3.133 \(\int \frac {(c+d x)^3}{(a+a \cos (e+f x))^2} \, dx\)

Optimal. Leaf size=271 \[ -\frac {4 i d^2 (c+d x) \text {Li}_2\left (-e^{i (e+f x)}\right )}{a^2 f^3}+\frac {2 d^2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {2 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}-\frac {d (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {i (c+d x)^3}{3 a^2 f}+\frac {4 d^3 \text {Li}_3\left (-e^{i (e+f x)}\right )}{a^2 f^4}+\frac {4 d^3 \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4} \]

[Out]

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

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Rubi [A]  time = 0.37, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3318, 4186, 4184, 3475, 3719, 2190, 2531, 2282, 6589} \[ -\frac {4 i d^2 (c+d x) \text {Li}_2\left (-e^{i (e+f x)}\right )}{a^2 f^3}+\frac {2 d^2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {2 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}-\frac {d (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {i (c+d x)^3}{3 a^2 f}+\frac {4 d^3 \text {Li}_3\left (-e^{i (e+f x)}\right )}{a^2 f^4}+\frac {4 d^3 \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(a + a*Cos[e + f*x])^2,x]

[Out]

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

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 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 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3719

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 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 4186

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(b^2*(c + d*x)^m*Cot[e
+ f*x]*(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*d^2*m*(m - 1))/(f^2*(n - 1)*(n - 2)), Int[(c + d
*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)^m*(b*Csc[e + f*x])^(n
- 2), x], x] - Simp[(b^2*d*m*(c + d*x)^(m - 1)*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[
{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

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 {(c+d x)^3}{(a+a \cos (e+f x))^2} \, dx &=\frac {\int (c+d x)^3 \csc ^4\left (\frac {e+\pi }{2}+\frac {f x}{2}\right ) \, dx}{4 a^2}\\ &=-\frac {d (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {(c+d x)^3 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\int (c+d x)^3 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{6 a^2}+\frac {d^2 \int (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a^2 f^2}\\ &=-\frac {d (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {2 d^2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {\left (2 d^3\right ) \int \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a^2 f^3}-\frac {d \int (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a^2 f}\\ &=-\frac {i (c+d x)^3}{3 a^2 f}+\frac {4 d^3 \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {d (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {2 d^2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {(2 i d) \int \frac {e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)^2}{1+e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{a^2 f}\\ &=-\frac {i (c+d x)^3}{3 a^2 f}+\frac {2 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {d (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {2 d^2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {\left (4 d^2\right ) \int (c+d x) \log \left (1+e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a^2 f^2}\\ &=-\frac {i (c+d x)^3}{3 a^2 f}+\frac {2 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 i d^2 (c+d x) \text {Li}_2\left (-e^{i (e+f x)}\right )}{a^2 f^3}-\frac {d (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {2 d^2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\left (4 i d^3\right ) \int \text {Li}_2\left (-e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a^2 f^3}\\ &=-\frac {i (c+d x)^3}{3 a^2 f}+\frac {2 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 i d^2 (c+d x) \text {Li}_2\left (-e^{i (e+f x)}\right )}{a^2 f^3}-\frac {d (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {2 d^2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\left (4 d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}{a^2 f^4}\\ &=-\frac {i (c+d x)^3}{3 a^2 f}+\frac {2 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 i d^2 (c+d x) \text {Li}_2\left (-e^{i (e+f x)}\right )}{a^2 f^3}+\frac {4 d^3 \text {Li}_3\left (-e^{i (e+f x)}\right )}{a^2 f^4}-\frac {d (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {2 d^2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}\\ \end {align*}

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Mathematica [A]  time = 1.05, size = 250, normalized size = 0.92 \[ \frac {2 \cos \left (\frac {1}{2} (e+f x)\right ) \left (-2 \cos ^3\left (\frac {1}{2} (e+f x)\right ) \left (-6 d \left (f^2 (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )-2 i d f (c+d x) \text {Li}_2\left (-e^{i (e+f x)}\right )+2 d^2 \text {Li}_3\left (-e^{i (e+f x)}\right )\right )-f^3 (c+d x)^3 \tan \left (\frac {1}{2} (e+f x)\right )+i f^3 (c+d x)^3\right )+12 d^2 \cos ^3\left (\frac {1}{2} (e+f x)\right ) \left (f (c+d x) \tan \left (\frac {1}{2} (e+f x)\right )+2 d \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )+f^3 (c+d x)^3 \sin \left (\frac {1}{2} (e+f x)\right )-3 d f^2 (c+d x)^2 \cos \left (\frac {1}{2} (e+f x)\right )\right )}{3 a^2 f^4 (\cos (e+f x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3/(a + a*Cos[e + f*x])^2,x]

