3.618 \(\int \frac {(1-\cos ^2(c+d x)) \sec ^4(c+d x)}{(a+b \cos (c+d x))^3} \, dx\)

Optimal. Leaf size=335 \[ -\frac {\left (4 a^2-5 b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{2 a^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {b \left (3 a^2-20 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac {b \left (9 a^2-10 b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a^4 d \left (a^2-b^2\right )}+\frac {\left (17 a^2-20 b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{6 a^3 d \left (a^2-b^2\right )}-\frac {b^2 \left (12 a^4-33 a^2 b^2+20 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 d (a-b)^{3/2} (a+b)^{3/2}}-\frac {\left (2 a^4-59 a^2 b^2+60 b^4\right ) \tan (c+d x)}{6 a^5 d \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x)}{2 a d (a+b \cos (c+d x))^2} \]

[Out]

-b^2*(12*a^4-33*a^2*b^2+20*b^4)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^6/(a-b)^(3/2)/(a+b)^(3/2)
/d+1/2*b*(3*a^2-20*b^2)*arctanh(sin(d*x+c))/a^6/d-1/6*(2*a^4-59*a^2*b^2+60*b^4)*tan(d*x+c)/a^5/(a^2-b^2)/d-1/2
*b*(9*a^2-10*b^2)*sec(d*x+c)*tan(d*x+c)/a^4/(a^2-b^2)/d+1/6*(17*a^2-20*b^2)*sec(d*x+c)^2*tan(d*x+c)/a^3/(a^2-b
^2)/d-1/2*sec(d*x+c)^2*tan(d*x+c)/a/d/(a+b*cos(d*x+c))^2-1/2*(4*a^2-5*b^2)*sec(d*x+c)^2*tan(d*x+c)/a^2/(a^2-b^
2)/d/(a+b*cos(d*x+c))

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Rubi [A]  time = 1.40, antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3056, 3055, 3001, 3770, 2659, 205} \[ -\frac {b^2 \left (-33 a^2 b^2+12 a^4+20 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 d (a-b)^{3/2} (a+b)^{3/2}}-\frac {\left (-59 a^2 b^2+2 a^4+60 b^4\right ) \tan (c+d x)}{6 a^5 d \left (a^2-b^2\right )}+\frac {b \left (3 a^2-20 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}+\frac {\left (17 a^2-20 b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{6 a^3 d \left (a^2-b^2\right )}-\frac {b \left (9 a^2-10 b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a^4 d \left (a^2-b^2\right )}-\frac {\left (4 a^2-5 b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{2 a^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\tan (c+d x) \sec ^2(c+d x)}{2 a d (a+b \cos (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

Int[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + b*Cos[c + d*x])^3,x]

[Out]

-((b^2*(12*a^4 - 33*a^2*b^2 + 20*b^4)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^6*(a - b)^(3/2)*(
a + b)^(3/2)*d)) + (b*(3*a^2 - 20*b^2)*ArcTanh[Sin[c + d*x]])/(2*a^6*d) - ((2*a^4 - 59*a^2*b^2 + 60*b^4)*Tan[c
+ d*x])/(6*a^5*(a^2 - b^2)*d) - (b*(9*a^2 - 10*b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*(a^2 - b^2)*d) + ((17*a
^2 - 20*b^2)*Sec[c + d*x]^2*Tan[c + d*x])/(6*a^3*(a^2 - b^2)*d) - (Sec[c + d*x]^2*Tan[c + d*x])/(2*a*d*(a + b*
Cos[c + d*x])^2) - ((4*a^2 - 5*b^2)*Sec[c + d*x]^2*Tan[c + d*x])/(2*a^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))

Rule 205

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

Rule 2659

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 3001

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

Rule 3055

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*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] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

