\(\int \frac {(e+f x)^3 \sec (c+d x)}{a+b \sin (c+d x)} \, dx\) [306]

Optimal result

Integrand size = 26, antiderivative size = 937 \[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 i a (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d}+\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^2}-\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}+\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^3}-\frac {6 i a f^3 \operatorname {PolyLog}\left (4,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}+\frac {6 i a f^3 \operatorname {PolyLog}\left (4,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^4}+\frac {3 i b f^3 \operatorname {PolyLog}\left (4,-e^{2 i (c+d x)}\right )}{4 \left (a^2-b^2\right ) d^4} \]

[Out]

3/4*I*b*f^3*polylog(4,-exp(2*I*(d*x+c)))/(a^2-b^2)/d^4+b*(f*x+e)^3*ln(1+exp(2*I*(d*x+c)))/(a^2-b^2)/d-b*(f*x+e
)^3*ln(1-I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)/d-b*(f*x+e)^3*ln(1-I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^
(1/2)))/(a^2-b^2)/d-3/2*I*b*f*(f*x+e)^2*polylog(2,-exp(2*I*(d*x+c)))/(a^2-b^2)/d^2-3*I*a*f*(f*x+e)^2*polylog(2
,I*exp(I*(d*x+c)))/(a^2-b^2)/d^2-6*I*b*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)/d^4-6*I
*b*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)/d^4+3*I*b*f*(f*x+e)^2*polylog(2,I*b*exp(I*(
d*x+c))/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)/d^2-6*a*f^2*(f*x+e)*polylog(3,-I*exp(I*(d*x+c)))/(a^2-b^2)/d^3+6*a*f^2*
(f*x+e)*polylog(3,I*exp(I*(d*x+c)))/(a^2-b^2)/d^3+3/2*b*f^2*(f*x+e)*polylog(3,-exp(2*I*(d*x+c)))/(a^2-b^2)/d^3
-6*b*f^2*(f*x+e)*polylog(3,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)/d^3-6*b*f^2*(f*x+e)*polylog(3,I*b
*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)/d^3+6*I*a*f^3*polylog(4,I*exp(I*(d*x+c)))/(a^2-b^2)/d^4+3*I*b*f
*(f*x+e)^2*polylog(2,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)/d^2-2*I*a*(f*x+e)^3*arctan(exp(I*(d*x+c
)))/(a^2-b^2)/d-6*I*a*f^3*polylog(4,-I*exp(I*(d*x+c)))/(a^2-b^2)/d^4+3*I*a*f*(f*x+e)^2*polylog(2,-I*exp(I*(d*x
+c)))/(a^2-b^2)/d^2

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 937, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {4629, 4615, 2221, 2611, 6744, 2320, 6724, 6874, 4266, 3800} \[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {6 i a \operatorname {PolyLog}\left (4,-i e^{i (c+d x)}\right ) f^3}{\left (a^2-b^2\right ) d^4}+\frac {6 i a \operatorname {PolyLog}\left (4,i e^{i (c+d x)}\right ) f^3}{\left (a^2-b^2\right ) d^4}-\frac {6 i b \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) f^3}{\left (a^2-b^2\right ) d^4}-\frac {6 i b \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) f^3}{\left (a^2-b^2\right ) d^4}+\frac {3 i b \operatorname {PolyLog}\left (4,-e^{2 i (c+d x)}\right ) f^3}{4 \left (a^2-b^2\right ) d^4}-\frac {6 a (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right ) f^2}{\left (a^2-b^2\right ) d^3}+\frac {6 a (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right ) f^2}{\left (a^2-b^2\right ) d^3}-\frac {6 b (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) f^2}{\left (a^2-b^2\right ) d^3}-\frac {6 b (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) f^2}{\left (a^2-b^2\right ) d^3}+\frac {3 b (e+f x) \operatorname {PolyLog}\left (3,-e^{2 i (c+d x)}\right ) f^2}{2 \left (a^2-b^2\right ) d^3}+\frac {3 i a (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right ) f}{\left (a^2-b^2\right ) d^2}-\frac {3 i a (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right ) f}{\left (a^2-b^2\right ) d^2}+\frac {3 i b (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) f}{\left (a^2-b^2\right ) d^2}+\frac {3 i b (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) f}{\left (a^2-b^2\right ) d^2}-\frac {3 i b (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right ) f}{2 \left (a^2-b^2\right ) d^2}-\frac {2 i a (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d} \]

