Integrand size = 23, antiderivative size = 227 \[ \int \frac {1}{(c+d x)^3 (a+i a \tan (e+f x))} \, dx=-\frac {i f}{2 a d^2 (c+d x)}-\frac {f^2 \cos \left (2 e-\frac {2 c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}+\frac {i f^2 \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{a d^3}+\frac {i f^2 \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}+\frac {f^2 \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}-\frac {1}{2 d (c+d x)^2 (a+i a \tan (e+f x))}+\frac {i f}{d^2 (c+d x) (a+i a \tan (e+f x))} \] Output:
-1/2*I*f/a/d^2/(d*x+c)-f^2*cos(-2*e+2*c*f/d)*Ci(2*c*f/d+2*f*x)/a/d^3-I*f^2 *Ci(2*c*f/d+2*f*x)*sin(-2*e+2*c*f/d)/a/d^3+I*f^2*cos(-2*e+2*c*f/d)*Si(2*c* f/d+2*f*x)/a/d^3-f^2*sin(-2*e+2*c*f/d)*Si(2*c*f/d+2*f*x)/a/d^3-1/2/d/(d*x+ c)^2/(a+I*a*tan(f*x+e))+I*f/d^2/(d*x+c)/(a+I*a*tan(f*x+e))
Time = 0.91 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.26 \[ \int \frac {1}{(c+d x)^3 (a+i a \tan (e+f x))} \, dx=\frac {\sec (e+f x) \left (\cos \left (\frac {c f}{d}\right )+i \sin \left (\frac {c f}{d}\right )\right ) \left (d \left (i d \cos \left (e+f \left (-\frac {c}{d}+x\right )\right )+(i d+2 c f+2 d f x) \cos \left (e+f \left (\frac {c}{d}+x\right )\right )-d \sin \left (e+f \left (-\frac {c}{d}+x\right )\right )+d \sin \left (e+f \left (\frac {c}{d}+x\right )\right )-2 i c f \sin \left (e+f \left (\frac {c}{d}+x\right )\right )-2 i d f x \sin \left (e+f \left (\frac {c}{d}+x\right )\right )\right )+4 f^2 (c+d x)^2 \operatorname {CosIntegral}\left (\frac {2 f (c+d x)}{d}\right ) \left (i \cos \left (e-\frac {f (c+d x)}{d}\right )+\sin \left (e-\frac {f (c+d x)}{d}\right )\right )+4 f^2 (c+d x)^2 \left (\cos \left (e-\frac {f (c+d x)}{d}\right )-i \sin \left (e-\frac {f (c+d x)}{d}\right )\right ) \text {Si}\left (\frac {2 f (c+d x)}{d}\right )\right )}{4 a d^3 (c+d x)^2 (-i+\tan (e+f x))} \] Input:
Integrate[1/((c + d*x)^3*(a + I*a*Tan[e + f*x])),x]
Output:
(Sec[e + f*x]*(Cos[(c*f)/d] + I*Sin[(c*f)/d])*(d*(I*d*Cos[e + f*(-(c/d) + x)] + (I*d + 2*c*f + 2*d*f*x)*Cos[e + f*(c/d + x)] - d*Sin[e + f*(-(c/d) + x)] + d*Sin[e + f*(c/d + x)] - (2*I)*c*f*Sin[e + f*(c/d + x)] - (2*I)*d*f *x*Sin[e + f*(c/d + x)]) + 4*f^2*(c + d*x)^2*CosIntegral[(2*f*(c + d*x))/d ]*(I*Cos[e - (f*(c + d*x))/d] + Sin[e - (f*(c + d*x))/d]) + 4*f^2*(c + d*x )^2*(Cos[e - (f*(c + d*x))/d] - I*Sin[e - (f*(c + d*x))/d])*SinIntegral[(2 *f*(c + d*x))/d]))/(4*a*d^3*(c + d*x)^2*(-I + Tan[e + f*x]))
Time = 0.98 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3042, 4208, 3042, 4207, 3042, 3784, 3042, 3780, 3783}
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.