Rule 3056

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c +
d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), I
nt[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*
(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d*(A*b^2 + a^2*C)*(m + n + 3)
*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
&& NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ
[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {\left (1-\cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+b \cos (c+d x))^3} \, dx &=-\frac {\sec ^2(c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^2}+\frac {\int \frac {\left (5 \left (a^2-b^2\right )-4 \left (a^2-b^2\right ) \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+b \cos (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=-\frac {\sec ^2(c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^2}-\frac {\left (4 a^2-5 b^2\right ) \sec ^2(c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {\left (17 a^4-37 a^2 b^2+20 b^4-a b \left (a^2-b^2\right ) \cos (c+d x)-3 \left (4 a^2-5 b^2\right ) \left (a^2-b^2\right ) \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (17 a^2-20 b^2\right ) \sec ^2(c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right ) d}-\frac {\sec ^2(c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^2}-\frac {\left (4 a^2-5 b^2\right ) \sec ^2(c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {\left (-6 b \left (9 a^4-19 a^2 b^2+10 b^4\right )-a \left (2 a^4-7 a^2 b^2+5 b^4\right ) \cos (c+d x)+2 b \left (17 a^4-37 a^2 b^2+20 b^4\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac {b \left (9 a^2-10 b^2\right ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}+\frac {\left (17 a^2-20 b^2\right ) \sec ^2(c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right ) d}-\frac {\sec ^2(c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^2}-\frac {\left (4 a^2-5 b^2\right ) \sec ^2(c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {\left (-2 \left (2 a^6-61 a^4 b^2+119 a^2 b^4-60 b^6\right )+2 a b \left (7 a^4-17 a^2 b^2+10 b^4\right ) \cos (c+d x)-6 b^2 \left (9 a^4-19 a^2 b^2+10 b^4\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{12 a^4 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (2 a^4-59 a^2 b^2+60 b^4\right ) \tan (c+d x)}{6 a^5 \left (a^2-b^2\right ) d}-\frac {b \left (9 a^2-10 b^2\right ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}+\frac {\left (17 a^2-20 b^2\right ) \sec ^2(c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right ) d}-\frac {\sec ^2(c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^2}-\frac {\left (4 a^2-5 b^2\right ) \sec ^2(c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {\left (6 b \left (3 a^2-20 b^2\right ) \left (a^2-b^2\right )^2-6 a b^2 \left (9 a^4-19 a^2 b^2+10 b^4\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{12 a^5 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (2 a^4-59 a^2 b^2+60 b^4\right ) \tan (c+d x)}{6 a^5 \left (a^2-b^2\right ) d}-\frac {b \left (9 a^2-10 b^2\right ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}+\frac {\left (17 a^2-20 b^2\right ) \sec ^2(c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right ) d}-\frac {\sec ^2(c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^2}-\frac {\left (4 a^2-5 b^2\right ) \sec ^2(c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\left (b \left (3 a^2-20 b^2\right )\right ) \int \sec (c+d x) \, dx}{2 a^6}-\frac {\left (b^2 \left (12 a^4-33 a^2 b^2+20 b^4\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{2 a^6 \left (a^2-b^2\right )}\\ &=\frac {b \left (3 a^2-20 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac {\left (2 a^4-59 a^2 b^2+60 b^4\right ) \tan (c+d x)}{6 a^5 \left (a^2-b^2\right ) d}-\frac {b \left (9 a^2-10 b^2\right ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}+\frac {\left (17 a^2-20 b^2\right ) \sec ^2(c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right ) d}-\frac {\sec ^2(c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^2}-\frac {\left (4 a^2-5 b^2\right ) \sec ^2(c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\left (b^2 \left (12 a^4-33 a^2 b^2+20 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^6 \left (a^2-b^2\right ) d}\\ &=-\frac {b^2 \left (12 a^4-33 a^2 b^2+20 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 (a-b)^{3/2} (a+b)^{3/2} d}+\frac {b \left (3 a^2-20 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac {\left (2 a^4-59 a^2 b^2+60 b^4\right ) \tan (c+d x)}{6 a^5 \left (a^2-b^2\right ) d}-\frac {b \left (9 a^2-10 b^2\right ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}+\frac {\left (17 a^2-20 b^2\right ) \sec ^2(c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right ) d}-\frac {\sec ^2(c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^2}-\frac {\left (4 a^2-5 b^2\right ) \sec ^2(c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 6.26, size = 563, normalized size = 1.68 \[ \frac {b^3 \sin (c+d x)}{2 a^4 d (a+b \cos (c+d x))^2}+\frac {a-9 b}{12 a^4 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {9 b-a}{12 a^4 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{6 a^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{6 a^3 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {\left (20 b^3-3 a^2 b\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac {\left (3 a^2 b-20 b^3\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac {18 b^2 \sin \left (\frac {1}{2} (c+d x)\right )-a^2 \sin \left (\frac {1}{2} (c+d x)\right )}{3 a^5 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {18 b^2 \sin \left (\frac {1}{2} (c+d x)\right )-a^2 \sin \left (\frac {1}{2} (c+d x)\right )}{3 a^5 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {7 a^2 b^3 \sin (c+d x)-8 b^5 \sin (c+d x)}{2 a^5 d (a-b) (a+b) (a+b \cos (c+d x))}+\frac {b^2 \left (12 a^4-33 a^2 b^2+20 b^4\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{a^6 d \left (a^2-b^2\right ) \sqrt {b^2-a^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + b*Cos[c + d*x])^3,x]

