Integrand size = 45, antiderivative size = 102 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx=-\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{7 f (c-i c \tan (e+f x))^{7/2}}-\frac {(i A-6 B) (a+i a \tan (e+f x))^{5/2}}{35 c f (c-i c \tan (e+f x))^{5/2}} \] Output:
-1/7*(I*A+B)*(a+I*a*tan(f*x+e))^(5/2)/f/(c-I*c*tan(f*x+e))^(7/2)-1/35*(I*A -6*B)*(a+I*a*tan(f*x+e))^(5/2)/c/f/(c-I*c*tan(f*x+e))^(5/2)
Time = 13.45 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.19 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx=\frac {a^2 \cos (e+f x) ((-6 i A+B) \cos (e+f x)-(A+6 i B) \sin (e+f x)) (\cos (6 e+8 f x)+i \sin (6 e+8 f x)) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{35 c^4 f (\cos (f x)+i \sin (f x))^2} \] Input:
Integrate[((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan [e + f*x])^(7/2),x]
Output:
(a^2*Cos[e + f*x]*(((-6*I)*A + B)*Cos[e + f*x] - (A + (6*I)*B)*Sin[e + f*x ])*(Cos[6*e + 8*f*x] + I*Sin[6*e + 8*f*x])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt [c - I*c*Tan[e + f*x]])/(35*c^4*f*(Cos[f*x] + I*Sin[f*x])^2)
Time = 0.63 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {3042, 4071, 87, 48}
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.
\(\displaystyle \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}}dx\) |
\(\Big \downarrow \) 4071 |
\(\displaystyle \frac {a c \int \frac {(i \tan (e+f x) a+a)^{3/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{9/2}}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {a c \left (\frac {(A+6 i B) \int \frac {(i \tan (e+f x) a+a)^{3/2}}{(c-i c \tan (e+f x))^{7/2}}d\tan (e+f x)}{7 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{5/2}}{7 a c (c-i c \tan (e+f x))^{7/2}}\right )}{f}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {a c \left (-\frac {i (A+6 i B) (a+i a \tan (e+f x))^{5/2}}{35 a c^2 (c-i c \tan (e+f x))^{5/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{5/2}}{7 a c (c-i c \tan (e+f x))^{7/2}}\right )}{f}\) |
Input:
Int[((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f *x])^(7/2),x]
Output:
(a*c*(-1/7*((I*A + B)*(a + I*a*Tan[e + f*x])^(5/2))/(a*c*(c - I*c*Tan[e + f*x])^(7/2)) - ((I/35)*(A + (6*I)*B)*(a + I*a*Tan[e + f*x])^(5/2))/(a*c^2* (c - I*c*Tan[e + f*x])^(5/2))))/f
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] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Si mp[a*(c/f) Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1)*(A + B*x), x], x , Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]
Time = 0.72 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.04
method | result | size |
risch | \(-\frac {a^{2} \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (5 i A \,{\mathrm e}^{6 i \left (f x +e \right )}+5 B \,{\mathrm e}^{6 i \left (f x +e \right )}+7 i A \,{\mathrm e}^{4 i \left (f x +e \right )}-7 B \,{\mathrm e}^{4 i \left (f x +e \right )}\right )}{70 c^{3} \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(106\) |
derivativedivides | \(-\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (i A \tan \left (f x +e \right )^{2}+5 i B \tan \left (f x +e \right )-6 B \tan \left (f x +e \right )^{2}+6 i A -5 A \tan \left (f x +e \right )-B \right )}{35 f \,c^{4} \left (i+\tan \left (f x +e \right )\right )^{5}}\) | \(115\) |