[Out]

(2*Cos[(e + f*x)/2]*(-3*d*f^2*(c + d*x)^2*Cos[(e + f*x)/2] + f^3*(c + d*x)^3*Sin[(e + f*x)/2] + 12*d^2*Cos[(e
+ f*x)/2]^3*(2*d*Log[Cos[(e + f*x)/2]] + f*(c + d*x)*Tan[(e + f*x)/2]) - 2*Cos[(e + f*x)/2]^3*(I*f^3*(c + d*x)
^3 - 6*d*(f^2*(c + d*x)^2*Log[1 + E^(I*(e + f*x))] - (2*I)*d*f*(c + d*x)*PolyLog[2, -E^(I*(e + f*x))] + 2*d^2*
PolyLog[3, -E^(I*(e + f*x))]) - f^3*(c + d*x)^3*Tan[(e + f*x)/2])))/(3*a^2*f^4*(1 + Cos[e + f*x])^2)

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fricas [C]  time = 0.66, size = 769, normalized size = 2.84 \[ -\frac {3 \, d^{3} f^{2} x^{2} + 6 \, c d^{2} f^{2} x + 3 \, c^{2} d f^{2} + 3 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2}\right )} \cos \left (f x + e\right ) - {\left (6 i \, d^{3} f x + 6 i \, c d^{2} f + {\left (6 i \, d^{3} f x + 6 i \, c d^{2} f\right )} \cos \left (f x + e\right )^{2} + {\left (12 i \, d^{3} f x + 12 i \, c d^{2} f\right )} \cos \left (f x + e\right )\right )} {\rm Li}_2\left (-\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) - {\left (-6 i \, d^{3} f x - 6 i \, c d^{2} f + {\left (-6 i \, d^{3} f x - 6 i \, c d^{2} f\right )} \cos \left (f x + e\right )^{2} + {\left (-12 i \, d^{3} f x - 12 i \, c d^{2} f\right )} \cos \left (f x + e\right )\right )} {\rm Li}_2\left (-\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 3 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2} + 2 \, d^{3} + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2} + 2 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2} + 2 \, d^{3}\right )} \cos \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2} + 2 \, d^{3} + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2} + 2 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2} + 2 \, d^{3}\right )} \cos \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + 1\right ) - 6 \, {\left (d^{3} \cos \left (f x + e\right )^{2} + 2 \, d^{3} \cos \left (f x + e\right ) + d^{3}\right )} {\rm polylog}\left (3, -\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) - 6 \, {\left (d^{3} \cos \left (f x + e\right )^{2} + 2 \, d^{3} \cos \left (f x + e\right ) + d^{3}\right )} {\rm polylog}\left (3, -\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - {\left (2 \, d^{3} f^{3} x^{3} + 6 \, c d^{2} f^{3} x^{2} + 2 \, c^{3} f^{3} + 6 \, c d^{2} f + 6 \, {\left (c^{2} d f^{3} + d^{3} f\right )} x + {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} + c^{3} f^{3} + 6 \, c d^{2} f + 3 \, {\left (c^{2} d f^{3} + 2 \, d^{3} f\right )} x\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{3 \, {\left (a^{2} f^{4} \cos \left (f x + e\right )^{2} + 2 \, a^{2} f^{4} \cos \left (f x + e\right ) + a^{2} f^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+a*cos(f*x+e))^2,x, algorithm="fricas")

[Out]

