3.1.54 \(\int \frac {(c+d x)^3}{a+b \tan (e+f x)} \, dx\) [54]

Optimal. Leaf size=243 \[ \frac {(c+d x)^4}{4 (a+i b) d}+\frac {b (c+d x)^3 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f}-\frac {3 i b d (c+d x)^2 \text {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^2}+\frac {3 b d^2 (c+d x) \text {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^3}+\frac {3 i b d^3 \text {PolyLog}\left (4,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{4 \left (a^2+b^2\right ) f^4} \]

[Out]

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

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Rubi [A]
time = 0.24, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3813, 2221, 2611, 6744, 2320, 6724} \begin {gather*} \frac {3 b d^2 (c+d x) \text {Li}_3\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 f^3 \left (a^2+b^2\right )}-\frac {3 i b d (c+d x)^2 \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 f^2 \left (a^2+b^2\right )}+\frac {b (c+d x)^3 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{f \left (a^2+b^2\right )}+\frac {3 i b d^3 \text {Li}_4\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{4 f^4 \left (a^2+b^2\right )}+\frac {(c+d x)^4}{4 d (a+i b)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c + d*x)^4/(4*(a + I*b)*d) + (b*(c + d*x)^3*Log[1 + ((a^2 + b^2)*E^((2*I)*(e + f*x)))/(a + I*b)^2])/((a^2 + b
^2)*f) - (((3*I)/2)*b*d*(c + d*x)^2*PolyLog[2, -(((a^2 + b^2)*E^((2*I)*(e + f*x)))/(a + I*b)^2)])/((a^2 + b^2)
*f^2) + (3*b*d^2*(c + d*x)*PolyLog[3, -(((a^2 + b^2)*E^((2*I)*(e + f*x)))/(a + I*b)^2)])/(2*(a^2 + b^2)*f^3) +
(((3*I)/4)*b*d^3*PolyLog[4, -(((a^2 + b^2)*E^((2*I)*(e + f*x)))/(a + I*b)^2)])/((a^2 + b^2)*f^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 3813

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

Rubi steps

\begin {align*} \int \frac {(c+d x)^3}{a+b \tan (e+f x)} \, dx &=\frac {(c+d x)^4}{4 (a+i b) d}+(2 i b) \int \frac {e^{2 i (e+f x)} (c+d x)^3}{(a+i b)^2+\left (a^2+b^2\right ) e^{2 i (e+f x)}} \, dx\\ &=\frac {(c+d x)^4}{4 (a+i b) d}+\frac {b (c+d x)^3 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f}-\frac {(3 b d) \int (c+d x)^2 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right ) \, dx}{\left (a^2+b^2\right ) f}\\ &=\frac {(c+d x)^4}{4 (a+i b) d}+\frac {b (c+d x)^3 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f}-\frac {3 i b d (c+d x)^2 \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^2}+\frac {\left (3 i b d^2\right ) \int (c+d x) \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right ) \, dx}{\left (a^2+b^2\right ) f^2}\\ &=\frac {(c+d x)^4}{4 (a+i b) d}+\frac {b (c+d x)^3 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f}-\frac {3 i b d (c+d x)^2 \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^2}+\frac {3 b d^2 (c+d x) \text {Li}_3\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^3}-\frac {\left (3 b d^3\right ) \int \text {Li}_3\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right ) \, dx}{2 \left (a^2+b^2\right ) f^3}\\ &=\frac {(c+d x)^4}{4 (a+i b) d}+\frac {b (c+d x)^3 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f}-\frac {3 i b d (c+d x)^2 \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^2}+\frac {3 b d^2 (c+d x) \text {Li}_3\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^3}+\frac {\left (3 i b d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {\left (a^2+b^2\right ) x}{(a+i b)^2}\right )}{x} \, dx,x,e^{2 i (e+f x)}\right )}{4 \left (a^2+b^2\right ) f^4}\\ &=\frac {(c+d x)^4}{4 (a+i b) d}+\frac {b (c+d x)^3 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f}-\frac {3 i b d (c+d x)^2 \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^2}+\frac {3 b d^2 (c+d x) \text {Li}_3\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^3}+\frac {3 i b d^3 \text {Li}_4\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{4 \left (a^2+b^2\right ) f^4}\\ \end {align*}

