3.11.76 \(\int \frac {(c+d \tan (e+f x))^2}{(a+i a \tan (e+f x))^3} \, dx\) [1076]

Optimal. Leaf size=129 \[ \frac {(c-i d)^2 x}{8 a^3}+\frac {i (c+i d)^2}{6 f (a+i a \tan (e+f x))^3}+\frac {(c+i d) (i c+3 d)}{8 a f (a+i a \tan (e+f x))^2}+\frac {i (c-i d)^2}{8 f \left (a^3+i a^3 \tan (e+f x)\right )} \]

[Out]

1/8*(c-I*d)^2*x/a^3+1/6*I*(c+I*d)^2/f/(a+I*a*tan(f*x+e))^3+1/8*(c+I*d)*(I*c+3*d)/a/f/(a+I*a*tan(f*x+e))^2+1/8*
I*(c-I*d)^2/f/(a^3+I*a^3*tan(f*x+e))

________________________________________________________________________________________

Rubi [A]
time = 0.12, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3621, 3607, 3560, 8} \begin {gather*} \frac {i (c-i d)^2}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {x (c-i d)^2}{8 a^3}+\frac {(c+i d) (3 d+i c)}{8 a f (a+i a \tan (e+f x))^2}+\frac {i (c+i d)^2}{6 f (a+i a \tan (e+f x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*Tan[e + f*x])^2/(a + I*a*Tan[e + f*x])^3,x]

[Out]

((c - I*d)^2*x)/(8*a^3) + ((I/6)*(c + I*d)^2)/(f*(a + I*a*Tan[e + f*x])^3) + ((c + I*d)*(I*c + 3*d))/(8*a*f*(a
+ I*a*Tan[e + f*x])^2) + ((I/8)*(c - I*d)^2)/(f*(a^3 + I*a^3*Tan[e + f*x]))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3560

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a*((a + b*Tan[c + d*x])^n/(2*b*d*n)), x] +
Dist[1/(2*a), Int[(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 0]

Rule 3607

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(
b*c - a*d))*((a + b*Tan[e + f*x])^m/(2*a*f*m)), x] + Dist[(b*c + a*d)/(2*a*b), Int[(a + b*Tan[e + f*x])^(m + 1
), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0]

Rule 3621

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(
-b)*(a*c + b*d)^2*((a + b*Tan[e + f*x])^m/(2*a^3*f*m)), x] + Dist[1/(2*a^2), Int[(a + b*Tan[e + f*x])^(m + 1)*
Simp[a*c^2 - 2*b*c*d + a*d^2 - 2*b*d^2*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a
*d, 0] && LeQ[m, -1] && EqQ[a^2 + b^2, 0]

Rubi steps

\begin {align*} \int \frac {(c+d \tan (e+f x))^2}{(a+i a \tan (e+f x))^3} \, dx &=\frac {i (c+i d)^2}{6 f (a+i a \tan (e+f x))^3}+\frac {\int \frac {a \left (c^2-2 i c d+d^2\right )-2 i a d^2 \tan (e+f x)}{(a+i a \tan (e+f x))^2} \, dx}{2 a^2}\\ &=\frac {i (c+i d)^2}{6 f (a+i a \tan (e+f x))^3}+\frac {(c+i d) (i c+3 d)}{8 a f (a+i a \tan (e+f x))^2}+\frac {(c-i d)^2 \int \frac {1}{a+i a \tan (e+f x)} \, dx}{4 a^2}\\ &=\frac {i (c+i d)^2}{6 f (a+i a \tan (e+f x))^3}+\frac {(c+i d) (i c+3 d)}{8 a f (a+i a \tan (e+f x))^2}+\frac {i (c-i d)^2}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {(c-i d)^2 \int 1 \, dx}{8 a^3}\\ &=\frac {(c-i d)^2 x}{8 a^3}+\frac {i (c+i d)^2}{6 f (a+i a \tan (e+f x))^3}+\frac {(c+i d) (i c+3 d)}{8 a f (a+i a \tan (e+f x))^2}+\frac {i (c-i d)^2}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.07, size = 256, normalized size = 1.98 \begin {gather*} \frac {\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left (3 (c+i d) (3 i c+d) \cos (4 f x) (\cos (e)-i \sin (e))+6 (3 c+i d) (i c+d) \cos (2 f x) (\cos (e)+i \sin (e))+12 (c-i d)^2 f x (\cos (3 e)+i \sin (3 e))+2 (c+i d)^2 \cos (6 f x) (i \cos (3 e)+\sin (3 e))+6 (c-i d) (3 c+i d) (\cos (e)+i \sin (e)) \sin (2 f x)+3 (3 c-i d) (c+i d) (\cos (e)-i \sin (e)) \sin (4 f x)+2 (c+i d)^2 (\cos (3 e)-i \sin (3 e)) \sin (6 f x)\right )}{96 f (a+i a \tan (e+f x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*Tan[e + f*x])^2/(a + I*a*Tan[e + f*x])^3,x]