[Out]

(b^2*(12*a^4 - 33*a^2*b^2 + 20*b^4)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(a^6*(a^2 - b^2)*Sqr
t[-a^2 + b^2]*d) + ((-3*a^2*b + 20*b^3)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(2*a^6*d) + ((3*a^2*b - 20*b
^3)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(2*a^6*d) + (a - 9*b)/(12*a^4*d*(Cos[(c + d*x)/2] - Sin[(c + d*x
)/2])^2) + Sin[(c + d*x)/2]/(6*a^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + Sin[(c + d*x)/2]/(6*a^3*d*(Cos
[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (-a + 9*b)/(12*a^4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (-(a^2*
Sin[(c + d*x)/2]) + 18*b^2*Sin[(c + d*x)/2])/(3*a^5*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (-(a^2*Sin[(c +
d*x)/2]) + 18*b^2*Sin[(c + d*x)/2])/(3*a^5*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (b^3*Sin[c + d*x])/(2*a
^4*d*(a + b*Cos[c + d*x])^2) + (7*a^2*b^3*Sin[c + d*x] - 8*b^5*Sin[c + d*x])/(2*a^5*(a - b)*(a + b)*d*(a + b*C
os[c + d*x]))

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fricas [B]  time = 1.49, size = 1447, normalized size = 4.32 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-cos(d*x+c)^2)*sec(d*x+c)^4/(a+b*cos(d*x+c))^3,x, algorithm="fricas")

[Out]

[1/12*(3*((12*a^4*b^4 - 33*a^2*b^6 + 20*b^8)*cos(d*x + c)^5 + 2*(12*a^5*b^3 - 33*a^3*b^5 + 20*a*b^7)*cos(d*x +
c)^4 + (12*a^6*b^2 - 33*a^4*b^4 + 20*a^2*b^6)*cos(d*x + c)^3)*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)) + 3*((3*a^6*b^3 - 26*a^4*b^5 + 43*a^2*b^7 - 20*b^9)*cos(d*x + c)^5 + 2*(3*a^
7*b^2 - 26*a^5*b^4 + 43*a^3*b^6 - 20*a*b^8)*cos(d*x + c)^4 + (3*a^8*b - 26*a^6*b^3 + 43*a^4*b^5 - 20*a^2*b^7)*
cos(d*x + c)^3)*log(sin(d*x + c) + 1) - 3*((3*a^6*b^3 - 26*a^4*b^5 + 43*a^2*b^7 - 20*b^9)*cos(d*x + c)^5 + 2*(
3*a^7*b^2 - 26*a^5*b^4 + 43*a^3*b^6 - 20*a*b^8)*cos(d*x + c)^4 + (3*a^8*b - 26*a^6*b^3 + 43*a^4*b^5 - 20*a^2*b
^7)*cos(d*x + c)^3)*log(-sin(d*x + c) + 1) + 2*(2*a^9 - 4*a^7*b^2 + 2*a^5*b^4 - (2*a^7*b^2 - 61*a^5*b^4 + 119*
a^3*b^6 - 60*a*b^8)*cos(d*x + c)^4 - (4*a^8*b - 95*a^6*b^3 + 181*a^4*b^5 - 90*a^2*b^7)*cos(d*x + c)^3 - 2*(a^9
- 12*a^7*b^2 + 21*a^5*b^4 - 10*a^3*b^6)*cos(d*x + c)^2 - 5*(a^8*b - 2*a^6*b^3 + a^4*b^5)*cos(d*x + c))*sin(d*
x + c))/((a^10*b^2 - 2*a^8*b^4 + a^6*b^6)*d*cos(d*x + c)^5 + 2*(a^11*b - 2*a^9*b^3 + a^7*b^5)*d*cos(d*x + c)^4
+ (a^12 - 2*a^10*b^2 + a^8*b^4)*d*cos(d*x + c)^3), -1/12*(6*((12*a^4*b^4 - 33*a^2*b^6 + 20*b^8)*cos(d*x + c)^
5 + 2*(12*a^5*b^3 - 33*a^3*b^5 + 20*a*b^7)*cos(d*x + c)^4 + (12*a^6*b^2 - 33*a^4*b^4 + 20*a^2*b^6)*cos(d*x + c
)^3)*sqrt(a^2 - b^2)*arctan(-(a*cos(d*x + c) + b)/(sqrt(a^2 - b^2)*sin(d*x + c))) - 3*((3*a^6*b^3 - 26*a^4*b^5
+ 43*a^2*b^7 - 20*b^9)*cos(d*x + c)^5 + 2*(3*a^7*b^2 - 26*a^5*b^4 + 43*a^3*b^6 - 20*a*b^8)*cos(d*x + c)^4 + (
3*a^8*b - 26*a^6*b^3 + 43*a^4*b^5 - 20*a^2*b^7)*cos(d*x + c)^3)*log(sin(d*x + c) + 1) + 3*((3*a^6*b^3 - 26*a^4
*b^5 + 43*a^2*b^7 - 20*b^9)*cos(d*x + c)^5 + 2*(3*a^7*b^2 - 26*a^5*b^4 + 43*a^3*b^6 - 20*a*b^8)*cos(d*x + c)^4
+ (3*a^8*b - 26*a^6*b^3 + 43*a^4*b^5 - 20*a^2*b^7)*cos(d*x + c)^3)*log(-sin(d*x + c) + 1) - 2*(2*a^9 - 4*a^7*
b^2 + 2*a^5*b^4 - (2*a^7*b^2 - 61*a^5*b^4 + 119*a^3*b^6 - 60*a*b^8)*cos(d*x + c)^4 - (4*a^8*b - 95*a^6*b^3 + 1
81*a^4*b^5 - 90*a^2*b^7)*cos(d*x + c)^3 - 2*(a^9 - 12*a^7*b^2 + 21*a^5*b^4 - 10*a^3*b^6)*cos(d*x + c)^2 - 5*(a
^8*b - 2*a^6*b^3 + a^4*b^5)*cos(d*x + c))*sin(d*x + c))/((a^10*b^2 - 2*a^8*b^4 + a^6*b^6)*d*cos(d*x + c)^5 + 2
*(a^11*b - 2*a^9*b^3 + a^7*b^5)*d*cos(d*x + c)^4 + (a^12 - 2*a^10*b^2 + a^8*b^4)*d*cos(d*x + c)^3)]