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Mathematica [A]
time = 2.27, size = 418, normalized size = 1.72 \begin {gather*} \frac {4 a c^3 f^4 x-4 i b c^3 f^4 x+6 a c^2 d f^4 x^2-6 i b c^2 d f^4 x^2+4 a c d^2 f^4 x^3-4 i b c d^2 f^4 x^3+a d^3 f^4 x^4-i b d^3 f^4 x^4+4 b c^3 f^3 \log \left (1+\frac {(a-i b) e^{2 i (e+f x)}}{a+i b}\right )+12 b c^2 d f^3 x \log \left (1+\frac {(a-i b) e^{2 i (e+f x)}}{a+i b}\right )+12 b c d^2 f^3 x^2 \log \left (1+\frac {(a-i b) e^{2 i (e+f x)}}{a+i b}\right )+4 b d^3 f^3 x^3 \log \left (1+\frac {(a-i b) e^{2 i (e+f x)}}{a+i b}\right )-6 i b d f^2 (c+d x)^2 \text {PolyLog}\left (2,-\frac {(a-i b) e^{2 i (e+f x)}}{a+i b}\right )+6 b d^2 f (c+d x) \text {PolyLog}\left (3,-\frac {(a-i b) e^{2 i (e+f x)}}{a+i b}\right )+3 i b d^3 \text {PolyLog}\left (4,-\frac {(a-i b) e^{2 i (e+f x)}}{a+i b}\right )}{4 \left (a^2+b^2\right ) f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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. 1467 vs. \(2 (220 ) = 440\).
time = 0.54, size = 1468, normalized size = 6.04

