3.267 \(\int \frac{1}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=229 \[ -\frac{b^3}{2 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}+\frac{b^2 \left (3 a^2+b^2\right )}{d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}+\frac{b \left (8 a^2 b^2+3 a^4+b^4\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4}+\frac{\csc ^2(c+d x) \left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right )}{2 d \left (a^2-b^2\right )^3}+\frac{(a-2 b) \log (1-\cos (c+d x))}{4 d (a+b)^4}-\frac{(a+2 b) \log (\cos (c+d x)+1)}{4 d (a-b)^4} \]

[Out]

-b^3/(2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2) + (b^2*(3*a^2 + b^2))/((a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) +
((b*(3*a^2 + b^2) - a*(a^2 + 3*b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)^3*d) + ((a - 2*b)*Log[1 - Co
s[c + d*x]])/(4*(a + b)^4*d) - ((a + 2*b)*Log[1 + Cos[c + d*x]])/(4*(a - b)^4*d) + (b*(3*a^4 + 8*a^2*b^2 + b^4
)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)

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Rubi [A]  time = 0.480423, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {4397, 2721, 1647, 1629} \[ -\frac{b^3}{2 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}+\frac{b^2 \left (3 a^2+b^2\right )}{d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}+\frac{b \left (8 a^2 b^2+3 a^4+b^4\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4}+\frac{\csc ^2(c+d x) \left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right )}{2 d \left (a^2-b^2\right )^3}+\frac{(a-2 b) \log (1-\cos (c+d x))}{4 d (a+b)^4}-\frac{(a+2 b) \log (\cos (c+d x)+1)}{4 d (a-b)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-b^3/(2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2) + (b^2*(3*a^2 + b^2))/((a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) +
((b*(3*a^2 + b^2) - a*(a^2 + 3*b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)^3*d) + ((a - 2*b)*Log[1 - Co
s[c + d*x]])/(4*(a + b)^4*d) - ((a + 2*b)*Log[1 + Cos[c + d*x]])/(4*(a - b)^4*d) + (b*(3*a^4 + 8*a^2*b^2 + b^4
)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)

Rule 4397

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

Rule 2721

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2
- b^2, 0] && IntegerQ[(p + 1)/2]

Rule 1647

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[((a*g - c*f*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rule 1629

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*
Pq*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int \frac{1}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx &=\int \frac{\cot ^3(c+d x)}{(b+a \cos (c+d x))^3} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x^3}{(b+x)^3 \left (a^2-x^2\right )^2} \, dx,x,a \cos (c+d x)\right )}{d}\\ &=\frac{\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{a^4 b^3 \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^3}-\frac{a^2 b^2 \left (3 a^4+3 a^2 b^2-2 b^4\right ) x}{\left (a^2-b^2\right )^3}-\frac{a^4 b \left (3 a^2-7 b^2\right ) x^2}{\left (a^2-b^2\right )^3}+\frac{a^4 \left (a^2+3 b^2\right ) x^3}{\left (a^2-b^2\right )^3}}{(b+x)^3 \left (a^2-x^2\right )} \, dx,x,a \cos (c+d x)\right )}{2 a^2 d}\\ &=\frac{\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}-\frac{\operatorname{Subst}\left (\int \left (\frac{a^2 (a-2 b)}{2 (a+b)^4 (a-x)}+\frac{a^2 (a+2 b)}{2 (a-b)^4 (a+x)}-\frac{2 a^2 b^3}{\left (a^2-b^2\right )^2 (b+x)^3}+\frac{2 a^2 b^2 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 (b+x)^2}-\frac{2 a^2 b \left (3 a^4+8 a^2 b^2+b^4\right )}{\left (a^2-b^2\right )^4 (b+x)}\right ) \, dx,x,a \cos (c+d x)\right )}{2 a^2 d}\\ &=-\frac{b^3}{2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}+\frac{b^2 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac{\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}+\frac{(a-2 b) \log (1-\cos (c+d x))}{4 (a+b)^4 d}-\frac{(a+2 b) \log (1+\cos (c+d x))}{4 (a-b)^4 d}+\frac{b \left (3 a^4+8 a^2 b^2+b^4\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4 d}\\ \end{align*}

