\(\int \frac {\sec (c+d x) (A+C \sec ^2(c+d x))}{(a+a \sec (c+d x))^2} \, dx\) [133]

Optimal result

Integrand size = 31, antiderivative size = 75 \[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {C \text {arctanh}(\sin (c+d x))}{a^2 d}+\frac {(A-5 C) \tan (c+d x)}{3 a^2 d (1+\sec (c+d x))}+\frac {(A+C) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2} \]

[Out]

C*arctanh(sin(d*x+c))/a^2/d+1/3*(A-5*C)*tan(d*x+c)/a^2/d/(1+sec(d*x+c))+1/3*(A+C)*tan(d*x+c)/d/(a+a*sec(d*x+c)
)^2

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {4164, 4083, 3855, 3879} \[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {2 (A-2 C) \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac {C \text {arctanh}(\sin (c+d x))}{a^2 d}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]

[In]

Int[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]

[Out]

(C*ArcTanh[Sin[c + d*x]])/(a^2*d) + (2*(A - 2*C)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c +
d*x]*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)

Rule 3855

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

Rule 3879

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[-Cot[e + f*x]/(f*(b + a*
Csc[e + f*x])), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]

Rule 4083

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x
_Symbol] :> Dist[B/b, Int[Csc[e + f*x], x], x] + Dist[(A*b - a*B)/b, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x]
, x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0]

Rule 4164

Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(
m_), x_Symbol] :> Simp[(-(A + C))*Cot[e + f*x]*Csc[e + f*x]*((a + b*Csc[e + f*x])^m/(f*(2*m + 1))), x] - Dist[
1/(a*b*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[(-b)*C - 2*A*b*(m + 1) + a*(A*(m + 2) -
C*(m - 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {(A+C) \sec (c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {\int \frac {\sec (c+d x) (a (2 A-C)+3 a C \sec (c+d x))}{a+a \sec (c+d x)} \, dx}{3 a^2} \\ & = -\frac {(A+C) \sec (c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {(2 (A-2 C)) \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{3 a}+\frac {C \int \sec (c+d x) \, dx}{a^2} \\ & = \frac {C \text {arctanh}(\sin (c+d x))}{a^2 d}-\frac {(A+C) \sec (c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {2 (A-2 C) \tan (c+d x)}{3 d \left (a^2+a^2 \sec (c+d x)\right )} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(377\) vs. \(2(75)=150\).

Time = 2.54 (sec) , antiderivative size = 377, normalized size of antiderivative = 5.03 \[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=-\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (3 C \cos \left (c+\frac {3 d x}{2}\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+3 C \cos \left (2 c+\frac {3 d x}{2}\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+9 C \cos \left (\frac {d x}{2}\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+9 C \cos \left (c+\frac {d x}{2}\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-3 C \cos \left (c+\frac {3 d x}{2}\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-3 C \cos \left (2 c+\frac {3 d x}{2}\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-6 A \sin \left (\frac {d x}{2}\right )+18 C \sin \left (\frac {d x}{2}\right )+6 A \sin \left (c+\frac {d x}{2}\right )-6 C \sin \left (c+\frac {d x}{2}\right )-4 A \sin \left (c+\frac {3 d x}{2}\right )+8 C \sin \left (c+\frac {3 d x}{2}\right )\right )}{6 a^2 d (1+\sec (c+d x))^2} \]

[In]

Integrate[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]

[Out]

-1/6*(Cos[(c + d*x)/2]*Sec[c/2]*Sec[c + d*x]^2*(3*C*Cos[c + (3*d*x)/2]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]
] + 3*C*Cos[2*c + (3*d*x)/2]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 9*C*Cos[(d*x)/2]*(Log[Cos[(c + d*x)/2]
- Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 9*C*Cos[c + (d*x)/2]*(Log[Cos[(c + d*x)/2]
- Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 3*C*Cos[c + (3*d*x)/2]*Log[Cos[(c + d*x)/2]
+ Sin[(c + d*x)/2]] - 3*C*Cos[2*c + (3*d*x)/2]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 6*A*Sin[(d*x)/2] + 1
8*C*Sin[(d*x)/2] + 6*A*Sin[c + (d*x)/2] - 6*C*Sin[c + (d*x)/2] - 4*A*Sin[c + (3*d*x)/2] + 8*C*Sin[c + (3*d*x)/
2]))/(a^2*d*(1 + Sec[c + d*x])^2)

