3.185 \(\int \frac {(e+f x)^3 \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx\)
Optimal. Leaf size=247 \[ \frac {12 f^3 \text {Li}_3\left (i e^{i (c+d x)}\right )}{a d^4}-\frac {6 f^3 \sin (c+d x)}{a d^4}-\frac {12 i f^2 (e+f x) \text {Li}_2\left (i e^{i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}-\frac {(e+f x)^3 \cos (c+d x)}{a d}-\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{a d}-\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f} \]
[Out]
-I*(f*x+e)^3/a/d-1/4*(f*x+e)^4/a/f+6*f^2*(f*x+e)*cos(d*x+c)/a/d^3-(f*x+e)^3*cos(d*x+c)/a/d-(f*x+e)^3*cot(1/2*c
+1/4*Pi+1/2*d*x)/a/d+6*f*(f*x+e)^2*ln(1-I*exp(I*(d*x+c)))/a/d^2-12*I*f^2*(f*x+e)*polylog(2,I*exp(I*(d*x+c)))/a
/d^3+12*f^3*polylog(3,I*exp(I*(d*x+c)))/a/d^4-6*f^3*sin(d*x+c)/a/d^4+3*f*(f*x+e)^2*sin(d*x+c)/a/d^2
________________________________________________________________________________________
Rubi [A] time = 0.47, antiderivative size = 247, normalized size of antiderivative =
1.00, number of steps used = 14, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) =
0.393, Rules used = {4515, 3296, 2637, 32, 3318, 4184, 3717, 2190, 2531, 2282, 6589}
\[ -\frac {12 i f^2 (e+f x) \text {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}+\frac {12 f^3 \text {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}-\frac {6 f^3 \sin (c+d x)}{a d^4}-\frac {(e+f x)^3 \cos (c+d x)}{a d}-\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{a d}-\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f} \]
Antiderivative was successfully verified.
[In]
Int[((e + f*x)^3*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]
[Out]
((-I)*(e + f*x)^3)/(a*d) - (e + f*x)^4/(4*a*f) + (6*f^2*(e + f*x)*Cos[c + d*x])/(a*d^3) - ((e + f*x)^3*Cos[c +
d*x])/(a*d) - ((e + f*x)^3*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) + (6*f*(e + f*x)^2*Log[1 - I*E^(I*(c + d*x))])/(a
*d^2) - ((12*I)*f^2*(e + f*x)*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^3) + (12*f^3*PolyLog[3, I*E^(I*(c + d*x))])/
(a*d^4) - (6*f^3*Sin[c + d*x])/(a*d^4) + (3*f*(e + f*x)^2*Sin[c + d*x])/(a*d^2)
Rule 32
Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]
Rule 2190
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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2282
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 2531
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 2637
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Rule 3296
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Rule 3318
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1*(e + (Pi*a)/(2*b)))/2 + (f*x)/2]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
Rule 3717
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
Rule 4184
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Rule 4515
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbo
l] :> Dist[1/b, Int[(e + f*x)^m*Sin[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Sin[c + d*x]^(n - 1)
)/(a + b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
Rule 6589
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]
Rubi steps
