\(\int \frac {\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [783]

Optimal result

Integrand size = 25, antiderivative size = 71 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\sec (c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac {2 \sec (c+d x)}{15 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {4 \tan (c+d x)}{15 a^2 d} \]

[Out]

1/5*sec(d*x+c)/d/(a+a*sin(d*x+c))^2-2/15*sec(d*x+c)/d/(a^2+a^2*sin(d*x+c))+4/15*tan(d*x+c)/a^2/d

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2938, 2751, 3852, 8} \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {4 \tan (c+d x)}{15 a^2 d}-\frac {2 \sec (c+d x)}{15 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {\sec (c+d x)}{5 d (a \sin (c+d x)+a)^2} \]

[In]

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

[Out]

Sec[c + d*x]/(5*d*(a + a*Sin[c + d*x])^2) - (2*Sec[c + d*x])/(15*d*(a^2 + a^2*Sin[c + d*x])) + (4*Tan[c + d*x]
)/(15*a^2*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2751

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[2*m + p + 1]), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m
, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]

Rule 2938

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(2*m + p
+ 1))), x] + Dist[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^
(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[
m + p], 0]) && NeQ[2*m + p + 1, 0]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sec (c+d x)}{5 d (a+a \sin (c+d x))^2}+\frac {2 \int \frac {\sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx}{5 a} \\ & = \frac {\sec (c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac {2 \sec (c+d x)}{15 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {4 \int \sec ^2(c+d x) \, dx}{15 a^2} \\ & = \frac {\sec (c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac {2 \sec (c+d x)}{15 d \left (a^2+a^2 \sin (c+d x)\right )}-\frac {4 \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 a^2 d} \\ & = \frac {\sec (c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac {2 \sec (c+d x)}{15 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {4 \tan (c+d x)}{15 a^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.15 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\sec (c+d x) (-80-5 \cos (c+d x)+64 \cos (2 (c+d x))+\cos (3 (c+d x))-80 \sin (c+d x)-4 \sin (2 (c+d x))+16 \sin (3 (c+d x)))}{240 a^2 d (1+\sin (c+d x))^2} \]

[In]

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

[Out]

-1/240*(Sec[c + d*x]*(-80 - 5*Cos[c + d*x] + 64*Cos[2*(c + d*x)] + Cos[3*(c + d*x)] - 80*Sin[c + d*x] - 4*Sin[
2*(c + d*x)] + 16*Sin[3*(c + d*x)]))/(a^2*d*(1 + Sin[c + d*x])^2)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.04

method	result	size
risch	\(-\frac {8 \left (5 i {\mathrm e}^{2 i \left (d x +c \right )}+5 \,{\mathrm e}^{3 i \left (d x +c \right )}-i-4 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{15 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{5} d \,a^{2}}\)	\(74\)
parallelrisch	\(\frac {-2-30 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{15 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\)	\(87\)
derivativedivides	\(\frac {-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {4}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {7}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {3}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {4}{16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+16}}{d \,a^{2}}\)	\(100\)
default	\(\frac {-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {4}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {7}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {3}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {4}{16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+16}}{d \,a^{2}}\)	\(100\)
norman	\(\frac {-\frac {2 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {2}{15 a d}-\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{15 d a}-\frac {8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5} a}\)	\(114\)

[In]

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

[Out]

-8/15*(5*I*exp(2*I*(d*x+c))+5*exp(3*I*(d*x+c))-I-4*exp(I*(d*x+c)))/(exp(I*(d*x+c))-I)/(exp(I*(d*x+c))+I)^5/d/a
^2

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.13 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {8 \, \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, \cos \left (d x + c\right )^{2} - 3\right )} \sin \left (d x + c\right ) - 9}{15 \, {\left (a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )\right )}} \]

[In]

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

[Out]

1/15*(8*cos(d*x + c)^2 + 2*(2*cos(d*x + c)^2 - 3)*sin(d*x + c) - 9)/(a^2*d*cos(d*x + c)^3 - 2*a^2*d*cos(d*x +
c)*sin(d*x + c) - 2*a^2*d*cos(d*x + c))

Sympy [F]

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

[In]

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

[Out]

Integral(sin(c + d*x)*sec(c + d*x)**2/(sin(c + d*x)**2 + 2*sin(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. 204 vs. \(2 (65) = 130\).

Time = 0.23 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.87 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {2 \, {\left (\frac {4 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {20 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 1\right )}}{15 \, {\left (a^{2} + \frac {4 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {5 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} d} \]

[In]

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

[Out]

2/15*(4*sin(d*x + c)/(cos(d*x + c) + 1) + 20*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 20*sin(d*x + c)^3/(cos(d*x
+ c) + 1)^3 + 15*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1)/((a^2 + 4*a^2*sin(d*x + c)/(cos(d*x + c) + 1) + 5*a^
2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 5*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 4*a^2*sin(d*x + c)^5/(cos(
d*x + c) + 1)^5 - a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6)*d)

Giac [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.32 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {15}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}} - \frac {15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 30 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 50 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 7}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{60 \, d} \]

[In]

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

[Out]

-1/60*(15/(a^2*(tan(1/2*d*x + 1/2*c) - 1)) - (15*tan(1/2*d*x + 1/2*c)^4 - 30*tan(1/2*d*x + 1/2*c)^3 - 40*tan(1
/2*d*x + 1/2*c)^2 - 50*tan(1/2*d*x + 1/2*c) - 7)/(a^2*(tan(1/2*d*x + 1/2*c) + 1)^5))/d

Mupad [B] (verification not implemented)

Time = 11.71 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.24 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{15}+\frac {8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{15}+\frac {8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\frac {8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{a^2\,d\,\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^5} \]

[In]

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

[Out]

((2*cos(c/2 + (d*x)/2)^6)/15 + (8*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2))/15 + 2*cos(c/2 + (d*x)/2)^2*sin(c/2
+ (d*x)/2)^4 + (8*cos(c/2 + (d*x)/2)^3*sin(c/2 + (d*x)/2)^3)/3 + (8*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^2
)/3)/(a^2*d*(cos(c/2 + (d*x)/2) - sin(c/2 + (d*x)/2))*(cos(c/2 + (d*x)/2) + sin(c/2 + (d*x)/2))^5)