*(a - b)) + (2*tan(c/2
+ (d*x)/2)^5*(8*a*b^4 + 6*a^4*b - 36*a^5 + b^5 - 7*a^2*b^3 + 19*a^3*b^2))/(3*b^4*(a + b)*(a - b)) + (2*tan(c/2
+ (d*x)/2)*(b^5 - 4*a^5 - 2*a^4*b + a^2*b^3 + 3*a^3*b^2))/(b^4*(a + b)*(a - b)))/(d*(a + b + tan(c/2 + (d*x)/
2)^8*(a - b) + tan(c/2 + (d*x)/2)^2*(4*a + 2*b) + tan(c/2 + (d*x)/2)^6*(4*a - 2*b) + 6*a*tan(c/2 + (d*x)/2)^4)
) - (2*a*atan(((a*(4*a^2 + b^2)*((32*tan(c/2 + (d*x)/2)*(32*a^12 - 32*a^11*b + a^2*b^10 - 2*a^3*b^9 + 7*a^4*b^
8 - 12*a^5*b^7 + 7*a^6*b^6 - 2*a^7*b^5 + 2*a^8*b^4 + 48*a^9*b^3 - 48*a^10*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3
*b^8) + (a*(4*a^2 + b^2)*((32*(a*b^17 + a^3*b^15 - 5*a^4*b^14 - 4*a^5*b^13 + 9*a^6*b^12 + 2*a^7*b^11 - 4*a^8*b
^10))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (a*tan(c/2 + (d*x)/2)*(4*a^2 + b^2)*(2*a*b^15 - 2*a^2*b^14 - 4*a
^3*b^13 + 4*a^4*b^12 + 2*a^5*b^11 - 2*a^6*b^10)*32i)/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))*1i)/b^5))/b^5
+ (a*(4*a^2 + b^2)*((32*tan(c/2 + (d*x)/2)*(32*a^12 - 32*a^11*b + a^2*b^10 - 2*a^3*b^9 + 7*a^4*b^8 - 12*a^5*b^
7 + 7*a^6*b^6 - 2*a^7*b^5 + 2*a^8*b^4 + 48*a^9*b^3 - 48*a^10*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) - (a*(4
*a^2 + b^2)*((32*(a*b^17 + a^3*b^15 - 5*a^4*b^14 - 4*a^5*b^13 + 9*a^6*b^12 + 2*a^7*b^11 - 4*a^8*b^10))/(a*b^14
+ b^15 - a^2*b^13 - a^3*b^12) + (a*tan(c/2 + (d*x)/2)*(4*a^2 + b^2)*(2*a*b^15 - 2*a^2*b^14 - 4*a^3*b^13 + 4*a
^4*b^12 + 2*a^5*b^11 - 2*a^6*b^10)*32i)/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))*1i)/b^5))/b^5)/((64*(64*a^1
4 - 32*a^13*b + 5*a^6*b^8 - 5*a^7*b^7 + 31*a^8*b^6 - 6*a^9*b^5 + 12*a^10*b^4 + 48*a^11*b^3 - 112*a^12*b^2))/(a
*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (a*(4*a^2 + b^2)*((32*tan(c/2 + (d*x)/2)*(32*a^12 - 32*a^11*b + a^2*b^10
- 2*a^3*b^9 + 7*a^4*b^8 - 12*a^5*b^7 + 7*a^6*b^6 - 2*a^7*b^5 + 2*a^8*b^4 + 48*a^9*b^3 - 48*a^10*b^2))/(a*b^10
+ b^11 - a^2*b^9 - a^3*b^8) + (a*(4*a^2 + b^2)*((32*(a*b^17 + a^3*b^15 - 5*a^4*b^14 - 4*a^5*b^13 + 9*a^6*b^12
+ 2*a^7*b^11 - 4*a^8*b^10))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (a*tan(c/2 + (d*x)/2)*(4*a^2 + b^2)*(2*a*
b^15 - 2*a^2*b^14 - 4*a^3*b^13 + 4*a^4*b^12 + 2*a^5*b^11 - 2*a^6*b^10)*32i)/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^
3*b^8)))*1i)/b^5)*1i)/b^5 + (a*(4*a^2 + b^2)*((32*tan(c/2 + (d*x)/2)*(32*a^12 - 32*a^11*b + a^2*b^10 - 2*a^3*b
^9 + 7*a^4*b^8 - 12*a^5*b^7 + 7*a^6*b^6 - 2*a^7*b^5 + 2*a^8*b^4 + 48*a^9*b^3 - 48*a^10*b^2))/(a*b^10 + b^11 -
a^2*b^9 - a^3*b^8) - (a*(4*a^2 + b^2)*((32*(a*b^17 + a^3*b^15 - 5*a^4*b^14 - 4*a^5*b^13 + 9*a^6*b^12 + 2*a^7*b