-1/3*(3*d^3*f^2*x^2 + 6*c*d^2*f^2*x + 3*c^2*d*f^2 + 3*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2)*cos(f*x + e) -
(6*I*d^3*f*x + 6*I*c*d^2*f + (6*I*d^3*f*x + 6*I*c*d^2*f)*cos(f*x + e)^2 + (12*I*d^3*f*x + 12*I*c*d^2*f)*cos(f
*x + e))*dilog(-cos(f*x + e) + I*sin(f*x + e)) - (-6*I*d^3*f*x - 6*I*c*d^2*f + (-6*I*d^3*f*x - 6*I*c*d^2*f)*co
s(f*x + e)^2 + (-12*I*d^3*f*x - 12*I*c*d^2*f)*cos(f*x + e))*dilog(-cos(f*x + e) - I*sin(f*x + e)) - 3*(d^3*f^2
*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + 2*d^3 + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + 2*d^3)*cos(f*x + e)^2 +
2*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + 2*d^3)*cos(f*x + e))*log(cos(f*x + e) + I*sin(f*x + e) + 1) - 3*(
d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + 2*d^3 + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + 2*d^3)*cos(f*x +
e)^2 + 2*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + 2*d^3)*cos(f*x + e))*log(cos(f*x + e) - I*sin(f*x + e) + 1
) - 6*(d^3*cos(f*x + e)^2 + 2*d^3*cos(f*x + e) + d^3)*polylog(3, -cos(f*x + e) + I*sin(f*x + e)) - 6*(d^3*cos(
f*x + e)^2 + 2*d^3*cos(f*x + e) + d^3)*polylog(3, -cos(f*x + e) - I*sin(f*x + e)) - (2*d^3*f^3*x^3 + 6*c*d^2*f
^3*x^2 + 2*c^3*f^3 + 6*c*d^2*f + 6*(c^2*d*f^3 + d^3*f)*x + (d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + c^3*f^3 + 6*c*d^2*
f + 3*(c^2*d*f^3 + 2*d^3*f)*x)*cos(f*x + e))*sin(f*x + e))/(a^2*f^4*cos(f*x + e)^2 + 2*a^2*f^4*cos(f*x + e) +
a^2*f^4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{3}}{{\left (a \cos \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+a*cos(f*x+e))^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^3/(a*cos(f*x + e) + a)^2, x)

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maple [B]  time = 0.46, size = 678, normalized size = 2.50 \[ -\frac {4 i d^{2} c e x}{a^{2} f^{2}}+\frac {4 d^{3} \polylog \left (3, -{\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{4}}+\frac {4 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{a^{2} f^{4}}-\frac {4 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{4}}-\frac {4 i d^{2} c \polylog \left (2, -{\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{3}}+\frac {2 d \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{a^{2} f^{2}}-\frac {2 d \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{2}}-\frac {2 d^{3} e^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{4}}-\frac {2 i d^{2} c \,x^{2}}{a^{2} f}-\frac {4 i d^{3} \polylog \left (2, -{\mathrm e}^{i \left (f x +e \right )}\right ) x}{a^{2} f^{3}}-\frac {2 i d^{3} x^{3}}{3 a^{2} f}+\frac {2 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) x^{2}}{a^{2} f^{2}}+\frac {4 i d^{3} e^{3}}{3 a^{2} f^{4}}+\frac {4 d^{2} c e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{3}}+\frac {2 i d^{3} e^{2} x}{a^{2} f^{3}}-\frac {2 i d^{2} c \,e^{2}}{a^{2} f^{3}}+\frac {4 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) c x}{a^{2} f^{2}}+\frac {2 i \left (3 i c^{2} d f \,{\mathrm e}^{i \left (f x +e \right )}+3 d^{3} f^{2} x^{3} {\mathrm e}^{i \left (f x +e \right )}+3 i d^{3} f \,x^{2} {\mathrm e}^{2 i \left (f x +e \right )}+6 i c \,d^{2} f x \,{\mathrm e}^{i \left (f x +e \right )}+9 c \,d^{2} f^{2} x^{2} {\mathrm e}^{i \left (f x +e \right )}+d^{3} f^{2} x^{3}+3 i d^{3} f \,x^{2} {\mathrm e}^{i \left (f x +e \right )}+3 i c^{2} d f \,{\mathrm e}^{2 i \left (f x +e \right )}+9 c^{2} d \,f^{2} x \,{\mathrm e}^{i \left (f x +e \right )}+3 c \,d^{2} f^{2} x^{2}+6 i c \,d^{2} f x \,{\mathrm e}^{2 i \left (f x +e \right )}+3 c^{3} f^{2} {\mathrm e}^{i \left (f x +e \right )}+3 c^{2} d \,f^{2} x +6 d^{3} x \,{\mathrm e}^{2 i \left (f x +e \right )}+c^{3} f^{2}+6 c \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+12 d^{3} x \,{\mathrm e}^{i \left (f x +e \right )}+12 c \,d^{2} {\mathrm e}^{i \left (f x +e \right )}+6 d^{3} x +6 c \,d^{2}\right )}{3 f^{3} a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3/(a+a*cos(f*x+e))^2,x)