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giac [A]  time = 1.44, size = 471, normalized size = 1.41 \[ \frac {\frac {6 \, {\left (12 \, a^{4} b^{2} - 33 \, a^{2} b^{4} + 20 \, b^{6}\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^{8} - a^{6} b^{2}\right )} \sqrt {a^{2} - b^{2}}} + \frac {6 \, {\left (8 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{7} - a^{5} b^{2}\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 )}^{2}} + \frac {3 \, {\left (3 \, a^{2} b - 20 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{6}} - \frac {3 \, {\left (3 \, a^{2} b - 20 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{6}} - \frac {2 \, {\left (9 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 8 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, 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} a^{5}}}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-cos(d*x+c)^2)*sec(d*x+c)^4/(a+b*cos(d*x+c))^3,x, algorithm="giac")

[Out]

1/6*(6*(12*a^4*b^2 - 33*a^2*b^4 + 20*b^6)*(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^8 - a^6*b^2)*sqrt(a^2 - b^2)) + 6*(8*a^3*b^3*t
an(1/2*d*x + 1/2*c)^3 - 7*a^2*b^4*tan(1/2*d*x + 1/2*c)^3 - 9*a*b^5*tan(1/2*d*x + 1/2*c)^3 + 8*b^6*tan(1/2*d*x
+ 1/2*c)^3 + 8*a^3*b^3*tan(1/2*d*x + 1/2*c) + 7*a^2*b^4*tan(1/2*d*x + 1/2*c) - 9*a*b^5*tan(1/2*d*x + 1/2*c) -
8*b^6*tan(1/2*d*x + 1/2*c))/((a^7 - a^5*b^2)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 + a + b)^2)
+ 3*(3*a^2*b - 20*b^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^6 - 3*(3*a^2*b - 20*b^3)*log(abs(tan(1/2*d*x + 1/2
*c) - 1))/a^6 - 2*(9*a*b*tan(1/2*d*x + 1/2*c)^5 + 36*b^2*tan(1/2*d*x + 1/2*c)^5 + 8*a^2*tan(1/2*d*x + 1/2*c)^3
- 72*b^2*tan(1/2*d*x + 1/2*c)^3 - 9*a*b*tan(1/2*d*x + 1/2*c) + 36*b^2*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1
/2*c)^2 - 1)^3*a^5))/d