method	result	size
risch	\(\text {Expression too large to display}\)	\(1468\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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. 1038 vs. \(2 (216) = 432\).
time = 0.75, size = 1038, normalized size = 4.27 \begin {gather*} -\frac {18 \, c^{2} d {\left (\frac {2 \, {\left (f x + e\right )} a}{{\left (a^{2} + b^{2}\right )} f} + \frac {2 \, b \log \left (b \tan \left (f x + e\right ) + a\right )}{{\left (a^{2} + b^{2}\right )} f} - \frac {b \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{{\left (a^{2} + b^{2}\right )} f}\right )} e - 6 \, {\left (\frac {2 \, {\left (f x + e\right )} a}{a^{2} + b^{2}} + \frac {2 \, b \log \left (b \tan \left (f x + e\right ) + a\right )}{a^{2} + b^{2}} - \frac {b \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{2} + b^{2}}\right )} c^{3} - \frac {3 \, {\left (f x + e\right )}^{4} {\left (a - i \, b\right )} d^{3} + 12 i \, b d^{3} {\rm Li}_{4}(\frac {{\left (i \, a + b\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{-i \, a + b}) + 12 \, {\left ({\left (a - i \, b\right )} c d^{2} f - {\left (a e - i \, b e\right )} d^{3}\right )} {\left (f x + e\right )}^{3} + 18 \, {\left ({\left (a - i \, b\right )} c^{2} d f^{2} - 2 \, {\left (a e - i \, b e\right )} c d^{2} f + {\left (a e^{2} - i \, b e^{2}\right )} d^{3}\right )} {\left (f x + e\right )}^{2} + 12 \, {\left (3 \, {\left (a e^{2} - i \, b e^{2}\right )} c d^{2} f - {\left (a e^{3} - i \, b e^{3}\right )} d^{3}\right )} {\left (f x + e\right )} - 12 \, {\left (-3 i \, b c d^{2} f e^{2} + i \, b d^{3} e^{3}\right )} \arctan \left (-b \cos \left (2 \, f x + 2 \, e\right ) + a \sin \left (2 \, f x + 2 \, e\right ) + b, a \cos \left (2 \, f x + 2 \, e\right ) + b \sin \left (2 \, f x + 2 \, e\right ) + a\right ) - 4 \, {\left (4 i \, {\left (f x + e\right )}^{3} b d^{3} + 9 \, {\left (i \, b c d^{2} f - i \, b d^{3} e\right )} {\left (f x + e\right )}^{2} + 9 \, {\left (i \, b c^{2} d f^{2} - 2 i \, b c d^{2} f e + i \, b d^{3} e^{2}\right )} {\left (f x + e\right )}\right )} \arctan \left (\frac {2 \, a b \cos \left (2 \, f x + 2 \, e\right ) - {\left (a^{2} - b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{a^{2} + b^{2}}, \frac {2 \, a b \sin \left (2 \, f x + 2 \, e\right ) + a^{2} + b^{2} + {\left (a^{2} - b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )}{a^{2} + b^{2}}\right ) - 6 \, {\left (4 i \, {\left (f x + e\right )}^{2} b d^{3} + 3 i \, b c^{2} d f^{2} - 6 i \, b c d^{2} f e + 3 i \, b d^{3} e^{2} + 6 \, {\left (i \, b c d^{2} f - i \, b d^{3} e\right )} {\left (f x + e\right )}\right )} {\rm Li}_2\left (\frac {{\left (i \, a + b\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{-i \, a + b}\right ) + 6 \, {\left (3 \, b c d^{2} f e^{2} - b d^{3} e^{3}\right )} \log \left ({\left (a^{2} + b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} + 4 \, a b \sin \left (2 \, f x + 2 \, e\right ) + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + a^{2} + b^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )\right ) + 2 \, {\left (4 \, {\left (f x + e\right )}^{3} b d^{3} + 9 \, {\left (b c d^{2} f - b d^{3} e\right )} {\left (f x + e\right )}^{2} + 9 \, {\left (b c^{2} d f^{2} - 2 \, b c d^{2} f e + b d^{3} e^{2}\right )} {\left (f x + e\right )}\right )} \log \left (\frac {{\left (a^{2} + b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} + 4 \, a b \sin \left (2 \, f x + 2 \, e\right ) + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + a^{2} + b^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )}{a^{2} + b^{2}}\right ) + 6 \, {\left (4 \, {\left (f x + e\right )} b d^{3} + 3 \, b c d^{2} f - 3 \, b d^{3} e\right )} {\rm Li}_{3}(\frac {{\left (i \, a + b\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{-i \, a + b})}{{\left (a^{2} + b^{2}\right )} f^{3}}}{12 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(18*c^2*d*(2*(f*x + e)*a/((a^2 + b^2)*f) + 2*b*log(b*tan(f*x + e) + a)/((a^2 + b^2)*f) - b*log(tan(f*x +
e)^2 + 1)/((a^2 + b^2)*f))*e - 6*(2*(f*x + e)*a/(a^2 + b^2) + 2*b*log(b*tan(f*x + e) + a)/(a^2 + b^2) - b*log
(tan(f*x + e)^2 + 1)/(a^2 + b^2))*c^3 - (3*(f*x + e)^4*(a - I*b)*d^3 + 12*I*b*d^3*polylog(4, (I*a + b)*e^(2*I*
f*x + 2*I*e)/(-I*a + b)) + 12*((a - I*b)*c*d^2*f - (a*e - I*b*e)*d^3)*(f*x + e)^3 + 18*((a - I*b)*c^2*d*f^2 -
2*(a*e - I*b*e)*c*d^2*f + (a*e^2 - I*b*e^2)*d^3)*(f*x + e)^2 + 12*(3*(a*e^2 - I*b*e^2)*c*d^2*f - (a*e^3 - I*b*
e^3)*d^3)*(f*x + e) - 12*(-3*I*b*c*d^2*f*e^2 + I*b*d^3*e^3)*arctan2(-b*cos(2*f*x + 2*e) + a*sin(2*f*x + 2*e) +
b, a*cos(2*f*x + 2*e) + b*sin(2*f*x + 2*e) + a) - 4*(4*I*(f*x + e)^3*b*d^3 + 9*(I*b*c*d^2*f - I*b*d^3*e)*(f*x
+ e)^2 + 9*(I*b*c^2*d*f^2 - 2*I*b*c*d^2*f*e + I*b*d^3*e^2)*(f*x + e))*arctan2((2*a*b*cos(2*f*x + 2*e) - (a^2
- b^2)*sin(2*f*x + 2*e))/(a^2 + b^2), (2*a*b*sin(2*f*x + 2*e) + a^2 + b^2 + (a^2 - b^2)*cos(2*f*x + 2*e))/(a^2
+ b^2)) - 6*(4*I*(f*x + e)^2*b*d^3 + 3*I*b*c^2*d*f^2 - 6*I*b*c*d^2*f*e + 3*I*b*d^3*e^2 + 6*(I*b*c*d^2*f - I*b
*d^3*e)*(f*x + e))*dilog((I*a + b)*e^(2*I*f*x + 2*I*e)/(-I*a + b)) + 6*(3*b*c*d^2*f*e^2 - b*d^3*e^3)*log((a^2
+ b^2)*cos(2*f*x + 2*e)^2 + 4*a*b*sin(2*f*x + 2*e) + (a^2 + b^2)*sin(2*f*x + 2*e)^2 + a^2 + b^2 + 2*(a^2 - b^2
)*cos(2*f*x + 2*e)) + 2*(4*(f*x + e)^3*b*d^3 + 9*(b*c*d^2*f - b*d^3*e)*(f*x + e)^2 + 9*(b*c^2*d*f^2 - 2*b*c*d^
2*f*e + b*d^3*e^2)*(f*x + e))*log(((a^2 + b^2)*cos(2*f*x + 2*e)^2 + 4*a*b*sin(2*f*x + 2*e) + (a^2 + b^2)*sin(2
*f*x + 2*e)^2 + a^2 + b^2 + 2*(a^2 - b^2)*cos(2*f*x + 2*e))/(a^2 + b^2)) + 6*(4*(f*x + e)*b*d^3 + 3*b*c*d^2*f
- 3*b*d^3*e)*polylog(3, (I*a + b)*e^(2*I*f*x + 2*I*e)/(-I*a + b)))/((a^2 + b^2)*f^3))/f

