Integrand size = 28, antiderivative size = 84 \[ \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {3 \operatorname {AppellF1}\left (\frac {7}{3},\frac {5}{2},1,\frac {10}{3},-i \tan (c+d x),i \tan (c+d x)\right ) \sqrt {1+i \tan (c+d x)} \tan ^{\frac {7}{3}}(c+d x)}{7 a d \sqrt {a+i a \tan (c+d x)}} \] Output:
3/7*AppellF1(7/3,5/2,1,10/3,-I*tan(d*x+c),I*tan(d*x+c))*(1+I*tan(d*x+c))^( 1/2)*tan(d*x+c)^(7/3)/a/d/(a+I*a*tan(d*x+c))^(1/2)
\[ \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx \] Input:
Integrate[Tan[c + d*x]^(4/3)/(a + I*a*Tan[c + d*x])^(3/2),x]
Output:
Integrate[Tan[c + d*x]^(4/3)/(a + I*a*Tan[c + d*x])^(3/2), x]
Time = 0.35 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 4047, 25, 27, 148, 27, 1013, 1012}
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.
\(\displaystyle \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\tan (c+d x)^{4/3}}{(a+i a \tan (c+d x))^{3/2}}dx\) |
\(\Big \downarrow \) 4047 |
\(\displaystyle \frac {i a^2 \int -\frac {\tan ^{\frac {4}{3}}(c+d x)}{a (a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{5/2}}d(i a \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {i a^2 \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{a (a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{5/2}}d(i a \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {i a \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{5/2}}d(i a \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 148 |
\(\displaystyle \frac {3 a^2 \int -\frac {a^5 \tan ^6(c+d x)}{\left (1-a^3 \tan ^3(c+d x)\right ) \left (\tan ^3(c+d x) a^4+a\right )^{5/2}}d\sqrt [3]{\tan (c+d x)}}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3 a \int -\frac {a^6 \tan ^6(c+d x)}{\left (1-a^3 \tan ^3(c+d x)\right ) \left (\tan ^3(c+d x) a^4+a\right )^{5/2}}d\sqrt [3]{\tan (c+d x)}}{d}\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {3 \sqrt {a^3 \tan ^3(c+d x)+1} \int -\frac {a^6 \tan ^6(c+d x)}{\left (1-a^3 \tan ^3(c+d x)\right ) \left (a^3 \tan ^3(c+d x)+1\right )^{5/2}}d\sqrt [3]{\tan (c+d x)}}{a d \sqrt {a^4 \tan ^3(c+d x)+a}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle -\frac {3 i a^6 \tan ^7(c+d x) \sqrt {a^3 \tan ^3(c+d x)+1} \operatorname {AppellF1}\left (\frac {7}{3},1,\frac {5}{2},\frac {10}{3},a^3 \tan ^3(c+d x),-a^3 \tan ^3(c+d x)\right )}{7 d \sqrt {a^4 \tan ^3(c+d x)+a}}\) |
Input:
Int[Tan[c + d*x]^(4/3)/(a + I*a*Tan[c + d*x])^(3/2),x]
Output:
(((-3*I)/7)*a^6*AppellF1[7/3, 1, 5/2, 10/3, a^3*Tan[c + d*x]^3, -(a^3*Tan[ c + d*x]^3)]*Tan[c + d*x]^7*Sqrt[1 + a^3*Tan[c + d*x]^3])/(d*Sqrt[a + a^4* Tan[c + d*x]^3])
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_] :> With[{k = Denominator[m]}, Simp[k/b Subst[Int[x^(k*(m + 1) - 1)*(c + d*(x^k/b))^n*(e + f*(x^k/b))^p, x], x, (b*x)^(1/k)], x]] /; FreeQ[{b, c, d, e, f, n, p}, x] && FractionQ[m] && IntegerQ[p]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^ n/a))^FracPart[p]) Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & & NeQ[m, n - 1] && !(IntegerQ[p] || GtQ[a, 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]
