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

[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]))

________________________________________________________________________________________

Rubi [A]  time = 1.39778, antiderivative size = 335, normalized size of antiderivative = 1., 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 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 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 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 3770

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

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 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]

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*}

Mathematica [A]  time = 6.24713, size = 563, normalized size = 1.68 \[ \frac{b^3 \sin (c+d x)}{2 a^4 d (a+b \cos (c+d x))^2}+\frac{18 b^2 \sin \left (\frac{1}{2} (c+d x)\right )-a^2 \sin \left (\frac{1}{2} (c+d x)\right )}{3 a^5 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{18 b^2 \sin \left (\frac{1}{2} (c+d x)\right )-a^2 \sin \left (\frac{1}{2} (c+d x)\right )}{3 a^5 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{7 a^2 b^3 \sin (c+d x)-8 b^5 \sin (c+d x)}{2 a^5 d (a-b) (a+b) (a+b \cos (c+d x))}+\frac{b^2 \left (-33 a^2 b^2+12 a^4+20 b^4\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{a^6 d \left (a^2-b^2\right ) \sqrt{b^2-a^2}}+\frac{\left (20 b^3-3 a^2 b\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac{\left (3 a^2 b-20 b^3\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac{a-9 b}{12 a^4 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{9 b-a}{12 a^4 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{\sin \left (\frac{1}{2} (c+d x)\right )}{6 a^3 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{\sin \left (\frac{1}{2} (c+d x)\right )}{6 a^3 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3} \]

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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Maple [B]  time = 0.08, size = 843, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

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((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*b^2+33/d/a^4/(a^
2-b^2)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((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((a-b)*tan(1/2*d*x+1/2*c)/((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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [B]  time = 7.26328, size = 3258, normalized size = 9.73 \begin{align*} \text{result too large to display} \end{align*}

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

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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Giac [A]  time = 1.54813, size = 636, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

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