3.2.28 \(\int \frac {(g \tan (e+f x))^p}{(a+a \sin (e+f x))^3} \, dx\) [128]

Optimal. Leaf size=248 \[ \frac {(g \tan (e+f x))^{1+p}}{a^3 f g (1+p)}-\frac {3 \cos ^2(e+f x)^{\frac {7+p}{2}} \, _2F_1\left (\frac {2+p}{2},\frac {7+p}{2};\frac {4+p}{2};\sin ^2(e+f x)\right ) \sec ^5(e+f x) (g \tan (e+f x))^{2+p}}{a^3 f g^2 (2+p)}+\frac {5 (g \tan (e+f x))^{3+p}}{a^3 f g^3 (3+p)}-\frac {\cos ^2(e+f x)^{\frac {7+p}{2}} \, _2F_1\left (\frac {4+p}{2},\frac {7+p}{2};\frac {6+p}{2};\sin ^2(e+f x)\right ) \sec ^3(e+f x) (g \tan (e+f x))^{4+p}}{a^3 f g^4 (4+p)}+\frac {4 (g \tan (e+f x))^{5+p}}{a^3 f g^5 (5+p)} \]

[Out]

(g*tan(f*x+e))^(1+p)/a^3/f/g/(1+p)-3*(cos(f*x+e)^2)^(7/2+1/2*p)*hypergeom([1+1/2*p, 7/2+1/2*p],[2+1/2*p],sin(f
*x+e)^2)*sec(f*x+e)^5*(g*tan(f*x+e))^(2+p)/a^3/f/g^2/(2+p)+5*(g*tan(f*x+e))^(3+p)/a^3/f/g^3/(3+p)-(cos(f*x+e)^
2)^(7/2+1/2*p)*hypergeom([2+1/2*p, 7/2+1/2*p],[3+1/2*p],sin(f*x+e)^2)*sec(f*x+e)^3*(g*tan(f*x+e))^(4+p)/a^3/f/
g^4/(4+p)+4*(g*tan(f*x+e))^(5+p)/a^3/f/g^5/(5+p)

________________________________________________________________________________________

Rubi [A]
time = 0.29, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2790, 2687, 276, 16, 2697, 14} \begin {gather*} \frac {4 (g \tan (e+f x))^{p+5}}{a^3 f g^5 (p+5)}-\frac {\sec ^3(e+f x) \cos ^2(e+f x)^{\frac {p+7}{2}} (g \tan (e+f x))^{p+4} \, _2F_1\left (\frac {p+4}{2},\frac {p+7}{2};\frac {p+6}{2};\sin ^2(e+f x)\right )}{a^3 f g^4 (p+4)}+\frac {5 (g \tan (e+f x))^{p+3}}{a^3 f g^3 (p+3)}-\frac {3 \sec ^5(e+f x) \cos ^2(e+f x)^{\frac {p+7}{2}} (g \tan (e+f x))^{p+2} \, _2F_1\left (\frac {p+2}{2},\frac {p+7}{2};\frac {p+4}{2};\sin ^2(e+f x)\right )}{a^3 f g^2 (p+2)}+\frac {(g \tan (e+f x))^{p+1}}{a^3 f g (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(g*Tan[e + f*x])^p/(a + a*Sin[e + f*x])^3,x]

[Out]

(g*Tan[e + f*x])^(1 + p)/(a^3*f*g*(1 + p)) - (3*(Cos[e + f*x]^2)^((7 + p)/2)*Hypergeometric2F1[(2 + p)/2, (7 +
p)/2, (4 + p)/2, Sin[e + f*x]^2]*Sec[e + f*x]^5*(g*Tan[e + f*x])^(2 + p))/(a^3*f*g^2*(2 + p)) + (5*(g*Tan[e +
f*x])^(3 + p))/(a^3*f*g^3*(3 + p)) - ((Cos[e + f*x]^2)^((7 + p)/2)*Hypergeometric2F1[(4 + p)/2, (7 + p)/2, (6
+ p)/2, Sin[e + f*x]^2]*Sec[e + f*x]^3*(g*Tan[e + f*x])^(4 + p))/(a^3*f*g^4*(4 + p)) + (4*(g*Tan[e + f*x])^(5
+ p))/(a^3*f*g^5*(5 + p))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 2697

