3.53 \(\int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=124 \[ \frac{17 B+i A}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{x (A-i B)}{8 a^3}+\frac{(-B+i A) \tan ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{-7 B+i A}{24 a d (a+i a \tan (c+d x))^2} \]

[Out]

-((A - I*B)*x)/(8*a^3) + ((I*A - B)*Tan[c + d*x]^2)/(6*d*(a + I*a*Tan[c + d*x])^3) + (I*A - 7*B)/(24*a*d*(a +
I*a*Tan[c + d*x])^2) + (I*A + 17*B)/(24*d*(a^3 + I*a^3*Tan[c + d*x]))

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Rubi [A]  time = 0.288754, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3595, 3590, 3526, 8} \[ \frac{17 B+i A}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{x (A-i B)}{8 a^3}+\frac{(-B+i A) \tan ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{-7 B+i A}{24 a d (a+i a \tan (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((A - I*B)*x)/(8*a^3) + ((I*A - B)*Tan[c + d*x]^2)/(6*d*(a + I*a*Tan[c + d*x])^3) + (I*A - 7*B)/(24*a*d*(a +
I*a*Tan[c + d*x])^2) + (I*A + 17*B)/(24*d*(a^3 + I*a^3*Tan[c + d*x]))

Rule 3595

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

Rule 3590

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

Rule 3526

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 8

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

Rubi steps

\begin{align*} \int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx &=\frac{(i A-B) \tan ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}-\frac{\int \frac{\tan (c+d x) (2 a (i A-B)-a (A-5 i B) \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx}{6 a^2}\\ &=\frac{(i A-B) \tan ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{i A-7 B}{24 a d (a+i a \tan (c+d x))^2}+\frac{i \int \frac{a^2 (i A-7 B)-2 a^2 (A-5 i B) \tan (c+d x)}{a+i a \tan (c+d x)} \, dx}{12 a^4}\\ &=\frac{(i A-B) \tan ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{i A-7 B}{24 a d (a+i a \tan (c+d x))^2}+\frac{i A+17 B}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{(A-i B) \int 1 \, dx}{8 a^3}\\ &=-\frac{(A-i B) x}{8 a^3}+\frac{(i A-B) \tan ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{i A-7 B}{24 a d (a+i a \tan (c+d x))^2}+\frac{i A+17 B}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 1.07636, size = 147, normalized size = 1.19 \[ \frac{\sec ^3(c+d x) (-9 (A-i B) \cos (c+d x)+2 (-6 i A d x+A-6 B d x+i B) \cos (3 (c+d x))-3 i A \sin (c+d x)-2 i A \sin (3 (c+d x))+12 A d x \sin (3 (c+d x))-27 B \sin (c+d x)+2 B \sin (3 (c+d x))-12 i B d x \sin (3 (c+d x)))}{96 a^3 d (\tan (c+d x)-i)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[c + d*x]^3*(-9*(A - I*B)*Cos[c + d*x] + 2*(A + I*B - (6*I)*A*d*x - 6*B*d*x)*Cos[3*(c + d*x)] - (3*I)*A*Si
n[c + d*x] - 27*B*Sin[c + d*x] - (2*I)*A*Sin[3*(c + d*x)] + 2*B*Sin[3*(c + d*x)] + 12*A*d*x*Sin[3*(c + d*x)] -
(12*I)*B*d*x*Sin[3*(c + d*x)]))/(96*a^3*d*(-I + Tan[c + d*x])^3)

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Maple [A]  time = 0.034, size = 203, normalized size = 1.6 \begin{align*}{\frac{A}{6\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}+{\frac{{\frac{i}{6}}B}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}-{\frac{A}{8\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) }}-{\frac{{\frac{7\,i}{8}}B}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{{\frac{i}{16}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) A}{{a}^{3}d}}+{\frac{\ln \left ( \tan \left ( dx+c \right ) -i \right ) B}{16\,{a}^{3}d}}-{\frac{{\frac{3\,i}{8}}A}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}+{\frac{5\,B}{8\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{B\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{16\,{a}^{3}d}}-{\frac{{\frac{i}{16}}A\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{{a}^{3}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^2*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^3,x)

[Out]