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maple [B]  time = 0.25, size = 843, normalized size = 2.52 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-cos(d*x+c)^2)*sec(d*x+c)^4/(a+b*cos(d*x+c))^3,x)

[Out]

8/d/a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2*b^3/(a+b)*tan(1/2*d*x+1/2*c)^3+1/d*b^4/a^4/(a*ta
n(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)*tan(1/2*d*x+1/2*c)^3-8/d*b^5/a^5/(a*tan(1/2*d*x+1/2*c)^
2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)*tan(1/2*d*x+1/2*c)^3+8/d/a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^
2*b+a+b)^2*b^3/(a-b)*tan(1/2*d*x+1/2*c)-1/d*b^4/a^4/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a-b
)*tan(1/2*d*x+1/2*c)-8/d*b^5/a^5/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a-b)*tan(1/2*d*x+1/2*c
)-12/d/a^2/(a^2-b^2)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*b^2+33/d/a^4/(a^
2-b^2)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*b^4-20/d*b^6/a^6/(a^2-b^2)/((a
-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))-1/3/d/a^3/(tan(1/2*d*x+1/2*c)-1)^3-1/2/d
/a^3/(tan(1/2*d*x+1/2*c)-1)^2-3/2/d/a^4/(tan(1/2*d*x+1/2*c)-1)^2*b-3/2/d/a^4/(tan(1/2*d*x+1/2*c)-1)*b-6/d*b^2/
a^5/(tan(1/2*d*x+1/2*c)-1)-3/2/d*b/a^4*ln(tan(1/2*d*x+1/2*c)-1)+10/d*b^3/a^6*ln(tan(1/2*d*x+1/2*c)-1)-1/3/d/a^
3/(tan(1/2*d*x+1/2*c)+1)^3+1/2/d/a^3/(tan(1/2*d*x+1/2*c)+1)^2+3/2/d/a^4/(tan(1/2*d*x+1/2*c)+1)^2*b-3/2/d/a^4/(
tan(1/2*d*x+1/2*c)+1)*b-6/d*b^2/a^5/(tan(1/2*d*x+1/2*c)+1)+3/2/d*b/a^4*ln(tan(1/2*d*x+1/2*c)+1)-10/d*b^3/a^6*l
n(tan(1/2*d*x+1/2*c)+1)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-cos(d*x+c)^2)*sec(d*x+c)^4/(a+b*cos(d*x+c))^3,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 details)Is 4*b^2-4*a^2 positive or negative?

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mupad [B]  time = 9.19, size = 4231, normalized size = 12.63 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(cos(c + d*x)^2 - 1)/(cos(c + d*x)^4*(a + b*cos(c + d*x))^3),x)

[Out]