\begin {align*} \int \frac {(e+f x)^3 \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int (e+f x)^3 \sin (c+d x) \, dx}{a}-\int \frac {(e+f x)^3 \sin (c+d x)}{a+a \sin (c+d x)} \, dx\\ &=-\frac {(e+f x)^3 \cos (c+d x)}{a d}-\frac {\int (e+f x)^3 \, dx}{a}+\frac {(3 f) \int (e+f x)^2 \cos (c+d x) \, dx}{a d}+\int \frac {(e+f x)^3}{a+a \sin (c+d x)} \, dx\\ &=-\frac {(e+f x)^4}{4 a f}-\frac {(e+f x)^3 \cos (c+d x)}{a d}+\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {\int (e+f x)^3 \csc ^2\left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {d x}{2}\right ) \, dx}{2 a}-\frac {\left (6 f^2\right ) \int (e+f x) \sin (c+d x) \, dx}{a d^2}\\ &=-\frac {(e+f x)^4}{4 a f}+\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}-\frac {(e+f x)^3 \cos (c+d x)}{a d}-\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {(3 f) \int (e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \, dx}{a d}-\frac {\left (6 f^3\right ) \int \cos (c+d x) \, dx}{a d^3}\\ &=-\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}+\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}-\frac {(e+f x)^3 \cos (c+d x)}{a d}-\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {6 f^3 \sin (c+d x)}{a d^4}+\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {(6 f) \int \frac {e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )} (e+f x)^2}{1-i e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}} \, dx}{a d}\\ &=-\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}+\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}-\frac {(e+f x)^3 \cos (c+d x)}{a d}-\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {6 f^3 \sin (c+d x)}{a d^4}+\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}-\frac {\left (12 f^2\right ) \int (e+f x) \log \left (1-i e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^2}\\ &=-\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}+\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}-\frac {(e+f x)^3 \cos (c+d x)}{a d}-\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {12 i f^2 (e+f x) \text {Li}_2\left (i e^{i (c+d x)}\right )}{a d^3}-\frac {6 f^3 \sin (c+d x)}{a d^4}+\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {\left (12 i f^3\right ) \int \text {Li}_2\left (i e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^3}\\ &=-\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}+\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}-\frac {(e+f x)^3 \cos (c+d x)}{a d}-\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {12 i f^2 (e+f x) \text {Li}_2\left (i e^{i (c+d x)}\right )}{a d^3}-\frac {6 f^3 \sin (c+d x)}{a d^4}+\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {\left (12 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right )}{a d^4}\\ &=-\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}+\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}-\frac {(e+f x)^3 \cos (c+d x)}{a d}-\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {12 i f^2 (e+f x) \text {Li}_2\left (i e^{i (c+d x)}\right )}{a d^3}+\frac {12 f^3 \text {Li}_3\left (i e^{i (c+d x)}\right )}{a d^4}-\frac {6 f^3 \sin (c+d x)}{a d^4}+\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 3.25, size = 1314, normalized size = 5.32 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