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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. 1209 vs. \(2 (216) = 432\).
time = 0.47, size = 1209, normalized size = 4.98 \begin {gather*} \frac {2 \, a d^{3} f^{4} x^{4} + 8 \, a c d^{2} f^{4} x^{3} + 12 \, a c^{2} d f^{4} x^{2} + 8 \, a c^{3} f^{4} x - 3 i \, b d^{3} {\rm polylog}\left (4, \frac {{\left (a^{2} + 2 i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} - 2 i \, a b + b^{2} - 2 \, {\left (-i \, a^{2} + 2 \, a b + i \, b^{2}\right )} \tan \left (f x + e\right )}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}}\right ) + 3 i \, b d^{3} {\rm polylog}\left (4, \frac {{\left (a^{2} - 2 i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} + 2 i \, a b + b^{2} - 2 \, {\left (i \, a^{2} + 2 \, a b - i \, b^{2}\right )} \tan \left (f x + e\right )}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}}\right ) - 6 \, {\left (-i \, b d^{3} f^{2} x^{2} - 2 i \, b c d^{2} f^{2} x - i \, b c^{2} d f^{2}\right )} {\rm Li}_2\left (\frac {2 \, {\left ({\left (i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} - i \, a b + {\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )} \tan \left (f x + e\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}} + 1\right ) - 6 \, {\left (i \, b d^{3} f^{2} x^{2} + 2 i \, b c d^{2} f^{2} x + i \, b c^{2} d f^{2}\right )} {\rm Li}_2\left (\frac {2 \, {\left ({\left (-i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} + i \, a b + {\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )} \tan \left (f x + e\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}} + 1\right ) + 4 \, {\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + 3 \, b c^{2} d f^{2} e - 3 \, b c d^{2} f e^{2} + b d^{3} e^{3}\right )} \log \left (-\frac {2 \, {\left ({\left (i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} - i \, a b + {\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )} \tan \left (f x + e\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}}\right ) + 4 \, {\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + 3 \, b c^{2} d f^{2} e - 3 \, b c d^{2} f e^{2} + b d^{3} e^{3}\right )} \log \left (-\frac {2 \, {\left ({\left (-i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} + i \, a b + {\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )} \tan \left (f x + e\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}}\right ) + 4 \, {\left (b c^{3} f^{3} - 3 \, b c^{2} d f^{2} e + 3 \, b c d^{2} f e^{2} - b d^{3} e^{3}\right )} \log \left (\frac {{\left (i \, a b + b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{2} + 1}\right ) + 4 \, {\left (b c^{3} f^{3} - 3 \, b c^{2} d f^{2} e + 3 \, b c d^{2} f e^{2} - b d^{3} e^{3}\right )} \log \left (\frac {{\left (i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{2} + 1}\right ) + 6 \, {\left (b d^{3} f x + b c d^{2} f\right )} {\rm polylog}\left (3, \frac {{\left (a^{2} + 2 i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} - 2 i \, a b + b^{2} - 2 \, {\left (-i \, a^{2} + 2 \, a b + i \, b^{2}\right )} \tan \left (f x + e\right )}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}}\right ) + 6 \, {\left (b d^{3} f x + b c d^{2} f\right )} {\rm polylog}\left (3, \frac {{\left (a^{2} - 2 i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} + 2 i \, a b + b^{2} - 2 \, {\left (i \, a^{2} + 2 \, a b - i \, b^{2}\right )} \tan \left (f x + e\right )}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}}\right )}{8 \, {\left (a^{2} + b^{2}\right )} f^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*a*d^3*f^4*x^4 + 8*a*c*d^2*f^4*x^3 + 12*a*c^2*d*f^4*x^2 + 8*a*c^3*f^4*x - 3*I*b*d^3*polylog(4, ((a^2 + 2
*I*a*b - b^2)*tan(f*x + e)^2 - a^2 - 2*I*a*b + b^2 - 2*(-I*a^2 + 2*a*b + I*b^2)*tan(f*x + e))/((a^2 + b^2)*tan
(f*x + e)^2 + a^2 + b^2)) + 3*I*b*d^3*polylog(4, ((a^2 - 2*I*a*b - b^2)*tan(f*x + e)^2 - a^2 + 2*I*a*b + b^2 -
2*(I*a^2 + 2*a*b - I*b^2)*tan(f*x + e))/((a^2 + b^2)*tan(f*x + e)^2 + a^2 + b^2)) - 6*(-I*b*d^3*f^2*x^2 - 2*I
*b*c*d^2*f^2*x - I*b*c^2*d*f^2)*dilog(2*((I*a*b - b^2)*tan(f*x + e)^2 - a^2 - I*a*b + (I*a^2 - 2*a*b - I*b^2)*
tan(f*x + e))/((a^2 + b^2)*tan(f*x + e)^2 + a^2 + b^2) + 1) - 6*(I*b*d^3*f^2*x^2 + 2*I*b*c*d^2*f^2*x + I*b*c^2
*d*f^2)*dilog(2*((-I*a*b - b^2)*tan(f*x + e)^2 - a^2 + I*a*b + (-I*a^2 - 2*a*b + I*b^2)*tan(f*x + e))/((a^2 +
b^2)*tan(f*x + e)^2 + a^2 + b^2) + 1) + 4*(b*d^3*f^3*x^3 + 3*b*c*d^2*f^3*x^2 + 3*b*c^2*d*f^3*x + 3*b*c^2*d*f^2
*e - 3*b*c*d^2*f*e^2 + b*d^3*e^3)*log(-2*((I*a*b - b^2)*tan(f*x + e)^2 - a^2 - I*a*b + (I*a^2 - 2*a*b - I*b^2)
*tan(f*x + e))/((a^2 + b^2)*tan(f*x + e)^2 + a^2 + b^2)) + 4*(b*d^3*f^3*x^3 + 3*b*c*d^2*f^3*x^2 + 3*b*c^2*d*f^
3*x + 3*b*c^2*d*f^2*e - 3*b*c*d^2*f*e^2 + b*d^3*e^3)*log(-2*((-I*a*b - b^2)*tan(f*x + e)^2 - a^2 + I*a*b + (-I
*a^2 - 2*a*b + I*b^2)*tan(f*x + e))/((a^2 + b^2)*tan(f*x + e)^2 + a^2 + b^2)) + 4*(b*c^3*f^3 - 3*b*c^2*d*f^2*e
+ 3*b*c*d^2*f*e^2 - b*d^3*e^3)*log(((I*a*b + b^2)*tan(f*x + e)^2 - a^2 + I*a*b + (I*a^2 + I*b^2)*tan(f*x + e)
)/(tan(f*x + e)^2 + 1)) + 4*(b*c^3*f^3 - 3*b*c^2*d*f^2*e + 3*b*c*d^2*f*e^2 - b*d^3*e^3)*log(((I*a*b - b^2)*tan
(f*x + e)^2 + a^2 + I*a*b + (I*a^2 + I*b^2)*tan(f*x + e))/(tan(f*x + e)^2 + 1)) + 6*(b*d^3*f*x + b*c*d^2*f)*po
lylog(3, ((a^2 + 2*I*a*b - b^2)*tan(f*x + e)^2 - a^2 - 2*I*a*b + b^2 - 2*(-I*a^2 + 2*a*b + I*b^2)*tan(f*x + e)
)/((a^2 + b^2)*tan(f*x + e)^2 + a^2 + b^2)) + 6*(b*d^3*f*x + b*c*d^2*f)*polylog(3, ((a^2 - 2*I*a*b - b^2)*tan(
f*x + e)^2 - a^2 + 2*I*a*b + b^2 - 2*(I*a^2 + 2*a*b - I*b^2)*tan(f*x + e))/((a^2 + b^2)*tan(f*x + e)^2 + a^2 +
b^2)))/((a^2 + b^2)*f^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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