Integrand size = 28, antiderivative size = 82 \[ \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\frac {3 \operatorname {AppellF1}\left (\frac {1}{3},\frac {5}{2},1,\frac {4}{3},-i \tan (c+d x),i \tan (c+d x)\right ) \sqrt {1+i \tan (c+d x)} \sqrt [3]{\tan (c+d x)}}{a d \sqrt {a+i a \tan (c+d x)}} \] Output:
3*AppellF1(1/3,5/2,1,4/3,-I*tan(d*x+c),I*tan(d*x+c))*(1+I*tan(d*x+c))^(1/2 )*tan(d*x+c)^(1/3)/a/d/(a+I*a*tan(d*x+c))^(1/2)
\[ \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {1}{\tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx \] Input:
Integrate[1/(Tan[c + d*x]^(2/3)*(a + I*a*Tan[c + d*x])^(3/2)),x]
Output:
Integrate[1/(Tan[c + d*x]^(2/3)*(a + I*a*Tan[c + d*x])^(3/2)), x]
Time = 0.32 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.04, 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, 937, 936}
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.
\(\displaystyle \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (c+d x)^{2/3} (a+i a \tan (c+d x))^{3/2}}dx\) |
\(\Big \downarrow \) 4047 |
\(\displaystyle \frac {i a^2 \int -\frac {1}{a \tan ^{\frac {2}{3}}(c+d x) (a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{5/2}}d(i a \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {i a^2 \int \frac {1}{a \tan ^{\frac {2}{3}}(c+d x) (a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{5/2}}d(i a \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {i a \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x) (a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{5/2}}d(i a \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 148 |
\(\displaystyle \frac {3 a^2 \int \frac {1}{a \left (1-a^3 \tan ^3(c+d x)\right ) \left (\tan ^3(c+d x) a^4+a\right )^{5/2}}d\sqrt [3]{\tan (c+d x)}}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3 a \int \frac {1}{\left (1-a^3 \tan ^3(c+d x)\right ) \left (\tan ^3(c+d x) a^4+a\right )^{5/2}}d\sqrt [3]{\tan (c+d x)}}{d}\) |
\(\Big \downarrow \) 937 |
\(\displaystyle \frac {3 \sqrt {a^3 \tan ^3(c+d x)+1} \int \frac {1}{\left (1-a^3 \tan ^3(c+d x)\right ) \left (a^3 \tan ^3(c+d x)+1\right )^{5/2}}d\sqrt [3]{\tan (c+d x)}}{a d \sqrt {a^4 \tan ^3(c+d x)+a}}\) |
\(\Big \downarrow \) 936 |
\(\displaystyle \frac {3 i \tan (c+d x) \sqrt {a^3 \tan ^3(c+d x)+1} \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {5}{2},\frac {4}{3},a^3 \tan ^3(c+d x),-a^3 \tan ^3(c+d x)\right )}{d \sqrt {a^4 \tan ^3(c+d x)+a}}\) |
Input:
Int[1/(Tan[c + d*x]^(2/3)*(a + I*a*Tan[c + d*x])^(3/2)),x]
Output:
((3*I)*AppellF1[1/3, 1, 5/2, 4/3, a^3*Tan[c + d*x]^3, -(a^3*Tan[c + d*x]^3 )]*Tan[c + d*x]*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[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) ], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((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[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q }, x] && NeQ[b*c - a*d, 0] && NeQ[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]
\[\int \frac {1}{\tan \left (d x +c \right )^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}d x\]
Input:
int(1/tan(d*x+c)^(2/3)/(a+I*a*tan(d*x+c))^(3/2),x)
Output:
int(1/tan(d*x+c)^(2/3)/(a+I*a*tan(d*x+c))^(3/2),x)
Timed out. \[ \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/tan(d*x+c)^(2/3)/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="fricas ")
Output:
Timed out
Timed out. \[ \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/tan(d*x+c)**(2/3)/(a+I*a*tan(d*x+c))**(3/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/tan(d*x+c)^(2/3)/(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. \[ \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/tan(d*x+c)^(2/3)/(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 ValueDegree mismat ch inside
Timed out. \[ \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {1}{{\mathrm {tan}\left (c+d\,x\right )}^{2/3}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \] Input:
int(1/(tan(c + d*x)^(2/3)*(a + a*tan(c + d*x)*1i)^(3/2)),x)
Output:
int(1/(tan(c + d*x)^(2/3)*(a + a*tan(c + d*x)*1i)^(3/2)), x)
\[ \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\frac {\int \frac {1}{\tan \left (d x +c \right )^{\frac {5}{3}} \sqrt {\tan \left (d x +c \right ) i +1}\, i +\tan \left (d x +c \right )^{\frac {2}{3}} \sqrt {\tan \left (d x +c \right ) i +1}}d x}{\sqrt {a}\, a} \] Input:
int(1/tan(d*x+c)^(2/3)/(a+I*a*tan(d*x+c))^(3/2),x)
Output:
int(1/(tan(c + d*x)**(2/3)*sqrt(tan(c + d*x)*i + 1)*tan(c + d*x)*i + tan(c + d*x)**(2/3)*sqrt(tan(c + d*x)*i + 1)),x)/(sqrt(a)*a)