3.1.35 \(\int \tan ^2(c+d x) (a+i a \tan (c+d x))^4 \, dx\) [35]

Optimal. Leaf size=116 \[ -8 a^4 x+\frac {8 i a^4 \log (\cos (c+d x))}{d}+\frac {4 a^4 \tan (c+d x)}{d}-\frac {i a (a+i a \tan (c+d x))^3}{3 d}-\frac {i (a+i a \tan (c+d x))^5}{5 a d}-\frac {i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d} \]

[Out]

-8*a^4*x+8*I*a^4*ln(cos(d*x+c))/d+4*a^4*tan(d*x+c)/d-1/3*I*a*(a+I*a*tan(d*x+c))^3/d-1/5*I*(a+I*a*tan(d*x+c))^5
/a/d-I*(a^2+I*a^2*tan(d*x+c))^2/d

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Rubi [A]
time = 0.06, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3624, 3559, 3558, 3556} \begin {gather*} \frac {4 a^4 \tan (c+d x)}{d}+\frac {8 i a^4 \log (\cos (c+d x))}{d}-8 a^4 x-\frac {i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\frac {i (a+i a \tan (c+d x))^5}{5 a d}-\frac {i a (a+i a \tan (c+d x))^3}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^4,x]

[Out]

-8*a^4*x + ((8*I)*a^4*Log[Cos[c + d*x]])/d + (4*a^4*Tan[c + d*x])/d - ((I/3)*a*(a + I*a*Tan[c + d*x])^3)/d - (
(I/5)*(a + I*a*Tan[c + d*x])^5)/(a*d) - (I*(a^2 + I*a^2*Tan[c + d*x])^2)/d

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 3624

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

Rubi steps

\begin {align*} \int \tan ^2(c+d x) (a+i a \tan (c+d x))^4 \, dx &=-\frac {i (a+i a \tan (c+d x))^5}{5 a d}-\int (a+i a \tan (c+d x))^4 \, dx\\ &=-\frac {i a (a+i a \tan (c+d x))^3}{3 d}-\frac {i (a+i a \tan (c+d x))^5}{5 a d}-(2 a) \int (a+i a \tan (c+d x))^3 \, dx\\ &=-\frac {i a (a+i a \tan (c+d x))^3}{3 d}-\frac {i (a+i a \tan (c+d x))^5}{5 a d}-\frac {i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\left (4 a^2\right ) \int (a+i a \tan (c+d x))^2 \, dx\\ &=-8 a^4 x+\frac {4 a^4 \tan (c+d x)}{d}-\frac {i a (a+i a \tan (c+d x))^3}{3 d}-\frac {i (a+i a \tan (c+d x))^5}{5 a d}-\frac {i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\left (8 i a^4\right ) \int \tan (c+d x) \, dx\\ &=-8 a^4 x+\frac {8 i a^4 \log (\cos (c+d x))}{d}+\frac {4 a^4 \tan (c+d x)}{d}-\frac {i a (a+i a \tan (c+d x))^3}{3 d}-\frac {i (a+i a \tan (c+d x))^5}{5 a d}-\frac {i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}\\ \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. \(294\) vs. \(2(116)=232\).
time = 2.18, size = 294, normalized size = 2.53 \begin {gather*} -\frac {a^4 \sec (c) \sec ^5(c+d x) \left (-90 i \cos (2 c+3 d x)+150 d x \cos (2 c+3 d x)-90 i \cos (4 c+3 d x)+150 d x \cos (4 c+3 d x)+30 d x \cos (4 c+5 d x)+30 d x \cos (6 c+5 d x)+30 \cos (d x) \left (-7 i+10 d x-5 i \log \left (\cos ^2(c+d x)\right )\right )+30 \cos (2 c+d x) \left (-7 i+10 d x-5 i \log \left (\cos ^2(c+d x)\right )\right )-75 i \cos (2 c+3 d x) \log \left (\cos ^2(c+d x)\right )-75 i \cos (4 c+3 d x) \log \left (\cos ^2(c+d x)\right )-15 i \cos (4 c+5 d x) \log \left (\cos ^2(c+d x)\right )-15 i \cos (6 c+5 d x) \log \left (\cos ^2(c+d x)\right )-445 \sin (d x)+345 \sin (2 c+d x)-275 \sin (2 c+3 d x)+120 \sin (4 c+3 d x)-79 \sin (4 c+5 d x)\right )}{120 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^4,x]

[Out]