\(\displaystyle \int \frac {1}{(c+d x)^3 (a+i a \tan (e+f x))} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(c+d x)^3 (a+i a \tan (e+f x))}dx\) |
\(\Big \downarrow \) 4208 |
\(\displaystyle -\frac {i f \int \frac {1}{(c+d x)^2 (i \tan (e+f x) a+a)}dx}{d}-\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \tan (e+f x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {i f \int \frac {1}{(c+d x)^2 (i \tan (e+f x) a+a)}dx}{d}-\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \tan (e+f x))}\) |
\(\Big \downarrow \) 4207 |
\(\displaystyle -\frac {i f \left (-\frac {f \int \frac {\sin (2 e+2 f x)}{c+d x}dx}{a d}-\frac {i f \int \frac {\cos (2 e+2 f x)}{c+d x}dx}{a d}-\frac {1}{d (c+d x) (a+i a \tan (e+f x))}\right )}{d}-\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \tan (e+f x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {i f \left (-\frac {f \int \frac {\sin (2 e+2 f x)}{c+d x}dx}{a d}-\frac {i f \int \frac {\sin \left (2 e+2 f x+\frac {\pi }{2}\right )}{c+d x}dx}{a d}-\frac {1}{d (c+d x) (a+i a \tan (e+f x))}\right )}{d}-\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \tan (e+f x))}\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle -\frac {i f \left (-\frac {f \left (\sin \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\cos \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx+\cos \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx\right )}{a d}-\frac {i f \left (\cos \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\cos \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx-\sin \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx\right )}{a d}-\frac {1}{d (c+d x) (a+i a \tan (e+f x))}\right )}{d}-\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \tan (e+f x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {i f \left (-\frac {i f \left (\cos \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 x f+\frac {2 c f}{d}+\frac {\pi }{2}\right )}{c+d x}dx-\sin \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx\right )}{a d}-\frac {f \left (\sin \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 x f+\frac {2 c f}{d}+\frac {\pi }{2}\right )}{c+d x}dx+\cos \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx\right )}{a d}-\frac {1}{d (c+d x) (a+i a \tan (e+f x))}\right )}{d}-\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \tan (e+f x))}\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle -\frac {i f \left (-\frac {i f \left (\cos \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 x f+\frac {2 c f}{d}+\frac {\pi }{2}\right )}{c+d x}dx-\frac {\sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{d}\right )}{a d}-\frac {f \left (\sin \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 x f+\frac {2 c f}{d}+\frac {\pi }{2}\right )}{c+d x}dx+\frac {\cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{d}\right )}{a d}-\frac {1}{d (c+d x) (a+i a \tan (e+f x))}\right )}{d}-\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \tan (e+f x))}\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle -\frac {i f \left (-\frac {f \left (\frac {\operatorname {CosIntegral}\left (2 x f+\frac {2 c f}{d}\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{d}+\frac {\cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{d}\right )}{a d}-\frac {i f \left (\frac {\operatorname {CosIntegral}\left (2 x f+\frac {2 c f}{d}\right ) \cos \left (2 e-\frac {2 c f}{d}\right )}{d}-\frac {\sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{d}\right )}{a d}-\frac {1}{d (c+d x) (a+i a \tan (e+f x))}\right )}{d}-\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \tan (e+f x))}\) |
Input:
Int[1/((c + d*x)^3*(a + I*a*Tan[e + f*x])),x]
Output:
((-1/2*I)*f)/(a*d^2*(c + d*x)) - 1/(2*d*(c + d*x)^2*(a + I*a*Tan[e + f*x]) ) - (I*f*(-((f*((CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/d + (Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/d))/(a*d)) - (I*f*(( Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/d - (Sin[2*e - (2*c*f )/d]*SinIntegral[(2*c*f)/d + 2*f*x])/d))/(a*d) - 1/(d*(c + d*x)*(a + I*a*T an[e + f*x]))))/d
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
