3.1.1 \(\int x^3 \tan (a+b x) \, dx\) [1]

Optimal. Leaf size=106 \[ \frac {i x^4}{4}-\frac {x^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {3 i x^2 \text {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 x \text {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 i \text {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{4 b^4} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3800, 2221, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {3 i \text {Li}_4\left (-e^{2 i (a+b x)}\right )}{4 b^4}-\frac {3 x \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 i x^2 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {x^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {i x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*Tan[a + b*x],x]

[Out]

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

Rule 2221

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

Rule 2320

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

Rule 2611

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

Rule 3800

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

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

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Mathematica [A]
time = 0.01, size = 106, normalized size = 1.00 \begin {gather*} \frac {i x^4}{4}-\frac {x^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {3 i x^2 \text {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 x \text {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 i \text {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{4 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*Tan[a + b*x],x]

[Out]

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

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Maple [A]
time = 0.16, size = 125, normalized size = 1.18

method	result	size
risch	\(\frac {i x^{4}}{4}+\frac {2 i a^{3} x}{b^{3}}+\frac {3 i a^{4}}{2 b^{4}}-\frac {x^{3} \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{b}+\frac {3 i x^{2} \polylog \left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{2}}-\frac {3 x \polylog \left (3, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{3}}-\frac {3 i \polylog \left (4, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{4 b^{4}}-\frac {2 a^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}\)	\(125\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*tan(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/4*I*x^4+2*I/b^3*a^3*x+3/2*I/b^4*a^4-x^3*ln(exp(2*I*(b*x+a))+1)/b+3/2*I*x^2*polylog(2,-exp(2*I*(b*x+a)))/b^2-
3/2*x*polylog(3,-exp(2*I*(b*x+a)))/b^3-3/4*I*polylog(4,-exp(2*I*(b*x+a)))/b^4-2/b^4*a^3*ln(exp(I*(b*x+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. 243 vs. \(2 (83) = 166\).
time = 0.53, size = 243, normalized size = 2.29 \begin {gather*} -\frac {-3 i \, {\left (b x + a\right )}^{4} + 12 i \, {\left (b x + a\right )}^{3} a - 18 i \, {\left (b x + a\right )}^{2} a^{2} + 12 \, a^{3} \log \left (\sec \left (b x + a\right )\right ) - 4 \, {\left (-4 i \, {\left (b x + a\right )}^{3} + 9 i \, {\left (b x + a\right )}^{2} a - 9 i \, {\left (b x + a\right )} a^{2}\right )} \arctan \left (\sin \left (2 \, b x + 2 \, a\right ), \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) - 6 \, {\left (4 i \, {\left (b x + a\right )}^{2} - 6 i \, {\left (b x + a\right )} a + 3 i \, a^{2}\right )} {\rm Li}_2\left (-e^{\left (2 i \, b x + 2 i \, a\right )}\right ) + 2 \, {\left (4 \, {\left (b x + a\right )}^{3} - 9 \, {\left (b x + a\right )}^{2} a + 9 \, {\left (b x + a\right )} a^{2}\right )} \log \left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + 6 \, {\left (4 \, b x + a\right )} {\rm Li}_{3}(-e^{\left (2 i \, b x + 2 i \, a\right )}) + 12 i \, {\rm Li}_{4}(-e^{\left (2 i \, b x + 2 i \, a\right )})}{12 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*tan(b*x+a),x, algorithm="maxima")

[Out]

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

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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. 286 vs. \(2 (83) = 166\).
time = 0.38, size = 286, normalized size = 2.70 \begin {gather*} -\frac {4 \, b^{3} x^{3} \log \left (-\frac {2 \, {\left (i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1}\right ) + 4 \, b^{3} x^{3} \log \left (-\frac {2 \, {\left (-i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1}\right ) + 6 i \, b^{2} x^{2} {\rm Li}_2\left (\frac {2 \, {\left (i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1} + 1\right ) - 6 i \, b^{2} x^{2} {\rm Li}_2\left (\frac {2 \, {\left (-i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1} + 1\right ) + 6 \, b x {\rm polylog}\left (3, \frac {\tan \left (b x + a\right )^{2} + 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) + 6 \, b x {\rm polylog}\left (3, \frac {\tan \left (b x + a\right )^{2} - 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) - 3 i \, {\rm polylog}\left (4, \frac {\tan \left (b x + a\right )^{2} + 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) + 3 i \, {\rm polylog}\left (4, \frac {\tan \left (b x + a\right )^{2} - 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right )}{8 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*tan(b*x+a),x, algorithm="fricas")

[Out]

-1/8*(4*b^3*x^3*log(-2*(I*tan(b*x + a) - 1)/(tan(b*x + a)^2 + 1)) + 4*b^3*x^3*log(-2*(-I*tan(b*x + a) - 1)/(ta
n(b*x + a)^2 + 1)) + 6*I*b^2*x^2*dilog(2*(I*tan(b*x + a) - 1)/(tan(b*x + a)^2 + 1) + 1) - 6*I*b^2*x^2*dilog(2*
(-I*tan(b*x + a) - 1)/(tan(b*x + a)^2 + 1) + 1) + 6*b*x*polylog(3, (tan(b*x + a)^2 + 2*I*tan(b*x + a) - 1)/(ta
n(b*x + a)^2 + 1)) + 6*b*x*polylog(3, (tan(b*x + a)^2 - 2*I*tan(b*x + a) - 1)/(tan(b*x + a)^2 + 1)) - 3*I*poly
log(4, (tan(b*x + a)^2 + 2*I*tan(b*x + a) - 1)/(tan(b*x + a)^2 + 1)) + 3*I*polylog(4, (tan(b*x + a)^2 - 2*I*ta
n(b*x + a) - 1)/(tan(b*x + a)^2 + 1)))/b^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*tan(b*x+a),x)

[Out]

Integral(x**3*tan(a + b*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(x^3*tan(b*x+a),x, algorithm="giac")

[Out]

integrate(x^3*tan(b*x + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,\mathrm {tan}\left (a+b\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*tan(a + b*x),x)

[Out]

int(x^3*tan(a + b*x), x)

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