-1/120*(a^4*Sec[c]*Sec[c + d*x]^5*((-90*I)*Cos[2*c + 3*d*x] + 150*d*x*Cos[2*c + 3*d*x] - (90*I)*Cos[4*c + 3*d*
x] + 150*d*x*Cos[4*c + 3*d*x] + 30*d*x*Cos[4*c + 5*d*x] + 30*d*x*Cos[6*c + 5*d*x] + 30*Cos[d*x]*(-7*I + 10*d*x
- (5*I)*Log[Cos[c + d*x]^2]) + 30*Cos[2*c + d*x]*(-7*I + 10*d*x - (5*I)*Log[Cos[c + d*x]^2]) - (75*I)*Cos[2*c
+ 3*d*x]*Log[Cos[c + d*x]^2] - (75*I)*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]^2] - (15*I)*Cos[4*c + 5*d*x]*Log[Cos[
c + d*x]^2] - (15*I)*Cos[6*c + 5*d*x]*Log[Cos[c + d*x]^2] - 445*Sin[d*x] + 345*Sin[2*c + d*x] - 275*Sin[2*c +
3*d*x] + 120*Sin[4*c + 3*d*x] - 79*Sin[4*c + 5*d*x]))/d

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Maple [A]
time = 0.05, size = 82, normalized size = 0.71

method	result	size
derivativedivides	\(\frac {a^{4} \left (8 \tan \left (d x +c \right )+\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}-i \left (\tan ^{4}\left (d x +c \right )\right )-\frac {7 \left (\tan ^{3}\left (d x +c \right )\right )}{3}+4 i \left (\tan ^{2}\left (d x +c \right )\right )-4 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )-8 \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\)	\(82\)
default	\(\frac {a^{4} \left (8 \tan \left (d x +c \right )+\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}-i \left (\tan ^{4}\left (d x +c \right )\right )-\frac {7 \left (\tan ^{3}\left (d x +c \right )\right )}{3}+4 i \left (\tan ^{2}\left (d x +c \right )\right )-4 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )-8 \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\)	\(82\)
risch	\(\frac {16 a^{4} c}{d}+\frac {4 i a^{4} \left (210 \,{\mathrm e}^{8 i \left (d x +c \right )}+555 \,{\mathrm e}^{6 i \left (d x +c \right )}+655 \,{\mathrm e}^{4 i \left (d x +c \right )}+365 \,{\mathrm e}^{2 i \left (d x +c \right )}+79\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}+\frac {8 i a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\)	\(100\)
norman	\(-8 a^{4} x +\frac {8 a^{4} \tan \left (d x +c \right )}{d}-\frac {7 a^{4} \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a^{4} \left (\tan ^{5}\left (d x +c \right )\right )}{5 d}+\frac {4 i a^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{d}-\frac {i a^{4} \left (\tan ^{4}\left (d x +c \right )\right )}{d}-\frac {4 i a^{4} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^2*(a+I*a*tan(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*a^4*(8*tan(d*x+c)+1/5*tan(d*x+c)^5-I*tan(d*x+c)^4-7/3*tan(d*x+c)^3+4*I*tan(d*x+c)^2-4*I*ln(1+tan(d*x+c)^2)
-8*arctan(tan(d*x+c)))

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Maxima [A]
time = 0.62, size = 95, normalized size = 0.82 \begin {gather*} \frac {3 \, a^{4} \tan \left (d x + c\right )^{5} - 15 i \, a^{4} \tan \left (d x + c\right )^{4} - 35 \, a^{4} \tan \left (d x + c\right )^{3} + 60 i \, a^{4} \tan \left (d x + c\right )^{2} - 120 \, {\left (d x + c\right )} a^{4} - 60 i \, a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 120 \, a^{4} \tan \left (d x + c\right )}{15 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^2*(a+I*a*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

1/15*(3*a^4*tan(d*x + c)^5 - 15*I*a^4*tan(d*x + c)^4 - 35*a^4*tan(d*x + c)^3 + 60*I*a^4*tan(d*x + c)^2 - 120*(
d*x + c)*a^4 - 60*I*a^4*log(tan(d*x + c)^2 + 1) + 120*a^4*tan(d*x + c))/d

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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. 217 vs. \(2 (98) = 196\).
time = 0.42, size = 217, normalized size = 1.87 \begin {gather*} -\frac {4 \, {\left (-210 i \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} - 555 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 655 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 365 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 79 i \, a^{4} + 30 \, {\left (-i \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} - 5 i \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} - 10 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 10 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 5 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{15 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^2*(a+I*a*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