[Out]

-4*I/a^2/f^2*d^2*c*e*x+4*d^3*polylog(3,-exp(I*(f*x+e)))/a^2/f^4+4/a^2/f^4*d^3*ln(exp(I*(f*x+e))+1)-4/a^2/f^4*d
^3*ln(exp(I*(f*x+e)))-2*I/a^2/f*d^2*c*x^2+2/a^2/f^2*d*c^2*ln(exp(I*(f*x+e))+1)-2/a^2/f^2*d*c^2*ln(exp(I*(f*x+e
)))-2/a^2/f^4*d^3*e^2*ln(exp(I*(f*x+e)))-4*I/a^2/f^3*d^2*c*polylog(2,-exp(I*(f*x+e)))-4*I/a^2/f^3*d^3*polylog(
2,-exp(I*(f*x+e)))*x-2/3*I/a^2/f*d^3*x^3+2/a^2/f^2*d^3*ln(exp(I*(f*x+e))+1)*x^2+4/3*I/a^2/f^4*d^3*e^3+4/a^2/f^
3*d^2*c*e*ln(exp(I*(f*x+e)))+2*I/a^2/f^3*d^3*e^2*x-2*I/a^2/f^3*d^2*c*e^2+4/a^2/f^2*d^2*ln(exp(I*(f*x+e))+1)*c*
x+2/3*I*(3*I*c^2*d*f*exp(I*(f*x+e))+3*d^3*f^2*x^3*exp(I*(f*x+e))+3*I*d^3*f*x^2*exp(2*I*(f*x+e))+6*I*c*d^2*f*x*
exp(I*(f*x+e))+9*c*d^2*f^2*x^2*exp(I*(f*x+e))+d^3*f^2*x^3+3*I*d^3*f*x^2*exp(I*(f*x+e))+3*I*c^2*d*f*exp(2*I*(f*
x+e))+9*c^2*d*f^2*x*exp(I*(f*x+e))+3*c*d^2*f^2*x^2+6*I*c*d^2*f*x*exp(2*I*(f*x+e))+3*c^3*f^2*exp(I*(f*x+e))+3*c
^2*d*f^2*x+6*d^3*x*exp(2*I*(f*x+e))+c^3*f^2+6*c*d^2*exp(2*I*(f*x+e))+12*d^3*x*exp(I*(f*x+e))+12*c*d^2*exp(I*(f
*x+e))+6*d^3*x+6*c*d^2)/f^3/a^2/(exp(I*(f*x+e))+1)^3

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maxima [B]  time = 3.96, size = 3274, normalized size = 12.08 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+a*cos(f*x+e))^2,x, algorithm="maxima")

[Out]

