3.11.65 \(\int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x)) \, dx\) [1065]

Optimal. Leaf size=110 \[ 4 a^3 (c-i d) x-\frac {4 a^3 (i c+d) \log (\cos (e+f x))}{f}-\frac {2 a^3 (c-i d) \tan (e+f x)}{f}+\frac {a (i c+d) (a+i a \tan (e+f x))^2}{2 f}+\frac {d (a+i a \tan (e+f x))^3}{3 f} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3608, 3559, 3558, 3556} \begin {gather*} -\frac {2 a^3 (c-i d) \tan (e+f x)}{f}-\frac {4 a^3 (d+i c) \log (\cos (e+f x))}{f}+4 a^3 x (c-i d)+\frac {a (d+i c) (a+i a \tan (e+f x))^2}{2 f}+\frac {d (a+i a \tan (e+f x))^3}{3 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4*a^3*(c - I*d)*x - (4*a^3*(I*c + d)*Log[Cos[e + f*x]])/f - (2*a^3*(c - I*d)*Tan[e + f*x])/f + (a*(I*c + d)*(a
+ I*a*Tan[e + f*x])^2)/(2*f) + (d*(a + I*a*Tan[e + f*x])^3)/(3*f)

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3558

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

Rule 3559

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1))
), x] + Dist[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] && G
tQ[n, 1]

Rule 3608

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

Rubi steps

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

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(331\) vs. \(2(110)=220\).
time = 3.40, size = 331, normalized size = 3.01 \begin {gather*} \frac {a^3 \sec (e) \sec ^3(e+f x) \left (6 c f x \cos (2 e+3 f x)-6 i d f x \cos (2 e+3 f x)+6 c f x \cos (4 e+3 f x)-6 i d f x \cos (4 e+3 f x)-3 i c \cos (2 e+3 f x) \log \left (\cos ^2(e+f x)\right )-3 d \cos (2 e+3 f x) \log \left (\cos ^2(e+f x)\right )-3 i c \cos (4 e+3 f x) \log \left (\cos ^2(e+f x)\right )-3 d \cos (4 e+3 f x) \log \left (\cos ^2(e+f x)\right )+3 \cos (f x) \left (-i c-3 d+6 c f x-6 i d f x+(-3 i c-3 d) \log \left (\cos ^2(e+f x)\right )\right )+3 \cos (2 e+f x) \left (-i c-3 d+6 c f x-6 i d f x+(-3 i c-3 d) \log \left (\cos ^2(e+f x)\right )\right )-18 c \sin (f x)+24 i d \sin (f x)+9 c \sin (2 e+f x)-15 i d \sin (2 e+f x)-9 c \sin (2 e+3 f x)+13 i d \sin (2 e+3 f x)\right )}{12 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*Sec[e]*Sec[e + f*x]^3*(6*c*f*x*Cos[2*e + 3*f*x] - (6*I)*d*f*x*Cos[2*e + 3*f*x] + 6*c*f*x*Cos[4*e + 3*f*x]
- (6*I)*d*f*x*Cos[4*e + 3*f*x] - (3*I)*c*Cos[2*e + 3*f*x]*Log[Cos[e + f*x]^2] - 3*d*Cos[2*e + 3*f*x]*Log[Cos[
e + f*x]^2] - (3*I)*c*Cos[4*e + 3*f*x]*Log[Cos[e + f*x]^2] - 3*d*Cos[4*e + 3*f*x]*Log[Cos[e + f*x]^2] + 3*Cos[
f*x]*((-I)*c - 3*d + 6*c*f*x - (6*I)*d*f*x + ((-3*I)*c - 3*d)*Log[Cos[e + f*x]^2]) + 3*Cos[2*e + f*x]*((-I)*c
- 3*d + 6*c*f*x - (6*I)*d*f*x + ((-3*I)*c - 3*d)*Log[Cos[e + f*x]^2]) - 18*c*Sin[f*x] + (24*I)*d*Sin[f*x] + 9*
c*Sin[2*e + f*x] - (15*I)*d*Sin[2*e + f*x] - 9*c*Sin[2*e + 3*f*x] + (13*I)*d*Sin[2*e + 3*f*x]))/(12*f)

