3.2.79 \(\int \frac {(e+f x)^3 \sin (c+d x)}{a+a \sin (c+d x)} \, dx\) [179]

Optimal. Leaf size=164 \[ \frac {i (e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \text {Li}_2\left (i e^{i (c+d x)}\right )}{a d^3}-\frac {12 f^3 \text {Li}_3\left (i e^{i (c+d x)}\right )}{a d^4} \]

[Out]

I*(f*x+e)^3/a/d+1/4*(f*x+e)^4/a/f+(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

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Rubi [A]
time = 0.22, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {4611, 32, 3399, 4269, 3798, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {12 f^3 \text {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}+\frac {12 i f^2 (e+f x) \text {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(I*(e + f*x)^3)/(a*d) + (e + f*x)^4/(4*a*f) + ((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)

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 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 3399

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + 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 3798

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 4269

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 4611

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 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]

Rubi steps

\begin {align*} \int \frac {(e+f x)^3 \sin (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int (e+f x)^3 \, dx}{a}-\int \frac {(e+f x)^3}{a+a \sin (c+d x)} \, dx\\ &=\frac {(e+f x)^4}{4 a f}-\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 {(e+f x)^4}{4 a f}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\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 {i (e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\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 {(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 {\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 {(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 {\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 {(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 {\left (12 f^3\right ) \text {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 {(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}\\ \end {align*}

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Mathematica [A]
time = 1.19, size = 240, normalized size = 1.46 \begin {gather*} \frac {x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )+\frac {8 f \left (-3 d^2 (e+f x)^2 \log (1-i \cos (c+d x)+\sin (c+d x))+6 i d f (e+f x) \text {Li}_2(i \cos (c+d x)-\sin (c+d x))-6 f^2 \text {Li}_3(i \cos (c+d x)-\sin (c+d x))+\frac {i d^3 x \left (3 e^2+3 e f x+f^2 x^2\right ) (\cos (c)+i \sin (c))}{\cos (c)+i (1+\sin (c))}\right )}{d^4}-\frac {8 (e+f x)^3 \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}}{4 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3) + (8*f*(-3*d^2*(e + f*x)^2*Log[1 - I*Cos[c + d*x] + Sin[c + d*x
]] + (6*I)*d*f*(e + f*x)*PolyLog[2, I*Cos[c + d*x] - Sin[c + d*x]] - 6*f^2*PolyLog[3, I*Cos[c + d*x] - Sin[c +
d*x]] + (I*d^3*x*(3*e^2 + 3*e*f*x + f^2*x^2)*(Cos[c] + I*Sin[c]))/(Cos[c] + I*(1 + Sin[c]))))/d^4 - (8*(e + f
*x)^3*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(4*a)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (145 ) = 290\).
time = 0.17, size = 537, normalized size = 3.27