Int[1/(((c_.) + (d_.)*(x_))^2*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])), x_Sy mbol] :> -Simp[(d*(c + d*x)*(a + b*Tan[e + f*x]))^(-1), x] + (-Simp[f/(a*d) Int[Sin[2*e + 2*f*x]/(c + d*x), x], x] + Simp[f/(b*d) Int[Cos[2*e + 2* f*x]/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0 ]
Int[((c_.) + (d_.)*(x_))^(m_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Sym bol] :> Simp[f*((c + d*x)^(m + 2)/(b*d^2*(m + 1)*(m + 2))), x] + (Simp[2*b* (f/(a*d*(m + 1))) Int[(c + d*x)^(m + 1)/(a + b*Tan[e + f*x]), x], x] + Si mp[(c + d*x)^(m + 1)/(d*(m + 1)*(a + b*Tan[e + f*x])), x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0] && LtQ[m, -1] && NeQ[m, -2]
Time = 0.61 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.95
method | result | size |
risch | \(-\frac {1}{4 a d \left (d x +c \right )^{2}}+\frac {i f^{3} {\mathrm e}^{-2 i \left (f x +e \right )} x}{2 a d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} {\mathrm e}^{-2 i \left (f x +e \right )}}{4 a d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {i f^{3} {\mathrm e}^{-2 i \left (f x +e \right )} c}{2 a \,d^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{2} {\mathrm e}^{\frac {2 i \left (c f -d e \right )}{d}} \operatorname {expIntegral}_{1}\left (2 i f x +2 i e +\frac {2 i \left (c f -d e \right )}{d}\right )}{a \,d^{3}}\) | \(216\) |
Input:
int(1/(d*x+c)^3/(a+I*a*tan(f*x+e)),x,method=_RETURNVERBOSE)
Output:
-1/4/a/d/(d*x+c)^2+1/2*I/a*f^3*exp(-2*I*(f*x+e))/d/(d^2*f^2*x^2+2*c*d*f^2* x+c^2*f^2)*x-1/4/a*f^2*exp(-2*I*(f*x+e))/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^ 2)+1/2*I/a*f^3*exp(-2*I*(f*x+e))/d^2/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*c+1 /a*f^2/d^3*exp(2*I*(c*f-d*e)/d)*Ei(1,2*I*f*x+2*I*e+2*I*(c*f-d*e)/d)
Time = 0.09 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.57 \[ \int \frac {1}{(c+d x)^3 (a+i a \tan (e+f x))} \, dx=\frac {{\left (2 i \, d^{2} f x + 2 i \, c d f - d^{2} - {\left (4 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (i \, d f x + i \, c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (i \, d e - i \, c f\right )}}{d}\right )} + d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{4 \, {\left (a d^{5} x^{2} + 2 \, a c d^{4} x + a c^{2} d^{3}\right )}} \] Input:
integrate(1/(d*x+c)^3/(a+I*a*tan(f*x+e)),x, algorithm="fricas")
Output:
1/4*(2*I*d^2*f*x + 2*I*c*d*f - d^2 - (4*(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f ^2)*Ei(-2*(I*d*f*x + I*c*f)/d)*e^(-2*(I*d*e - I*c*f)/d) + d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/(a*d^5*x^2 + 2*a*c*d^4*x + a*c^2*d^3)
\[ \int \frac {1}{(c+d x)^3 (a+i a \tan (e+f x))} \, dx=- \frac {i \int \frac {1}{c^{3} \tan {\left (e + f x \right )} - i c^{3} + 3 c^{2} d x \tan {\left (e + f x \right )} - 3 i c^{2} d x + 3 c d^{2} x^{2} \tan {\left (e + f x \right )} - 3 i c d^{2} x^{2} + d^{3} x^{3} \tan {\left (e + f x \right )} - i d^{3} x^{3}}\, dx}{a} \] Input:
integrate(1/(d*x+c)**3/(a+I*a*tan(f*x+e)),x)
Output:
-I*Integral(1/(c**3*tan(e + f*x) - I*c**3 + 3*c**2*d*x*tan(e + f*x) - 3*I* c**2*d*x + 3*c*d**2*x**2*tan(e + f*x) - 3*I*c*d**2*x**2 + d**3*x**3*tan(e + f*x) - I*d**3*x**3), x)/a
Time = 0.23 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(c+d x)^3 (a+i a \tan (e+f x))} \, dx=-\frac {2 \, f^{3} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) E_{3}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) + 2 i \, f^{3} E_{3}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + f^{3}}{4 \, {\left ({\left (f x + e\right )}^{2} a d^{3} + a d^{3} e^{2} - 2 \, a c d^{2} e f + a c^{2} d f^{2} - 2 \, {\left (a d^{3} e - a c d^{2} f\right )} {\left (f x + e\right )}\right )} f} \] Input:
integrate(1/(d*x+c)^3/(a+I*a*tan(f*x+e)),x, algorithm="maxima")
Output:
-1/4*(2*f^3*cos(-2*(d*e - c*f)/d)*exp_integral_e(3, -2*(-I*(f*x + e)*d + I *d*e - I*c*f)/d) + 2*I*f^3*exp_integral_e(3, -2*(-I*(f*x + e)*d + I*d*e - I*c*f)/d)*sin(-2*(d*e - c*f)/d) + f^3)/(((f*x + e)^2*a*d^3 + a*d^3*e^2 - 2 *a*c*d^2*e*f + a*c^2*d*f^2 - 2*(a*d^3*e - a*c*d^2*f)*(f*x + e))*f)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (212) = 424\).