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Maple [A]
time = 0.12, size = 100, normalized size = 0.91

method	result	size
derivativedivides	\(\frac {a^{3} \left (-\frac {i d \left (\tan ^{3}\left (f x +e \right )\right )}{3}-\frac {i c \left (\tan ^{2}\left (f x +e \right )\right )}{2}+4 i \tan \left (f x +e \right ) d -\frac {3 d \left (\tan ^{2}\left (f x +e \right )\right )}{2}-3 c \tan \left (f x +e \right )+\frac {\left (4 i c +4 d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (-4 i d +4 c \right ) \arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\)	\(100\)
default	\(\frac {a^{3} \left (-\frac {i d \left (\tan ^{3}\left (f x +e \right )\right )}{3}-\frac {i c \left (\tan ^{2}\left (f x +e \right )\right )}{2}+4 i \tan \left (f x +e \right ) d -\frac {3 d \left (\tan ^{2}\left (f x +e \right )\right )}{2}-3 c \tan \left (f x +e \right )+\frac {\left (4 i c +4 d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (-4 i d +4 c \right ) \arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\)	\(100\)
norman	\(\left (-4 i a^{3} d +4 a^{3} c \right ) x -\frac {\left (i a^{3} c +3 a^{3} d \right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2 f}-\frac {\left (-4 i a^{3} d +3 a^{3} c \right ) \tan \left (f x +e \right )}{f}-\frac {i a^{3} d \left (\tan ^{3}\left (f x +e \right )\right )}{3 f}+\frac {2 \left (i a^{3} c +a^{3} d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{f}\)	\(117\)
risch	\(\frac {8 i a^{3} d e}{f}-\frac {8 a^{3} c e}{f}-\frac {2 a^{3} \left (12 i c \,{\mathrm e}^{4 i \left (f x +e \right )}+24 d \,{\mathrm e}^{4 i \left (f x +e \right )}+21 i c \,{\mathrm e}^{2 i \left (f x +e \right )}+33 \,{\mathrm e}^{2 i \left (f x +e \right )} d +9 i c +13 d \right )}{3 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{3}}-\frac {4 a^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) d}{f}-\frac {4 i a^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) c}{f}\)	\(145\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.51, size = 114, normalized size = 1.04 \begin {gather*} -\frac {2 i \, a^{3} d \tan \left (f x + e\right )^{3} + 3 \, {\left (i \, a^{3} c + 3 \, a^{3} d\right )} \tan \left (f x + e\right )^{2} - 24 \, {\left (a^{3} c - i \, a^{3} d\right )} {\left (f x + e\right )} + 12 \, {\left (-i \, a^{3} c - a^{3} d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + 6 \, {\left (3 \, a^{3} c - 4 i \, a^{3} d\right )} \tan \left (f x + e\right )}{6 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*I*a^3*d*tan(f*x + e)^3 + 3*(I*a^3*c + 3*a^3*d)*tan(f*x + e)^2 - 24*(a^3*c - I*a^3*d)*(f*x + e) + 12*(-
I*a^3*c - a^3*d)*log(tan(f*x + e)^2 + 1) + 6*(3*a^3*c - 4*I*a^3*d)*tan(f*x + e))/f

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (98) = 196\).
time = 0.96, size = 205, normalized size = 1.86 \begin {gather*} -\frac {2 \, {\left (9 i \, a^{3} c + 13 \, a^{3} d + 12 \, {\left (i \, a^{3} c + 2 \, a^{3} d\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, {\left (7 i \, a^{3} c + 11 \, a^{3} d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 6 \, {\left (i \, a^{3} c + a^{3} d + {\left (i \, a^{3} c + a^{3} d\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, {\left (i \, a^{3} c + a^{3} d\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, {\left (i \, a^{3} c + a^{3} d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )\right )}}{3 \, {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(9*I*a^3*c + 13*a^3*d + 12*(I*a^3*c + 2*a^3*d)*e^(4*I*f*x + 4*I*e) + 3*(7*I*a^3*c + 11*a^3*d)*e^(2*I*f*x
+ 2*I*e) + 6*(I*a^3*c + a^3*d + (I*a^3*c + a^3*d)*e^(6*I*f*x + 6*I*e) + 3*(I*a^3*c + a^3*d)*e^(4*I*f*x + 4*I*e
) + 3*(I*a^3*c + a^3*d)*e^(2*I*f*x + 2*I*e))*log(e^(2*I*f*x + 2*I*e) + 1))/(f*e^(6*I*f*x + 6*I*e) + 3*f*e^(4*I
*f*x + 4*I*e) + 3*f*e^(2*I*f*x + 2*I*e) + f)