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*Sec[e + f
*x])^m*(b*Tan[e + f*x])^(n + 1)*((Cos[e + f*x]^2)^((m + n + 1)/2)/(b*f*(n + 1)))*Hypergeometric2F1[(n + 1)/2,
(m + n + 1)/2, (n + 3)/2, Sin[e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[(n - 1)/2] &&  !In
tegerQ[m/2]

Rule 2790

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.), x_Symbol] :> Dist[a^(2*
m), Int[ExpandIntegrand[(g*Tan[e + f*x])^p/Sec[e + f*x]^m, (a*Sec[e + f*x] - b*Tan[e + f*x])^(-m), x], x], x]
/; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(g \tan (e+f x))^p}{(a+a \sin (e+f x))^3} \, dx &=\frac {\int \left (a^3 \sec ^6(e+f x) (g \tan (e+f x))^p-3 a^3 \sec ^5(e+f x) \tan (e+f x) (g \tan (e+f x))^p+3 a^3 \sec ^4(e+f x) \tan ^2(e+f x) (g \tan (e+f x))^p-a^3 \sec ^3(e+f x) \tan ^3(e+f x) (g \tan (e+f x))^p\right ) \, dx}{a^6}\\ &=\frac {\int \sec ^6(e+f x) (g \tan (e+f x))^p \, dx}{a^3}-\frac {\int \sec ^3(e+f x) \tan ^3(e+f x) (g \tan (e+f x))^p \, dx}{a^3}-\frac {3 \int \sec ^5(e+f x) \tan (e+f x) (g \tan (e+f x))^p \, dx}{a^3}+\frac {3 \int \sec ^4(e+f x) \tan ^2(e+f x) (g \tan (e+f x))^p \, dx}{a^3}\\ &=\frac {\text {Subst}\left (\int (g x)^p \left (1+x^2\right )^2 \, dx,x,\tan (e+f x)\right )}{a^3 f}-\frac {\int \sec ^3(e+f x) (g \tan (e+f x))^{3+p} \, dx}{a^3 g^3}+\frac {3 \int \sec ^4(e+f x) (g \tan (e+f x))^{2+p} \, dx}{a^3 g^2}-\frac {3 \int \sec ^5(e+f x) (g \tan (e+f x))^{1+p} \, dx}{a^3 g}\\ &=-\frac {3 \cos ^2(e+f x)^{\frac {7+p}{2}} \, _2F_1\left (\frac {2+p}{2},\frac {7+p}{2};\frac {4+p}{2};\sin ^2(e+f x)\right ) \sec ^5(e+f x) (g \tan (e+f x))^{2+p}}{a^3 f g^2 (2+p)}-\frac {\cos ^2(e+f x)^{\frac {7+p}{2}} \, _2F_1\left (\frac {4+p}{2},\frac {7+p}{2};\frac {6+p}{2};\sin ^2(e+f x)\right ) \sec ^3(e+f x) (g \tan (e+f x))^{4+p}}{a^3 f g^4 (4+p)}+\frac {\text {Subst}\left (\int \left ((g x)^p+\frac {2 (g x)^{2+p}}{g^2}+\frac {(g x)^{4+p}}{g^4}\right ) \, dx,x,\tan (e+f x)\right )}{a^3 f}+\frac {3 \text {Subst}\left (\int (g x)^{2+p} \left (1+x^2\right ) \, dx,x,\tan (e+f x)\right )}{a^3 f g^2}\\ &=\frac {(g \tan (e+f x))^{1+p}}{a^3 f g (1+p)}-\frac {3 \cos ^2(e+f x)^{\frac {7+p}{2}} \, _2F_1\left (\frac {2+p}{2},\frac {7+p}{2};\frac {4+p}{2};\sin ^2(e+f x)\right ) \sec ^5(e+f x) (g \tan (e+f x))^{2+p}}{a^3 f g^2 (2+p)}+\frac {2 (g \tan (e+f x))^{3+p}}{a^3 f g^3 (3+p)}-\frac {\cos ^2(e+f x)^{\frac {7+p}{2}} \, _2F_1\left (\frac {4+p}{2},\frac {7+p}{2};\frac {6+p}{2};\sin ^2(e+f x)\right ) \sec ^3(e+f x) (g \tan (e+f x))^{4+p}}{a^3 f g^4 (4+p)}+\frac {(g \tan (e+f x))^{5+p}}{a^3 f g^5 (5+p)}+\frac {3 \text {Subst}\left (\int \left ((g x)^{2+p}+\frac {(g x)^{4+p}}{g^2}\right ) \, dx,x,\tan (e+f x)\right )}{a^3 f g^2}\\ &=\frac {(g \tan (e+f x))^{1+p}}{a^3 f g (1+p)}-\frac {3 \cos ^2(e+f x)^{\frac {7+p}{2}} \, _2F_1\left (\frac {2+p}{2},\frac {7+p}{2};\frac {4+p}{2};\sin ^2(e+f x)\right ) \sec ^5(e+f x) (g \tan (e+f x))^{2+p}}{a^3 f g^2 (2+p)}+\frac {5 (g \tan (e+f x))^{3+p}}{a^3 f g^3 (3+p)}-\frac {\cos ^2(e+f x)^{\frac {7+p}{2}} \, _2F_1\left (\frac {4+p}{2},\frac {7+p}{2};\frac {6+p}{2};\sin ^2(e+f x)\right ) \sec ^3(e+f x) (g \tan (e+f x))^{4+p}}{a^3 f g^4 (4+p)}+\frac {4 (g \tan (e+f x))^{5+p}}{a^3 f g^5 (5+p)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
