Integrand size = 25, antiderivative size = 154 \[ \int \frac {1}{(a-i a x)^{17/4} (a+i a x)^{9/4}} \, dx=-\frac {2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}}-\frac {2 i}{13 a^3 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac {14 x}{65 a^6 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac {42 \sqrt [4]{1+x^2} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{65 a^6 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
-2/13*I/a^2/(a-I*a*x)^(13/4)/(a+I*a*x)^(5/4)-2/13*I/a^3/(a-I*a*x)^(9/4)/(a +I*a*x)^(5/4)+14/65*x/a^6/(a-I*a*x)^(1/4)/(a+I*a*x)^(1/4)/(x^2+1)+42/65*(x ^2+1)^(1/4)*(cos(1/2*arctan(x))^2)^(1/2)/cos(1/2*arctan(x))*EllipticE(sin( 1/2*arctan(x)),2^(1/2))/a^6/(a-I*a*x)^(1/4)/(a+I*a*x)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.45 \[ \int \frac {1}{(a-i a x)^{17/4} (a+i a x)^{9/4}} \, dx=-\frac {i \sqrt [4]{1+i x} \operatorname {Hypergeometric2F1}\left (-\frac {13}{4},\frac {9}{4},-\frac {9}{4},\frac {1}{2}-\frac {i x}{2}\right )}{13 \sqrt [4]{2} a^3 (a-i a x)^{13/4} \sqrt [4]{a+i a x}} \]
((-1/13*I)*(1 + I*x)^(1/4)*Hypergeometric2F1[-13/4, 9/4, -9/4, 1/2 - (I/2) *x])/(2^(1/4)*a^3*(a - I*a*x)^(13/4)*(a + I*a*x)^(1/4))
Time = 0.22 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.19, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {61, 61, 46, 215, 213, 212}
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.
\(\displaystyle \int \frac {1}{(a-i a x)^{17/4} (a+i a x)^{9/4}} \, dx\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {9 \int \frac {1}{(a-i a x)^{13/4} (i x a+a)^{9/4}}dx}{13 a}-\frac {2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {9 \left (\frac {7 \int \frac {1}{(a-i a x)^{9/4} (i x a+a)^{9/4}}dx}{9 a}-\frac {2 i}{9 a^2 (a-i a x)^{9/4} (a+i a x)^{5/4}}\right )}{13 a}-\frac {2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}}\) |
\(\Big \downarrow \) 46 |
\(\displaystyle \frac {9 \left (\frac {7 \sqrt [4]{a^2 x^2+a^2} \int \frac {1}{\left (x^2 a^2+a^2\right )^{9/4}}dx}{9 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^2 (a-i a x)^{9/4} (a+i a x)^{5/4}}\right )}{13 a}-\frac {2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {9 \left (\frac {7 \sqrt [4]{a^2 x^2+a^2} \left (\frac {3 \int \frac {1}{\left (x^2 a^2+a^2\right )^{5/4}}dx}{5 a^2}+\frac {2 x}{5 a^2 \left (a^2 x^2+a^2\right )^{5/4}}\right )}{9 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^2 (a-i a x)^{9/4} (a+i a x)^{5/4}}\right )}{13 a}-\frac {2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}}\) |
\(\Big \downarrow \) 213 |
\(\displaystyle \frac {9 \left (\frac {7 \sqrt [4]{a^2 x^2+a^2} \left (\frac {3 \sqrt [4]{x^2+1} \int \frac {1}{\left (x^2+1\right )^{5/4}}dx}{5 a^4 \sqrt [4]{a^2 x^2+a^2}}+\frac {2 x}{5 a^2 \left (a^2 x^2+a^2\right )^{5/4}}\right )}{9 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^2 (a-i a x)^{9/4} (a+i a x)^{5/4}}\right )}{13 a}-\frac {2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}}\) |
\(\Big \downarrow \) 212 |
\(\displaystyle \frac {9 \left (\frac {7 \sqrt [4]{a^2 x^2+a^2} \left (\frac {2 x}{5 a^2 \left (a^2 x^2+a^2\right )^{5/4}}+\frac {6 \sqrt [4]{x^2+1} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{5 a^4 \sqrt [4]{a^2 x^2+a^2}}\right )}{9 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^2 (a-i a x)^{9/4} (a+i a x)^{5/4}}\right )}{13 a}-\frac {2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}}\) |
((-2*I)/13)/(a^2*(a - I*a*x)^(13/4)*(a + I*a*x)^(5/4)) + (9*(((-2*I)/9)/(a ^2*(a - I*a*x)^(9/4)*(a + I*a*x)^(5/4)) + (7*(a^2 + a^2*x^2)^(1/4)*((2*x)/ (5*a^2*(a^2 + a^2*x^2)^(5/4)) + (6*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(5*a^4*(a^2 + a^2*x^2)^(1/4))))/(9*a*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/ 4))))/(13*a)
