Integrand size = 26, antiderivative size = 86 \[ \int \frac {\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\frac {\operatorname {AppellF1}\left (1+m,\frac {7}{3},1,2+m,-i \tan (c+d x),i \tan (c+d x)\right ) \sqrt [3]{1+i \tan (c+d x)} \tan ^{1+m}(c+d x)}{a d (1+m) \sqrt [3]{a+i a \tan (c+d x)}} \]
AppellF1(1+m,7/3,1,2+m,-I*tan(d*x+c),I*tan(d*x+c))*(1+I*tan(d*x+c))^(1/3)* tan(d*x+c)^(1+m)/a/d/(1+m)/(a+I*a*tan(d*x+c))^(1/3)
\[ \int \frac {\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\int \frac {\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx \]
Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3042, 4047, 25, 27, 152, 150}
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.
\(\displaystyle \int \frac {\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\tan (c+d x)^m}{(a+i a \tan (c+d x))^{4/3}}dx\) |
\(\Big \downarrow \) 4047 |
\(\displaystyle \frac {i a^2 \int -\frac {\tan ^m(c+d x)}{a (a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{7/3}}d(i a \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {i a^2 \int \frac {\tan ^m(c+d x)}{a (a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{7/3}}d(i a \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {i a \int \frac {\tan ^m(c+d x)}{(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{7/3}}d(i a \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 152 |
\(\displaystyle -\frac {i \sqrt [3]{1+i \tan (c+d x)} \int \frac {\tan ^m(c+d x)}{(i \tan (c+d x)+1)^{7/3} (a-i a \tan (c+d x))}d(i a \tan (c+d x))}{a d \sqrt [3]{a+i a \tan (c+d x)}}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {\sqrt [3]{1+i \tan (c+d x)} \tan ^{m+1}(c+d x) \operatorname {AppellF1}\left (m+1,\frac {7}{3},1,m+2,-i \tan (c+d x),i \tan (c+d x)\right )}{a d (m+1) \sqrt [3]{a+i a \tan (c+d x)}}\) |
(AppellF1[1 + m, 7/3, 1, 2 + m, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*(1 + I* Tan[c + d*x])^(1/3)*Tan[c + d*x]^(1 + m))/(a*d*(1 + m)*(a + I*a*Tan[c + d* x])^(1/3))
3.3.98.3.1 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[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 , (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]) Int[(b*x)^m*(1 + d*(x/c))^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !GtQ[c, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(b/f) Subst[Int[(a + x)^(m - 1)*(( c + (d/b)*x)^n/(b^2 + a*x)), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d ^2, 0]
\[\int \frac {\tan ^{m}\left (d x +c \right )}{\left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}d x\]
\[ \int \frac {\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\int { \frac {\tan \left (d x + c\right )^{m}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}}} \,d x } \]
integral(1/4*2^(2/3)*((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))^m*(a/(e^(2*I*d*x + 2*I*c) + 1))^(2/3)*(e^(4*I*d*x + 4*I*c) + 2*e^(2*I* d*x + 2*I*c) + 1)*e^(-8/3*I*d*x - 8/3*I*c)/a^2, x)
\[ \int \frac {\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\int \frac {\tan ^{m}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {4}{3}}}\, dx \]
Exception generated. \[ \int \frac {\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\int { \frac {\tan \left (d x + c\right )^{m}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}}} \,d x } \]
Timed out. \[ \int \frac {\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^m}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{4/3}} \,d x \]