[In]

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

[Out]

((-2*I)*a*(e + f*x)^3*ArcTan[E^(I*(c + d*x))])/((a^2 - b^2)*d) - (b*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/
(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*d) - (b*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])
/((a^2 - b^2)*d) + (b*(e + f*x)^3*Log[1 + E^((2*I)*(c + d*x))])/((a^2 - b^2)*d) + ((3*I)*a*f*(e + f*x)^2*PolyL
og[2, (-I)*E^(I*(c + d*x))])/((a^2 - b^2)*d^2) - ((3*I)*a*f*(e + f*x)^2*PolyLog[2, I*E^(I*(c + d*x))])/((a^2 -
b^2)*d^2) + ((3*I)*b*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^2)
+ ((3*I)*b*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^2) - (((3*I)
/2)*b*f*(e + f*x)^2*PolyLog[2, -E^((2*I)*(c + d*x))])/((a^2 - b^2)*d^2) - (6*a*f^2*(e + f*x)*PolyLog[3, (-I)*E
^(I*(c + d*x))])/((a^2 - b^2)*d^3) + (6*a*f^2*(e + f*x)*PolyLog[3, I*E^(I*(c + d*x))])/((a^2 - b^2)*d^3) - (6*
b*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^3) - (6*b*f^2*(e + f*x
)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^3) + (3*b*f^2*(e + f*x)*PolyLog[3, -
E^((2*I)*(c + d*x))])/(2*(a^2 - b^2)*d^3) - ((6*I)*a*f^3*PolyLog[4, (-I)*E^(I*(c + d*x))])/((a^2 - b^2)*d^4) +
((6*I)*a*f^3*PolyLog[4, I*E^(I*(c + d*x))])/((a^2 - b^2)*d^4) - ((6*I)*b*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))
/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^4) - ((6*I)*b*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2
])])/((a^2 - b^2)*d^4) + (((3*I)/4)*b*f^3*PolyLog[4, -E^((2*I)*(c + d*x))])/((a^2 - b^2)*d^4)

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x
] - Dist[2*I, Int[(c + d*x)^m*(E^(2*I*(e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]

Rule 4266

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 4615

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1))), x] + (Int[(e + f*x)^m*(E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2] - I*
b*E^(I*(c + d*x)))), x] + Int[(e + f*x)^m*(E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x)))), x])
/; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]