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97

method	result	size
parallelrisch	\(\frac {-6 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+6 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-3 A +9 C \right )}{6 a^{2} d}\)	\(73\)
derivativedivides	\(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -2 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+2 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d \,a^{2}}\)	\(91\)
default	\(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -2 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+2 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d \,a^{2}}\)	\(91\)
risch	\(\frac {2 i \left (3 A \,{\mathrm e}^{2 i \left (d x +c \right )}-3 C \,{\mathrm e}^{2 i \left (d x +c \right )}+3 A \,{\mathrm e}^{i \left (d x +c \right )}-9 C \,{\mathrm e}^{i \left (d x +c \right )}+2 A -4 C \right )}{3 d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{a^{2} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{a^{2} d}\)	\(122\)
norman	\(\frac {\frac {\left (A -3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d}-\frac {\left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 a d}+\frac {\left (5 A -7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{6 a d}-\frac {\left (7 A -17 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 a d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2} a}+\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2} d}-\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2} d}\)	\(159\)

[In]

int(sec(d*x+c)*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/6*(-6*C*ln(tan(1/2*d*x+1/2*c)-1)+6*C*ln(tan(1/2*d*x+1/2*c)+1)-tan(1/2*d*x+1/2*c)*((A+C)*tan(1/2*d*x+1/2*c)^2
-3*A+9*C))/a^2/d

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.71 \[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {3 \, {\left (C \cos \left (d x + c\right )^{2} + 2 \, C \cos \left (d x + c\right ) + C\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (C \cos \left (d x + c\right )^{2} + 2 \, C \cos \left (d x + c\right ) + C\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, {\left (A - 2 \, C\right )} \cos \left (d x + c\right ) + A - 5 \, C\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]

[In]

integrate(sec(d*x+c)*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

1/6*(3*(C*cos(d*x + c)^2 + 2*C*cos(d*x + c) + C)*log(sin(d*x + c) + 1) - 3*(C*cos(d*x + c)^2 + 2*C*cos(d*x + c
) + C)*log(-sin(d*x + c) + 1) + 2*(2*(A - 2*C)*cos(d*x + c) + A - 5*C)*sin(d*x + c))/(a^2*d*cos(d*x + c)^2 + 2
*a^2*d*cos(d*x + c) + a^2*d)

Sympy [F]

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

[In]

integrate(sec(d*x+c)*(A+C*sec(d*x+c)**2)/(a+a*sec(d*x+c))**2,x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (71) = 142\).

Time = 0.21 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.95 \[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=-\frac {C {\left (\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {6 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac {6 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}}\right )} - \frac {A {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2}}}{6 \, d} \]

[In]

integrate(sec(d*x+c)*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/6*(C*((9*sin(d*x + c)/(cos(d*x + c) + 1) + sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/a^2 - 6*log(sin(d*x + c)/(c
os(d*x + c) + 1) + 1)/a^2 + 6*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^2) - A*(3*sin(d*x + c)/(cos(d*x + c)
+ 1) - sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/a^2)/d

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.49 \[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {\frac {6 \, C \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{2}} - \frac {6 \, C \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{2}} - \frac {A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \]

[In]

integrate(sec(d*x+c)*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^2,x, algorithm="giac")

[Out]

1/6*(6*C*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^2 - 6*C*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^2 - (A*a^4*tan(1/2*
d*x + 1/2*c)^3 + C*a^4*tan(1/2*d*x + 1/2*c)^3 - 3*A*a^4*tan(1/2*d*x + 1/2*c) + 9*C*a^4*tan(1/2*d*x + 1/2*c))/a
^6)/d

Mupad [B] (verification not implemented)

Time = 14.96 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03 \[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {2\,C\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {A+C}{2\,a^2}-\frac {2\,A-2\,C}{2\,a^2}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (A+C\right )}{6\,a^2\,d} \]

[In]

int((A + C/cos(c + d*x)^2)/(cos(c + d*x)*(a + a/cos(c + d*x))^2),x)

[Out]

(2*C*atanh(tan(c/2 + (d*x)/2)))/(a^2*d) - (tan(c/2 + (d*x)/2)*((A + C)/(2*a^2) - (2*A - 2*C)/(2*a^2)))/d - (ta
n(c/2 + (d*x)/2)^3*(A + C))/(6*a^2*d)