Integrate[((e + f*x)^3*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]
[Out]
((-6 + 4*I)*d^3*e^3*Cos[(c + d*x)/2] + 6*d^2*e^2*f*Cos[(c + d*x)/2] + 12*d*e*f^2*Cos[(c + d*x)/2] - 12*f^3*Cos
[(c + d*x)/2] - 4*d^4*e^3*x*Cos[(c + d*x)/2] - (18 - 12*I)*d^3*e^2*f*x*Cos[(c + d*x)/2] + 12*d^2*e*f^2*x*Cos[(
c + d*x)/2] + 12*d*f^3*x*Cos[(c + d*x)/2] - 6*d^4*e^2*f*x^2*Cos[(c + d*x)/2] - (18 - 12*I)*d^3*e*f^2*x^2*Cos[(
c + d*x)/2] + 6*d^2*f^3*x^2*Cos[(c + d*x)/2] - 4*d^4*e*f^2*x^3*Cos[(c + d*x)/2] - (6 - 4*I)*d^3*f^3*x^3*Cos[(c
+ d*x)/2] - d^4*f^3*x^4*Cos[(c + d*x)/2] - 2*d^3*e^3*Cos[(3*(c + d*x))/2] - 6*d^2*e^2*f*Cos[(3*(c + d*x))/2]
+ 12*d*e*f^2*Cos[(3*(c + d*x))/2] + 12*f^3*Cos[(3*(c + d*x))/2] - 6*d^3*e^2*f*x*Cos[(3*(c + d*x))/2] - 12*d^2*
e*f^2*x*Cos[(3*(c + d*x))/2] + 12*d*f^3*x*Cos[(3*(c + d*x))/2] - 6*d^3*e*f^2*x^2*Cos[(3*(c + d*x))/2] - 6*d^2*
f^3*x^2*Cos[(3*(c + d*x))/2] - 2*d^3*f^3*x^3*Cos[(3*(c + d*x))/2] + 24*d^2*e^2*f*Cos[(c + d*x)/2]*Log[1 + I*Co
s[c + d*x] + Sin[c + d*x]] + 48*d^2*e*f^2*x*Cos[(c + d*x)/2]*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]] + 24*d^2*f
^3*x^2*Cos[(c + d*x)/2]*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]] + (6 + 4*I)*d^3*e^3*Sin[(c + d*x)/2] + 6*d^2*e^
2*f*Sin[(c + d*x)/2] - 12*d*e*f^2*Sin[(c + d*x)/2] - 12*f^3*Sin[(c + d*x)/2] - 4*d^4*e^3*x*Sin[(c + d*x)/2] +
(18 + 12*I)*d^3*e^2*f*x*Sin[(c + d*x)/2] + 12*d^2*e*f^2*x*Sin[(c + d*x)/2] - 12*d*f^3*x*Sin[(c + d*x)/2] - 6*d
^4*e^2*f*x^2*Sin[(c + d*x)/2] + (18 + 12*I)*d^3*e*f^2*x^2*Sin[(c + d*x)/2] + 6*d^2*f^3*x^2*Sin[(c + d*x)/2] -
4*d^4*e*f^2*x^3*Sin[(c + d*x)/2] + (6 + 4*I)*d^3*f^3*x^3*Sin[(c + d*x)/2] - d^4*f^3*x^4*Sin[(c + d*x)/2] + 24*
d^2*e^2*f*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*Sin[(c + d*x)/2] + 48*d^2*e*f^2*x*Log[1 + I*Cos[c + d*x] + Si
n[c + d*x]]*Sin[(c + d*x)/2] + 24*d^2*f^3*x^2*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*Sin[(c + d*x)/2] + (48*I)
*d*f^2*(e + f*x)*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 48*f^3*P
olyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 2*d^3*e^3*Sin[(3*(c + d*x)
)/2] + 6*d^2*e^2*f*Sin[(3*(c + d*x))/2] + 12*d*e*f^2*Sin[(3*(c + d*x))/2] - 12*f^3*Sin[(3*(c + d*x))/2] - 6*d^
3*e^2*f*x*Sin[(3*(c + d*x))/2] + 12*d^2*e*f^2*x*Sin[(3*(c + d*x))/2] + 12*d*f^3*x*Sin[(3*(c + d*x))/2] - 6*d^3
*e*f^2*x^2*Sin[(3*(c + d*x))/2] + 6*d^2*f^3*x^2*Sin[(3*(c + d*x))/2] - 2*d^3*f^3*x^3*Sin[(3*(c + d*x))/2])/(4*
a*d^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))
________________________________________________________________________________________
fricas [C] time = 0.57, size = 1313, normalized size = 5.32 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^3*sin(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="fricas")
[Out]
-1/4*(d^4*f^3*x^4 + 4*d^3*e^3 - 12*d^2*e^2*f + 4*(d^4*e*f^2 + d^3*f^3)*x^3 + 24*f^3 + 6*(d^4*e^2*f + 2*d^3*e*f
^2 - 2*d^2*f^3)*x^2 + 4*(d^3*f^3*x^3 + d^3*e^3 + 3*d^2*e^2*f - 6*d*e*f^2 - 6*f^3 + 3*(d^3*e*f^2 + d^2*f^3)*x^2
+ 3*(d^3*e^2*f + 2*d^2*e*f^2 - 2*d*f^3)*x)*cos(d*x + c)^2 + 4*(d^4*e^3 + 3*d^3*e^2*f - 6*d^2*e*f^2)*x + (d^4*
f^3*x^4 + 8*d^3*e^3 - 24*d*e*f^2 + 4*(d^4*e*f^2 + 2*d^3*f^3)*x^3 + 6*(d^4*e^2*f + 4*d^3*e*f^2)*x^2 + 4*(d^4*e^