Rule 4629

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sec[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbo
l] :> Dist[-b^2/(a^2 - b^2), Int[(e + f*x)^m*(Sec[c + d*x]^(n - 2)/(a + b*Sin[c + d*x])), x], x] + Dist[1/(a^2
- b^2), Int[(e + f*x)^m*Sec[c + d*x]^n*(a - b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m
, 0] && NeQ[a^2 - b^2, 0] && IGtQ[n, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^3 \sec (c+d x) (a-b \sin (c+d x)) \, dx}{a^2-b^2}-\frac {b^2 \int \frac {(e+f x)^3 \cos (c+d x)}{a+b \sin (c+d x)} \, dx}{a^2-b^2} \\ & = \frac {i b (e+f x)^4}{4 \left (a^2-b^2\right ) f}+\frac {\int \left (a (e+f x)^3 \sec (c+d x)-b (e+f x)^3 \tan (c+d x)\right ) \, dx}{a^2-b^2}-\frac {b^2 \int \frac {e^{i (c+d x)} (e+f x)^3}{a-\sqrt {a^2-b^2}-i b e^{i (c+d x)}} \, dx}{a^2-b^2}-\frac {b^2 \int \frac {e^{i (c+d x)} (e+f x)^3}{a+\sqrt {a^2-b^2}-i b e^{i (c+d x)}} \, dx}{a^2-b^2} \\ & = \frac {i b (e+f x)^4}{4 \left (a^2-b^2\right ) f}-\frac {b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac {a \int (e+f x)^3 \sec (c+d x) \, dx}{a^2-b^2}-\frac {b \int (e+f x)^3 \tan (c+d x) \, dx}{a^2-b^2}+\frac {(3 b f) \int (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) d}+\frac {(3 b f) \int (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) d} \\ & = -\frac {2 i a (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {(2 i b) \int \frac {e^{2 i (c+d x)} (e+f x)^3}{1+e^{2 i (c+d x)}} \, dx}{a^2-b^2}-\frac {(3 a f) \int (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d}+\frac {(3 a f) \int (e+f x)^2 \log \left (1+i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d}-\frac {\left (6 i b f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) d^2}-\frac {\left (6 i b f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) d^2} \\ & = -\frac {2 i a (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d}+\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}-\frac {(3 b f) \int (e+f x)^2 \log \left (1+e^{2 i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d}-\frac {\left (6 i a f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^2}+\frac {\left (6 i a f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^2}+\frac {\left (6 b f^3\right ) \int \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) d^3}+\frac {\left (6 b f^3\right ) \int \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) d^3} \\ & = -\frac {2 i a (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d}+\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^2}-\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}+\frac {\left (3 i b f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^2}-\frac {\left (6 i b f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {i b x}{a-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}-\frac {\left (6 i b f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {i b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}+\frac {\left (6 a f^3\right ) \int \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^3}-\frac {\left (6 a f^3\right ) \int \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^3} \\ & = -\frac {2 i a (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d}+\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^2}-\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}+\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^3}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^4}-\frac {\left (6 i a f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}+\frac {\left (6 i a f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}-\frac {\left (3 b f^3\right ) \int \operatorname {PolyLog}\left (3,-e^{2 i (c+d x)}\right ) \, dx}{2 \left (a^2-b^2\right ) d^3} \\ & = -\frac {2 i a (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d}+\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^2}-\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}+\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^3}-\frac {6 i a f^3 \operatorname {PolyLog}\left (4,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}+\frac {6 i a f^3 \operatorname {PolyLog}\left (4,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^4}+\frac {\left (3 i b f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{4 \left (a^2-b^2\right ) d^4} \\ & = -\frac {2 i a (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d}+\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^2}-\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}+\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^3}-\frac {6 i a f^3 \operatorname {PolyLog}\left (4,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}+\frac {6 i a f^3 \operatorname {PolyLog}\left (4,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^4}+\frac {3 i b f^3 \operatorname {PolyLog}\left (4,-e^{2 i (c+d x)}\right )}{4 \left (a^2-b^2\right ) d^4} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2496\) vs. \(2(937)=1874\).

Time = 6.88 (sec) , antiderivative size = 2496, normalized size of antiderivative = 2.66 \[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