\[\int \frac {\tan \left (d x +c \right )^{\frac {4}{3}}}{\left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}d x\]
Input:
int(tan(d*x+c)^(4/3)/(a+I*a*tan(d*x+c))^(3/2),x)
Output:
int(tan(d*x+c)^(4/3)/(a+I*a*tan(d*x+c))^(3/2),x)
Timed out. \[ \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(tan(d*x+c)^(4/3)/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {\tan ^{\frac {4}{3}}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(tan(d*x+c)**(4/3)/(a+I*a*tan(d*x+c))**(3/2),x)
Output:
Integral(tan(c + d*x)**(4/3)/(I*a*(tan(c + d*x) - I))**(3/2), x)
Exception generated. \[ \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(tan(d*x+c)^(4/3)/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Exception generated. \[ \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(tan(d*x+c)^(4/3)/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Degree mismatch inside factorisatio n over extensionUnable to transpose Error: Bad Argument ValueDone
Timed out. \[ \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^{4/3}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \] Input:
int(tan(c + d*x)^(4/3)/(a + a*tan(c + d*x)*1i)^(3/2),x)
Output:
int(tan(c + d*x)^(4/3)/(a + a*tan(c + d*x)*1i)^(3/2), x)
\[ \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx =\text {Too large to display} \] Input:
int(tan(d*x+c)^(4/3)/(a+I*a*tan(d*x+c))^(3/2),x)
Output:
(sqrt(a)*( - 3*tan(c + d*x)**(1/3)*sqrt(tan(c + d*x)*i + 1) + 118211949363 2000*int(( - tan(c + d*x)**(1/3)*sqrt(tan(c + d*x)*i + 1)*tan(c + d*x)**2) /(337748426752000*tan(c + d*x)**3*i + 337748426752000*tan(c + d*x)**2 + 33 7748426752000*tan(c + d*x)*i + 337748426752000),x)*tan(c + d*x)**2*d*i + 1 182119493632000*int(( - tan(c + d*x)**(1/3)*sqrt(tan(c + d*x)*i + 1)*tan(c + d*x)**2)/(337748426752000*tan(c + d*x)**3*i + 337748426752000*tan(c + d *x)**2 + 337748426752000*tan(c + d*x)*i + 337748426752000),x)*d*i - 337748 426752000*int(( - tan(c + d*x)**(1/3)*sqrt(tan(c + d*x)*i + 1))/(337748426 752000*tan(c + d*x)**4*i + 337748426752000*tan(c + d*x)**3 + 3377484267520 00*tan(c + d*x)**2*i + 337748426752000*tan(c + d*x)),x)*tan(c + d*x)**2*d - 337748426752000*int(( - tan(c + d*x)**(1/3)*sqrt(tan(c + d*x)*i + 1))/(3 37748426752000*tan(c + d*x)**4*i + 337748426752000*tan(c + d*x)**3 + 33774 8426752000*tan(c + d*x)**2*i + 337748426752000*tan(c + d*x)),x)*d - 844371 066880000*int(( - tan(c + d*x)**(1/3)*sqrt(tan(c + d*x)*i + 1))/(337748426 752000*tan(c + d*x)**3*i + 337748426752000*tan(c + d*x)**2 + 3377484267520 00*tan(c + d*x)*i + 337748426752000),x)*tan(c + d*x)**2*d*i - 844371066880 000*int(( - tan(c + d*x)**(1/3)*sqrt(tan(c + d*x)*i + 1))/(337748426752000 *tan(c + d*x)**3*i + 337748426752000*tan(c + d*x)**2 + 337748426752000*tan (c + d*x)*i + 337748426752000),x)*d*i - 5*int((tan(c + d*x)**(1/3)*sqrt(ta n(c + d*x)*i + 1)*tan(c + d*x)**2)/(tan(c + d*x)**3*i + tan(c + d*x)**2...