\(\int \frac {\sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx\) [69]

Optimal result

Integrand size = 19, antiderivative size = 97 \[ \int \frac {\sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx=\frac {\text {arctanh}(\sin (c+d x))}{a^3 d}-\frac {\sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {7 \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {22 \sin (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )} \]

[Out]

arctanh(sin(d*x+c))/a^3/d-1/5*sin(d*x+c)/d/(a+a*cos(d*x+c))^3-7/15*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^2-22/15*sin
(d*x+c)/d/(a^3+a^3*cos(d*x+c))

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2845, 3057, 12, 3855} \[ \int \frac {\sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx=\frac {\text {arctanh}(\sin (c+d x))}{a^3 d}-\frac {22 \sin (c+d x)}{15 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac {7 \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac {\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3} \]

[In]

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

[Out]

ArcTanh[Sin[c + d*x]]/(a^3*d) - Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3) - (7*Sin[c + d*x])/(15*a*d*(a + a*Co
s[c + d*x])^2) - (22*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2845

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

Rule 3057

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

Rule 3855

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {\int \frac {(5 a-2 a \cos (c+d x)) \sec (c+d x)}{(a+a \cos (c+d x))^2} \, dx}{5 a^2} \\ & = -\frac {\sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {7 \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {\int \frac {\left (15 a^2-7 a^2 \cos (c+d x)\right ) \sec (c+d x)}{a+a \cos (c+d x)} \, dx}{15 a^4} \\ & = -\frac {\sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {7 \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {22 \sin (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac {\int 15 a^3 \sec (c+d x) \, dx}{15 a^6} \\ & = -\frac {\sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {7 \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {22 \sin (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac {\int \sec (c+d x) \, dx}{a^3} \\ & = \frac {\text {arctanh}(\sin (c+d x))}{a^3 d}-\frac {\sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {7 \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {22 \sin (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(201\) vs. \(2(97)=194\).

Time = 0.55 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.07 \[ \int \frac {\sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx=-\frac {2 \cos \left (\frac {1}{2} (c+d x)\right ) \left (60 \cos ^5\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+3 \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )+14 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )+88 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )+3 \cos \left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {c}{2}\right )+14 \cos ^3\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {c}{2}\right )\right )}{15 a^3 d (1+\cos (c+d x))^3} \]

[In]

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

[Out]

(-2*Cos[(c + d*x)/2]*(60*Cos[(c + d*x)/2]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] +
Sin[(c + d*x)/2]]) + 3*Sec[c/2]*Sin[(d*x)/2] + 14*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 88*Cos[(c + d*x)
/2]^4*Sec[c/2]*Sin[(d*x)/2] + 3*Cos[(c + d*x)/2]*Tan[c/2] + 14*Cos[(c + d*x)/2]^3*Tan[c/2]))/(15*a^3*d*(1 + Co
s[c + d*x])^3)

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(\frac {-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{4 d \,a^{3}}\)	\(75\)
default	\(\frac {-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{4 d \,a^{3}}\)	\(75\)
parallelrisch	\(\frac {-3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-20 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-60 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+60 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-105 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{60 a^{3} d}\)	\(75\)
norman	\(\frac {-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d a}-\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d a}-\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{20 d a}}{a^{2}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3} d}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3} d}\)	\(101\)
risch	\(-\frac {2 i \left (15 \,{\mathrm e}^{4 i \left (d x +c \right )}+75 \,{\mathrm e}^{3 i \left (d x +c \right )}+145 \,{\mathrm e}^{2 i \left (d x +c \right )}+95 \,{\mathrm e}^{i \left (d x +c \right )}+22\right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{a^{3} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,a^{3}}\)	\(111\)

[In]

int(sec(d*x+c)/(a+cos(d*x+c)*a)^3,x,method=_RETURNVERBOSE)

[Out]

1/4/d/a^3*(-1/5*tan(1/2*d*x+1/2*c)^5-4/3*tan(1/2*d*x+1/2*c)^3-7*tan(1/2*d*x+1/2*c)-4*ln(tan(1/2*d*x+1/2*c)-1)+
4*ln(tan(1/2*d*x+1/2*c)+1))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.63 \[ \int \frac {\sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx=\frac {15 \, {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (22 \, \cos \left (d x + c\right )^{2} + 51 \, \cos \left (d x + c\right ) + 32\right )} \sin \left (d x + c\right )}{30 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]

[In]

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

[Out]

1/30*(15*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*log(sin(d*x + c) + 1) - 15*(cos(d*x + c)^3 +
3*cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*log(-sin(d*x + c) + 1) - 2*(22*cos(d*x + c)^2 + 51*cos(d*x + c) + 32)*
sin(d*x + c))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)

Sympy [F]

\[ \int \frac {\sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx=\frac {\int \frac {\sec {\left (c + d x \right )}}{\cos ^{3}{\left (c + d x \right )} + 3 \cos ^{2}{\left (c + d x \right )} + 3 \cos {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]

[In]

integrate(sec(d*x+c)/(a+a*cos(d*x+c))**3,x)

[Out]

Integral(sec(c + d*x)/(cos(c + d*x)**3 + 3*cos(c + d*x)**2 + 3*cos(c + d*x) + 1), x)/a**3

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.23 \[ \int \frac {\sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx=-\frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {60 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} + \frac {60 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}}{60 \, d} \]

[In]

integrate(sec(d*x+c)/(a+a*cos(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/60*((105*sin(d*x + c)/(cos(d*x + c) + 1) + 20*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*sin(d*x + c)^5/(cos(d
*x + c) + 1)^5)/a^3 - 60*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^3 + 60*log(sin(d*x + c)/(cos(d*x + c) + 1)
- 1)/a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \frac {\sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx=\frac {\frac {60 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac {60 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} - \frac {3 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 20 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]

[In]

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

[Out]

1/60*(60*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^3 - 60*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^3 - (3*a^12*tan(1/2*
d*x + 1/2*c)^5 + 20*a^12*tan(1/2*d*x + 1/2*c)^3 + 105*a^12*tan(1/2*d*x + 1/2*c))/a^15)/d

Mupad [B] (verification not implemented)

Time = 14.18 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.60 \[ \int \frac {\sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx=-\frac {105\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-120\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{60\,a^3\,d} \]

[In]

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

[Out]

-(105*tan(c/2 + (d*x)/2) - 120*atanh(tan(c/2 + (d*x)/2)) + 20*tan(c/2 + (d*x)/2)^3 + 3*tan(c/2 + (d*x)/2)^5)/(
60*a^3*d)