[Out]

(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*(3*(c + I*d)*((3*I)*c + d)*Cos[4*f*x]*(Cos[e] - I*Sin[e]) + 6*(3*c +
I*d)*(I*c + d)*Cos[2*f*x]*(Cos[e] + I*Sin[e]) + 12*(c - I*d)^2*f*x*(Cos[3*e] + I*Sin[3*e]) + 2*(c + I*d)^2*Co
s[6*f*x]*(I*Cos[3*e] + Sin[3*e]) + 6*(c - I*d)*(3*c + I*d)*(Cos[e] + I*Sin[e])*Sin[2*f*x] + 3*(3*c - I*d)*(c +
I*d)*(Cos[e] - I*Sin[e])*Sin[4*f*x] + 2*(c + I*d)^2*(Cos[3*e] - I*Sin[3*e])*Sin[6*f*x]))/(96*f*(a + I*a*Tan[e
+ f*x])^3)

________________________________________________________________________________________

Maple [A]
time = 0.23, size = 152, normalized size = 1.18

method	result	size
derivativedivides	\(\frac {-\frac {i \left (2 i c d -c^{2}+d^{2}\right ) \ln \left (\tan \left (f x +e \right )+i\right )}{16}-\frac {\frac {1}{2} c d +\frac {1}{4} i c^{2}+\frac {3}{4} i d^{2}}{2 \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {i c d +\frac {1}{2} c^{2}-\frac {1}{2} d^{2}}{3 \left (\tan \left (f x +e \right )-i\right )^{3}}+\left (-\frac {1}{16} i c^{2}+\frac {1}{16} i d^{2}-\frac {1}{8} c d \right ) \ln \left (\tan \left (f x +e \right )-i\right )-\frac {\frac {1}{4} i c d -\frac {1}{8} c^{2}+\frac {1}{8} d^{2}}{\tan \left (f x +e \right )-i}}{f \,a^{3}}\)	\(152\)
default	\(\frac {-\frac {i \left (2 i c d -c^{2}+d^{2}\right ) \ln \left (\tan \left (f x +e \right )+i\right )}{16}-\frac {\frac {1}{2} c d +\frac {1}{4} i c^{2}+\frac {3}{4} i d^{2}}{2 \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {i c d +\frac {1}{2} c^{2}-\frac {1}{2} d^{2}}{3 \left (\tan \left (f x +e \right )-i\right )^{3}}+\left (-\frac {1}{16} i c^{2}+\frac {1}{16} i d^{2}-\frac {1}{8} c d \right ) \ln \left (\tan \left (f x +e \right )-i\right )-\frac {\frac {1}{4} i c d -\frac {1}{8} c^{2}+\frac {1}{8} d^{2}}{\tan \left (f x +e \right )-i}}{f \,a^{3}}\)	\(152\)
risch	\(-\frac {i x c d}{4 a^{3}}+\frac {x \,c^{2}}{8 a^{3}}-\frac {x \,d^{2}}{8 a^{3}}+\frac {{\mathrm e}^{-2 i \left (f x +e \right )} c d}{8 a^{3} f}+\frac {3 i {\mathrm e}^{-2 i \left (f x +e \right )} c^{2}}{16 a^{3} f}+\frac {i {\mathrm e}^{-2 i \left (f x +e \right )} d^{2}}{16 a^{3} f}-\frac {{\mathrm e}^{-4 i \left (f x +e \right )} c d}{16 a^{3} f}+\frac {3 i {\mathrm e}^{-4 i \left (f x +e \right )} c^{2}}{32 a^{3} f}+\frac {i {\mathrm e}^{-4 i \left (f x +e \right )} d^{2}}{32 a^{3} f}-\frac {{\mathrm e}^{-6 i \left (f x +e \right )} c d}{24 a^{3} f}+\frac {i {\mathrm e}^{-6 i \left (f x +e \right )} c^{2}}{48 a^{3} f}-\frac {i {\mathrm e}^{-6 i \left (f x +e \right )} d^{2}}{48 a^{3} f}\)	\(212\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*tan(f*x+e))^2/(a+I*a*tan(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

