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

Optimal. Leaf size=160 \[ -\frac {3 b \tan (c+d x)}{a^3 d}+\frac {3 \tan (c+d x) \sec (c+d x)}{2 a^2 d}+\frac {2 b \left (2 a^2-3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d \sqrt {a-b} \sqrt {a+b}}-\frac {\left (a^2-6 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac {\tan (c+d x) \sec (c+d x)}{a d (a+b \cos (c+d x))} \]

[Out]

-1/2*(a^2-6*b^2)*arctanh(sin(d*x+c))/a^4/d+2*b*(2*a^2-3*b^2)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2)
)/a^4/d/(a-b)^(1/2)/(a+b)^(1/2)-3*b*tan(d*x+c)/a^3/d+3/2*sec(d*x+c)*tan(d*x+c)/a^2/d-sec(d*x+c)*tan(d*x+c)/a/d
/(a+b*cos(d*x+c))

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Rubi [A]  time = 0.67, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ \frac {2 b \left (2 a^2-3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d \sqrt {a-b} \sqrt {a+b}}-\frac {\left (a^2-6 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac {3 b \tan (c+d x)}{a^3 d}+\frac {3 \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac {\tan (c+d x) \sec (c+d x)}{a d (a+b \cos (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*(2*a^2 - 3*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) - ((a
^2 - 6*b^2)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (3*b*Tan[c + d*x])/(a^3*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(2*a
^2*d) - (Sec[c + d*x]*Tan[c + d*x])/(a*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 ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx &=-\frac {\sec (c+d x) \tan (c+d x)}{a d (a+b \cos (c+d x))}+\frac {\int \frac {\left (3 \left (a^2-b^2\right )-2 \left (a^2-b^2\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx}{a \left (a^2-b^2\right )}\\ &=\frac {3 \sec (c+d x) \tan (c+d x)}{2 a^2 d}-\frac {\sec (c+d x) \tan (c+d x)}{a d (a+b \cos (c+d x))}+\frac {\int \frac {\left (-6 b \left (a^2-b^2\right )-a \left (a^2-b^2\right ) \cos (c+d x)+3 b \left (a^2-b^2\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )}\\ &=-\frac {3 b \tan (c+d x)}{a^3 d}+\frac {3 \sec (c+d x) \tan (c+d x)}{2 a^2 d}-\frac {\sec (c+d x) \tan (c+d x)}{a d (a+b \cos (c+d x))}+\frac {\int \frac {\left (-a^4+7 a^2 b^2-6 b^4+3 a b \left (a^2-b^2\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )}\\ &=-\frac {3 b \tan (c+d x)}{a^3 d}+\frac {3 \sec (c+d x) \tan (c+d x)}{2 a^2 d}-\frac {\sec (c+d x) \tan (c+d x)}{a d (a+b \cos (c+d x))}-\frac {\left (a^2-6 b^2\right ) \int \sec (c+d x) \, dx}{2 a^4}+\frac {\left (b \left (2 a^2-3 b^2\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{a^4}\\ &=-\frac {\left (a^2-6 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac {3 b \tan (c+d x)}{a^3 d}+\frac {3 \sec (c+d x) \tan (c+d x)}{2 a^2 d}-\frac {\sec (c+d x) \tan (c+d x)}{a d (a+b \cos (c+d x))}+\frac {\left (2 b \left (2 a^2-3 b^2\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^4 d}\\ &=\frac {2 b \left (2 a^2-3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 \sqrt {a-b} \sqrt {a+b} d}-\frac {\left (a^2-6 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac {3 b \tan (c+d x)}{a^3 d}+\frac {3 \sec (c+d x) \tan (c+d x)}{2 a^2 d}-\frac {\sec (c+d x) \tan (c+d x)}{a d (a+b \cos (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 3.48, size = 271, normalized size = 1.69 \[ \frac {\frac {8 b \left (3 b^2-2 a^2\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\sqrt {b^2-a^2}}+\frac {a^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {a^2}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}+2 a^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-2 a^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )-\frac {4 a b^2 \sin (c+d x)}{a+b \cos (c+d x)}-8 a b \tan (c+d x)-12 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{4 a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*b*(-2*a^2 + 3*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 2*a^2*Log[Cos[
(c + d*x)/2] - Sin[(c + d*x)/2]] - 12*b^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2*a^2*Log[Cos[(c + d*x)/2
] + Sin[(c + d*x)/2]] + 12*b^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + a^2/(Cos[(c + d*x)/2] - Sin[(c + d*x
)/2])^2 - a^2/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (4*a*b^2*Sin[c + d*x])/(a + b*Cos[c + d*x]) - 8*a*b*Ta
n[c + d*x])/(4*a^4*d)