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Sympy [A]
time = 0.39, size = 184, normalized size = 1.67 \begin {gather*} - \frac {4 i a^{3} \left (c - i d\right ) \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{f} + \frac {- 18 i a^{3} c - 26 a^{3} d + \left (- 42 i a^{3} c e^{2 i e} - 66 a^{3} d e^{2 i e}\right ) e^{2 i f x} + \left (- 24 i a^{3} c e^{4 i e} - 48 a^{3} d e^{4 i e}\right ) e^{4 i f x}}{3 f e^{6 i e} e^{6 i f x} + 9 f e^{4 i e} e^{4 i f x} + 9 f e^{2 i e} e^{2 i f x} + 3 f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*I*a**3*(c - I*d)*log(exp(2*I*f*x) + exp(-2*I*e))/f + (-18*I*a**3*c - 26*a**3*d + (-42*I*a**3*c*exp(2*I*e) -
66*a**3*d*exp(2*I*e))*exp(2*I*f*x) + (-24*I*a**3*c*exp(4*I*e) - 48*a**3*d*exp(4*I*e))*exp(4*I*f*x))/(3*f*exp(
6*I*e)*exp(6*I*f*x) + 9*f*exp(4*I*e)*exp(4*I*f*x) + 9*f*exp(2*I*e)*exp(2*I*f*x) + 3*f)

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (98) = 196\).
time = 0.60, size = 333, normalized size = 3.03 \begin {gather*} -\frac {2 \, {\left (6 i \, a^{3} c e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 6 \, a^{3} d e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 18 i \, a^{3} c e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 18 \, a^{3} d e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 18 i \, a^{3} c e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 18 \, a^{3} d e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 12 i \, a^{3} c e^{\left (4 i \, f x + 4 i \, e\right )} + 24 \, a^{3} d e^{\left (4 i \, f x + 4 i \, e\right )} + 21 i \, a^{3} c e^{\left (2 i \, f x + 2 i \, e\right )} + 33 \, a^{3} d e^{\left (2 i \, f x + 2 i \, e\right )} + 6 i \, a^{3} c \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 6 \, a^{3} d \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 9 i \, a^{3} c + 13 \, a^{3} d\right )}}{3 \, {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(6*I*a^3*c*e^(6*I*f*x + 6*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 6*a^3*d*e^(6*I*f*x + 6*I*e)*log(e^(2*I*f*x
+ 2*I*e) + 1) + 18*I*a^3*c*e^(4*I*f*x + 4*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 18*a^3*d*e^(4*I*f*x + 4*I*e)*log
(e^(2*I*f*x + 2*I*e) + 1) + 18*I*a^3*c*e^(2*I*f*x + 2*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 18*a^3*d*e^(2*I*f*x
+ 2*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 12*I*a^3*c*e^(4*I*f*x + 4*I*e) + 24*a^3*d*e^(4*I*f*x + 4*I*e) + 21*I*a
^3*c*e^(2*I*f*x + 2*I*e) + 33*a^3*d*e^(2*I*f*x + 2*I*e) + 6*I*a^3*c*log(e^(2*I*f*x + 2*I*e) + 1) + 6*a^3*d*log
(e^(2*I*f*x + 2*I*e) + 1) + 9*I*a^3*c + 13*a^3*d)/(f*e^(6*I*f*x + 6*I*e) + 3*f*e^(4*I*f*x + 4*I*e) + 3*f*e^(2*
I*f*x + 2*I*e) + f)

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Mupad [B]
time = 5.42, size = 125, normalized size = 1.14 \begin {gather*} \frac {\mathrm {tan}\left (e+f\,x\right )\,\left (-a^3\,\left (2\,c-d\,1{}\mathrm {i}\right )+a^3\,\left (2\,d+c\,1{}\mathrm {i}\right )\,1{}\mathrm {i}+a^3\,d\,1{}\mathrm {i}\right )}{f}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (4\,a^3\,d+a^3\,c\,4{}\mathrm {i}\right )}{f}-\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {a^3\,\left (2\,d+c\,1{}\mathrm {i}\right )}{2}+\frac {a^3\,d}{2}\right )}{f}-\frac {a^3\,d\,{\mathrm {tan}\left (e+f\,x\right )}^3\,1{}\mathrm {i}}{3\,f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(tan(e + f*x)*(a^3*(c*1i + 2*d)*1i - a^3*(2*c - d*1i) + a^3*d*1i))/f + (log(tan(e + f*x) + 1i)*(a^3*c*4i + 4*a
^3*d))/f - (tan(e + f*x)^2*((a^3*(c*1i + 2*d))/2 + (a^3*d)/2))/f - (a^3*d*tan(e + f*x)^3*1i)/(3*f)

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