time = 5.44, size = 1200, normalized size = 4.84 \begin {gather*} \frac {(2+p) \left (F_1\left (1+p;p,2+p;2+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right )-4 F_1\left (1+p;p,3+p;2+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right )+8 F_1\left (1+p;p,4+p;2+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right )-8 F_1\left (1+p;p,5+p;2+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right )+4 F_1\left (1+p;p,6+p;2+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right )\right ) \sin (e+f x) (g \tan (e+f x))^p}{a^3 f (1+p) (1+\sin (e+f x))^3 \left ((2+p) F_1\left (1+p;p,2+p;2+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right )-4 (2+p) F_1\left (1+p;p,3+p;2+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right )+16 F_1\left (1+p;p,4+p;2+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right )+8 p F_1\left (1+p;p,4+p;2+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right )-16 F_1\left (1+p;p,5+p;2+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right )-8 p F_1\left (1+p;p,5+p;2+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right )+8 F_1\left (1+p;p,6+p;2+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right )+4 p F_1\left (1+p;p,6+p;2+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right )-2 F_1\left (2+p;p,3+p;3+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right )-p F_1\left (2+p;p,3+p;3+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right )+12 F_1\left (2+p;p,4+p;3+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right )+4 p F_1\left (2+p;p,4+p;3+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right )-32 F_1\left (2+p;p,5+p;3+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right )-8 p F_1\left (2+p;p,5+p;3+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right )+40 F_1\left (2+p;p,6+p;3+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right )+8 p F_1\left (2+p;p,6+p;3+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right )-24 F_1\left (2+p;p,7+p;3+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right )-4 p F_1\left (2+p;p,7+p;3+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right )+p F_1\left (2+p;1+p,2+p;3+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right )-4 p F_1\left (2+p;1+p,3+p;3+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right )+8 p F_1\left (2+p;1+p,4+p;3+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right )-8 p F_1\left (2+p;1+p,5+p;3+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right )+4 p F_1\left (2+p;1+p,6+p;3+p;\tan \left (\frac {1}{2} (e+f x)\right ),-\tan \left (\frac {1}{2} (e+f x)\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(g*Tan[e + f*x])^p/(a + a*Sin[e + f*x])^3,x]