^11 - 4*a^8*b^10))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + (a*tan(c/2 + (d*x)/2)*(4*a^2 + b^2)*(2*a*b^15 - 2*a
^2*b^14 - 4*a^3*b^13 + 4*a^4*b^12 + 2*a^5*b^11 - 2*a^6*b^10)*32i)/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))*1
i)/b^5)*1i)/b^5))*(4*a^2 + b^2))/(b^5*d) - (a^4*atan(((a^4*(4*a^2 - 5*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((32*t
an(c/2 + (d*x)/2)*(32*a^12 - 32*a^11*b + a^2*b^10 - 2*a^3*b^9 + 7*a^4*b^8 - 12*a^5*b^7 + 7*a^6*b^6 - 2*a^7*b^5
+ 2*a^8*b^4 + 48*a^9*b^3 - 48*a^10*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) + (a^4*((32*(a*b^17 + a^3*b^15 -
5*a^4*b^14 - 4*a^5*b^13 + 9*a^6*b^12 + 2*a^7*b^11 - 4*a^8*b^10))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (32*
a^4*tan(c/2 + (d*x)/2)*(4*a^2 - 5*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*(2*a*b^15 - 2*a^2*b^14 - 4*a^3*b^13 + 4*a^
4*b^12 + 2*a^5*b^11 - 2*a^6*b^10))/((a*b^10 + b^11 - a^2*b^9 - a^3*b^8)*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^
5)))*(4*a^2 - 5*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5))*1i)/(b^11 - 3*a^2
*b^9 + 3*a^4*b^7 - a^6*b^5) + (a^4*(4*a^2 - 5*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((32*tan(c/2 + (d*x)/2)*(32*a^
12 - 32*a^11*b + a^2*b^10 - 2*a^3*b^9 + 7*a^4*b^8 - 12*a^5*b^7 + 7*a^6*b^6 - 2*a^7*b^5 + 2*a^8*b^4 + 48*a^9*b^
3 - 48*a^10*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) - (a^4*((32*(a*b^17 + a^3*b^15 - 5*a^4*b^14 - 4*a^5*b^13
+ 9*a^6*b^12 + 2*a^7*b^11 - 4*a^8*b^10))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + (32*a^4*tan(c/2 + (d*x)/2)*(
4*a^2 - 5*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*(2*a*b^15 - 2*a^2*b^14 - 4*a^3*b^13 + 4*a^4*b^12 + 2*a^5*b^11 - 2*
a^6*b^10))/((a*b^10 + b^11 - a^2*b^9 - a^3*b^8)*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*(4*a^2 - 5*b^2)*(-(
a + b)^3*(a - b)^3)^(1/2))/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5))*1i)/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b
^5))/((64*(64*a^14 - 32*a^13*b + 5*a^6*b^8 - 5*a^7*b^7 + 31*a^8*b^6 - 6*a^9*b^5 + 12*a^10*b^4 + 48*a^11*b^3 -
112*a^12*b^2))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (a^4*(4*a^2 - 5*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((32*
tan(c/2 + (d*x)/2)*(32*a^12 - 32*a^11*b + a^2*b^10 - 2*a^3*b^9 + 7*a^4*b^8 - 12*a^5*b^7 + 7*a^6*b^6 - 2*a^7*b^
5 + 2*a^8*b^4 + 48*a^9*b^3 - 48*a^10*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) + (a^4*((32*(a*b^17 + a^3*b^15
- 5*a^4*b^14 - 4*a^5*b^13 + 9*a^6*b^12 + 2*a^7*b^11 - 4*a^8*b^10))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (32
*a^4*tan(c/2 + (d*x)/2)*(4*a^2 - 5*b^2)*(-(a + ...

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