1/f/a^3*(-1/16*I*(2*I*c*d-c^2+d^2)*ln(tan(f*x+e)+I)-1/2*(1/2*c*d+1/4*I*c^2+3/4*I*d^2)/(tan(f*x+e)-I)^2-1/3*(I*
c*d+1/2*c^2-1/2*d^2)/(tan(f*x+e)-I)^3+(-1/16*I*c^2+1/16*I*d^2-1/8*c*d)*ln(tan(f*x+e)-I)-(1/4*I*c*d-1/8*c^2+1/8
*d^2)/(tan(f*x+e)-I))

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^2/(a+I*a*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

________________________________________________________________________________________

Fricas [A]
time = 0.74, size = 114, normalized size = 0.88 \begin {gather*} \frac {{\left (12 \, {\left (c^{2} - 2 i \, c d - d^{2}\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} + 2 i \, c^{2} - 4 \, c d - 2 i \, d^{2} - 6 \, {\left (-3 i \, c^{2} - 2 \, c d - i \, d^{2}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 3 \, {\left (-3 i \, c^{2} + 2 \, c d - i \, d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{96 \, a^{3} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^2/(a+I*a*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

1/96*(12*(c^2 - 2*I*c*d - d^2)*f*x*e^(6*I*f*x + 6*I*e) + 2*I*c^2 - 4*c*d - 2*I*d^2 - 6*(-3*I*c^2 - 2*c*d - I*d
^2)*e^(4*I*f*x + 4*I*e) - 3*(-3*I*c^2 + 2*c*d - I*d^2)*e^(2*I*f*x + 2*I*e))*e^(-6*I*f*x - 6*I*e)/(a^3*f)

________________________________________________________________________________________

Sympy [A]
time = 0.39, size = 401, normalized size = 3.11 \begin {gather*} \begin {cases} \frac {\left (\left (512 i a^{6} c^{2} f^{2} e^{6 i e} - 1024 a^{6} c d f^{2} e^{6 i e} - 512 i a^{6} d^{2} f^{2} e^{6 i e}\right ) e^{- 6 i f x} + \left (2304 i a^{6} c^{2} f^{2} e^{8 i e} - 1536 a^{6} c d f^{2} e^{8 i e} + 768 i a^{6} d^{2} f^{2} e^{8 i e}\right ) e^{- 4 i f x} + \left (4608 i a^{6} c^{2} f^{2} e^{10 i e} + 3072 a^{6} c d f^{2} e^{10 i e} + 1536 i a^{6} d^{2} f^{2} e^{10 i e}\right ) e^{- 2 i f x}\right ) e^{- 12 i e}}{24576 a^{9} f^{3}} & \text {for}\: a^{9} f^{3} e^{12 i e} \neq 0 \\x \left (- \frac {c^{2} - 2 i c d - d^{2}}{8 a^{3}} + \frac {\left (c^{2} e^{6 i e} + 3 c^{2} e^{4 i e} + 3 c^{2} e^{2 i e} + c^{2} - 2 i c d e^{6 i e} - 2 i c d e^{4 i e} + 2 i c d e^{2 i e} + 2 i c d - d^{2} e^{6 i e} + d^{2} e^{4 i e} + d^{2} e^{2 i e} - d^{2}\right ) e^{- 6 i e}}{8 a^{3}}\right ) & \text {otherwise} \end {cases} + \frac {x \left (c^{2} - 2 i c d - d^{2}\right )}{8 a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))**2/(a+I*a*tan(f*x+e))**3,x)

[Out]

Piecewise((((512*I*a**6*c**2*f**2*exp(6*I*e) - 1024*a**6*c*d*f**2*exp(6*I*e) - 512*I*a**6*d**2*f**2*exp(6*I*e)
)*exp(-6*I*f*x) + (2304*I*a**6*c**2*f**2*exp(8*I*e) - 1536*a**6*c*d*f**2*exp(8*I*e) + 768*I*a**6*d**2*f**2*exp
(8*I*e))*exp(-4*I*f*x) + (4608*I*a**6*c**2*f**2*exp(10*I*e) + 3072*a**6*c*d*f**2*exp(10*I*e) + 1536*I*a**6*d**
2*f**2*exp(10*I*e))*exp(-2*I*f*x))*exp(-12*I*e)/(24576*a**9*f**3), Ne(a**9*f**3*exp(12*I*e), 0)), (x*(-(c**2 -
2*I*c*d - d**2)/(8*a**3) + (c**2*exp(6*I*e) + 3*c**2*exp(4*I*e) + 3*c**2*exp(2*I*e) + c**2 - 2*I*c*d*exp(6*I*
e) - 2*I*c*d*exp(4*I*e) + 2*I*c*d*exp(2*I*e) + 2*I*c*d - d**2*exp(6*I*e) + d**2*exp(4*I*e) + d**2*exp(2*I*e) -
d**2)*exp(-6*I*e)/(8*a**3)), True)) + x*(c**2 - 2*I*c*d - d**2)/(8*a**3)