default | \(-\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (i A \tan \left (f x +e \right )^{2}+5 i B \tan \left (f x +e \right )-6 B \tan \left (f x +e \right )^{2}+6 i A -5 A \tan \left (f x +e \right )-B \right )}{35 f \,c^{4} \left (i+\tan \left (f x +e \right )\right )^{5}}\) | \(115\) |
parts | \(\frac {A \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (5 i \tan \left (f x +e \right )+\tan \left (f x +e \right )^{2}+6\right )}{35 f \,c^{4} \left (i+\tan \left (f x +e \right )\right )^{5}}-\frac {i B \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (-1+5 i \tan \left (f x +e \right )-6 \tan \left (f x +e \right )^{2}\right )}{35 f \,c^{4} \left (i+\tan \left (f x +e \right )\right )^{5}}\) | \(171\) |
Input:
int((a+I*a*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(7/2),x,m ethod=_RETURNVERBOSE)
Output:
-1/70*a^2/c^3*(a*exp(2*I*(f*x+e))/(exp(2*I*(f*x+e))+1))^(1/2)/(c/(exp(2*I* (f*x+e))+1))^(1/2)/f*(5*I*A*exp(6*I*(f*x+e))+5*B*exp(6*I*(f*x+e))+7*I*A*ex p(4*I*(f*x+e))-7*B*exp(4*I*(f*x+e)))
Time = 0.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.02 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx=-\frac {{\left (5 \, {\left (i \, A + B\right )} a^{2} e^{\left (9 i \, f x + 9 i \, e\right )} + 2 \, {\left (6 i \, A - B\right )} a^{2} e^{\left (7 i \, f x + 7 i \, e\right )} + 7 \, {\left (i \, A - B\right )} a^{2} e^{\left (5 i \, f x + 5 i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{70 \, c^{4} f} \] Input:
integrate((a+I*a*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(7/ 2),x, algorithm="fricas")
Output:
-1/70*(5*(I*A + B)*a^2*e^(9*I*f*x + 9*I*e) + 2*(6*I*A - B)*a^2*e^(7*I*f*x + 7*I*e) + 7*(I*A - B)*a^2*e^(5*I*f*x + 5*I*e))*sqrt(a/(e^(2*I*f*x + 2*I*e ) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1))/(c^4*f)
Timed out. \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((a+I*a*tan(f*x+e))**(5/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))**( 7/2),x)
Output:
Timed out
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (78) = 156\).
Time = 0.21 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.63 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx=-\frac {70 \, {\left (5 \, {\left (A - i \, B\right )} a^{2} \cos \left (9 \, f x + 9 \, e\right ) + 2 \, {\left (6 \, A + i \, B\right )} a^{2} \cos \left (7 \, f x + 7 \, e\right ) + 7 \, {\left (A + i \, B\right )} a^{2} \cos \left (5 \, f x + 5 \, e\right ) - 5 \, {\left (-i \, A - B\right )} a^{2} \sin \left (9 \, f x + 9 \, e\right ) - 2 \, {\left (-6 i \, A + B\right )} a^{2} \sin \left (7 \, f x + 7 \, e\right ) - 7 \, {\left (-i \, A + B\right )} a^{2} \sin \left (5 \, f x + 5 \, e\right )\right )} \sqrt {a} \sqrt {c}}{-4900 \, {\left (i \, c^{4} \cos \left (2 \, f x + 2 \, e\right ) - c^{4} \sin \left (2 \, f x + 2 \, e\right ) + i \, c^{4}\right )} f} \] Input:
integrate((a+I*a*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(7/ 2),x, algorithm="maxima")
Output:
-70*(5*(A - I*B)*a^2*cos(9*f*x + 9*e) + 2*(6*A + I*B)*a^2*cos(7*f*x + 7*e) + 7*(A + I*B)*a^2*cos(5*f*x + 5*e) - 5*(-I*A - B)*a^2*sin(9*f*x + 9*e) - 2*(-6*I*A + B)*a^2*sin(7*f*x + 7*e) - 7*(-I*A + B)*a^2*sin(5*f*x + 5*e))*s qrt(a)*sqrt(c)/((-4900*I*c^4*cos(2*f*x + 2*e) + 4900*c^4*sin(2*f*x + 2*e) - 4900*I*c^4)*f)