((tan(c/2 + (d*x)/2)^9*(3*a^4*b - 10*a*b^4 + 20*b^5 - 23*a^2*b^3 + 9*a^3*b^2))/(a^5*(a + b)) - (tan(c/2 + (d*x
)/2)*(10*a*b^4 + 3*a^4*b + 20*b^5 - 23*a^2*b^3 - 9*a^3*b^2))/(a^5*(a - b)) + (2*tan(c/2 + (d*x)/2)^5*(8*a^6 -
180*b^6 + 197*a^2*b^4 - 34*a^4*b^2))/(3*a^5*(a + b)*(a - b)) + (2*tan(c/2 + (d*x)/2)^3*(90*a*b^5 - a^5*b + 4*a
^6 + 120*b^6 - 118*a^2*b^4 - 86*a^3*b^3))/(3*a^5*(a + b)*(a - b)) + (2*tan(c/2 + (d*x)/2)^7*(a^5*b - 90*a*b^5
+ 4*a^6 + 120*b^6 - 118*a^2*b^4 + 86*a^3*b^3))/(3*a^5*(a + b)*(a - b)))/(d*(2*a*b + tan(c/2 + (d*x)/2)^4*(4*a*
b - 2*a^2 + 10*b^2) + tan(c/2 + (d*x)/2)^6*(4*a*b + 2*a^2 - 10*b^2) - tan(c/2 + (d*x)/2)^10*(a^2 - 2*a*b + b^2
) + a^2 + b^2 - tan(c/2 + (d*x)/2)^2*(6*a*b + a^2 + 5*b^2) + tan(c/2 + (d*x)/2)^8*(a^2 - 6*a*b + 5*b^2))) + (a
tan((((3*a^2*b - 20*b^3)*(((3*a^2*b - 20*b^3)*((4*(12*a^19*b + 80*a^12*b^8 - 40*a^13*b^7 - 212*a^14*b^6 + 96*a
^15*b^5 + 180*a^16*b^4 - 68*a^17*b^3 - 48*a^18*b^2))/(a^17*b + a^18 - a^15*b^3 - a^16*b^2) - (4*tan(c/2 + (d*x
)/2)*(3*a^2*b - 20*b^3)*(8*a^17*b - 8*a^12*b^6 + 8*a^13*b^5 + 16*a^14*b^4 - 16*a^15*b^3 - 8*a^16*b^2))/(a^6*(a
^12*b + a^13 - a^10*b^3 - a^11*b^2))))/(2*a^6) - (8*tan(c/2 + (d*x)/2)*(800*b^12 - 800*a*b^11 - 1840*a^2*b^10
+ 1840*a^3*b^9 + 1298*a^4*b^8 - 1298*a^5*b^7 - 281*a^6*b^6 + 276*a^7*b^5 + 15*a^8*b^4 - 18*a^9*b^3 + 9*a^10*b^
2))/(a^12*b + a^13 - a^10*b^3 - a^11*b^2))*1i)/(2*a^6) - ((3*a^2*b - 20*b^3)*(((3*a^2*b - 20*b^3)*((4*(12*a^19
*b + 80*a^12*b^8 - 40*a^13*b^7 - 212*a^14*b^6 + 96*a^15*b^5 + 180*a^16*b^4 - 68*a^17*b^3 - 48*a^18*b^2))/(a^17
*b + a^18 - a^15*b^3 - a^16*b^2) + (4*tan(c/2 + (d*x)/2)*(3*a^2*b - 20*b^3)*(8*a^17*b - 8*a^12*b^6 + 8*a^13*b^
5 + 16*a^14*b^4 - 16*a^15*b^3 - 8*a^16*b^2))/(a^6*(a^12*b + a^13 - a^10*b^3 - a^11*b^2))))/(2*a^6) + (8*tan(c/
2 + (d*x)/2)*(800*b^12 - 800*a*b^11 - 1840*a^2*b^10 + 1840*a^3*b^9 + 1298*a^4*b^8 - 1298*a^5*b^7 - 281*a^6*b^6
+ 276*a^7*b^5 + 15*a^8*b^4 - 18*a^9*b^3 + 9*a^10*b^2))/(a^12*b + a^13 - a^10*b^3 - a^11*b^2))*1i)/(2*a^6))/((
8*(4000*a*b^13 - 8000*b^14 + 23600*a^2*b^12 - 10800*a^3*b^11 - 24540*a^4*b^10 + 9870*a^5*b^9 + 10677*a^6*b^8 -
3411*a^7*b^7 - 1845*a^8*b^6 + 324*a^9*b^5 + 108*a^10*b^4))/(a^17*b + a^18 - a^15*b^3 - a^16*b^2) + ((3*a^2*b
- 20*b^3)*(((3*a^2*b - 20*b^3)*((4*(12*a^19*b + 80*a^12*b^8 - 40*a^13*b^7 - 212*a^14*b^6 + 96*a^15*b^5 + 180*a
^16*b^4 - 68*a^17*b^3 - 48*a^18*b^2))/(a^17*b + a^18 - a^15*b^3 - a^16*b^2) - (4*tan(c/2 + (d*x)/2)*(3*a^2*b -