1/6/d/a^3/(tan(d*x+c)-I)^3*A+1/6*I/d/a^3/(tan(d*x+c)-I)^3*B-1/8/d/a^3/(tan(d*x+c)-I)*A-7/8*I/d/a^3/(tan(d*x+c)
-I)*B+1/16*I/d/a^3*ln(tan(d*x+c)-I)*A+1/16/d/a^3*ln(tan(d*x+c)-I)*B-3/8*I/d/a^3/(tan(d*x+c)-I)^2*A+5/8/d/a^3/(
tan(d*x+c)-I)^2*B-1/16/d/a^3*B*ln(tan(d*x+c)+I)-1/16*I/d/a^3*A*ln(tan(d*x+c)+I)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.38038, size = 219, normalized size = 1.77 \begin{align*} -\frac{{\left (12 \,{\left (A - i \, B\right )} d x e^{\left (6 i \, d x + 6 i \, c\right )} -{\left (6 i \, A + 18 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} -{\left (3 i \, A - 9 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i \, A - 2 \, B\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{96 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(12*(A - I*B)*d*x*e^(6*I*d*x + 6*I*c) - (6*I*A + 18*B)*e^(4*I*d*x + 4*I*c) - (3*I*A - 9*B)*e^(2*I*d*x +
2*I*c) + 2*I*A - 2*B)*e^(-6*I*d*x - 6*I*c)/(a^3*d)

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Sympy [A]  time = 5.45332, size = 260, normalized size = 2.1 \begin{align*} \begin{cases} \frac{\left (\left (- 512 i A a^{6} d^{2} e^{6 i c} + 512 B a^{6} d^{2} e^{6 i c}\right ) e^{- 6 i d x} + \left (768 i A a^{6} d^{2} e^{8 i c} - 2304 B a^{6} d^{2} e^{8 i c}\right ) e^{- 4 i d x} + \left (1536 i A a^{6} d^{2} e^{10 i c} + 4608 B a^{6} d^{2} e^{10 i c}\right ) e^{- 2 i d x}\right ) e^{- 12 i c}}{24576 a^{9} d^{3}} & \text{for}\: 24576 a^{9} d^{3} e^{12 i c} \neq 0 \\x \left (\frac{A - i B}{8 a^{3}} - \frac{\left (A e^{6 i c} - A e^{4 i c} - A e^{2 i c} + A - i B e^{6 i c} + 3 i B e^{4 i c} - 3 i B e^{2 i c} + i B\right ) e^{- 6 i c}}{8 a^{3}}\right ) & \text{otherwise} \end{cases} + \frac{x \left (- A + i B\right )}{8 a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**2*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))**3,x)

[Out]

Piecewise((((-512*I*A*a**6*d**2*exp(6*I*c) + 512*B*a**6*d**2*exp(6*I*c))*exp(-6*I*d*x) + (768*I*A*a**6*d**2*ex
p(8*I*c) - 2304*B*a**6*d**2*exp(8*I*c))*exp(-4*I*d*x) + (1536*I*A*a**6*d**2*exp(10*I*c) + 4608*B*a**6*d**2*exp
(10*I*c))*exp(-2*I*d*x))*exp(-12*I*c)/(24576*a**9*d**3), Ne(24576*a**9*d**3*exp(12*I*c), 0)), (x*((A - I*B)/(8
*a**3) - (A*exp(6*I*c) - A*exp(4*I*c) - A*exp(2*I*c) + A - I*B*exp(6*I*c) + 3*I*B*exp(4*I*c) - 3*I*B*exp(2*I*c
) + I*B)*exp(-6*I*c)/(8*a**3)), True)) + x*(-A + I*B)/(8*a**3)

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Giac [A]  time = 1.58729, size = 177, normalized size = 1.43 \begin{align*} -\frac{\frac{6 \,{\left (-i \, A - B\right )} \log \left (\tan \left (d x + c\right ) - i\right )}{a^{3}} + \frac{6 \,{\left (i \, A + B\right )} \log \left (i \, \tan \left (d x + c\right ) - 1\right )}{a^{3}} + \frac{11 i \, A \tan \left (d x + c\right )^{3} + 11 \, B \tan \left (d x + c\right )^{3} + 45 \, A \tan \left (d x + c\right )^{2} + 51 i \, B \tan \left (d x + c\right )^{2} - 21 i \, A \tan \left (d x + c\right ) + 75 \, B \tan \left (d x + c\right ) - 3 \, A - 29 i \, B}{a^{3}{\left (\tan \left (d x + c\right ) - i\right )}^{3}}}{96 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(6*(-I*A - B)*log(tan(d*x + c) - I)/a^3 + 6*(I*A + B)*log(I*tan(d*x + c) - 1)/a^3 + (11*I*A*tan(d*x + c)
^3 + 11*B*tan(d*x + c)^3 + 45*A*tan(d*x + c)^2 + 51*I*B*tan(d*x + c)^2 - 21*I*A*tan(d*x + c) + 75*B*tan(d*x +
c) - 3*A - 29*I*B)/(a^3*(tan(d*x + c) - I)^3))/d