3 + 6*d^3*e^2*f - 6*d*f^3)*x)*cos(d*x + c) - (-24*I*d*f^3*x - 24*I*d*e*f^2 + (-24*I*d*f^3*x - 24*I*d*e*f^2)*co
s(d*x + c) + (-24*I*d*f^3*x - 24*I*d*e*f^2)*sin(d*x + c))*dilog(I*cos(d*x + c) - sin(d*x + c)) - (24*I*d*f^3*x
+ 24*I*d*e*f^2 + (24*I*d*f^3*x + 24*I*d*e*f^2)*cos(d*x + c) + (24*I*d*f^3*x + 24*I*d*e*f^2)*sin(d*x + c))*dil
og(-I*cos(d*x + c) - sin(d*x + c)) - 12*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3 + (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^
3)*cos(d*x + c) + (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*sin(d*x + c))*log(cos(d*x + c) + I*sin(d*x + c) + I) - 1
2*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3 + (d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)
*cos(d*x + c) + (d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*sin(d*x + c))*log(I*cos(d*x + c) + sin(d
*x + c) + 1) - 12*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3 + (d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*
e*f^2 - c^2*f^3)*cos(d*x + c) + (d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*sin(d*x + c))*log(-I*cos
(d*x + c) + sin(d*x + c) + 1) - 12*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3 + (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*co
s(d*x + c) + (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*sin(d*x + c))*log(-cos(d*x + c) + I*sin(d*x + c) + I) - 24*(f
^3*cos(d*x + c) + f^3*sin(d*x + c) + f^3)*polylog(3, I*cos(d*x + c) - sin(d*x + c)) - 24*(f^3*cos(d*x + c) + f
^3*sin(d*x + c) + f^3)*polylog(3, -I*cos(d*x + c) - sin(d*x + c)) + (d^4*f^3*x^4 - 4*d^3*e^3 - 12*d^2*e^2*f +
4*(d^4*e*f^2 - d^3*f^3)*x^3 + 24*f^3 + 6*(d^4*e^2*f - 2*d^3*e*f^2 - 2*d^2*f^3)*x^2 + 4*(d^4*e^3 - 3*d^3*e^2*f
- 6*d^2*e*f^2)*x + 4*(d^3*f^3*x^3 + d^3*e^3 - 3*d^2*e^2*f - 6*d*e*f^2 + 6*f^3 + 3*(d^3*e*f^2 - d^2*f^3)*x^2 +
3*(d^3*e^2*f - 2*d^2*e*f^2 - 2*d*f^3)*x)*cos(d*x + c))*sin(d*x + c))/(a*d^4*cos(d*x + c) + a*d^4*sin(d*x + c)
+ a*d^4)
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (f x + e\right )}^{3} \sin \left (d x + c\right )^{2}}{a \sin \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^3*sin(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="giac")
[Out]
integrate((f*x + e)^3*sin(d*x + c)^2/(a*sin(d*x + c) + a), x)
________________________________________________________________________________________
maple [B] time = 0.44, size = 748, normalized size = 3.03 \[ -\frac {f^{3} x^{4}}{4 a}-\frac {e^{3} x}{a}-\frac {\left (f^{3} x^{3} d^{3}+3 d^{3} e \,f^{2} x^{2}+3 i d^{2} f^{3} x^{2}+3 d^{3} e^{2} f x +6 i d^{2} e \,f^{2} x +d^{3} e^{3}+3 i d^{2} e^{2} f -6 d \,f^{3} x -6 f^{2} e d -6 i f^{3}\right ) {\mathrm e}^{i \left (d x +c \right )}}{2 a \,d^{4}}-\frac {\left (f^{3} x^{3} d^{3}+3 d^{3} e \,f^{2} x^{2}-3 i d^{2} f^{3} x^{2}+3 d^{3} e^{2} f x -6 i d^{2} e \,f^{2} x +d^{3} e^{3}-3 i d^{2} e^{2} f -6 d \,f^{3} x -6 f^{2} e d +6 i f^{3}\right ) {\mathrm e}^{-i \left (d x +c \right )}}{2 a \,d^{4}}-\frac {2 \left (f^{3} x^{3}+3 e \,f^{2} x^{2}+3 e^{2} f x +e^{3}\right )}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}+\frac {12 f^{3} \polylog \left (3, i {\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{4}}-\frac {12 i f^{2} e c x}{a \,d^{2}}-\frac {e \,f^{2} x^{3}}{a}-\frac {3 e^{2} f \,x^{2}}{2 a}+\frac {6 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) e^{2}}{a \,d^{2}}+\frac {6 f^{3} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) x^{2}}{a \,d^{2}}-\frac {6 f^{3} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) c^{2}}{a \,d^{4}}-\frac {6 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right ) e^{2}}{a \,d^{2}}-\frac {2 i f^{3} x^{3}}{a d}-\frac {6 f^{3} c^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{4}}+\frac {4 i f^{3} c^{3}}{a \,d^{4}}+\frac {6 f^{3} c^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a \,d^{4}}-\frac {12 i f^{3} \polylog \left (2, i {\mathrm e}^{i \left (d x +c \right )}\right ) x}{a \,d^{3}}-\frac {6 i f^{2} e \,c^{2}}{a \,d^{3}}+\frac {12 f^{2} e \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) x}{a \,d^{2}}+\frac {12 f^{2} e \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) c}{a \,d^{3}}+\frac {6 i f^{3} c^{2} x}{a \,d^{3}}-\frac {12 i f^{2} e \polylog \left (2, i {\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}-\frac {12 f^{2} e c \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a \,d^{3}}+\frac {12 f^{2} e c \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}-\frac {6 i f^{2} e \,x^{2}}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f*x+e)^3*sin(d*x+c)^2/(a+a*sin(d*x+c)),x)
[Out]
12/a/d^2*f^2*e*ln(1-I*exp(I*(d*x+c)))*x+12/a/d^3*f^2*e*ln(1-I*exp(I*(d*x+c)))*c+12/a/d^3*f^2*e*c*ln(exp(I*(d*x
+c)))-12*I/a/d^2*e*f^2*c*x-2*I/a/d*f^3*x^3+4*I/a/d^4*f^3*c^3-1/4/a*f^3*x^4-1/a*e^3*x-2*(f^3*x^3+3*e*f^2*x^2+3*
e^2*f*x+e^3)/d/a/(exp(I*(d*x+c))+I)-1/a*e*f^2*x^3-3/2/a*e^2*f*x^2-6/a/d^2*f*ln(exp(I*(d*x+c)))*e^2-6/a/d^4*f^3
*c^2*ln(exp(I*(d*x+c)))+6/a/d^4*f^3*c^2*ln(exp(I*(d*x+c))+I)+6/a/d^2*f*ln(exp(I*(d*x+c))+I)*e^2-1/2*(f^3*x^3*d
^3+3*I*d^2*f^3*x^2+3*d^3*e*f^2*x^2+6*I*d^2*e*f^2*x+3*d^3*e^2*f*x+3*I*d^2*e^2*f+d^3*e^3-6*d*f^3*x-6*I*f^3-6*f^2
*e*d)/a/d^4*exp(I*(d*x+c))+6*I/a/d^3*f^3*c^2*x-12*I/a/d^3*f^3*polylog(2,I*exp(I*(d*x+c)))*x-6*I/a/d*e*f^2*x^2-
6*I/a/d^3*e*f^2*c^2-12*I/a/d^3*e*f^2*polylog(2,I*exp(I*(d*x+c)))+6/a/d^2*f^3*ln(1-I*exp(I*(d*x+c)))*x^2-1/2*(f
^3*x^3*d^3-3*I*d^2*f^3*x^2+3*d^3*e*f^2*x^2-6*I*d^2*e*f^2*x+3*d^3*e^2*f*x-3*I*d^2*e^2*f+d^3*e^3-6*d*f^3*x+6*I*f
^3-6*f^2*e*d)/a/d^4*exp(-I*(d*x+c))+12*f^3*polylog(3,I*exp(I*(d*x+c)))/a/d^4-6/a/d^4*f^3*ln(1-I*exp(I*(d*x+c))
)*c^2-12/a/d^3*f^2*e*c*ln(exp(I*(d*x+c))+I)
________________________________________________________________________________________
maxima [B] time = 1.97, size = 4598, normalized size = 18.62 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^3*sin(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="maxima")
[Out]
-1/2*(12*c^2*e*f^2*((sin(d*x + c)/(cos(d*x + c) + 1) + sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 2)/(a*d^2 + a*d^2
*sin(d*x + c)/(cos(d*x + c) + 1) + a*d^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a*d^2*sin(d*x + c)^3/(cos(d*x +
c) + 1)^3) + arctan(sin(d*x + c)/(cos(d*x + c) + 1))/(a*d^2)) - 12*c*e^2*f*((sin(d*x + c)/(cos(d*x + c) + 1)
+ sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 2)/(a*d + a*d*sin(d*x + c)/(cos(d*x + c) + 1) + a*d*sin(d*x + c)^2/(co