Mathematica [C]  time = 6.3109, size = 696, normalized size = 3.04 \[ -\frac{2 i \left (8 a^2 b^3+3 a^4 b+b^5\right ) (c+d x) \tan ^3(c+d x) (a \cos (c+d x)+b)^3}{d (a-b)^4 (a+b)^4 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{b^2 \left (3 a^2+b^2\right ) \tan ^3(c+d x) (a \cos (c+d x)+b)^2}{d (b-a)^3 (a+b)^3 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{\left (8 a^2 b^3+3 a^4 b+b^5\right ) \tan ^3(c+d x) (a \cos (c+d x)+b)^3 \log (a \cos (c+d x)+b)}{d \left (b^2-a^2\right )^4 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{b^3 \tan ^3(c+d x) (a \cos (c+d x)+b)}{2 d (b-a)^2 (a+b)^2 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{i (-a-2 b) \tan ^{-1}(\tan (c+d x)) \tan ^3(c+d x) (a \cos (c+d x)+b)^3}{2 d (b-a)^4 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{i (a-2 b) \tan ^{-1}(\tan (c+d x)) \tan ^3(c+d x) (a \cos (c+d x)+b)^3}{2 d (a+b)^4 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{(-a-2 b) \tan ^3(c+d x) \log \left (\cos ^2\left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)^3}{4 d (b-a)^4 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{(a-2 b) \tan ^3(c+d x) \log \left (\sin ^2\left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)^3}{4 d (a+b)^4 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{\tan ^3(c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)^3}{8 d (a+b)^3 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{\tan ^3(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)^3}{8 d (b-a)^3 (a \sin (c+d x)+b \tan (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^3*(b + a*Cos[c + d*x])*Tan[c + d*x]^3)/(2*(-a + b)^2*(a + b)^2*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - (b
^2*(3*a^2 + b^2)*(b + a*Cos[c + d*x])^2*Tan[c + d*x]^3)/((-a + b)^3*(a + b)^3*d*(a*Sin[c + d*x] + b*Tan[c + d*
x])^3) - ((2*I)*(3*a^4*b + 8*a^2*b^3 + b^5)*(c + d*x)*(b + a*Cos[c + d*x])^3*Tan[c + d*x]^3)/((a - b)^4*(a + b
)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - ((I/2)*(-a - 2*b)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x])^3*Tan
[c + d*x]^3)/((-a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - ((I/2)*(a - 2*b)*ArcTan[Tan[c + d*x]]*(b + a
*Cos[c + d*x])^3*Tan[c + d*x]^3)/((a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - ((b + a*Cos[c + d*x])^3*C
sc[(c + d*x)/2]^2*Tan[c + d*x]^3)/(8*(a + b)^3*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) + ((-a - 2*b)*(b + a*Cos
[c + d*x])^3*Log[Cos[(c + d*x)/2]^2]*Tan[c + d*x]^3)/(4*(-a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) + ((
3*a^4*b + 8*a^2*b^3 + b^5)*(b + a*Cos[c + d*x])^3*Log[b + a*Cos[c + d*x]]*Tan[c + d*x]^3)/((-a^2 + b^2)^4*d*(a
*Sin[c + d*x] + b*Tan[c + d*x])^3) + ((a - 2*b)*(b + a*Cos[c + d*x])^3*Log[Sin[(c + d*x)/2]^2]*Tan[c + d*x]^3)
/(4*(a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - ((b + a*Cos[c + d*x])^3*Sec[(c + d*x)/2]^2*Tan[c + d*x]
^3)/(8*(-a + b)^3*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3)