Time = 0.24 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.34 \[ \int \frac {1}{(c+d x)^3 (a+i a \tan (e+f x))} \, dx =\text {Too large to display} \] Input:
integrate(1/(d*x+c)^3/(a+I*a*tan(f*x+e)),x, algorithm="giac")
Output:
-1/4*(4*d^2*f^2*x^2*cos(2*c*f/d)*cos_integral(-2*(d*f*x + c*f)/d) + 4*I*d^ 2*f^2*x^2*cos_integral(-2*(d*f*x + c*f)/d)*sin(2*c*f/d) - 4*I*d^2*f^2*x^2* cos(2*c*f/d)*sin_integral(2*(d*f*x + c*f)/d) + 4*d^2*f^2*x^2*sin(2*c*f/d)* sin_integral(2*(d*f*x + c*f)/d) + 8*c*d*f^2*x*cos(2*c*f/d)*cos_integral(-2 *(d*f*x + c*f)/d) + 8*I*c*d*f^2*x*cos_integral(-2*(d*f*x + c*f)/d)*sin(2*c *f/d) - 8*I*c*d*f^2*x*cos(2*c*f/d)*sin_integral(2*(d*f*x + c*f)/d) + 8*c*d *f^2*x*sin(2*c*f/d)*sin_integral(2*(d*f*x + c*f)/d) + 4*c^2*f^2*cos(2*c*f/ d)*cos_integral(-2*(d*f*x + c*f)/d) + 4*I*c^2*f^2*cos_integral(-2*(d*f*x + c*f)/d)*sin(2*c*f/d) - 4*I*c^2*f^2*cos(2*c*f/d)*sin_integral(2*(d*f*x + c *f)/d) + 4*c^2*f^2*sin(2*c*f/d)*sin_integral(2*(d*f*x + c*f)/d) - 2*I*d^2* f*x*cos(2*f*x) - 2*d^2*f*x*sin(2*f*x) - 2*I*c*d*f*cos(2*f*x) - 2*c*d*f*sin (2*f*x) + d^2*cos(2*f*x) + d^2*cos(2*e) - I*d^2*sin(2*f*x) + I*d^2*sin(2*e ))/(a*d^5*x^2*cos(2*e) + I*a*d^5*x^2*sin(2*e) + 2*a*c*d^4*x*cos(2*e) + 2*I *a*c*d^4*x*sin(2*e) + a*c^2*d^3*cos(2*e) + I*a*c^2*d^3*sin(2*e))
Timed out. \[ \int \frac {1}{(c+d x)^3 (a+i a \tan (e+f x))} \, dx=\int \frac {1}{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,{\left (c+d\,x\right )}^3} \,d x \] Input:
int(1/((a + a*tan(e + f*x)*1i)*(c + d*x)^3),x)
Output:
int(1/((a + a*tan(e + f*x)*1i)*(c + d*x)^3), x)
\[ \int \frac {1}{(c+d x)^3 (a+i a \tan (e+f x))} \, dx=\frac {\int \frac {1}{\tan \left (f x +e \right ) c^{3} i +3 \tan \left (f x +e \right ) c^{2} d i x +3 \tan \left (f x +e \right ) c \,d^{2} i \,x^{2}+\tan \left (f x +e \right ) d^{3} i \,x^{3}+c^{3}+3 c^{2} d x +3 c \,d^{2} x^{2}+d^{3} x^{3}}d x}{a} \] Input:
int(1/(d*x+c)^3/(a+I*a*tan(f*x+e)),x)
Output:
int(1/(tan(e + f*x)*c**3*i + 3*tan(e + f*x)*c**2*d*i*x + 3*tan(e + f*x)*c* d**2*i*x**2 + tan(e + f*x)*d**3*i*x**3 + c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)/a