Exception generated. \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+I*a*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(7/ 2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad Argument TypeError: Bad Argument TypeError: Bad Argument TypeDone
Time = 8.54 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.88 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx=-\frac {a^2\,\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (A\,\cos \left (4\,e+4\,f\,x\right )\,7{}\mathrm {i}+A\,\cos \left (6\,e+6\,f\,x\right )\,5{}\mathrm {i}-7\,B\,\cos \left (4\,e+4\,f\,x\right )+5\,B\,\cos \left (6\,e+6\,f\,x\right )-7\,A\,\sin \left (4\,e+4\,f\,x\right )-5\,A\,\sin \left (6\,e+6\,f\,x\right )-B\,\sin \left (4\,e+4\,f\,x\right )\,7{}\mathrm {i}+B\,\sin \left (6\,e+6\,f\,x\right )\,5{}\mathrm {i}\right )}{70\,c^3\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \] Input:
int(((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^(5/2))/(c - c*tan(e + f* x)*1i)^(7/2),x)
Output:
-(a^2*((a*(cos(2*e + 2*f*x) + sin(2*e + 2*f*x)*1i + 1))/(cos(2*e + 2*f*x) + 1))^(1/2)*(A*cos(4*e + 4*f*x)*7i + A*cos(6*e + 6*f*x)*5i - 7*B*cos(4*e + 4*f*x) + 5*B*cos(6*e + 6*f*x) - 7*A*sin(4*e + 4*f*x) - 5*A*sin(6*e + 6*f* x) - B*sin(4*e + 4*f*x)*7i + B*sin(6*e + 6*f*x)*5i))/(70*c^3*f*((c*(cos(2* e + 2*f*x) - sin(2*e + 2*f*x)*1i + 1))/(cos(2*e + 2*f*x) + 1))^(1/2))
\[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx =\text {Too large to display} \] Input:
int((a+I*a*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(7/2),x)
Output:
(sqrt(c)*sqrt(a)*a**2*(sqrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x)*i + 1 )*tan(e + f*x)*a + sqrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x)*i + 1)*b + 2*int(( - sqrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x)*i + 1)*tan(e + f *x)**3)/(tan(e + f*x)**5 + 3*tan(e + f*x)**4*i - 2*tan(e + f*x)**3 + 2*tan (e + f*x)**2*i - 3*tan(e + f*x) - i),x)*tan(e + f*x)**2*a*f - 6*int(( - sq rt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x)**3)/(tan(e + f*x)**5 + 3*tan(e + f*x)**4*i - 2*tan(e + f*x)**3 + 2*tan(e + f*x)**2*i - 3*tan(e + f*x) - i),x)*tan(e + f*x)**2*b*f*i + 2*int(( - sqrt(tan(e + f *x)*i + 1)*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x)**3)/(tan(e + f*x)**5 + 3*tan(e + f*x)**4*i - 2*tan(e + f*x)**3 + 2*tan(e + f*x)**2*i - 3*tan(e + f*x) - i),x)*a*f - 6*int(( - sqrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x )*i + 1)*tan(e + f*x)**3)/(tan(e + f*x)**5 + 3*tan(e + f*x)**4*i - 2*tan(e + f*x)**3 + 2*tan(e + f*x)**2*i - 3*tan(e + f*x) - i),x)*b*f*i + 6*int((s qrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x))/(tan(e + f*x)**5 + 3*tan(e + f*x)**4*i - 2*tan(e + f*x)**3 + 2*tan(e + f*x)**2*i - 3*tan(e + f*x) - i),x)*tan(e + f*x)**2*a*f - 2*int((sqrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x))/(tan(e + f*x)**5 + 3*tan(e + f*x)**4*i - 2*tan(e + f*x)**3 + 2*tan(e + f*x)**2*i - 3*tan(e + f*x) - i) ,x)*tan(e + f*x)**2*b*f*i + 6*int((sqrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x))/(tan(e + f*x)**5 + 3*tan(e + f*x)**4*i - 2*...