s(d*x + c) + 1)^2 + a*d*sin(d*x + c)^3/(cos(d*x + c) + 1)^3) + arctan(sin(d*x + c)/(cos(d*x + c) + 1))/(a*d))
- 6*(((d*x + c)^2 - 1)*cos(d*x + c)^4 + ((d*x + c)^2 - 1)*sin(d*x + c)^4 + ((d*x + c)*cos(d*x + c) + sin(d*x +
c) + 1)*cos(2*d*x + 2*c)^3 + 7*(d*x + c)*cos(d*x + c)^3 + (d*x + (d*x + c)*sin(d*x + c) + c - cos(d*x + c))*s
in(2*d*x + 2*c)^3 + (2*(d*x + c)^2 - 3)*sin(d*x + c)^3 + (((d*x + c)^2 - 1)*cos(d*x + c)^2 + ((d*x + c)^2 - 3)
*sin(d*x + c)^2 + (d*x + c)^2 + 6*(d*x + c)*cos(d*x + c) + 2*((d*x + c)^2 - (d*x + c)*cos(d*x + c) - 2)*sin(d*
x + c) - 1)*cos(2*d*x + 2*c)^2 + ((d*x + c)^2 - 1)*cos(d*x + c)^2 + (((d*x + c)^2 - 3)*cos(d*x + c)^2 + ((d*x
+ c)^2 - 1)*sin(d*x + c)^2 + (d*x + c)^2 + ((d*x + c)*cos(d*x + c) + sin(d*x + c) + 1)*cos(2*d*x + 2*c) + 8*(d
*x + c)*cos(d*x + c) + 2*((d*x + c)^2 + (d*x + c)*cos(d*x + c) - 1)*sin(d*x + c) - 1)*sin(2*d*x + 2*c)^2 + (2*
((d*x + c)^2 - 1)*cos(d*x + c)^2 + (d*x + c)^2 + 7*(d*x + c)*cos(d*x + c) - 3)*sin(d*x + c)^2 + ((d*x + c)*cos
(d*x + c)^3 - (2*(d*x + c)^2 - 3)*sin(d*x + c)^3 - (4*(d*x + c)^2 - (d*x + c)*cos(d*x + c) - 6)*sin(d*x + c)^2
+ 2*cos(d*x + c)^2 - ((2*(d*x + c)^2 - 3)*cos(d*x + c)^2 + 2*(d*x + c)^2 + 12*(d*x + c)*cos(d*x + c) - 4)*sin
(d*x + c) + 1)*cos(2*d*x + 2*c) + (d*x + c)*cos(d*x + c) - 2*(cos(d*x + c)^4 + sin(d*x + c)^4 + (cos(d*x + c)^
2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1)*cos(2*d*x + 2*c)^2 + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c
) + 1)*sin(2*d*x + 2*c)^2 + 2*cos(d*x + c)^2*sin(d*x + c) + (2*cos(d*x + c)^2 + 1)*sin(d*x + c)^2 + 2*sin(d*x
+ c)^3 - 2*(sin(d*x + c)^3 + (cos(d*x + c)^2 + 1)*sin(d*x + c) + 2*sin(d*x + c)^2)*cos(2*d*x + 2*c) + cos(d*x
+ c)^2 + 2*(cos(d*x + c)^3 + cos(d*x + c)*sin(d*x + c)^2 + 2*cos(d*x + c)*sin(d*x + c) + cos(d*x + c))*sin(2*d
*x + 2*c))*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1) + ((2*(d*x + c)^2 - 3)*cos(d*x + c)^3 + (
d*x + c)*sin(d*x + c)^3 + (d*x + (d*x + c)*sin(d*x + c) + c - cos(d*x + c))*cos(2*d*x + 2*c)^2 + 14*(d*x + c)*
cos(d*x + c)^2 + (2*d*x + (2*(d*x + c)^2 - 3)*cos(d*x + c) + 2*c)*sin(d*x + c)^2 + d*x + 2*((d*x + c)*cos(d*x
+ c)^2 - (d*x + c)*sin(d*x + c)^2 - (d*x + c - 2*cos(d*x + c))*sin(d*x + c) + cos(d*x + c))*cos(2*d*x + 2*c) +
2*((d*x + c)^2 - 1)*cos(d*x + c) + ((d*x + c)*cos(d*x + c)^2 + 2*d*x + 4*((d*x + c)^2 - 1)*cos(d*x + c) + 2*c
)*sin(d*x + c) + c)*sin(2*d*x + 2*c) + ((2*(d*x + c)^2 - 3)*cos(d*x + c)^2 + 2*(d*x + c)*cos(d*x + c) - 1)*sin
(d*x + c))*c*e*f^2/(a*d^2*cos(d*x + c)^4 + a*d^2*sin(d*x + c)^4 + 2*a*d^2*cos(d*x + c)^2*sin(d*x + c) + 2*a*d^
2*sin(d*x + c)^3 + a*d^2*cos(d*x + c)^2 + (a*d^2*cos(d*x + c)^2 + a*d^2*sin(d*x + c)^2 + 2*a*d^2*sin(d*x + c)
+ a*d^2)*cos(2*d*x + 2*c)^2 + (a*d^2*cos(d*x + c)^2 + a*d^2*sin(d*x + c)^2 + 2*a*d^2*sin(d*x + c) + a*d^2)*sin
(2*d*x + 2*c)^2 + (2*a*d^2*cos(d*x + c)^2 + a*d^2)*sin(d*x + c)^2 - 2*(a*d^2*sin(d*x + c)^3 + 2*a*d^2*sin(d*x
+ c)^2 + (a*d^2*cos(d*x + c)^2 + a*d^2)*sin(d*x + c))*cos(2*d*x + 2*c) + 2*(a*d^2*cos(d*x + c)^3 + a*d^2*cos(d
*x + c)*sin(d*x + c)^2 + 2*a*d^2*cos(d*x + c)*sin(d*x + c) + a*d^2*cos(d*x + c))*sin(2*d*x + 2*c)) + 4*e^3*((s
