\(\int \frac {\sec ^3(c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx\) [1355]

Optimal result

Integrand size = 27, antiderivative size = 138 \[ \int \frac {\sec ^3(c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 a b^3 \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2} d}+\frac {\sec ^3(c+d x) (a-b \sin (c+d x))}{3 \left (a^2-b^2\right ) d}-\frac {\sec (c+d x) \left (3 a b^2-b \left (a^2+2 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d} \]

[Out]

-2*a*b^3*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/(a^2-b^2)^(5/2)/d+1/3*sec(d*x+c)^3*(a-b*sin(d*x+c))/
(a^2-b^2)/d-1/3*sec(d*x+c)*(3*a*b^2-b*(a^2+2*b^2)*sin(d*x+c))/(a^2-b^2)^2/d

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2945, 12, 2739, 632, 210} \[ \int \frac {\sec ^3(c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 a b^3 \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{d \left (a^2-b^2\right )^{5/2}}+\frac {\sec ^3(c+d x) (a-b \sin (c+d x))}{3 d \left (a^2-b^2\right )}-\frac {\sec (c+d x) \left (3 a b^2-b \left (a^2+2 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )^2} \]

[In]

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

[Out]

(-2*a*b^3*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) + (Sec[c + d*x]^3*(a - b*Sin
[c + d*x]))/(3*(a^2 - b^2)*d) - (Sec[c + d*x]*(3*a*b^2 - b*(a^2 + 2*b^2)*Sin[c + d*x]))/(3*(a^2 - b^2)^2*d)

Rule 12

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

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*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 2945

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c - a*d - (a*c -
b*d)*Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Dist[1/(g^2*(a^2 - b^2)*(p + 1)), Int[(g*Cos[e + f*x])^(p
+ 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e
+ f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegerQ[2*m]

Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^3(c+d x) (a-b \sin (c+d x))}{3 \left (a^2-b^2\right ) d}-\frac {\int \frac {\sec ^2(c+d x) \left (-a b+2 b^2 \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{3 \left (a^2-b^2\right )} \\ & = \frac {\sec ^3(c+d x) (a-b \sin (c+d x))}{3 \left (a^2-b^2\right ) d}-\frac {\sec (c+d x) \left (3 a b^2-b \left (a^2+2 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {\int -\frac {3 a b^3}{a+b \sin (c+d x)} \, dx}{3 \left (a^2-b^2\right )^2} \\ & = \frac {\sec ^3(c+d x) (a-b \sin (c+d x))}{3 \left (a^2-b^2\right ) d}-\frac {\sec (c+d x) \left (3 a b^2-b \left (a^2+2 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}-\frac {\left (a b^3\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{\left (a^2-b^2\right )^2} \\ & = \frac {\sec ^3(c+d x) (a-b \sin (c+d x))}{3 \left (a^2-b^2\right ) d}-\frac {\sec (c+d x) \left (3 a b^2-b \left (a^2+2 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}-\frac {\left (2 a b^3\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{\left (a^2-b^2\right )^2 d} \\ & = \frac {\sec ^3(c+d x) (a-b \sin (c+d x))}{3 \left (a^2-b^2\right ) d}-\frac {\sec (c+d x) \left (3 a b^2-b \left (a^2+2 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {\left (4 a b^3\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{\left (a^2-b^2\right )^2 d} \\ & = -\frac {2 a b^3 \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2} d}+\frac {\sec ^3(c+d x) (a-b \sin (c+d x))}{3 \left (a^2-b^2\right ) d}-\frac {\sec (c+d x) \left (3 a b^2-b \left (a^2+2 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.12 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.47 \[ \int \frac {\sec ^3(c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-\frac {24 a b^3 \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {\sec ^3(c+d x) \left (4 a^3-10 a b^2-\frac {3}{2} a \left (a^2-7 b^2\right ) \cos (c+d x)-6 a b^2 \cos (2 (c+d x))-\frac {1}{2} a^3 \cos (3 (c+d x))+\frac {7}{2} a b^2 \cos (3 (c+d x))-3 a^2 b \sin (c+d x)+6 b^3 \sin (c+d x)+a^2 b \sin (3 (c+d x))+2 b^3 \sin (3 (c+d x))\right )}{(a-b)^2 (a+b)^2}}{12 d} \]

[In]

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

[Out]

((-24*a*b^3*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (Sec[c + d*x]^3*(4*a^3 - 10*
a*b^2 - (3*a*(a^2 - 7*b^2)*Cos[c + d*x])/2 - 6*a*b^2*Cos[2*(c + d*x)] - (a^3*Cos[3*(c + d*x)])/2 + (7*a*b^2*Co
s[3*(c + d*x)])/2 - 3*a^2*b*Sin[c + d*x] + 6*b^3*Sin[c + d*x] + a^2*b*Sin[3*(c + d*x)] + 2*b^3*Sin[3*(c + d*x)
]))/((a - b)^2*(a + b)^2))/(12*d)