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Maple [A]  time = 0.151, size = 322, normalized size = 1.4 \begin{align*}{\frac{1}{4\,d \left ( a-b \right ) ^{3} \left ( \cos \left ( dx+c \right ) +1 \right ) }}-{\frac{a\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{4\, \left ( a-b \right ) ^{4}d}}-{\frac{b\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{2\, \left ( a-b \right ) ^{4}d}}+{\frac{1}{4\,d \left ( a+b \right ) ^{3} \left ( -1+\cos \left ( dx+c \right ) \right ) }}+{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) a}{4\,d \left ( a+b \right ) ^{4}}}-{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) b}{2\,d \left ( a+b \right ) ^{4}}}-{\frac{{b}^{3}}{2\,d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2} \left ( b+a\cos \left ( dx+c \right ) \right ) ^{2}}}+3\,{\frac{b\ln \left ( b+a\cos \left ( dx+c \right ) \right ){a}^{4}}{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}+8\,{\frac{{b}^{3}\ln \left ( b+a\cos \left ( dx+c \right ) \right ){a}^{2}}{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}+{\frac{{b}^{5}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}+3\,{\frac{{a}^{2}{b}^{2}}{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3} \left ( b+a\cos \left ( dx+c \right ) \right ) }}+{\frac{{b}^{4}}{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3} \left ( b+a\cos \left ( dx+c \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*sin(d*x+c)+b*tan(d*x+c))^3,x)

[Out]

1/4/d/(a-b)^3/(cos(d*x+c)+1)-1/4*a*ln(cos(d*x+c)+1)/(a-b)^4/d-1/2*b*ln(cos(d*x+c)+1)/(a-b)^4/d+1/4/d/(a+b)^3/(
-1+cos(d*x+c))+1/4/d/(a+b)^4*ln(-1+cos(d*x+c))*a-1/2/d/(a+b)^4*ln(-1+cos(d*x+c))*b-1/2/d*b^3/(a+b)^2/(a-b)^2/(
b+a*cos(d*x+c))^2+3/d*b/(a+b)^4/(a-b)^4*ln(b+a*cos(d*x+c))*a^4+8/d*b^3/(a+b)^4/(a-b)^4*ln(b+a*cos(d*x+c))*a^2+
1/d*b^5/(a+b)^4/(a-b)^4*ln(b+a*cos(d*x+c))+3/d*b^2/(a+b)^3/(a-b)^3/(b+a*cos(d*x+c))*a^2+1/d*b^4/(a+b)^3/(a-b)^
3/(b+a*cos(d*x+c))

________________________________________________________________________________________

Maxima [B]  time = 1.21293, size = 811, normalized size = 3.54 \begin{align*} \frac{\frac{8 \,{\left (3 \, a^{4} b + 8 \, a^{2} b^{3} + b^{5}\right )} \log \left (a + b - \frac{{\left (a - b\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} + \frac{4 \,{\left (a - 2 \, b\right )} \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac{a^{6} - 2 \, a^{5} b - a^{4} b^{2} + 4 \, a^{3} b^{3} - a^{2} b^{4} - 2 \, a b^{5} + b^{6} - \frac{2 \,{\left (a^{6} - 4 \, a^{5} b + 29 \, a^{4} b^{2} + 24 \, a^{3} b^{3} + 11 \, a^{2} b^{4} + 20 \, a b^{5} - b^{6}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{{\left (a^{6} - 6 \, a^{5} b + 63 \, a^{4} b^{2} - 52 \, a^{3} b^{3} + 31 \, a^{2} b^{4} - 38 \, a b^{5} + b^{6}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{\frac{{\left (a^{9} + a^{8} b - 4 \, a^{7} b^{2} - 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} + 6 \, a^{4} b^{5} - 4 \, a^{3} b^{6} - 4 \, a^{2} b^{7} + a b^{8} + b^{9}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{2 \,{\left (a^{9} - a^{8} b - 4 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} - 6 \, a^{4} b^{5} - 4 \, a^{3} b^{6} + 4 \, a^{2} b^{7} + a b^{8} - b^{9}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{{\left (a^{9} - 3 \, a^{8} b + 8 \, a^{6} b^{3} - 6 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + 8 \, a^{3} b^{6} - 3 \, a b^{8} + b^{9}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac{\sin \left (d x + c\right )^{2}}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sin(d*x+c)+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

