Integrand size = 15, antiderivative size = 154 \[ \int \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(i-b d n) x}{b d n}+\frac {i x \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac {2 i x \operatorname {Hypergeometric2F1}\left (1,-\frac {i}{2 b d n},1-\frac {i}{2 b d n},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n} \]
(I-b*d*n)*x/b/d/n+I*x*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/b/d/n/(1+exp(2*I* a*d)*(c*x^n)^(2*I*b*d))-2*I*x*hypergeom([1, -1/2*I/b/d/n],[1-1/2*I/b/d/n], -exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/b/d/n
Time = 8.14 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.20 \[ \int \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {e^{2 i d \left (a+b \log \left (c x^n\right )\right )} x \operatorname {Hypergeometric2F1}\left (1,1-\frac {i}{2 b d n},2-\frac {i}{2 b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-(-i+2 b d n) x \left (b d n+i \operatorname {Hypergeometric2F1}\left (1,-\frac {i}{2 b d n},1-\frac {i}{2 b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-\tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{b d n (-i+2 b d n)} \]
(E^((2*I)*d*(a + b*Log[c*x^n]))*x*Hypergeometric2F1[1, 1 - (I/2)/(b*d*n), 2 - (I/2)/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] - (-I + 2*b*d*n)*x*(b* d*n + I*Hypergeometric2F1[1, (-1/2*I)/(b*d*n), 1 - (I/2)/(b*d*n), -E^((2*I )*d*(a + b*Log[c*x^n]))] - Tan[d*(a + b*Log[c*x^n])]))/(b*d*n*(-I + 2*b*d* n))
Time = 0.43 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.34, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5004, 5006, 1004, 27, 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.
\(\displaystyle \int \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 5004 |
\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \int \left (c x^n\right )^{\frac {1}{n}-1} \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 5006 |
\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \int \frac {\left (c x^n\right )^{\frac {1}{n}-1} \left (i-i e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^2}{\left (e^{2 i a d} \left (c x^n\right )^{2 i b d}+1\right )^2}d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 1004 |
\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \left (\frac {i e^{-2 i a d} \int -\frac {2 \left (c x^n\right )^{\frac {1}{n}-1} \left (\frac {e^{2 i a d} (1-i b d n)}{n}-\frac {e^{4 i a d} (i b d n+1) \left (c x^n\right )^{2 i b d}}{n}\right )}{e^{2 i a d} \left (c x^n\right )^{2 i b d}+1}d\left (c x^n\right )}{2 b d}+\frac {i \left (c x^n\right )^{\frac {1}{n}} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}\right )}{n}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \left (\frac {i \left (c x^n\right )^{\frac {1}{n}} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac {i e^{-2 i a d} \int \frac {\left (c x^n\right )^{\frac {1}{n}-1} \left (\frac {e^{2 i a d} (1-i b d n)}{n}-\frac {e^{4 i a d} (i b d n+1) \left (c x^n\right )^{2 i b d}}{n}\right )}{e^{2 i a d} \left (c x^n\right )^{2 i b d}+1}d\left (c x^n\right )}{b d}\right )}{n}\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \left (\frac {i \left (c x^n\right )^{\frac {1}{n}} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac {i e^{-2 i a d} \left (\frac {2 e^{2 i a d} \int \frac {\left (c x^n\right )^{\frac {1}{n}-1}}{e^{2 i a d} \left (c x^n\right )^{2 i b d}+1}d\left (c x^n\right )}{n}-e^{2 i a d} (1+i b d n) \left (c x^n\right )^{\frac {1}{n}}\right )}{b d}\right )}{n}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \left (\frac {i \left (c x^n\right )^{\frac {1}{n}} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac {i e^{-2 i a d} \left (2 e^{2 i a d} \left (c x^n\right )^{\frac {1}{n}} \operatorname {Hypergeometric2F1}\left (1,-\frac {i}{2 b d n},1-\frac {i}{2 b d n},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )-e^{2 i a d} (1+i b d n) \left (c x^n\right )^{\frac {1}{n}}\right )}{b d}\right )}{n}\) |