________________________________________________________________________________________

Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (104) = 208\).
time = 0.78, size = 215, normalized size = 1.67 \begin {gather*} -\frac {\frac {6 \, {\left (i \, c^{2} + 2 \, c d - i \, d^{2}\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a^{3}} + \frac {6 \, {\left (-i \, c^{2} - 2 \, c d + i \, d^{2}\right )} \log \left (i \, \tan \left (f x + e\right ) - 1\right )}{a^{3}} + \frac {-11 i \, c^{2} \tan \left (f x + e\right )^{3} - 22 \, c d \tan \left (f x + e\right )^{3} + 11 i \, d^{2} \tan \left (f x + e\right )^{3} - 45 \, c^{2} \tan \left (f x + e\right )^{2} + 90 i \, c d \tan \left (f x + e\right )^{2} + 45 \, d^{2} \tan \left (f x + e\right )^{2} + 69 i \, c^{2} \tan \left (f x + e\right ) + 138 \, c d \tan \left (f x + e\right ) - 21 i \, d^{2} \tan \left (f x + e\right ) + 51 \, c^{2} - 38 i \, c d - 3 \, d^{2}}{a^{3} {\left (\tan \left (f x + e\right ) - i\right )}^{3}}}{96 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^2/(a+I*a*tan(f*x+e))^3,x, algorithm="giac")

[Out]

-1/96*(6*(I*c^2 + 2*c*d - I*d^2)*log(tan(f*x + e) - I)/a^3 + 6*(-I*c^2 - 2*c*d + I*d^2)*log(I*tan(f*x + e) - 1
)/a^3 + (-11*I*c^2*tan(f*x + e)^3 - 22*c*d*tan(f*x + e)^3 + 11*I*d^2*tan(f*x + e)^3 - 45*c^2*tan(f*x + e)^2 +
90*I*c*d*tan(f*x + e)^2 + 45*d^2*tan(f*x + e)^2 + 69*I*c^2*tan(f*x + e) + 138*c*d*tan(f*x + e) - 21*I*d^2*tan(
f*x + e) + 51*c^2 - 38*I*c*d - 3*d^2)/(a^3*(tan(f*x + e) - I)^3))/f

________________________________________________________________________________________

Mupad [B]
time = 5.44, size = 148, normalized size = 1.15 \begin {gather*} \frac {\frac {c\,d}{6\,a^3}-\mathrm {tan}\left (e+f\,x\right )\,\left (\frac {3\,c^2}{8\,a^3}+\frac {d^2}{8\,a^3}-\frac {c\,d\,3{}\mathrm {i}}{4\,a^3}\right )-{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {c\,d}{4\,a^3}+\frac {c^2\,1{}\mathrm {i}}{8\,a^3}-\frac {d^2\,1{}\mathrm {i}}{8\,a^3}\right )+\frac {c^2\,5{}\mathrm {i}}{12\,a^3}+\frac {d^2\,1{}\mathrm {i}}{12\,a^3}}{f\,\left (-{\mathrm {tan}\left (e+f\,x\right )}^3\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,3{}\mathrm {i}+1\right )}-\frac {x\,{\left (d+c\,1{}\mathrm {i}\right )}^2}{8\,a^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*tan(e + f*x))^2/(a + a*tan(e + f*x)*1i)^3,x)

[Out]

((c^2*5i)/(12*a^3) - tan(e + f*x)*((3*c^2)/(8*a^3) + d^2/(8*a^3) - (c*d*3i)/(4*a^3)) - tan(e + f*x)^2*((c^2*1i
)/(8*a^3) - (d^2*1i)/(8*a^3) + (c*d)/(4*a^3)) + (d^2*1i)/(12*a^3) + (c*d)/(6*a^3))/(f*(tan(e + f*x)*3i - 3*tan
(e + f*x)^2 - tan(e + f*x)^3*1i + 1)) - (x*(c*1i + d)^2)/(8*a^3)

________________________________________________________________________________________