1/6*(12*(2*(3*(f*x + e)*sin(f*x + e) + cos(2*f*x + 2*e) + cos(f*x + e))*cos(3*f*x + 3*e) + 2*(9*(f*x + e)*sin(
f*x + e) + 6*cos(f*x + e) + 1)*cos(2*f*x + 2*e) + 6*cos(2*f*x + 2*e)^2 + 6*cos(f*x + e)^2 - (2*(3*cos(2*f*x +
2*e) + 3*cos(f*x + e) + 1)*cos(3*f*x + 3*e) + cos(3*f*x + 3*e)^2 + 6*(3*cos(f*x + e) + 1)*cos(2*f*x + 2*e) + 9
*cos(2*f*x + 2*e)^2 + 9*cos(f*x + e)^2 + 6*(sin(2*f*x + 2*e) + sin(f*x + e))*sin(3*f*x + 3*e) + sin(3*f*x + 3*
e)^2 + 9*sin(2*f*x + 2*e)^2 + 18*sin(2*f*x + 2*e)*sin(f*x + e) + 9*sin(f*x + e)^2 + 6*cos(f*x + e) + 1)*log(co
s(f*x + e)^2 + sin(f*x + e)^2 + 2*cos(f*x + e) + 1) - 2*(f*x + 3*(f*x + e)*cos(f*x + e) + e - sin(2*f*x + 2*e)
- sin(f*x + e))*sin(3*f*x + 3*e) - 6*(f*x + 3*(f*x + e)*cos(f*x + e) + e - 2*sin(f*x + e))*sin(2*f*x + 2*e) +
6*sin(2*f*x + 2*e)^2 + 6*sin(f*x + e)^2 + 2*cos(f*x + e))*c*d^2*e/(a^2*f^2*cos(3*f*x + 3*e)^2 + 9*a^2*f^2*cos
(2*f*x + 2*e)^2 + 9*a^2*f^2*cos(f*x + e)^2 + a^2*f^2*sin(3*f*x + 3*e)^2 + 9*a^2*f^2*sin(2*f*x + 2*e)^2 + 18*a^
2*f^2*sin(2*f*x + 2*e)*sin(f*x + e) + 9*a^2*f^2*sin(f*x + e)^2 + 6*a^2*f^2*cos(f*x + e) + a^2*f^2 + 2*(3*a^2*f
^2*cos(2*f*x + 2*e) + 3*a^2*f^2*cos(f*x + e) + a^2*f^2)*cos(3*f*x + 3*e) + 6*(3*a^2*f^2*cos(f*x + e) + a^2*f^2
)*cos(2*f*x + 2*e) + 6*(a^2*f^2*sin(2*f*x + 2*e) + a^2*f^2*sin(f*x + e))*sin(3*f*x + 3*e)) - 6*(2*(3*(f*x + e)
*sin(f*x + e) + cos(2*f*x + 2*e) + cos(f*x + e))*cos(3*f*x + 3*e) + 2*(9*(f*x + e)*sin(f*x + e) + 6*cos(f*x +
e) + 1)*cos(2*f*x + 2*e) + 6*cos(2*f*x + 2*e)^2 + 6*cos(f*x + e)^2 - (2*(3*cos(2*f*x + 2*e) + 3*cos(f*x + e) +
1)*cos(3*f*x + 3*e) + cos(3*f*x + 3*e)^2 + 6*(3*cos(f*x + e) + 1)*cos(2*f*x + 2*e) + 9*cos(2*f*x + 2*e)^2 + 9
*cos(f*x + e)^2 + 6*(sin(2*f*x + 2*e) + sin(f*x + e))*sin(3*f*x + 3*e) + sin(3*f*x + 3*e)^2 + 9*sin(2*f*x + 2*
e)^2 + 18*sin(2*f*x + 2*e)*sin(f*x + e) + 9*sin(f*x + e)^2 + 6*cos(f*x + e) + 1)*log(cos(f*x + e)^2 + sin(f*x
+ e)^2 + 2*cos(f*x + e) + 1) - 2*(f*x + 3*(f*x + e)*cos(f*x + e) + e - sin(2*f*x + 2*e) - sin(f*x + e))*sin(3*
f*x + 3*e) - 6*(f*x + 3*(f*x + e)*cos(f*x + e) + e - 2*sin(f*x + e))*sin(2*f*x + 2*e) + 6*sin(2*f*x + 2*e)^2 +
6*sin(f*x + e)^2 + 2*cos(f*x + e))*c^2*d/(a^2*f*cos(3*f*x + 3*e)^2 + 9*a^2*f*cos(2*f*x + 2*e)^2 + 9*a^2*f*cos
(f*x + e)^2 + a^2*f*sin(3*f*x + 3*e)^2 + 9*a^2*f*sin(2*f*x + 2*e)^2 + 18*a^2*f*sin(2*f*x + 2*e)*sin(f*x + e) +