(x*((I*(c*x^n)^n^(-1)*(1 - E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))/(b*d*(1 + E ^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))) - (I*(-(E^((2*I)*a*d)*(1 + I*b*d*n)*(c* x^n)^n^(-1)) + 2*E^((2*I)*a*d)*(c*x^n)^n^(-1)*Hypergeometric2F1[1, (-1/2*I )/(b*d*n), 1 - (I/2)/(b*d*n), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))]))/(b*d *E^((2*I)*a*d))))/(n*(c*x^n)^n^(-1))
3.2.68.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[(-(c*b - a*d))*(e*x)^(m + 1)*(a + b*x^n)^(p + 1) *((c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1))), x] + Simp[1/(a*b*n*(p + 1)) Int [(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c *b - a*d)*(m + 1)) + d*(c*b*n*(p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^ n, x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && Lt Q[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[Tan[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Si mp[x/(n*(c*x^n)^(1/n)) Subst[Int[x^(1/n - 1)*Tan[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
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]
\[\int {\tan \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{2}d x\]
\[ \int \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \tan \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \]
\[ \int \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \tan ^{2}{\left (d \left (a + b \log {\left (c x^{n} \right )}\right ) \right )}\, dx \]
\[ \int \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \tan \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \]
-((b*d*cos(2*b*d*log(c))^2 + b*d*sin(2*b*d*log(c))^2)*n*x*cos(2*b*d*log(x^ n) + 2*a*d)^2 + (b*d*cos(2*b*d*log(c))^2 + b*d*sin(2*b*d*log(c))^2)*n*x*si n(2*b*d*log(x^n) + 2*a*d)^2 + b*d*n*x + 2*(b*d*n*cos(2*b*d*log(c)) - sin(2 *b*d*log(c)))*x*cos(2*b*d*log(x^n) + 2*a*d) - 2*(b*d*n*sin(2*b*d*log(c)) + cos(2*b*d*log(c)))*x*sin(2*b*d*log(x^n) + 2*a*d) + 2*(2*b^2*d^2*n^2*cos(2 *b*d*log(c))*cos(2*b*d*log(x^n) + 2*a*d) - 2*b^2*d^2*n^2*sin(2*b*d*log(c)) *sin(2*b*d*log(x^n) + 2*a*d) + b^2*d^2*n^2 + (b^2*d^2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d*log(c))^2)*n^2*cos(2*b*d*log(x^n) + 2*a*d)^2 + (b^2*d^ 2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d*log(c))^2)*n^2*sin(2*b*d*log(x^n ) + 2*a*d)^2)*integrate((cos(2*b*d*log(x^n) + 2*a*d)*sin(2*b*d*log(c)) + c os(2*b*d*log(c))*sin(2*b*d*log(x^n) + 2*a*d))/(2*b^2*d^2*n^2*cos(2*b*d*log (c))*cos(2*b*d*log(x^n) + 2*a*d) - 2*b^2*d^2*n^2*sin(2*b*d*log(c))*sin(2*b *d*log(x^n) + 2*a*d) + b^2*d^2*n^2 + (b^2*d^2*cos(2*b*d*log(c))^2 + b^2*d^ 2*sin(2*b*d*log(c))^2)*n^2*cos(2*b*d*log(x^n) + 2*a*d)^2 + (b^2*d^2*cos(2* b*d*log(c))^2 + b^2*d^2*sin(2*b*d*log(c))^2)*n^2*sin(2*b*d*log(x^n) + 2*a* d)^2), x))/(2*b*d*n*cos(2*b*d*log(c))*cos(2*b*d*log(x^n) + 2*a*d) - 2*b*d* n*sin(2*b*d*log(c))*sin(2*b*d*log(x^n) + 2*a*d) + (b*d*cos(2*b*d*log(c))^2 + b*d*sin(2*b*d*log(c))^2)*n*cos(2*b*d*log(x^n) + 2*a*d)^2 + (b*d*cos(2*b *d*log(c))^2 + b*d*sin(2*b*d*log(c))^2)*n*sin(2*b*d*log(x^n) + 2*a*d)^2 + b*d*n)
Timed out. \[ \int \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Timed out} \]
Timed out. \[ \int \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int {\mathrm {tan}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2 \,d x \]