((4*((I*b*(e + f*x)^4)/f - (2*(a - b)*(1 + E^((2*I)*c))*(e + f*x)^3*Log[1 - I/E^(I*(c + d*x))])/d + (2*(a + b)
*(1 + E^((2*I)*c))*(e + f*x)^3*Log[1 + I/E^(I*(c + d*x))])/d + (6*(a + b)*(1 + E^((2*I)*c))*f*(I*d^2*(e + f*x)
^2*PolyLog[2, (-I)/E^(I*(c + d*x))] + 2*f*(d*(e + f*x)*PolyLog[3, (-I)/E^(I*(c + d*x))] - I*f*PolyLog[4, (-I)/
E^(I*(c + d*x))])))/d^4 - ((6*I)*(a - b)*(1 + E^((2*I)*c))*f*(d^2*(e + f*x)^2*PolyLog[2, I/E^(I*(c + d*x))] -
(2*I)*d*f*(e + f*x)*PolyLog[3, I/E^(I*(c + d*x))] - 2*f^2*PolyLog[4, I/E^(I*(c + d*x))]))/d^4))/((a^2 - b^2)*(
1 + E^((2*I)*c))) + (4*b*((-4*I)*d^4*e^3*E^((2*I)*c)*x - (6*I)*d^4*e^2*E^((2*I)*c)*f*x^2 - (4*I)*d^4*e*E^((2*I
)*c)*f^2*x^3 - I*d^4*E^((2*I)*c)*f^3*x^4 - (2*I)*d^3*e^3*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d
*x))))] + (2*I)*d^3*e^3*E^((2*I)*c)*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] - d^3*e^3*Log
[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] + d^3*e^3*E^((2*I)*c)*Log[4*a^2*E^((2*I)*(c + d
*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] - 6*d^3*e^2*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a
^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e^2*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 +
b^2)*E^((2*I)*c)])] - 6*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)
*c)])] + 6*d^3*e*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c
)])] - 2*d^3*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 2*d^3*E^(
(2*I)*c)*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e^2*f*x
*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e^2*E^((2*I)*c)*f*x*Log
[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(
2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*
c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 2*d^3*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*
E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 2*d^3*E^((2*I)*c)*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I
*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (6*I)*d^2*(-1 + E^((2*I)*c))*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(2*c
+ d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (6*I)*d^2*(-1 + E^((2*I)*c))*f*(e + f*x)^2*PolyLog[
2, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*d*e*f^2*PolyLog[3, (I*b*E^(I*
(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*d*e*E^((2*I)*c)*f^2*PolyLog[3, (I*b*E^(I*(2
*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*d*f^3*x*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a
*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*d*E^((2*I)*c)*f^3*x*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E
^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*d*e*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqr
t[(-a^2 + b^2)*E^((2*I)*c)]))] + 12*d*e*E^((2*I)*c)*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt
[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*d*f^3*x*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)
*E^((2*I)*c)]))] + 12*d*E^((2*I)*c)*f^3*x*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*
E^((2*I)*c)]))] - (12*I)*f^3*PolyLog[4, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])
] + (12*I)*E^((2*I)*c)*f^3*PolyLog[4, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])]
- (12*I)*f^3*PolyLog[4, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (12*I)*E^((
2*I)*c)*f^3*PolyLog[4, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))]))/((-a^2 + b^2
)*d^4*(-1 + E^((2*I)*c))) - (8*b*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3)*Csc[c]^3)/((a - b)*(a + b)*(Csc
[c/2] - Sec[c/2])*(Csc[c/2] + Sec[c/2])))/8

Maple [F]

\[\int \frac {\left (f x +e \right )^{3} \sec \left (d x +c \right )}{a +b \sin \left (d x +c \right )}d x\]

[In]

int((f*x+e)^3*sec(d*x+c)/(a+b*sin(d*x+c)),x)

[Out]

int((f*x+e)^3*sec(d*x+c)/(a+b*sin(d*x+c)),x)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3069 vs. \(2 (822) = 1644\).