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.56

method	result	size
derivativedivides	\(\frac {-\frac {2}{\left (4 a -4 b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {4}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3} \left (4 a -4 b \right )}-\frac {-a +2 b}{2 \left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 a \,b^{3} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2} \sqrt {a^{2}-b^{2}}}-\frac {4}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (4 a +4 b \right )}-\frac {2}{\left (4 a +4 b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a +2 b}{2 \left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(215\)
default	\(\frac {-\frac {2}{\left (4 a -4 b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {4}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3} \left (4 a -4 b \right )}-\frac {-a +2 b}{2 \left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 a \,b^{3} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2} \sqrt {a^{2}-b^{2}}}-\frac {4}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (4 a +4 b \right )}-\frac {2}{\left (4 a +4 b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a +2 b}{2 \left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(215\)
risch	\(\frac {2 i \left (3 i b^{2} a \,{\mathrm e}^{5 i \left (d x +c \right )}-4 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+10 i b^{2} a \,{\mathrm e}^{3 i \left (d x +c \right )}+3 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 i a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}+6 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+a^{2} b +2 b^{3}\right )}{3 d \left (-a^{2}+b^{2}\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {i b^{3} a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}+\frac {i b^{3} a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}\)	\(301\)

[In]

int(sec(d*x+c)^4*sin(d*x+c)/(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 469, normalized size of antiderivative = 3.40 \[ \int \frac {\sec ^3(c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx=\left [-\frac {3 \, \sqrt {-a^{2} + b^{2}} a b^{3} \cos \left (d x + c\right )^{3} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 2 \, a^{5} + 4 \, a^{3} b^{2} - 2 \, a b^{4} + 6 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5} - {\left (a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} d \cos \left (d x + c\right )^{3}}, \frac {3 \, \sqrt {a^{2} - b^{2}} a b^{3} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) \cos \left (d x + c\right )^{3} + a^{5} - 2 \, a^{3} b^{2} + a b^{4} - 3 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{2} - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5} - {\left (a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{3 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} d \cos \left (d x + c\right )^{3}}\right ] \]

[In]

integrate(sec(d*x+c)^4*sin(d*x+c)/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\sec ^3(c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\sin {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]

[In]

integrate(sec(d*x+c)**4*sin(d*x+c)/(a+b*sin(d*x+c)),x)

[Out]

Integral(sin(c + d*x)*sec(c + d*x)**4/(a + b*sin(c + d*x)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sec ^3(c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(sec(d*x+c)^4*sin(d*x+c)/(a+b*sin(d*x+c)),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 de

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.74 \[ \int \frac {\sec ^3(c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 \, {\left (\frac {3 \, {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} a b^{3}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} - b^{2}}} + \frac {3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 6 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 4 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3} - 4 \, a b^{2}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}\right )}}{3 \, d} \]

[In]

integrate(sec(d*x+c)^4*sin(d*x+c)/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

-2/3*(3*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))*a*b^3
/((a^4 - 2*a^2*b^2 + b^4)*sqrt(a^2 - b^2)) + (3*b^3*tan(1/2*d*x + 1/2*c)^5 + 3*a^3*tan(1/2*d*x + 1/2*c)^4 - 6*
a*b^2*tan(1/2*d*x + 1/2*c)^4 - 4*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 2*b^3*tan(1/2*d*x + 1/2*c)^3 + 6*a*b^2*tan(1/2
*d*x + 1/2*c)^2 + 3*b^3*tan(1/2*d*x + 1/2*c) + a^3 - 4*a*b^2)/((a^4 - 2*a^2*b^2 + b^4)*(tan(1/2*d*x + 1/2*c)^2
- 1)^3))/d

Mupad [B] (verification not implemented)

Time = 17.38 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.74 \[ \int \frac {\sec ^3(c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {2\,\left (4\,a\,b^2-a^3\right )}{3\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {2\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^4-2\,a^2\,b^2+b^4}-\frac {2\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{a^4-2\,a^2\,b^2+b^4}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (2\,a\,b^2-a^3\right )}{a^4-2\,a^2\,b^2+b^4}+\frac {4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,a^2+b^2\right )}{3\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {4\,a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{a^4-2\,a^2\,b^2+b^4}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {2\,a\,b^3\,\mathrm {atan}\left (\frac {\frac {a\,b^3\,\left (2\,a^4\,b-4\,a^2\,b^3+2\,b^5\right )}{{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}}+\frac {2\,a^2\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}{{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}}}{2\,a\,b^3}\right )}{d\,{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}} \]

[In]

int(sin(c + d*x)/(cos(c + d*x)^4*(a + b*sin(c + d*x))),x)

[Out]

((2*(4*a*b^2 - a^3))/(3*(a^4 + b^4 - 2*a^2*b^2)) - (2*b^3*tan(c/2 + (d*x)/2))/(a^4 + b^4 - 2*a^2*b^2) - (2*b^3
*tan(c/2 + (d*x)/2)^5)/(a^4 + b^4 - 2*a^2*b^2) + (2*tan(c/2 + (d*x)/2)^4*(2*a*b^2 - a^3))/(a^4 + b^4 - 2*a^2*b
^2) + (4*b*tan(c/2 + (d*x)/2)^3*(2*a^2 + b^2))/(3*(a^4 + b^4 - 2*a^2*b^2)) - (4*a*b^2*tan(c/2 + (d*x)/2)^2)/(a
^4 + b^4 - 2*a^2*b^2))/(d*(3*tan(c/2 + (d*x)/2)^2 - 3*tan(c/2 + (d*x)/2)^4 + tan(c/2 + (d*x)/2)^6 - 1)) - (2*a
*b^3*atan(((a*b^3*(2*a^4*b + 2*b^5 - 4*a^2*b^3))/((a + b)^(5/2)*(a - b)^(5/2)) + (2*a^2*b^3*tan(c/2 + (d*x)/2)
*(a^4 + b^4 - 2*a^2*b^2))/((a + b)^(5/2)*(a - b)^(5/2)))/(2*a*b^3)))/(d*(a + b)^(5/2)*(a - b)^(5/2))