\(\int x^3 \tan (d (a+b \log (c x^n))) \, dx\) [158]

Optimal result

Integrand size = 17, antiderivative size = 71 \[ \int x^3 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {i x^4}{4}+\frac {1}{2} i x^4 \operatorname {Hypergeometric2F1}\left (1,-\frac {2 i}{b d n},1-\frac {2 i}{b d n},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \] Output:

-1/4*I*x^4+1/2*I*x^4*hypergeom([1, -2*I/b/d/n],[1-2*I/b/d/n],-exp(2*I*a*d) 
*(c*x^n)^(2*I*b*d))
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(146\) vs. \(2(71)=142\).

Time = 5.07 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.06 \[ \int x^3 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x^4 \left (2 i e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1-\frac {2 i}{b d n},2-\frac {2 i}{b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(-2 i+b d n) \operatorname {Hypergeometric2F1}\left (1,-\frac {2 i}{b d n},1-\frac {2 i}{b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{-8-4 i b d n} \] Input:

Integrate[x^3*Tan[d*(a + b*Log[c*x^n])],x]
 

Output:

(x^4*((2*I)*E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - (2*I)/ 
(b*d*n), 2 - (2*I)/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] + (-2*I + b*d 
*n)*Hypergeometric2F1[1, (-2*I)/(b*d*n), 1 - (2*I)/(b*d*n), -E^((2*I)*d*(a 
 + b*Log[c*x^n]))]))/(-8 - (4*I)*b*d*n)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.51, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {5008, 5006, 959, 888}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^3 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\)

\(\Big \downarrow \) 5008

\(\displaystyle \frac {x^4 \left (c x^n\right )^{-4/n} \int \left (c x^n\right )^{\frac {4}{n}-1} \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )d\left (c x^n\right )}{n}\)

\(\Big \downarrow \) 5006

\(\displaystyle \frac {x^4 \left (c x^n\right )^{-4/n} \int \frac {\left (c x^n\right )^{\frac {4}{n}-1} \left (i-i e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e^{2 i a d} \left (c x^n\right )^{2 i b d}+1}d\left (c x^n\right )}{n}\)

\(\Big \downarrow \) 959

\(\displaystyle \frac {x^4 \left (c x^n\right )^{-4/n} \left (2 i \int \frac {\left (c x^n\right )^{\frac {4}{n}-1}}{e^{2 i a d} \left (c x^n\right )^{2 i b d}+1}d\left (c x^n\right )-\frac {1}{4} i n \left (c x^n\right )^{4/n}\right )}{n}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {x^4 \left (c x^n\right )^{-4/n} \left (\frac {1}{2} i n \left (c x^n\right )^{4/n} \operatorname {Hypergeometric2F1}\left (1,-\frac {2 i}{b d n},1-\frac {2 i}{b d n},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )-\frac {1}{4} i n \left (c x^n\right )^{4/n}\right )}{n}\)

Input:

Int[x^3*Tan[d*(a + b*Log[c*x^n])],x]
 

Output:

(x^4*((-1/4*I)*n*(c*x^n)^(4/n) + (I/2)*n*(c*x^n)^(4/n)*Hypergeometric2F1[1 
, (-2*I)/(b*d*n), 1 - (2*I)/(b*d*n), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))] 
))/(n*(c*x^n)^(4/n))
 

Defintions of rubi rules used

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 959

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p 
+ 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p 
 + 1) + 1))   Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, 
 n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
 

rule 5006

Int[((e_.)*(x_))^(m_.)*Tan[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] 
:> Int[(e*x)^m*((I - I*E^(2*I*a*d)*x^(2*I*b*d))/(1 + E^(2*I*a*d)*x^(2*I*b*d 
)))^p, x] /; FreeQ[{a, b, d, e, m, p}, x]
 

rule 5008

Int[((e_.)*(x_))^(m_.)*Tan[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_ 
.), x_Symbol] :> Simp[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))   Subst[Int[x 
^((m + 1)/n - 1)*Tan[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, 
 c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
 

Maple [F]

\[\int x^{3} \tan \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]

Input:

int(x^3*tan(d*(a+b*ln(c*x^n))),x)
 

Output:

int(x^3*tan(d*(a+b*ln(c*x^n))),x)
 

Fricas [F]

\[ \int x^3 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{3} \tan \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \] Input:

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

Output:

integral(x^3*tan(b*d*log(c*x^n) + a*d), x)
 

Sympy [F]

\[ \int x^3 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^{3} \tan {\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \] Input:

integrate(x**3*tan(d*(a+b*ln(c*x**n))),x)
 

Output:

Integral(x**3*tan(a*d + b*d*log(c*x**n)), x)
 

Maxima [F]

\[ \int x^3 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{3} \tan \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \] Input:

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

Output:

integrate(x^3*tan((b*log(c*x^n) + a)*d), x)
 

Giac [F]

\[ \int x^3 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{3} \tan \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \] Input:

integrate(x^3*tan(d*(a+b*log(c*x^n))),x, algorithm="giac")
 

Output:

integrate(x^3*tan((b*log(c*x^n) + a)*d), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^3 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^3\,\mathrm {tan}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \] Input:

int(x^3*tan(d*(a + b*log(c*x^n))),x)
 

Output:

int(x^3*tan(d*(a + b*log(c*x^n))), x)
 

Reduce [F]

\[ \int x^3 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \tan \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right ) x^{3}d x \] Input:

int(x^3*tan(d*(a+b*log(c*x^n))),x)
 

Output:

int(tan(log(x**n*c)*b*d + a*d)*x**3,x)