Time = 0.62 (sec) , antiderivative size = 3069, normalized size of antiderivative = 3.28 \[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate((f*x+e)^3*sec(d*x+c)/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/2*(-6*I*b*f^3*polylog(4, -(I*a*cos(d*x + c) + a*sin(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a
^2 - b^2)/b^2))/b) - 6*I*b*f^3*polylog(4, -(I*a*cos(d*x + c) + a*sin(d*x + c) - (b*cos(d*x + c) - I*b*sin(d*x
+ c))*sqrt(-(a^2 - b^2)/b^2))/b) + 6*I*b*f^3*polylog(4, -(-I*a*cos(d*x + c) + a*sin(d*x + c) + (b*cos(d*x + c)
+ I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b) + 6*I*b*f^3*polylog(4, -(-I*a*cos(d*x + c) + a*sin(d*x + c) -
(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b) - 6*I*(a - b)*f^3*polylog(4, I*cos(d*x + c) + s
in(d*x + c)) - 6*I*(a + b)*f^3*polylog(4, I*cos(d*x + c) - sin(d*x + c)) + 6*I*(a - b)*f^3*polylog(4, -I*cos(d
*x + c) + sin(d*x + c)) + 6*I*(a + b)*f^3*polylog(4, -I*cos(d*x + c) - sin(d*x + c)) - 3*(I*b*d^2*f^3*x^2 + 2*
I*b*d^2*e*f^2*x + I*b*d^2*e^2*f)*dilog((I*a*cos(d*x + c) - a*sin(d*x + c) + (b*cos(d*x + c) + I*b*sin(d*x + c)
)*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1) - 3*(I*b*d^2*f^3*x^2 + 2*I*b*d^2*e*f^2*x + I*b*d^2*e^2*f)*dilog((I*a*cos(
d*x + c) - a*sin(d*x + c) - (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1) - 3*(-I*b*d
^2*f^3*x^2 - 2*I*b*d^2*e*f^2*x - I*b*d^2*e^2*f)*dilog((-I*a*cos(d*x + c) - a*sin(d*x + c) + (b*cos(d*x + c) -
I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1) - 3*(-I*b*d^2*f^3*x^2 - 2*I*b*d^2*e*f^2*x - I*b*d^2*e^2*f
)*dilog((-I*a*cos(d*x + c) - a*sin(d*x + c) - (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/
b + 1) - 3*(-I*(a - b)*d^2*f^3*x^2 - 2*I*(a - b)*d^2*e*f^2*x - I*(a - b)*d^2*e^2*f)*dilog(I*cos(d*x + c) + sin
(d*x + c)) - 3*(-I*(a + b)*d^2*f^3*x^2 - 2*I*(a + b)*d^2*e*f^2*x - I*(a + b)*d^2*e^2*f)*dilog(I*cos(d*x + c) -
sin(d*x + c)) - 3*(I*(a - b)*d^2*f^3*x^2 + 2*I*(a - b)*d^2*e*f^2*x + I*(a - b)*d^2*e^2*f)*dilog(-I*cos(d*x +
c) + sin(d*x + c)) - 3*(I*(a + b)*d^2*f^3*x^2 + 2*I*(a + b)*d^2*e*f^2*x + I*(a + b)*d^2*e^2*f)*dilog(-I*cos(d*
x + c) - sin(d*x + c)) + (b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*log(2*b*cos(d*x + c) + 2*
I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a) + (b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^
3*f^3)*log(2*b*cos(d*x + c) - 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*a) + (b*d^3*e^3 - 3*b*c*d^
2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*log(-2*b*cos(d*x + c) + 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2)
+ 2*I*a) + (b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*log(-2*b*cos(d*x + c) - 2*I*b*sin(d*x
+ c) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*a) + (b*d^3*f^3*x^3 + 3*b*d^3*e*f^2*x^2 + 3*b*d^3*e^2*f*x + 3*b*c*d^2*
e^2*f - 3*b*c^2*d*e*f^2 + b*c^3*f^3)*log(-(I*a*cos(d*x + c) - a*sin(d*x + c) + (b*cos(d*x + c) + I*b*sin(d*x +
c))*sqrt(-(a^2 - b^2)/b^2) - b)/b) + (b*d^3*f^3*x^3 + 3*b*d^3*e*f^2*x^2 + 3*b*d^3*e^2*f*x + 3*b*c*d^2*e^2*f -
3*b*c^2*d*e*f^2 + b*c^3*f^3)*log(-(I*a*cos(d*x + c) - a*sin(d*x + c) - (b*cos(d*x + c) + I*b*sin(d*x + c))*sq
rt(-(a^2 - b^2)/b^2) - b)/b) + (b*d^3*f^3*x^3 + 3*b*d^3*e*f^2*x^2 + 3*b*d^3*e^2*f*x + 3*b*c*d^2*e^2*f - 3*b*c^