method	result	size
risch	\(\frac {f^{3} x^{4}}{4 a}+\frac {e^{4}}{4 a f}-\frac {6 i f^{3} c^{2} x}{a \,d^{3}}-\frac {6 f^{3} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) x^{2}}{a \,d^{2}}+\frac {6 f^{3} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) c^{2}}{a \,d^{4}}+\frac {6 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right ) e^{2}}{a \,d^{2}}+\frac {6 f^{3} c^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{4}}-\frac {6 f^{3} c^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a \,d^{4}}-\frac {6 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) e^{2}}{a \,d^{2}}+\frac {2 i f^{3} x^{3}}{a d}-\frac {4 i f^{3} c^{3}}{a \,d^{4}}+\frac {12 i f^{2} e c x}{a \,d^{2}}+\frac {f^{2} e \,x^{3}}{a}+\frac {3 f \,e^{2} x^{2}}{2 a}+\frac {e^{3} x}{a}+\frac {6 i f^{2} e \,x^{2}}{a d}+\frac {2 f^{3} x^{3}+6 e \,f^{2} x^{2}+6 e^{2} f x +2 e^{3}}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}-\frac {12 f^{2} e \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) c}{a \,d^{3}}-\frac {12 f^{2} e c \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}-\frac {12 f^{3} \polylog \left (3, i {\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{4}}-\frac {12 f^{2} e \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) x}{a \,d^{2}}+\frac {12 i f^{2} e \polylog \left (2, i {\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}+\frac {6 i f^{2} e \,c^{2}}{a \,d^{3}}+\frac {12 i f^{3} \polylog \left (2, i {\mathrm e}^{i \left (d x +c \right )}\right ) x}{a \,d^{3}}+\frac {12 f^{2} e c \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a \,d^{3}}\)	\(537\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^3*sin(d*x+c)/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/4/a*f^3*x^4+1/4/a/f*e^4-6/a/d^2*f^3*ln(1-I*exp(I*(d*x+c)))*x^2+6/a/d^4*f^3*ln(1-I*exp(I*(d*x+c)))*c^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*l
n(exp(I*(d*x+c))+I)*e^2+2*I/a/d*f^3*x^3-4*I/a/d^4*f^3*c^3+12*I/a/d^2*f^2*e*c*x+1/a*f^2*e*x^3+3/2/a*f*e^2*x^2+1
/a*e^3*x-12/a/d^3*f^2*e*c*ln(exp(I*(d*x+c)))+12*I/a/d^3*f^3*polylog(2,I*exp(I*(d*x+c)))*x-6*I/a/d^3*f^3*c^2*x+
6*I/a/d*f^2*e*x^2+6*I/a/d^3*f^2*e*c^2+12*I/a/d^3*f^2*e*polylog(2,I*exp(I*(d*x+c)))+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)-12*f^3*polylog(3,I*exp(I*(d*x+c)))/a/d^4+12/a/d^3*f^2*e*c*ln(exp(I*(d*x+c))+
I)-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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1316 vs. \(2 (144) = 288\).
time = 0.63, size = 1316, normalized size = 8.02 \begin {gather*} \frac {12 \, c^{2} f^{2} {\left (\frac {1}{a d^{2} + \frac {a d^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}} + \frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a d^{2}}\right )} e - 12 \, c f {\left (\frac {1}{a d + \frac {a d \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}} + \frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a d}\right )} e^{2} - \frac {6 \, {\left ({\left (d x + c\right )}^{2} \cos \left (d x + c\right )^{2} + {\left (d x + c\right )}^{2} \sin \left (d x + c\right )^{2} + 2 \, {\left (d x + c\right )}^{2} \sin \left (d x + c\right ) + {\left (d x + c\right )}^{2} + 4 \, {\left (d x + c\right )} \cos \left (d x + c\right ) - 2 \, {\left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right )\right )} c f^{2} e}{a d^{2} \cos \left (d x + c\right )^{2} + a d^{2} \sin \left (d x + c\right )^{2} + 2 \, a d^{2} \sin \left (d x + c\right ) + a d^{2}} + 4 \, {\left (\frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {1}{a + \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}}\right )} e^{3} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} \cos \left (d x + c\right )^{2} + {\left (d x + c\right )}^{2} \sin \left (d x + c\right )^{2} + 2 \, {\left (d x + c\right )}^{2} \sin \left (d x + c\right ) + {\left (d x + c\right )}^{2} + 4 \, {\left (d x + c\right )} \cos \left (d x + c\right ) - 2 \, {\left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right )\right )} f e^{2}}{a d \cos \left (d x + c\right )^{2} + a d \sin \left (d x + c\right )^{2} + 2 \, a d \sin \left (d x + c\right ) + a d} + \frac {2 \, {\left ({\left (d x + c\right )}^{4} f^{3} + 6 \, {\left (d x + c\right )}^{2} c^{2} f^{3} - 4 \, {\left (d x + c\right )} c^{3} f^{3} + 8 i \, c^{3} f^{3} - 4 \, {\left (c f^{3} - d f^{2} e\right )} {\left (d x + c\right )}^{3} - 24 \, {\left (c^{2} f^{3} \cos \left (d x + c\right ) + i \, c^{2} f^{3} \sin \left (d x + c\right ) + i \, c^{2} f^{3}\right )} \arctan \left (\sin \left (d x + c\right ) + 1, \cos \left (d x + c\right )\right ) + 24 \, {\left (i \, {\left (d x + c\right )}^{2} f^{3} + 2 \, {\left (-i \, c f^{3} + i \, d f^{2} e\right )} {\left (d x + c\right )} + {\left ({\left (d x + c\right )}^{2} f^{3} - 2 \, {\left (c f^{3} - d f^{2} e\right )} {\left (d x + c\right )}\right )} \cos \left (d x + c\right ) + {\left (i \, {\left (d x + c\right )}^{2} f^{3} + 2 \, {\left (-i \, c f^{3} + i \, d f^{2} e\right )} {\left (d x + c\right )}\right )} \sin \left (d x + c\right )\right )} \arctan \left (\cos \left (d x + c\right ), \sin \left (d x + c\right ) + 1\right ) - {\left (i \, {\left (d x + c\right )}^{4} f^{3} - 4 \, {\left (i \, c^{3} + 6 \, c^{2}\right )} {\left (d x + c\right )} f^{3} - 4 \, {\left ({\left (i \, c + 2\right )} f^{3} - i \, d f^{2} e\right )} {\left (d x + c\right )}^{3} - 6 \, {\left ({\left (-i \, c^{2} - 4 \, c\right )} f^{3} + 4 \, d f^{2} e\right )} {\left (d x + c\right )}^{2}\right )} \cos \left (d x + c\right ) + 48 \, {\left (i \, {\left (d x + c\right )} f^{3} - i \, c f^{3} + i \, d f^{2} e + {\left ({\left (d x + c\right )} f^{3} - c f^{3} + d f^{2} e\right )} \cos \left (d x + c\right ) + {\left (i \, {\left (d x + c\right )} f^{3} - i \, c f^{3} + i \, d f^{2} e\right )} \sin \left (d x + c\right )\right )} {\rm Li}_2\left (i \, e^{\left (i \, d x + i \, c\right )}\right ) - 12 \, {\left ({\left (d x + c\right )}^{2} f^{3} + c^{2} f^{3} - 2 \, {\left (c f^{3} - d f^{2} e\right )} {\left (d x + c\right )} - {\left (i \, {\left (d x + c\right )}^{2} f^{3} + i \, c^{2} f^{3} + 2 \, {\left (-i \, c f^{3} + i \, d f^{2} e\right )} {\left (d x + c\right )}\right )} \cos \left (d x + c\right ) + {\left ({\left (d x + c\right )}^{2} f^{3} + c^{2} f^{3} - 2 \, {\left (c f^{3} - d f^{2} e\right )} {\left (d x + c\right )}\right )} \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) + 48 \, {\left (i \, f^{3} \cos \left (d x + c\right ) - f^{3} \sin \left (d x + c\right ) - f^{3}\right )} {\rm Li}_{3}(i \, e^{\left (i \, d x + i \, c\right )}) + {\left ({\left (d x + c\right )}^{4} f^{3} - 4 \, {\left (c^{3} - 6 i \, c^{2}\right )} {\left (d x + c\right )} f^{3} - 4 \, {\left ({\left (c - 2 i\right )} f^{3} - d f^{2} e\right )} {\left (d x + c\right )}^{3} + 6 \, {\left ({\left (c^{2} - 4 i \, c\right )} f^{3} + 4 i \, d f^{2} e\right )} {\left (d x + c\right )}^{2}\right )} \sin \left (d x + c\right )\right )}}{-4 i \, a d^{3} \cos \left (d x + c\right ) + 4 \, a d^{3} \sin \left (d x + c\right ) + 4 \, a d^{3}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(12*c^2*f^2*(1/(a*d^2 + a*d^2*sin(d*x + c)/(cos(d*x + c) + 1)) + arctan(sin(d*x + c)/(cos(d*x + c) + 1))/(
a*d^2))*e - 12*c*f*(1/(a*d + a*d*sin(d*x + c)/(cos(d*x + c) + 1)) + arctan(sin(d*x + c)/(cos(d*x + c) + 1))/(a
*d))*e^2 - 6*((d*x + c)^2*cos(d*x + c)^2 + (d*x + c)^2*sin(d*x + c)^2 + 2*(d*x + c)^2*sin(d*x + c) + (d*x + c)
^2 + 4*(d*x + c)*cos(d*x + c) - 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1)*log(cos(d*x + c)^2 +
sin(d*x + c)^2 + 2*sin(d*x + c) + 1))*c*f^2*e/(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) + 4*(arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + 1/(a + a*sin(d*x + c)/(cos(d*x + c) + 1)))*e^3 +
3*((d*x + c)^2*cos(d*x + c)^2 + (d*x + c)^2*sin(d*x + c)^2 + 2*(d*x + c)^2*sin(d*x + c) + (d*x + c)^2 + 4*(d*
x + c)*cos(d*x + c) - 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1)*log(cos(d*x + c)^2 + sin(d*x +
c)^2 + 2*sin(d*x + c) + 1))*f*e^2/(a*d*cos(d*x + c)^2 + a*d*sin(d*x + c)^2 + 2*a*d*sin(d*x + c) + a*d) + 2*((d
*x + c)^4*f^3 + 6*(d*x + c)^2*c^2*f^3 - 4*(d*x + c)*c^3*f^3 + 8*I*c^3*f^3 - 4*(c*f^3 - d*f^2*e)*(d*x + c)^3 -
24*(c^2*f^3*cos(d*x + c) + I*c^2*f^3*sin(d*x + c) + I*c^2*f^3)*arctan2(sin(d*x + c) + 1, cos(d*x + c)) + 24*(I
*(d*x + c)^2*f^3 + 2*(-I*c*f^3 + I*d*f^2*e)*(d*x + c) + ((d*x + c)^2*f^3 - 2*(c*f^3 - d*f^2*e)*(d*x + c))*cos(
d*x + c) + (I*(d*x + c)^2*f^3 + 2*(-I*c*f^3 + I*d*f^2*e)*(d*x + c))*sin(d*x + c))*arctan2(cos(d*x + c), sin(d*
x + c) + 1) - (I*(d*x + c)^4*f^3 - 4*(I*c^3 + 6*c^2)*(d*x + c)*f^3 - 4*((I*c + 2)*f^3 - I*d*f^2*e)*(d*x + c)^3
- 6*((-I*c^2 - 4*c)*f^3 + 4*d*f^2*e)*(d*x + c)^2)*cos(d*x + c) + 48*(I*(d*x + c)*f^3 - I*c*f^3 + I*d*f^2*e +
((d*x + c)*f^3 - c*f^3 + d*f^2*e)*cos(d*x + c) + (I*(d*x + c)*f^3 - I*c*f^3 + I*d*f^2*e)*sin(d*x + c))*dilog(I
*e^(I*d*x + I*c)) - 12*((d*x + c)^2*f^3 + c^2*f^3 - 2*(c*f^3 - d*f^2*e)*(d*x + c) - (I*(d*x + c)^2*f^3 + I*c^2