in(d*x + c)/(cos(d*x + c) + 1) + sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 2)/(a + a*sin(d*x + c)/(cos(d*x + c) +
1) + a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a*sin(d*x + c)^3/(cos(d*x + c) + 1)^3) + arctan(sin(d*x + c)/(cos
(d*x + c) + 1))/a) + 3*(((d*x + c)^2 - 1)*cos(d*x + c)^4 + ((d*x + c)^2 - 1)*sin(d*x + c)^4 + ((d*x + c)*cos(d
*x + c) + sin(d*x + c) + 1)*cos(2*d*x + 2*c)^3 + 7*(d*x + c)*cos(d*x + c)^3 + (d*x + (d*x + c)*sin(d*x + c) +
c - cos(d*x + c))*sin(2*d*x + 2*c)^3 + (2*(d*x + c)^2 - 3)*sin(d*x + c)^3 + (((d*x + c)^2 - 1)*cos(d*x + c)^2
+ ((d*x + c)^2 - 3)*sin(d*x + c)^2 + (d*x + c)^2 + 6*(d*x + c)*cos(d*x + c) + 2*((d*x + c)^2 - (d*x + c)*cos(d
*x + c) - 2)*sin(d*x + c) - 1)*cos(2*d*x + 2*c)^2 + ((d*x + c)^2 - 1)*cos(d*x + c)^2 + (((d*x + c)^2 - 3)*cos(
d*x + c)^2 + ((d*x + c)^2 - 1)*sin(d*x + c)^2 + (d*x + c)^2 + ((d*x + c)*cos(d*x + c) + sin(d*x + c) + 1)*cos(
2*d*x + 2*c) + 8*(d*x + c)*cos(d*x + c) + 2*((d*x + c)^2 + (d*x + c)*cos(d*x + c) - 1)*sin(d*x + c) - 1)*sin(2
*d*x + 2*c)^2 + (2*((d*x + c)^2 - 1)*cos(d*x + c)^2 + (d*x + c)^2 + 7*(d*x + c)*cos(d*x + c) - 3)*sin(d*x + c)
^2 + ((d*x + c)*cos(d*x + c)^3 - (2*(d*x + c)^2 - 3)*sin(d*x + c)^3 - (4*(d*x + c)^2 - (d*x + c)*cos(d*x + c)
- 6)*sin(d*x + c)^2 + 2*cos(d*x + c)^2 - ((2*(d*x + c)^2 - 3)*cos(d*x + c)^2 + 2*(d*x + c)^2 + 12*(d*x + c)*co
s(d*x + c) - 4)*sin(d*x + c) + 1)*cos(2*d*x + 2*c) + (d*x + c)*cos(d*x + c) - 2*(cos(d*x + c)^4 + sin(d*x + c)
^4 + (cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1)*cos(2*d*x + 2*c)^2 + (cos(d*x + c)^2 + sin(d*x + c
)^2 + 2*sin(d*x + c) + 1)*sin(2*d*x + 2*c)^2 + 2*cos(d*x + c)^2*sin(d*x + c) + (2*cos(d*x + c)^2 + 1)*sin(d*x
+ c)^2 + 2*sin(d*x + c)^3 - 2*(sin(d*x + c)^3 + (cos(d*x + c)^2 + 1)*sin(d*x + c) + 2*sin(d*x + c)^2)*cos(2*d*
x + 2*c) + cos(d*x + c)^2 + 2*(cos(d*x + c)^3 + cos(d*x + c)*sin(d*x + c)^2 + 2*cos(d*x + c)*sin(d*x + c) + co
s(d*x + c))*sin(2*d*x + 2*c))*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1) + ((2*(d*x + c)^2 - 3)
*cos(d*x + c)^3 + (d*x + c)*sin(d*x + c)^3 + (d*x + (d*x + c)*sin(d*x + c) + c - cos(d*x + c))*cos(2*d*x + 2*c
)^2 + 14*(d*x + c)*cos(d*x + c)^2 + (2*d*x + (2*(d*x + c)^2 - 3)*cos(d*x + c) + 2*c)*sin(d*x + c)^2 + d*x + 2*
((d*x + c)*cos(d*x + c)^2 - (d*x + c)*sin(d*x + c)^2 - (d*x + c - 2*cos(d*x + c))*sin(d*x + c) + cos(d*x + c))
*cos(2*d*x + 2*c) + 2*((d*x + c)^2 - 1)*cos(d*x + c) + ((d*x + c)*cos(d*x + c)^2 + 2*d*x + 4*((d*x + c)^2 - 1)
*cos(d*x + c) + 2*c)*sin(d*x + c) + c)*sin(2*d*x + 2*c) + ((2*(d*x + c)^2 - 3)*cos(d*x + c)^2 + 2*(d*x + c)*co
s(d*x + c) - 1)*sin(d*x + c))*e^2*f/(a*d*cos(d*x + c)^4 + a*d*sin(d*x + c)^4 + 2*a*d*cos(d*x + c)^2*sin(d*x +
c) + 2*a*d*sin(d*x + c)^3 + a*d*cos(d*x + c)^2 + (a*d*cos(d*x + c)^2 + a*d*sin(d*x + c)^2 + 2*a*d*sin(d*x + c)
+ a*d)*cos(2*d*x + 2*c)^2 + (a*d*cos(d*x + c)^2 + a*d*sin(d*x + c)^2 + 2*a*d*sin(d*x + c) + a*d)*sin(2*d*x +
2*c)^2 + (2*a*d*cos(d*x + c)^2 + a*d)*sin(d*x + c)^2 - 2*(a*d*sin(d*x + c)^3 + 2*a*d*sin(d*x + c)^2 + (a*d*cos
(d*x + c)^2 + a*d)*sin(d*x + c))*cos(2*d*x + 2*c) + 2*(a*d*cos(d*x + c)^3 + a*d*cos(d*x + c)*sin(d*x + c)^2 +
2*a*d*cos(d*x + c)*sin(d*x + c) + a*d*cos(d*x + c))*sin(2*d*x + 2*c)) + 2*((d*x + c)^4*f^3 + (4*d*e*f^2 - (4*c