2*d*e*f^2 + b*c^3*f^3)*log(-(-I*a*cos(d*x + c) - a*sin(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a
^2 - b^2)/b^2) - b)/b) + (b*d^3*f^3*x^3 + 3*b*d^3*e*f^2*x^2 + 3*b*d^3*e^2*f*x + 3*b*c*d^2*e^2*f - 3*b*c^2*d*e*
f^2 + b*c^3*f^3)*log(-(-I*a*cos(d*x + c) - a*sin(d*x + c) - (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b
^2)/b^2) - b)/b) - ((a + b)*d^3*e^3 - 3*(a + b)*c*d^2*e^2*f + 3*(a + b)*c^2*d*e*f^2 - (a + b)*c^3*f^3)*log(cos
(d*x + c) + I*sin(d*x + c) + I) + ((a - b)*d^3*e^3 - 3*(a - b)*c*d^2*e^2*f + 3*(a - b)*c^2*d*e*f^2 - (a - b)*c
^3*f^3)*log(cos(d*x + c) - I*sin(d*x + c) + I) - ((a + b)*d^3*f^3*x^3 + 3*(a + b)*d^3*e*f^2*x^2 + 3*(a + b)*d^
3*e^2*f*x + 3*(a + b)*c*d^2*e^2*f - 3*(a + b)*c^2*d*e*f^2 + (a + b)*c^3*f^3)*log(I*cos(d*x + c) + sin(d*x + c)
+ 1) + ((a - b)*d^3*f^3*x^3 + 3*(a - b)*d^3*e*f^2*x^2 + 3*(a - b)*d^3*e^2*f*x + 3*(a - b)*c*d^2*e^2*f - 3*(a
- b)*c^2*d*e*f^2 + (a - b)*c^3*f^3)*log(I*cos(d*x + c) - sin(d*x + c) + 1) - ((a + b)*d^3*f^3*x^3 + 3*(a + b)*
d^3*e*f^2*x^2 + 3*(a + b)*d^3*e^2*f*x + 3*(a + b)*c*d^2*e^2*f - 3*(a + b)*c^2*d*e*f^2 + (a + b)*c^3*f^3)*log(-
I*cos(d*x + c) + sin(d*x + c) + 1) + ((a - b)*d^3*f^3*x^3 + 3*(a - b)*d^3*e*f^2*x^2 + 3*(a - b)*d^3*e^2*f*x +
3*(a - b)*c*d^2*e^2*f - 3*(a - b)*c^2*d*e*f^2 + (a - b)*c^3*f^3)*log(-I*cos(d*x + c) - sin(d*x + c) + 1) - ((a
+ b)*d^3*e^3 - 3*(a + b)*c*d^2*e^2*f + 3*(a + b)*c^2*d*e*f^2 - (a + b)*c^3*f^3)*log(-cos(d*x + c) + I*sin(d*x
+ c) + I) + ((a - b)*d^3*e^3 - 3*(a - b)*c*d^2*e^2*f + 3*(a - b)*c^2*d*e*f^2 - (a - b)*c^3*f^3)*log(-cos(d*x
+ c) - I*sin(d*x + c) + I) + 6*(b*d*f^3*x + b*d*e*f^2)*polylog(3, -(I*a*cos(d*x + c) + a*sin(d*x + c) + (b*cos
(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b) + 6*(b*d*f^3*x + b*d*e*f^2)*polylog(3, -(I*a*cos(d*x
+ c) + a*sin(d*x + c) - (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b) + 6*(b*d*f^3*x + b*d*e*
f^2)*polylog(3, -(-I*a*cos(d*x + c) + a*sin(d*x + c) + (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b
^2))/b) + 6*(b*d*f^3*x + b*d*e*f^2)*polylog(3, -(-I*a*cos(d*x + c) + a*sin(d*x + c) - (b*cos(d*x + c) + I*b*si
n(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b) + 6*((a - b)*d*f^3*x + (a - b)*d*e*f^2)*polylog(3, I*cos(d*x + c) + sin
(d*x + c)) - 6*((a + b)*d*f^3*x + (a + b)*d*e*f^2)*polylog(3, I*cos(d*x + c) - sin(d*x + c)) + 6*((a - b)*d*f^
3*x + (a - b)*d*e*f^2)*polylog(3, -I*cos(d*x + c) + sin(d*x + c)) - 6*((a + b)*d*f^3*x + (a + b)*d*e*f^2)*poly
log(3, -I*cos(d*x + c) - sin(d*x + c)))/((a^2 - b^2)*d^4)

Sympy [F]

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

[In]

integrate((f*x+e)**3*sec(d*x+c)/(a+b*sin(d*x+c)),x)

[Out]

Integral((e + f*x)**3*sec(c + d*x)/(a + b*sin(c + d*x)), x)

Maxima [F(-2)]

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

[In]

integrate((f*x+e)^3*sec(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 [F]

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

[In]

integrate((f*x+e)^3*sec(d*x+c)/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)^3*sec(d*x + c)/(b*sin(d*x + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \]

[In]

int((e + f*x)^3/(cos(c + d*x)*(a + b*sin(c + d*x))),x)

[Out]

\text{Hanged}