[Out]

((2 + p)*(AppellF1[1 + p, p, 2 + p, 2 + p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]] - 4*AppellF1[1 + p, p, 3 + p,
2 + p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]] + 8*AppellF1[1 + p, p, 4 + p, 2 + p, Tan[(e + f*x)/2], -Tan[(e + f
*x)/2]] - 8*AppellF1[1 + p, p, 5 + p, 2 + p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]] + 4*AppellF1[1 + p, p, 6 + p
, 2 + p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]])*Sin[e + f*x]*(g*Tan[e + f*x])^p)/(a^3*f*(1 + p)*(1 + Sin[e + f*
x])^3*((2 + p)*AppellF1[1 + p, p, 2 + p, 2 + p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]] - 4*(2 + p)*AppellF1[1 +
p, p, 3 + p, 2 + p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]] + 16*AppellF1[1 + p, p, 4 + p, 2 + p, Tan[(e + f*x)/2
], -Tan[(e + f*x)/2]] + 8*p*AppellF1[1 + p, p, 4 + p, 2 + p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]] - 16*AppellF
1[1 + p, p, 5 + p, 2 + p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]] - 8*p*AppellF1[1 + p, p, 5 + p, 2 + p, Tan[(e +
f*x)/2], -Tan[(e + f*x)/2]] + 8*AppellF1[1 + p, p, 6 + p, 2 + p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]] + 4*p*A
ppellF1[1 + p, p, 6 + p, 2 + p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]] - 2*AppellF1[2 + p, p, 3 + p, 3 + p, Tan[
(e + f*x)/2], -Tan[(e + f*x)/2]]*Tan[(e + f*x)/2] - p*AppellF1[2 + p, p, 3 + p, 3 + p, Tan[(e + f*x)/2], -Tan[
(e + f*x)/2]]*Tan[(e + f*x)/2] + 12*AppellF1[2 + p, p, 4 + p, 3 + p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]]*Tan[
(e + f*x)/2] + 4*p*AppellF1[2 + p, p, 4 + p, 3 + p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]]*Tan[(e + f*x)/2] - 32
*AppellF1[2 + p, p, 5 + p, 3 + p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]]*Tan[(e + f*x)/2] - 8*p*AppellF1[2 + p,
p, 5 + p, 3 + p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]]*Tan[(e + f*x)/2] + 40*AppellF1[2 + p, p, 6 + p, 3 + p, T
an[(e + f*x)/2], -Tan[(e + f*x)/2]]*Tan[(e + f*x)/2] + 8*p*AppellF1[2 + p, p, 6 + p, 3 + p, Tan[(e + f*x)/2],
-Tan[(e + f*x)/2]]*Tan[(e + f*x)/2] - 24*AppellF1[2 + p, p, 7 + p, 3 + p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]]
*Tan[(e + f*x)/2] - 4*p*AppellF1[2 + p, p, 7 + p, 3 + p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]]*Tan[(e + f*x)/2]
+ p*AppellF1[2 + p, 1 + p, 2 + p, 3 + p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]]*Tan[(e + f*x)/2] - 4*p*AppellF1
[2 + p, 1 + p, 3 + p, 3 + p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]]*Tan[(e + f*x)/2] + 8*p*AppellF1[2 + p, 1 + p
, 4 + p, 3 + p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]]*Tan[(e + f*x)/2] - 8*p*AppellF1[2 + p, 1 + p, 5 + p, 3 +
p, Tan[(e + f*x)/2], -Tan[(e + f*x)/2]]*Tan[(e + f*x)/2] + 4*p*AppellF1[2 + p, 1 + p, 6 + p, 3 + p, Tan[(e + f
*x)/2], -Tan[(e + f*x)/2]]*Tan[(e + f*x)/2]))

________________________________________________________________________________________

Maple [F]
time = 0.67, size = 0, normalized size = 0.00 \[\int \frac {\left (g \tan \left (f x +e \right )\right )^{p}}{\left (a +a \sin \left (f x +e \right )\right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*tan(f*x+e))^p/(a+a*sin(f*x+e))^3,x)

[Out]

int((g*tan(f*x+e))^p/(a+a*sin(f*x+e))^3,x)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*tan(f*x+e))^p/(a+a*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

integrate((g*tan(f*x + e))^p/(a*sin(f*x + e) + a)^3, x)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*tan(f*x+e))^p/(a+a*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

integral(-(g*tan(f*x + e))^p/(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e)), x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\left (g \tan {\left (e + f x \right )}\right )^{p}}{\sin ^{3}{\left (e + f x \right )} + 3 \sin ^{2}{\left (e + f x \right )} + 3 \sin {\left (e + f x \right )} + 1}\, dx}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*tan(f*x+e))**p/(a+a*sin(f*x+e))**3,x)

[Out]

Integral((g*tan(e + f*x))**p/(sin(e + f*x)**3 + 3*sin(e + f*x)**2 + 3*sin(e + f*x) + 1), x)/a**3

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*tan(f*x+e))^p/(a+a*sin(f*x+e))^3,x, algorithm="giac")

[Out]

integrate((g*tan(f*x + e))^p/(a*sin(f*x + e) + a)^3, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (g\,\mathrm {tan}\left (e+f\,x\right )\right )}^p}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*tan(e + f*x))^p/(a + a*sin(e + f*x))^3,x)

[Out]

int((g*tan(e + f*x))^p/(a + a*sin(e + f*x))^3, x)

________________________________________________________________________________________