9*a^2*f*sin(f*x + e)^2 + 6*a^2*f*cos(f*x + e) + a^2*f + 2*(3*a^2*f*cos(2*f*x + 2*e) + 3*a^2*f*cos(f*x + e) +
a^2*f)*cos(3*f*x + 3*e) + 6*(3*a^2*f*cos(f*x + e) + a^2*f)*cos(2*f*x + 2*e) + 6*(a^2*f*sin(2*f*x + 2*e) + a^2*
f*sin(f*x + e))*sin(3*f*x + 3*e)) + c^3*(3*sin(f*x + e)/(cos(f*x + e) + 1) + sin(f*x + e)^3/(cos(f*x + e) + 1)
^3)/a^2 + 3*c*d^2*e^2*(3*sin(f*x + e)/(cos(f*x + e) + 1) + sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/(a^2*f^2) - 3*
c^2*d*e*(3*sin(f*x + e)/(cos(f*x + e) + 1) + sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/(a^2*f) - 6*(2*d^3*e^3 + 12*
d^3*e - 12*c*d^2*f - (6*(f*x + e)^2*d^3 + 6*d^3*e^2 + 12*d^3 - 12*(d^3*e - c*d^2*f)*(f*x + e) + 6*((f*x + e)^2
*d^3 + d^3*e^2 + 2*d^3 - 2*(d^3*e - c*d^2*f)*(f*x + e))*cos(3*f*x + 3*e) + 18*((f*x + e)^2*d^3 + d^3*e^2 + 2*d
^3 - 2*(d^3*e - c*d^2*f)*(f*x + e))*cos(2*f*x + 2*e) + 18*((f*x + e)^2*d^3 + d^3*e^2 + 2*d^3 - 2*(d^3*e - c*d^
2*f)*(f*x + e))*cos(f*x + e) + (6*I*(f*x + e)^2*d^3 + 6*I*d^3*e^2 + 12*I*d^3 + (-12*I*d^3*e + 12*I*c*d^2*f)*(f
*x + e))*sin(3*f*x + 3*e) + (18*I*(f*x + e)^2*d^3 + 18*I*d^3*e^2 + 36*I*d^3 + (-36*I*d^3*e + 36*I*c*d^2*f)*(f*
x + e))*sin(2*f*x + 2*e) + (18*I*(f*x + e)^2*d^3 + 18*I*d^3*e^2 + 36*I*d^3 + (-36*I*d^3*e + 36*I*c*d^2*f)*(f*x
+ e))*sin(f*x + e))*arctan2(sin(f*x + e), cos(f*x + e) + 1) + 2*((f*x + e)^3*d^3 - 3*(d^3*e - c*d^2*f)*(f*x +
e)^2 + 3*(d^3*e^2 + 2*d^3)*(f*x + e))*cos(3*f*x + 3*e) + (6*(f*x + e)^3*d^3 - 6*I*d^3*e^2 + 12*d^3*e - 12*c*d
^2*f - 6*(3*d^3*e - 3*c*d^2*f + I*d^3)*(f*x + e)^2 + (18*d^3*e^2 + 12*I*d^3*e - 12*I*c*d^2*f + 24*d^3)*(f*x +
e))*cos(2*f*x + 2*e) + (6*d^3*e^3 - 6*I*(f*x + e)^2*d^3 - 6*I*d^3*e^2 + 24*d^3*e - 24*c*d^2*f - (-12*I*d^3*e +
12*I*c*d^2*f - 12*d^3)*(f*x + e))*cos(f*x + e) + (12*(f*x + e)*d^3 - 12*d^3*e + 12*c*d^2*f + 12*((f*x + e)*d^
3 - d^3*e + c*d^2*f)*cos(3*f*x + 3*e) + 36*((f*x + e)*d^3 - d^3*e + c*d^2*f)*cos(2*f*x + 2*e) + 36*((f*x + e)*
d^3 - d^3*e + c*d^2*f)*cos(f*x + e) - (-12*I*(f*x + e)*d^3 + 12*I*d^3*e - 12*I*c*d^2*f)*sin(3*f*x + 3*e) - (-3
6*I*(f*x + e)*d^3 + 36*I*d^3*e - 36*I*c*d^2*f)*sin(2*f*x + 2*e) - (-36*I*(f*x + e)*d^3 + 36*I*d^3*e - 36*I*c*d
^2*f)*sin(f*x + e))*dilog(-e^(I*f*x + I*e)) - (-3*I*(f*x + e)^2*d^3 - 3*I*d^3*e^2 - 6*I*d^3 + (6*I*d^3*e - 6*I