-4/15*(-210*I*a^4*e^(8*I*d*x + 8*I*c) - 555*I*a^4*e^(6*I*d*x + 6*I*c) - 655*I*a^4*e^(4*I*d*x + 4*I*c) - 365*I*
a^4*e^(2*I*d*x + 2*I*c) - 79*I*a^4 + 30*(-I*a^4*e^(10*I*d*x + 10*I*c) - 5*I*a^4*e^(8*I*d*x + 8*I*c) - 10*I*a^4
*e^(6*I*d*x + 6*I*c) - 10*I*a^4*e^(4*I*d*x + 4*I*c) - 5*I*a^4*e^(2*I*d*x + 2*I*c) - I*a^4)*log(e^(2*I*d*x + 2*
I*c) + 1))/(d*e^(10*I*d*x + 10*I*c) + 5*d*e^(8*I*d*x + 8*I*c) + 10*d*e^(6*I*d*x + 6*I*c) + 10*d*e^(4*I*d*x + 4
*I*c) + 5*d*e^(2*I*d*x + 2*I*c) + d)

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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (97) = 194\).
time = 0.33, size = 218, normalized size = 1.88 \begin {gather*} \frac {8 i a^{4} \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {840 i a^{4} e^{8 i c} e^{8 i d x} + 2220 i a^{4} e^{6 i c} e^{6 i d x} + 2620 i a^{4} e^{4 i c} e^{4 i d x} + 1460 i a^{4} e^{2 i c} e^{2 i d x} + 316 i a^{4}}{15 d e^{10 i c} e^{10 i d x} + 75 d e^{8 i c} e^{8 i d x} + 150 d e^{6 i c} e^{6 i d x} + 150 d e^{4 i c} e^{4 i d x} + 75 d e^{2 i c} e^{2 i d x} + 15 d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**2*(a+I*a*tan(d*x+c))**4,x)

[Out]

8*I*a**4*log(exp(2*I*d*x) + exp(-2*I*c))/d + (840*I*a**4*exp(8*I*c)*exp(8*I*d*x) + 2220*I*a**4*exp(6*I*c)*exp(
6*I*d*x) + 2620*I*a**4*exp(4*I*c)*exp(4*I*d*x) + 1460*I*a**4*exp(2*I*c)*exp(2*I*d*x) + 316*I*a**4)/(15*d*exp(1
0*I*c)*exp(10*I*d*x) + 75*d*exp(8*I*c)*exp(8*I*d*x) + 150*d*exp(6*I*c)*exp(6*I*d*x) + 150*d*exp(4*I*c)*exp(4*I
*d*x) + 75*d*exp(2*I*c)*exp(2*I*d*x) + 15*d)

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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. 274 vs. \(2 (98) = 196\).
time = 0.85, size = 274, normalized size = 2.36 \begin {gather*} -\frac {4 \, {\left (-30 i \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 150 i \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 300 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 300 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 150 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 210 i \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} - 555 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 655 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 365 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 30 i \, a^{4} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 79 i \, a^{4}\right )}}{15 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^2*(a+I*a*tan(d*x+c))^4,x, algorithm="giac")

[Out]

-4/15*(-30*I*a^4*e^(10*I*d*x + 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 150*I*a^4*e^(8*I*d*x + 8*I*c)*log(e^(2*I
*d*x + 2*I*c) + 1) - 300*I*a^4*e^(6*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 300*I*a^4*e^(4*I*d*x + 4*I*c
)*log(e^(2*I*d*x + 2*I*c) + 1) - 150*I*a^4*e^(2*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 210*I*a^4*e^(8*I
*d*x + 8*I*c) - 555*I*a^4*e^(6*I*d*x + 6*I*c) - 655*I*a^4*e^(4*I*d*x + 4*I*c) - 365*I*a^4*e^(2*I*d*x + 2*I*c)
- 30*I*a^4*log(e^(2*I*d*x + 2*I*c) + 1) - 79*I*a^4)/(d*e^(10*I*d*x + 10*I*c) + 5*d*e^(8*I*d*x + 8*I*c) + 10*d*
e^(6*I*d*x + 6*I*c) + 10*d*e^(4*I*d*x + 4*I*c) + 5*d*e^(2*I*d*x + 2*I*c) + d)

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Mupad [B]
time = 3.73, size = 87, normalized size = 0.75 \begin {gather*} -\frac {\frac {7\,a^4\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}-8\,a^4\,\mathrm {tan}\left (c+d\,x\right )-\frac {a^4\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5}+a^4\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,8{}\mathrm {i}-a^4\,{\mathrm {tan}\left (c+d\,x\right )}^2\,4{}\mathrm {i}+a^4\,{\mathrm {tan}\left (c+d\,x\right )}^4\,1{}\mathrm {i}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^2*(a + a*tan(c + d*x)*1i)^4,x)

[Out]

-(a^4*log(tan(c + d*x) + 1i)*8i - 8*a^4*tan(c + d*x) - a^4*tan(c + d*x)^2*4i + (7*a^4*tan(c + d*x)^3)/3 + a^4*
tan(c + d*x)^4*1i - (a^4*tan(c + d*x)^5)/5)/d

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