+ 2*I)*f^3)*(d*x + c)^3 + 12*I*d*e*f^2 - (-10*I*c^3 + 6*c^2 + 12*I*c - 12)*f^3 + 6*(-I*d*e*f^2 + (c^2 + I*c -
1)*f^3)*(d*x + c)^2 - (12*d*e*f^2 + (4*c^3 + 6*I*c^2 - 12*c - 12*I)*f^3)*(d*x + c) - (24*c^2*f^3*cos(d*x + c)
+ 24*I*c^2*f^3*sin(d*x + c) + 24*I*c^2*f^3)*arctan2(sin(d*x + c) + 1, cos(d*x + c)) - (-24*I*(d*x + c)^2*f^3
+ (-48*I*d*e*f^2 + 48*I*c*f^3)*(d*x + c) - 24*((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*cos(d*x + c) +
(-24*I*(d*x + c)^2*f^3 + (-48*I*d*e*f^2 + 48*I*c*f^3)*(d*x + c))*sin(d*x + c))*arctan2(cos(d*x + c), sin(d*x
+ c) + 1) - (2*I*(d*x + c)^3*f^3 - 12*I*d*e*f^2 + (-2*I*c^3 - 6*c^2 + 12*I*c + 12)*f^3 + (6*I*d*e*f^2 - 6*(I*c
+ 1)*f^3)*(d*x + c)^2 - (12*d*e*f^2 - (6*I*c^2 + 12*c - 12*I)*f^3)*(d*x + c))*cos(2*d*x + 2*c) - (I*(d*x + c)
^4*f^3 - 2*(-2*I*d*e*f^2 + (2*I*c + 5)*f^3)*(d*x + c)^3 + 12*d*e*f^2 + (2*c^3 - 6*I*c^2 - 12*c + 12*I)*f^3 - (
30*d*e*f^2 - (6*I*c^2 + 30*c - 6*I)*f^3)*(d*x + c)^2 + (-12*I*d*e*f^2 + (-4*I*c^3 - 30*c^2 + 12*I*c + 12)*f^3)
*(d*x + c))*cos(d*x + c) - (-48*I*d*e*f^2 - 48*I*(d*x + c)*f^3 + 48*I*c*f^3 - 48*(d*e*f^2 + (d*x + c)*f^3 - c*
f^3)*cos(d*x + c) + (-48*I*d*e*f^2 - 48*I*(d*x + c)*f^3 + 48*I*c*f^3)*sin(d*x + c))*dilog(I*e^(I*d*x + I*c)) -
(12*(d*x + c)^2*f^3 + 12*c^2*f^3 + 24*(d*e*f^2 - c*f^3)*(d*x + c) + (-12*I*(d*x + c)^2*f^3 - 12*I*c^2*f^3 + (
-24*I*d*e*f^2 + 24*I*c*f^3)*(d*x + c))*cos(d*x + c) + 12*((d*x + c)^2*f^3 + c^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x
+ c))*sin(d*x + c))*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1) + 48*(I*f^3*cos(d*x + c) - f^3*
sin(d*x + c) - f^3)*polylog(3, I*e^(I*d*x + I*c)) + (2*(d*x + c)^3*f^3 - 12*d*e*f^2 - (2*c^3 - 6*I*c^2 - 12*c
+ 12*I)*f^3 + (6*d*e*f^2 - (6*c - 6*I)*f^3)*(d*x + c)^2 + 6*(2*I*d*e*f^2 + (c^2 - 2*I*c - 2)*f^3)*(d*x + c))*s
in(2*d*x + 2*c) + ((d*x + c)^4*f^3 + (4*d*e*f^2 - (4*c - 10*I)*f^3)*(d*x + c)^3 - 12*I*d*e*f^2 - (2*I*c^3 + 6*
c^2 - 12*I*c - 12)*f^3 + 6*(5*I*d*e*f^2 + (c^2 - 5*I*c - 1)*f^3)*(d*x + c)^2 - (12*d*e*f^2 + (4*c^3 - 30*I*c^2
- 12*c + 12*I)*f^3)*(d*x + c))*sin(d*x + c))/(-4*I*a*d^3*cos(d*x + c) + 4*a*d^3*sin(d*x + c) + 4*a*d^3))/d
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\sin \left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^3}{a+a\,\sin \left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((sin(c + d*x)^2*(e + f*x)^3)/(a + a*sin(c + d*x)),x)
[Out]
int((sin(c + d*x)^2*(e + f*x)^3)/(a + a*sin(c + d*x)), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {e^{3} \sin ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{3} x^{3} \sin ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e f^{2} x^{2} \sin ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e^{2} f x \sin ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)**3*sin(d*x+c)**2/(a+a*sin(d*x+c)),x)
[Out]
(Integral(e**3*sin(c + d*x)**2/(sin(c + d*x) + 1), x) + Integral(f**3*x**3*sin(c + d*x)**2/(sin(c + d*x) + 1),
x) + Integral(3*e*f**2*x**2*sin(c + d*x)**2/(sin(c + d*x) + 1), x) + Integral(3*e**2*f*x*sin(c + d*x)**2/(sin
(c + d*x) + 1), x))/a
________________________________________________________________________________________