1/8*(8*(3*a^4*b + 8*a^2*b^3 + b^5)*log(a + b - (a - b)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)/(a^8 - 4*a^6*b^2 +
6*a^4*b^4 - 4*a^2*b^6 + b^8) + 4*(a - 2*b)*log(sin(d*x + c)/(cos(d*x + c) + 1))/(a^4 + 4*a^3*b + 6*a^2*b^2 +
4*a*b^3 + b^4) - (a^6 - 2*a^5*b - a^4*b^2 + 4*a^3*b^3 - a^2*b^4 - 2*a*b^5 + b^6 - 2*(a^6 - 4*a^5*b + 29*a^4*b^
2 + 24*a^3*b^3 + 11*a^2*b^4 + 20*a*b^5 - b^6)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + (a^6 - 6*a^5*b + 63*a^4*b^
2 - 52*a^3*b^3 + 31*a^2*b^4 - 38*a*b^5 + b^6)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4)/((a^9 + a^8*b - 4*a^7*b^2 -
4*a^6*b^3 + 6*a^5*b^4 + 6*a^4*b^5 - 4*a^3*b^6 - 4*a^2*b^7 + a*b^8 + b^9)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2
- 2*(a^9 - a^8*b - 4*a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 - 6*a^4*b^5 - 4*a^3*b^6 + 4*a^2*b^7 + a*b^8 - b^9)*sin(d*
x + c)^4/(cos(d*x + c) + 1)^4 + (a^9 - 3*a^8*b + 8*a^6*b^3 - 6*a^5*b^4 - 6*a^4*b^5 + 8*a^3*b^6 - 3*a*b^8 + b^9
)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + sin(d*x + c)^2/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*(cos(d*x + c) + 1)^2)
)/d