20*b^3)*(8*a^17*b - 8*a^12*b^6 + 8*a^13*b^5 + 16*a^14*b^4 - 16*a^15*b^3 - 8*a^16*b^2))/(a^6*(a^12*b + a^13 -
a^10*b^3 - a^11*b^2))))/(2*a^6) - (8*tan(c/2 + (d*x)/2)*(800*b^12 - 800*a*b^11 - 1840*a^2*b^10 + 1840*a^3*b^9
+ 1298*a^4*b^8 - 1298*a^5*b^7 - 281*a^6*b^6 + 276*a^7*b^5 + 15*a^8*b^4 - 18*a^9*b^3 + 9*a^10*b^2))/(a^12*b + a
^13 - a^10*b^3 - a^11*b^2)))/(2*a^6) + ((3*a^2*b - 20*b^3)*(((3*a^2*b - 20*b^3)*((4*(12*a^19*b + 80*a^12*b^8 -
40*a^13*b^7 - 212*a^14*b^6 + 96*a^15*b^5 + 180*a^16*b^4 - 68*a^17*b^3 - 48*a^18*b^2))/(a^17*b + a^18 - a^15*b
^3 - a^16*b^2) + (4*tan(c/2 + (d*x)/2)*(3*a^2*b - 20*b^3)*(8*a^17*b - 8*a^12*b^6 + 8*a^13*b^5 + 16*a^14*b^4 -
16*a^15*b^3 - 8*a^16*b^2))/(a^6*(a^12*b + a^13 - a^10*b^3 - a^11*b^2))))/(2*a^6) + (8*tan(c/2 + (d*x)/2)*(800*
b^12 - 800*a*b^11 - 1840*a^2*b^10 + 1840*a^3*b^9 + 1298*a^4*b^8 - 1298*a^5*b^7 - 281*a^6*b^6 + 276*a^7*b^5 + 1
5*a^8*b^4 - 18*a^9*b^3 + 9*a^10*b^2))/(a^12*b + a^13 - a^10*b^3 - a^11*b^2)))/(2*a^6)))*(3*a^2*b - 20*b^3)*1i)
/(a^6*d) - (b^2*atan(((b^2*((8*tan(c/2 + (d*x)/2)*(800*b^12 - 800*a*b^11 - 1840*a^2*b^10 + 1840*a^3*b^9 + 1298
*a^4*b^8 - 1298*a^5*b^7 - 281*a^6*b^6 + 276*a^7*b^5 + 15*a^8*b^4 - 18*a^9*b^3 + 9*a^10*b^2))/(a^12*b + a^13 -
a^10*b^3 - a^11*b^2) - (b^2*((4*(12*a^19*b + 80*a^12*b^8 - 40*a^13*b^7 - 212*a^14*b^6 + 96*a^15*b^5 + 180*a^16
*b^4 - 68*a^17*b^3 - 48*a^18*b^2))/(a^17*b + a^18 - a^15*b^3 - a^16*b^2) - (4*b^2*tan(c/2 + (d*x)/2)*(-(a + b)
^3*(a - b)^3)^(1/2)*(12*a^4 + 20*b^4 - 33*a^2*b^2)*(8*a^17*b - 8*a^12*b^6 + 8*a^13*b^5 + 16*a^14*b^4 - 16*a^15
*b^3 - 8*a^16*b^2))/((a^12*b + a^13 - a^10*b^3 - a^11*b^2)*(a^12 - a^6*b^6 + 3*a^8*b^4 - 3*a^10*b^2)))*(-(a +
b)^3*(a - b)^3)^(1/2)*(12*a^4 + 20*b^4 - 33*a^2*b^2))/(2*(a^12 - a^6*b^6 + 3*a^8*b^4 - 3*a^10*b^2)))*(-(a + b)
^3*(a - b)^3)^(1/2)*(12*a^4 + 20*b^4 - 33*a^2*b^2)*1i)/(2*(a^12 - a^6*b^6 + 3*a^8*b^4 - 3*a^10*b^2)) + (b^2*((
8*tan(c/2 + (d*x)/2)*(800*b^12 - 800*a*b^11 - 1840*a^2*b^10 + 1840*a^3*b^9 + 1298*a^4*b^8 - 1298*a^5*b^7 - 281
*a^6*b^6 + 276*a^7*b^5 + 15*a^8*b^4 - 18*a^9*b^3 + 9*a^10*b^2))/(a^12*b + a^13 - a^10*b^3 - a^11*b^2) + (b^2*(
(4*(12*a^19*b + 80*a^12*b^8 - 40*a^13*b^7 - 212*a^14*b^6 + 96*a^15*b^5 + 180*a^16*b^4 - 68*a^17*b^3 - 48*a^18*
b^2))/(a^17*b + a^18 - a^15*b^3 - a^16*b^2) + (4*b^2*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a - b)^3)^(1/2)*(12*a^4 +
20*b^4 - 33*a^2*b^2)*(8*a^17*b - 8*a^12*b^6 + 8*a^13*b^5 + 16*a^14*b^4 - 16*a^15*b^3 - 8*a^16*b^2))/((a^12*b