3.13.23.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^FracPart[m]*((c + d*x)^FracPart[m]/(a*c + b*d*x^2)^FracPart[m]) I nt[(a*c + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && !IntegerQ[2*m]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) )*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(1/4)/( a*(a + b*x^2)^(1/4)) Int[1/(1 + b*(x^2/a))^(5/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a] && PosQ[b/a]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.84
method | result | size |
risch | \(\frac {\frac {42}{65} x^{5}+\frac {84}{65} i x^{4}+\frac {14}{65} x^{3}+\frac {112}{65} i x^{2}-\frac {46}{65} x +\frac {4}{13} i}{\left (x -i\right ) \left (x +i\right )^{3} a^{6} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}-\frac {21 x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-x^{2}\right ) \left (-a^{2} \left (i x -1\right ) \left (i x +1\right )\right )^{\frac {1}{4}}}{65 \left (a^{2}\right )^{\frac {1}{4}} a^{6} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\) | \(130\) |
2/65*(42*I*x^4+21*x^5+56*I*x^2-23*x+7*x^3+10*I)/(x-I)/(x+I)^3/a^6/(-a*(I*x -1))^(1/4)/(a*(I*x+1))^(1/4)-21/65/(a^2)^(1/4)*x*hypergeom([1/4,1/2],[3/2] ,-x^2)/a^6*(-a^2*(I*x-1)*(I*x+1))^(1/4)/(-a*(I*x-1))^(1/4)/(a*(I*x+1))^(1/ 4)
\[ \int \frac {1}{(a-i a x)^{17/4} (a+i a x)^{9/4}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {9}{4}} {\left (-i \, a x + a\right )}^{\frac {17}{4}}} \,d x } \]
1/65*(2*(21*x^5 + 42*I*x^4 + 7*x^3 + 56*I*x^2 - 23*x + 10*I)*(I*a*x + a)^( 3/4)*(-I*a*x + a)^(3/4) + 65*(a^8*x^6 + 2*I*a^8*x^5 + a^8*x^4 + 4*I*a^8*x^ 3 - a^8*x^2 + 2*I*a^8*x - a^8)*integral(-21/65*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)/(a^8*x^2 + a^8), x))/(a^8*x^6 + 2*I*a^8*x^5 + a^8*x^4 + 4*I*a^8* x^3 - a^8*x^2 + 2*I*a^8*x - a^8)
Timed out. \[ \int \frac {1}{(a-i a x)^{17/4} (a+i a x)^{9/4}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{(a-i a x)^{17/4} (a+i a x)^{9/4}} \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int \frac {1}{(a-i a x)^{17/4} (a+i a x)^{9/4}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:The choice was done assuming 0=[0,0 ]ext_reduce Error: Bad Argument TypeDone
Timed out. \[ \int \frac {1}{(a-i a x)^{17/4} (a+i a x)^{9/4}} \, dx=\int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{17/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{9/4}} \,d x \]
\[ \int \frac {1}{(a-i a x)^{17/4} (a+i a x)^{9/4}} \, dx=-\frac {\int \frac {1}{2 \left (i x +1\right )^{\frac {1}{4}} \left (-i x +1\right )^{\frac {1}{4}} i \,x^{5}+4 \left (i x +1\right )^{\frac {1}{4}} \left (-i x +1\right )^{\frac {1}{4}} i \,x^{3}+2 \left (i x +1\right )^{\frac {1}{4}} \left (-i x +1\right )^{\frac {1}{4}} i x +\left (i x +1\right )^{\frac {1}{4}} \left (-i x +1\right )^{\frac {1}{4}} x^{6}+\left (i x +1\right )^{\frac {1}{4}} \left (-i x +1\right )^{\frac {1}{4}} x^{4}-\left (i x +1\right )^{\frac {1}{4}} \left (-i x +1\right )^{\frac {1}{4}} x^{2}-\left (i x +1\right )^{\frac {1}{4}} \left (-i x +1\right )^{\frac {1}{4}}}d x}{\sqrt {a}\, a^{6}} \]
int(( - 1)/((a*i*x + a)**(1/4)*( - a*i*x + a)**(1/4)*a**6*(2*i*x**5 + 4*i* x**3 + 2*i*x + x**6 + x**4 - x**2 - 1)),x)
( - int(1/(2*(i*x + 1)**(1/4)*( - i*x + 1)**(1/4)*i*x**5 + 4*(i*x + 1)**(1 /4)*( - i*x + 1)**(1/4)*i*x**3 + 2*(i*x + 1)**(1/4)*( - i*x + 1)**(1/4)*i* x + (i*x + 1)**(1/4)*( - i*x + 1)**(1/4)*x**6 + (i*x + 1)**(1/4)*( - i*x + 1)**(1/4)*x**4 - (i*x + 1)**(1/4)*( - i*x + 1)**(1/4)*x**2 - (i*x + 1)**( 1/4)*( - i*x + 1)**(1/4)),x))/(sqrt(a)*a**6)