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Fricas [B]  time = 1.22552, size = 2344, normalized size = 10.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sin(d*x+c)+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/4*(16*a^4*b^3 - 8*a^2*b^5 - 8*b^7 - 2*(a^7 + 8*a^5*b^2 - 7*a^3*b^4 - 2*a*b^6)*cos(d*x + c)^3 + 2*(a^6*b - 1
1*a^4*b^3 + 7*a^2*b^5 + 3*b^7)*cos(d*x + c)^2 + 2*(11*a^5*b^2 - 10*a^3*b^4 - a*b^6)*cos(d*x + c) + 4*(3*a^4*b^
3 + 8*a^2*b^5 + b^7 - (3*a^6*b + 8*a^4*b^3 + a^2*b^5)*cos(d*x + c)^4 - 2*(3*a^5*b^2 + 8*a^3*b^4 + a*b^6)*cos(d
*x + c)^3 + (3*a^6*b + 5*a^4*b^3 - 7*a^2*b^5 - b^7)*cos(d*x + c)^2 + 2*(3*a^5*b^2 + 8*a^3*b^4 + a*b^6)*cos(d*x
+ c))*log(a*cos(d*x + c) + b) - (a^5*b^2 + 6*a^4*b^3 + 14*a^3*b^4 + 16*a^2*b^5 + 9*a*b^6 + 2*b^7 - (a^7 + 6*a
^6*b + 14*a^5*b^2 + 16*a^4*b^3 + 9*a^3*b^4 + 2*a^2*b^5)*cos(d*x + c)^4 - 2*(a^6*b + 6*a^5*b^2 + 14*a^4*b^3 + 1
6*a^3*b^4 + 9*a^2*b^5 + 2*a*b^6)*cos(d*x + c)^3 + (a^7 + 6*a^6*b + 13*a^5*b^2 + 10*a^4*b^3 - 5*a^3*b^4 - 14*a^
2*b^5 - 9*a*b^6 - 2*b^7)*cos(d*x + c)^2 + 2*(a^6*b + 6*a^5*b^2 + 14*a^4*b^3 + 16*a^3*b^4 + 9*a^2*b^5 + 2*a*b^6
)*cos(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + (a^5*b^2 - 6*a^4*b^3 + 14*a^3*b^4 - 16*a^2*b^5 + 9*a*b^6 - 2*b^7
- (a^7 - 6*a^6*b + 14*a^5*b^2 - 16*a^4*b^3 + 9*a^3*b^4 - 2*a^2*b^5)*cos(d*x + c)^4 - 2*(a^6*b - 6*a^5*b^2 + 1
4*a^4*b^3 - 16*a^3*b^4 + 9*a^2*b^5 - 2*a*b^6)*cos(d*x + c)^3 + (a^7 - 6*a^6*b + 13*a^5*b^2 - 10*a^4*b^3 - 5*a^
3*b^4 + 14*a^2*b^5 - 9*a*b^6 + 2*b^7)*cos(d*x + c)^2 + 2*(a^6*b - 6*a^5*b^2 + 14*a^4*b^3 - 16*a^3*b^4 + 9*a^2*
b^5 - 2*a*b^6)*cos(d*x + c))*log(-1/2*cos(d*x + c) + 1/2))/((a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^
8)*d*cos(d*x + c)^4 + 2*(a^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^9)*d*cos(d*x + c)^3 - (a^10 - 5*a^8*b
^2 + 10*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10)*d*cos(d*x + c)^2 - 2*(a^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b
^7 + a*b^9)*d*cos(d*x + c) - (a^8*b^2 - 4*a^6*b^4 + 6*a^4*b^6 - 4*a^2*b^8 + b^10)*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \sin{\left (c + d x \right )} + b \tan{\left (c + d x \right )}\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sin(d*x+c)+b*tan(d*x+c))**3,x)

[Out]

Integral((a*sin(c + d*x) + b*tan(c + d*x))**(-3), x)

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Giac [B]  time = 1.23048, size = 1080, normalized size = 4.72 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sin(d*x+c)+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

1/8*(2*(a - 2*b)*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)
+ 8*(3*a^4*b + 8*a^2*b^3 + b^5)*log(abs(-a - b - a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + b*(cos(d*x + c) -
1)/(cos(d*x + c) + 1)))/(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8) + (a + b - 2*a*(cos(d*x + c) - 1)/(cos
(d*x + c) + 1) + 4*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))*(cos(d*x + c) + 1)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4
*a*b^3 + b^4)*(cos(d*x + c) - 1)) - (cos(d*x + c) - 1)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*(cos(d*x + c) + 1)) -
4*(9*a^6*b + 6*a^5*b^2 + 9*a^4*b^3 + 28*a^3*b^4 + 11*a^2*b^5 - 2*a*b^6 + 3*b^7 + 18*a^6*b*(cos(d*x + c) - 1)/(
cos(d*x + c) + 1) - 12*a^5*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 26*a^4*b^3*(cos(d*x + c) - 1)/(cos(d*x
+ c) + 1) + 4*a^3*b^4*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 38*a^2*b^5*(cos(d*x + c) - 1)/(cos(d*x + c) + 1)
+ 8*a*b^6*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 6*b^7*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 9*a^6*b*(cos(
d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 18*a^5*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 33*a^4*b^3*(cos(
d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 48*a^3*b^4*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 27*a^2*b^5*(cos(
d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 6*a*b^6*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 3*b^7*(cos(d*x + c)
- 1)^2/(cos(d*x + c) + 1)^2)/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*(a + b + a*(cos(d*x + c) - 1)/(
cos(d*x + c) + 1) - b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))^2))/d