+ a^13 - a^10*b^3 - a^11*b^2)*(a^12 - a^6*b^6 + 3*a^8*b^4 - 3*a^10*b^2)))*(-(a + b)^3*(a - b)^3)^(1/2)*(12*a^4
+ 20*b^4 - 33*a^2*b^2))/(2*(a^12 - a^6*b^6 + 3*a^8*b^4 - 3*a^10*b^2)))*(-(a + b)^3*(a - b)^3)^(1/2)*(12*a^4 +
20*b^4 - 33*a^2*b^2)*1i)/(2*(a^12 - a^6*b^6 + 3*a^8*b^4 - 3*a^10*b^2)))/((8*(4000*a*b^13 - 8000*b^14 + 23600*
a^2*b^12 - 10800*a^3*b^11 - 24540*a^4*b^10 + 9870*a^5*b^9 + 10677*a^6*b^8 - 3411*a^7*b^7 - 1845*a^8*b^6 + 324*
a^9*b^5 + 108*a^10*b^4))/(a^17*b + a^18 - a^15*b^3 - a^16*b^2) - (b^2*((8*tan(c/2 + (d*x)/2)*(800*b^12 - 800*a
*b^11 - 1840*a^2*b^10 + 1840*a^3*b^9 + 1298*a^4*b^8 - 1298*a^5*b^7 - 281*a^6*b^6 + 276*a^7*b^5 + 15*a^8*b^4 -
18*a^9*b^3 + 9*a^10*b^2))/(a^12*b + a^13 - a^10*b^3 - a^11*b^2) - (b^2*((4*(12*a^19*b + 80*a^12*b^8 - 40*a^13*
b^7 - 212*a^14*b^6 + 96*a^15*b^5 + 180*a^16*b^4 - 68*a^17*b^3 - 48*a^18*b^2))/(a^17*b + a^18 - a^15*b^3 - a^16
*b^2) - (4*b^2*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a - b)^3)^(1/2)*(12*a^4 + 20*b^4 - 33*a^2*b^2)*(8*a^17*b - 8*a^
12*b^6 + 8*a^13*b^5 + 16*a^14*b^4 - 16*a^15*b^3 - 8*a^16*b^2))/((a^12*b + a^13 - a^10*b^3 - a^11*b^2)*(a^12 -
a^6*b^6 + 3*a^8*b^4 - 3*a^10*b^2)))*(-(a + b)^3*(a - b)^3)^(1/2)*(12*a^4 + 20*b^4 - 33*a^2*b^2))/(2*(a^12 - a^
6*b^6 + 3*a^8*b^4 - 3*a^10*b^2)))*(-(a + b)^3*(a - b)^3)^(1/2)*(12*a^4 + 20*b^4 - 33*a^2*b^2))/(2*(a^12 - a^6*
b^6 + 3*a^8*b^4 - 3*a^10*b^2)) + (b^2*((8*tan(c/2 + (d*x)/2)*(800*b^12 - 800*a*b^11 - 1840*a^2*b^10 + 1840*a^3
*b^9 + 1298*a^4*b^8 - 1298*a^5*b^7 - 281*a^6*b^6 + 276*a^7*b^5 + 15*a^8*b^4 - 18*a^9*b^3 + 9*a^10*b^2))/(a^12*
b + a^13 - a^10*b^3 - a^11*b^2) + (b^2*((4*(12*a^19*b + 80*a^12*b^8 - 40*a^13*b^7 - 212*a^14*b^6 + 96*a^15*b^5
+ 180*a^16*b^4 - 68*a^17*b^3 - 48*a^18*b^2))/(a^17*b + a^18 - a^15*b^3 - a^16*b^2) + (4*b^2*tan(c/2 + (d*x)/2
)*(-(a + b)^3*(a - b)^3)^(1/2)*(12*a^4 + 20*b^4 - 33*a^2*b^2)*(8*a^17*b - 8*a^12*b^6 + 8*a^13*b^5 + 16*a^14*b^
4 - 16*a^15*b^3 - 8*a^16*b^2))/((a^12*b + a^13 - a^10*b^3 - a^11*b^2)*(a^12 - a^6*b^6 + 3*a^8*b^4 - 3*a^10*b^2
)))*(-(a + b)^3*(a - b)^3)^(1/2)*(12*a^4 + 20*b^4 - 33*a^2*b^2))/(2*(a^12 - a^6*b^6 + 3*a^8*b^4 - 3*a^10*b^2))
)*(-(a + b)^3*(a - b)^3)^(1/2)*(12*a^4 + 20*b^4 - 33*a^2*b^2))/(2*(a^12 - a^6*b^6 + 3*a^8*b^4 - 3*a^10*b^2))))
*(-(a + b)^3*(a - b)^3)^(1/2)*(12*a^4 + 20*b^4 - 33*a^2*b^2)*1i)/(d*(a^12 - a^6*b^6 + 3*a^8*b^4 - 3*a^10*b^2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-cos(d*x+c)**2)*sec(d*x+c)**4/(a+b*cos(d*x+c))**3,x)

[Out]

Timed out

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