*c*d^2*f)*(f*x + e) + (-3*I*(f*x + e)^2*d^3 - 3*I*d^3*e^2 - 6*I*d^3 + (6*I*d^3*e - 6*I*c*d^2*f)*(f*x + e))*cos
(3*f*x + 3*e) + (-9*I*(f*x + e)^2*d^3 - 9*I*d^3*e^2 - 18*I*d^3 + (18*I*d^3*e - 18*I*c*d^2*f)*(f*x + e))*cos(2*
f*x + 2*e) + (-9*I*(f*x + e)^2*d^3 - 9*I*d^3*e^2 - 18*I*d^3 + (18*I*d^3*e - 18*I*c*d^2*f)*(f*x + e))*cos(f*x +
e) + 3*((f*x + e)^2*d^3 + d^3*e^2 + 2*d^3 - 2*(d^3*e - c*d^2*f)*(f*x + e))*sin(3*f*x + 3*e) + 9*((f*x + e)^2*
d^3 + d^3*e^2 + 2*d^3 - 2*(d^3*e - c*d^2*f)*(f*x + e))*sin(2*f*x + 2*e) + 9*((f*x + e)^2*d^3 + d^3*e^2 + 2*d^3
- 2*(d^3*e - c*d^2*f)*(f*x + e))*sin(f*x + e))*log(cos(f*x + e)^2 + sin(f*x + e)^2 + 2*cos(f*x + e) + 1) - (-
12*I*d^3*cos(3*f*x + 3*e) - 36*I*d^3*cos(2*f*x + 2*e) - 36*I*d^3*cos(f*x + e) + 12*d^3*sin(3*f*x + 3*e) + 36*d
^3*sin(2*f*x + 2*e) + 36*d^3*sin(f*x + e) - 12*I*d^3)*polylog(3, -e^(I*f*x + I*e)) - (-2*I*(f*x + e)^3*d^3 + (
6*I*d^3*e - 6*I*c*d^2*f)*(f*x + e)^2 + (-6*I*d^3*e^2 - 12*I*d^3)*(f*x + e))*sin(3*f*x + 3*e) - (-6*I*(f*x + e)
^3*d^3 - 6*d^3*e^2 - 12*I*d^3*e + 12*I*c*d^2*f + (18*I*d^3*e - 18*I*c*d^2*f - 6*d^3)*(f*x + e)^2 + (-18*I*d^3*
e^2 + 12*d^3*e - 12*c*d^2*f - 24*I*d^3)*(f*x + e))*sin(2*f*x + 2*e) - (-6*I*d^3*e^3 - 6*(f*x + e)^2*d^3 - 6*d^
3*e^2 - 24*I*d^3*e + 24*I*c*d^2*f + 12*(d^3*e - c*d^2*f - I*d^3)*(f*x + e))*sin(f*x + e))/(-3*I*a^2*f^3*cos(3*
f*x + 3*e) - 9*I*a^2*f^3*cos(2*f*x + 2*e) - 9*I*a^2*f^3*cos(f*x + e) + 3*a^2*f^3*sin(3*f*x + 3*e) + 9*a^2*f^3*
sin(2*f*x + 2*e) + 9*a^2*f^3*sin(f*x + e) - 3*I*a^2*f^3))/f

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(a + a*cos(e + f*x))^2,x)

[Out]

\text{Hanged}

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {c^{3}}{\cos ^{2}{\left (e + f x \right )} + 2 \cos {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{3} x^{3}}{\cos ^{2}{\left (e + f x \right )} + 2 \cos {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c d^{2} x^{2}}{\cos ^{2}{\left (e + f x \right )} + 2 \cos {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c^{2} d x}{\cos ^{2}{\left (e + f x \right )} + 2 \cos {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3/(a+a*cos(f*x+e))**2,x)

[Out]

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

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