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fricas [B]  time = 0.76, size = 757, normalized size = 4.73 \[ \left [\frac {2 \, {\left ({\left (2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (d x + c\right )^{3} + {\left (2 \, a^{3} b - 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}\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 ) - {\left ({\left (a^{4} b - 7 \, a^{2} b^{3} + 6 \, b^{5}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{5} - 7 \, a^{3} b^{2} + 6 \, a b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (a^{4} b - 7 \, a^{2} b^{3} + 6 \, b^{5}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{5} - 7 \, a^{3} b^{2} + 6 \, a b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (a^{5} - a^{3} b^{2} - 6 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (a^{4} b - a^{2} b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{6} b - a^{4} b^{3}\right )} d \cos \left (d x + c\right )^{3} + {\left (a^{7} - a^{5} b^{2}\right )} d \cos \left (d x + c\right )^{2}\right )}}, \frac {4 \, {\left ({\left (2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (d x + c\right )^{3} + {\left (2 \, a^{3} b - 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}\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 ({\left (a^{4} b - 7 \, a^{2} b^{3} + 6 \, b^{5}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{5} - 7 \, a^{3} b^{2} + 6 \, a b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (a^{4} b - 7 \, a^{2} b^{3} + 6 \, b^{5}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{5} - 7 \, a^{3} b^{2} + 6 \, a b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (a^{5} - a^{3} b^{2} - 6 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (a^{4} b - a^{2} b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{6} b - a^{4} b^{3}\right )} d \cos \left (d x + c\right )^{3} + {\left (a^{7} - a^{5} b^{2}\right )} d \cos \left (d x + c\right )^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.58, size = 269, normalized size = 1.68 \[ -\frac {\frac {4 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )} a^{3}} + \frac {{\left (a^{2} - 6 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {{\left (a^{2} - 6 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {4 \, {\left (2 \, a^{2} b - 3 \, b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{4}} - \frac {2 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{3}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*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)*a^3) + (a^2 -
6*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^4 - (a^2 - 6*b^2)*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^4 + 4*(2*a^
2*b - 3*b^3)*(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)))/(sqrt(a^2 - b^2)*a^4) - 2*(a*tan(1/2*d*x + 1/2*c)^3 + 4*b*tan(1/2*d*x + 1/2*c)^
3 + a*tan(1/2*d*x + 1/2*c) - 4*b*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 - 1)^2*a^3))/d

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maple [B]  time = 0.19, size = 364, normalized size = 2.28 \[ -\frac {2 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{3} \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}+\frac {4 b \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d \,a^{2} \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {6 b^{3} \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d \,a^{4} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {1}{2 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2 b}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d \,a^{2}}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b^{2}}{d \,a^{4}}-\frac {1}{2 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {2 b}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d \,a^{2}}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b^{2}}{d \,a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^3/(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 details)Is 4*b^2-4*a^2 positive or negative?