*f^3 + 2*(-I*c*f^3 + I*d*f^2*e)*(d*x + c))*cos(d*x + c) + ((d*x + c)^2*f^3 + c^2*f^3 - 2*(c*f^3 - d*f^2*e)*(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)) + ((d*x + c)^4*f^3 - 4*(c^3 - 6*I*c^2)*(d*x + c)*f^3 - 4*((
c - 2*I)*f^3 - d*f^2*e)*(d*x + c)^3 + 6*((c^2 - 4*I*c)*f^3 + 4*I*d*f^2*e)*(d*x + c)^2)*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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1051 vs. \(2 (144) = 288\).
time = 0.39, size = 1051, normalized size = 6.41 \begin {gather*} \frac {d^{4} f^{3} x^{4} + 4 \, d^{3} f^{3} x^{3} + {\left (d^{4} f^{3} x^{4} + 4 \, d^{3} f^{3} x^{3} + 4 \, {\left (d^{4} x + d^{3}\right )} e^{3} + 6 \, {\left (d^{4} f x^{2} + 2 \, d^{3} f x\right )} e^{2} + 4 \, {\left (d^{4} f^{2} x^{3} + 3 \, d^{3} f^{2} x^{2}\right )} e\right )} \cos \left (d x + c\right ) - 24 \, {\left (-i \, d f^{3} x - i \, d f^{2} e + {\left (-i \, d f^{3} x - i \, d f^{2} e\right )} \cos \left (d x + c\right ) + {\left (-i \, d f^{3} x - i \, d f^{2} e\right )} \sin \left (d x + c\right )\right )} {\rm Li}_2\left (i \, \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right ) - 24 \, {\left (i \, d f^{3} x + i \, d f^{2} e + {\left (i \, d f^{3} x + i \, d f^{2} e\right )} \cos \left (d x + c\right ) + {\left (i \, d f^{3} x + i \, d f^{2} e\right )} \sin \left (d x + c\right )\right )} {\rm Li}_2\left (-i \, \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right ) + 4 \, {\left (d^{4} x + d^{3}\right )} e^{3} + 6 \, {\left (d^{4} f x^{2} + 2 \, d^{3} f x\right )} e^{2} + 4 \, {\left (d^{4} f^{2} x^{3} + 3 \, d^{3} f^{2} x^{2}\right )} e - 12 \, {\left (c^{2} f^{3} - 2 \, c d f^{2} e + d^{2} f e^{2} + {\left (c^{2} f^{3} - 2 \, c d f^{2} e + d^{2} f e^{2}\right )} \cos \left (d x + c\right ) + {\left (c^{2} f^{3} - 2 \, c d f^{2} e + d^{2} f e^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + i\right ) - 12 \, {\left (d^{2} f^{3} x^{2} - c^{2} f^{3} + {\left (d^{2} f^{3} x^{2} - c^{2} f^{3} + 2 \, {\left (d^{2} f^{2} x + c d f^{2}\right )} e\right )} \cos \left (d x + c\right ) + 2 \, {\left (d^{2} f^{2} x + c d f^{2}\right )} e + {\left (d^{2} f^{3} x^{2} - c^{2} f^{3} + 2 \, {\left (d^{2} f^{2} x + c d f^{2}\right )} e\right )} \sin \left (d x + c\right )\right )} \log \left (i \, \cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right ) - 12 \, {\left (d^{2} f^{3} x^{2} - c^{2} f^{3} + {\left (d^{2} f^{3} x^{2} - c^{2} f^{3} + 2 \, {\left (d^{2} f^{2} x + c d f^{2}\right )} e\right )} \cos \left (d x + c\right ) + 2 \, {\left (d^{2} f^{2} x + c d f^{2}\right )} e + {\left (d^{2} f^{3} x^{2} - c^{2} f^{3} + 2 \, {\left (d^{2} f^{2} x + c d f^{2}\right )} e\right )} \sin \left (d x + c\right )\right )} \log \left (-i \, \cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right ) - 12 \, {\left (c^{2} f^{3} - 2 \, c d f^{2} e + d^{2} f e^{2} + {\left (c^{2} f^{3} - 2 \, c d f^{2} e + d^{2} f e^{2}\right )} \cos \left (d x + c\right ) + {\left (c^{2} f^{3} - 2 \, c d f^{2} e + d^{2} f e^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + i\right ) - 24 \, {\left (f^{3} \cos \left (d x + c\right ) + f^{3} \sin \left (d x + c\right ) + f^{3}\right )} {\rm polylog}\left (3, i \, \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right ) - 24 \, {\left (f^{3} \cos \left (d x + c\right ) + f^{3} \sin \left (d x + c\right ) + f^{3}\right )} {\rm polylog}\left (3, -i \, \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right ) + {\left (d^{4} f^{3} x^{4} - 4 \, d^{3} f^{3} x^{3} + 4 \, {\left (d^{4} x - d^{3}\right )} e^{3} + 6 \, {\left (d^{4} f x^{2} - 2 \, d^{3} f x\right )} e^{2} + 4 \, {\left (d^{4} f^{2} x^{3} - 3 \, d^{3} f^{2} x^{2}\right )} e\right )} \sin \left (d x + c\right )}{4 \, {\left (a d^{4} \cos \left (d x + c\right ) + a d^{4} \sin \left (d x + c\right ) + a d^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(d^4*f^3*x^4 + 4*d^3*f^3*x^3 + (d^4*f^3*x^4 + 4*d^3*f^3*x^3 + 4*(d^4*x + d^3)*e^3 + 6*(d^4*f*x^2 + 2*d^3*f
*x)*e^2 + 4*(d^4*f^2*x^3 + 3*d^3*f^2*x^2)*e)*cos(d*x + c) - 24*(-I*d*f^3*x - I*d*f^2*e + (-I*d*f^3*x - I*d*f^2
*e)*cos(d*x + c) + (-I*d*f^3*x - I*d*f^2*e)*sin(d*x + c))*dilog(I*cos(d*x + c) - sin(d*x + c)) - 24*(I*d*f^3*x
+ I*d*f^2*e + (I*d*f^3*x + I*d*f^2*e)*cos(d*x + c) + (I*d*f^3*x + I*d*f^2*e)*sin(d*x + c))*dilog(-I*cos(d*x +
c) - sin(d*x + c)) + 4*(d^4*x + d^3)*e^3 + 6*(d^4*f*x^2 + 2*d^3*f*x)*e^2 + 4*(d^4*f^2*x^3 + 3*d^3*f^2*x^2)*e
- 12*(c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2 + (c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2)*cos(d*x + c) + (c^2*f^3 - 2*c*d*
f^2*e + d^2*f*e^2)*sin(d*x + c))*log(cos(d*x + c) + I*sin(d*x + c) + I) - 12*(d^2*f^3*x^2 - c^2*f^3 + (d^2*f^3
*x^2 - c^2*f^3 + 2*(d^2*f^2*x + c*d*f^2)*e)*cos(d*x + c) + 2*(d^2*f^2*x + c*d*f^2)*e + (d^2*f^3*x^2 - c^2*f^3
+ 2*(d^2*f^2*x + c*d*f^2)*e)*sin(d*x + c))*log(I*cos(d*x + c) + sin(d*x + c) + 1) - 12*(d^2*f^3*x^2 - c^2*f^3
+ (d^2*f^3*x^2 - c^2*f^3 + 2*(d^2*f^2*x + c*d*f^2)*e)*cos(d*x + c) + 2*(d^2*f^2*x + c*d*f^2)*e + (d^2*f^3*x^2
- c^2*f^3 + 2*(d^2*f^2*x + c*d*f^2)*e)*sin(d*x + c))*log(-I*cos(d*x + c) + sin(d*x + c) + 1) - 12*(c^2*f^3 - 2
*c*d*f^2*e + d^2*f*e^2 + (c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2)*cos(d*x + c) + (c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2
)*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)*polyl
og(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*f^3*x^3 + 4*(d^4*x - d^3)*e^3 + 6*(d^4*f*x^2 - 2*d^3*f*x)*e^2 + 4*(d
^4*f^2*x^3 - 3*d^3*f^2*x^2)*e)*sin(d*x + c))/(a*d^4*cos(d*x + c) + a*d^4*sin(d*x + c) + a*d^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(e**3*sin(c + d*x)/(sin(c + d*x) + 1), x) + Integral(f**3*x**3*sin(c + d*x)/(sin(c + d*x) + 1), x) +
Integral(3*e*f**2*x**2*sin(c + d*x)/(sin(c + d*x) + 1), x) + Integral(3*e**2*f*x*sin(c + d*x)/(sin(c + d*x) +
1), x))/a

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sin \left (c+d\,x\right )\,{\left (e+f\,x\right )}^3}{a+a\,\sin \left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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