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mupad [B]  time = 3.32, size = 1662, normalized size = 10.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*tan(c/2 + (d*x)/2)^3*(a^2 + 6*b^2))/a^3 - (tan(c/2 + (d*x)/2)*(3*a*b - a^2 + 6*b^2))/a^3 + (tan(c/2 + (d*x
)/2)^5*(3*a*b + a^2 - 6*b^2))/a^3)/(d*(a + b - tan(c/2 + (d*x)/2)^2*(a + 3*b) - tan(c/2 + (d*x)/2)^4*(a - 3*b)
+ tan(c/2 + (d*x)/2)^6*(a - b))) + (atanh((8*tan(c/2 + (d*x)/2))/((8*b)/a + (24*b^2)/a^2 - (24*b^3)/a^3 + (14
4*b^4)/a^4 - (144*b^5)/a^5 - 8) + (8*b*tan(c/2 + (d*x)/2))/(8*a - 8*b - (24*b^2)/a + (24*b^3)/a^2 - (144*b^4)/
a^3 + (144*b^5)/a^4) - (24*b^2*tan(c/2 + (d*x)/2))/(8*a*b - 8*a^2 + 24*b^2 - (24*b^3)/a + (144*b^4)/a^2 - (144
*b^5)/a^3) + (24*b^3*tan(c/2 + (d*x)/2))/(24*a*b^2 + 8*a^2*b - 8*a^3 - 24*b^3 + (144*b^4)/a - (144*b^5)/a^2) +
(144*b^4*tan(c/2 + (d*x)/2))/(24*a*b^3 - 8*a^3*b + 8*a^4 - 144*b^4 - 24*a^2*b^2 + (144*b^5)/a) + (144*b^5*tan
(c/2 + (d*x)/2))/(144*a*b^4 + 8*a^4*b - 8*a^5 - 144*b^5 - 24*a^2*b^3 + 24*a^3*b^2))*(a^2 - 6*b^2))/(a^4*d) - (
b*atan(((b*(-(a + b)*(a - b))^(1/2)*(2*a^2 - 3*b^2)*((8*tan(c/2 + (d*x)/2)*(144*a*b^6 - 3*a^6*b + a^7 - 72*b^7
- 48*a^2*b^5 - 48*a^3*b^4 + 19*a^4*b^3 + 7*a^5*b^2))/a^6 + (b*(-(a + b)*(a - b))^(1/2)*((8*(2*a^12 - 10*a^11*
b - 12*a^8*b^4 + 18*a^9*b^3 + 2*a^10*b^2))/a^9 + (8*b*tan(c/2 + (d*x)/2)*(-(a + b)*(a - b))^(1/2)*(2*a^2 - 3*b
^2)*(8*a^10*b + 8*a^8*b^3 - 16*a^9*b^2))/(a^6*(a^6 - a^4*b^2)))*(2*a^2 - 3*b^2))/(a^6 - a^4*b^2))*1i)/(a^6 - a
^4*b^2) + (b*(-(a + b)*(a - b))^(1/2)*(2*a^2 - 3*b^2)*((8*tan(c/2 + (d*x)/2)*(144*a*b^6 - 3*a^6*b + a^7 - 72*b
^7 - 48*a^2*b^5 - 48*a^3*b^4 + 19*a^4*b^3 + 7*a^5*b^2))/a^6 - (b*(-(a + b)*(a - b))^(1/2)*((8*(2*a^12 - 10*a^1
1*b - 12*a^8*b^4 + 18*a^9*b^3 + 2*a^10*b^2))/a^9 - (8*b*tan(c/2 + (d*x)/2)*(-(a + b)*(a - b))^(1/2)*(2*a^2 - 3
*b^2)*(8*a^10*b + 8*a^8*b^3 - 16*a^9*b^2))/(a^6*(a^6 - a^4*b^2)))*(2*a^2 - 3*b^2))/(a^6 - a^4*b^2))*1i)/(a^6 -
a^4*b^2))/((16*(162*a*b^7 - 2*a^7*b - 108*b^8 + 54*a^2*b^6 - 153*a^3*b^5 + 18*a^4*b^4 + 33*a^5*b^3 - 4*a^6*b^
2))/a^9 - (b*(-(a + b)*(a - b))^(1/2)*(2*a^2 - 3*b^2)*((8*tan(c/2 + (d*x)/2)*(144*a*b^6 - 3*a^6*b + a^7 - 72*b
^7 - 48*a^2*b^5 - 48*a^3*b^4 + 19*a^4*b^3 + 7*a^5*b^2))/a^6 + (b*(-(a + b)*(a - b))^(1/2)*((8*(2*a^12 - 10*a^1
1*b - 12*a^8*b^4 + 18*a^9*b^3 + 2*a^10*b^2))/a^9 + (8*b*tan(c/2 + (d*x)/2)*(-(a + b)*(a - b))^(1/2)*(2*a^2 - 3
*b^2)*(8*a^10*b + 8*a^8*b^3 - 16*a^9*b^2))/(a^6*(a^6 - a^4*b^2)))*(2*a^2 - 3*b^2))/(a^6 - a^4*b^2)))/(a^6 - a^
4*b^2) + (b*(-(a + b)*(a - b))^(1/2)*(2*a^2 - 3*b^2)*((8*tan(c/2 + (d*x)/2)*(144*a*b^6 - 3*a^6*b + a^7 - 72*b^
7 - 48*a^2*b^5 - 48*a^3*b^4 + 19*a^4*b^3 + 7*a^5*b^2))/a^6 - (b*(-(a + b)*(a - b))^(1/2)*((8*(2*a^12 - 10*a^11
*b - 12*a^8*b^4 + 18*a^9*b^3 + 2*a^10*b^2))/a^9 - (8*b*tan(c/2 + (d*x)/2)*(-(a + b)*(a - b))^(1/2)*(2*a^2 - 3*
b^2)*(8*a^10*b + 8*a^8*b^3 - 16*a^9*b^2))/(a^6*(a^6 - a^4*b^2)))*(2*a^2 - 3*b^2))/(a^6 - a^4*b^2)))/(a^6 - a^4
*b^2)))*(-(a + b)*(a - b))^(1/2)*(2*a^2 - 3*b^2)*2i)/(d*(a^6 - a^4*b^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {\sec ^{3}{\left (c + d x \right )}}{a^{2} + 2 a b \cos {\left (c + d x \right )} + b^{2} \cos ^{2}{\left (c + d x \right )}}\right )\, dx - \int \frac {\cos ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{a^{2} + 2 a b \cos {\left (c + d x \right )} + b^{2} \cos ^{2}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-sec(c + d*x)**3/(a**2 + 2*a*b*cos(c + d*x) + b**2*cos(c + d*x)**2), x) - Integral(cos(c + d*x)**2*s
ec(c + d*x)**3/(a**2 + 2*a*b*cos(c + d*x) + b**2*cos(c + d*x)**2), x)

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