3.1.37 \(\int (a+i a \tan (c+d x))^4 \, dx\) [37]

Optimal. Leaf size=89 \[ 8 a^4 x-\frac {8 i a^4 \log (\cos (c+d x))}{d}-\frac {4 a^4 \tan (c+d x)}{d}+\frac {i a (a+i a \tan (c+d x))^3}{3 d}+\frac {i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d} \]

[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+I*(a^2+I*a^2*tan(d*x+c))^2/
d

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Rubi [A]
time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {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 (a+i a \tan (c+d x))^3}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(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
*(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]

Rubi steps

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

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Mathematica [A]
time = 0.91, size = 176, normalized size = 1.98 \begin {gather*} \frac {a^4 \sec (c) \sec ^3(c+d x) \left (6 d x \cos (2 c+3 d x)+6 d x \cos (4 c+3 d x)+3 \cos (d x) \left (-2 i+6 d x-3 i \log \left (\cos ^2(c+d x)\right )\right )+3 \cos (2 c+d x) \left (-2 i+6 d x-3 i \log \left (\cos ^2(c+d x)\right )\right )-3 i \cos (2 c+3 d x) \log \left (\cos ^2(c+d x)\right )-3 i \cos (4 c+3 d x) \log \left (\cos ^2(c+d x)\right )-21 \sin (d x)+12 \sin (2 c+d x)-11 \sin (2 c+3 d x)\right )}{6 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[c + d*x])^4,x]

[Out]

(a^4*Sec[c]*Sec[c + d*x]^3*(6*d*x*Cos[2*c + 3*d*x] + 6*d*x*Cos[4*c + 3*d*x] + 3*Cos[d*x]*(-2*I + 6*d*x - (3*I)
*Log[Cos[c + d*x]^2]) + 3*Cos[2*c + d*x]*(-2*I + 6*d*x - (3*I)*Log[Cos[c + d*x]^2]) - (3*I)*Cos[2*c + 3*d*x]*L
og[Cos[c + d*x]^2] - (3*I)*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]^2] - 21*Sin[d*x] + 12*Sin[2*c + d*x] - 11*Sin[2*c
+ 3*d*x]))/(6*d)

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Maple [A]
time = 0.04, size = 61, normalized size = 0.69

method	result	size
derivativedivides	\(\frac {a^{4} \left (-7 \tan \left (d x +c \right )+\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-2 i \left (\tan ^{2}\left (d x +c \right )\right )+4 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+8 \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\)	\(61\)
default	\(\frac {a^{4} \left (-7 \tan \left (d x +c \right )+\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-2 i \left (\tan ^{2}\left (d x +c \right )\right )+4 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+8 \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\)	\(61\)
norman	\(8 a^{4} x -\frac {7 a^{4} \tan \left (d x +c \right )}{d}+\frac {a^{4} \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {2 i a^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {4 i a^{4} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\)	\(75\)
risch	\(-\frac {16 a^{4} c}{d}-\frac {4 i a^{4} \left (18 \,{\mathrm e}^{4 i \left (d x +c \right )}+27 \,{\mathrm e}^{2 i \left (d x +c \right )}+11\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {8 i a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*a^4*(-7*tan(d*x+c)+1/3*tan(d*x+c)^3-2*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 = 0.57, size = 108, normalized size = 1.21 \begin {gather*} a^{4} x + \frac {{\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{4}}{3 \, d} + \frac {6 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{4}}{d} + \frac {2 i \, a^{4} {\left (\frac {1}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )}}{d} + \frac {4 i \, a^{4} \log \left (\sec \left (d x + c\right )\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^4*x + 1/3*(tan(d*x + c)^3 + 3*d*x + 3*c - 3*tan(d*x + c))*a^4/d + 6*(d*x + c - tan(d*x + c))*a^4/d + 2*I*a^4
*(1/(sin(d*x + c)^2 - 1) - log(sin(d*x + c)^2 - 1))/d + 4*I*a^4*log(sec(d*x + c))/d

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Fricas [A]
time = 0.43, size = 137, normalized size = 1.54 \begin {gather*} -\frac {4 \, {\left (18 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 27 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 11 i \, a^{4} + 6 \, {\left (i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 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 )}}{3 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, 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((a+I*a*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

-4/3*(18*I*a^4*e^(4*I*d*x + 4*I*c) + 27*I*a^4*e^(2*I*d*x + 2*I*c) + 11*I*a^4 + 6*(I*a^4*e^(6*I*d*x + 6*I*c) +
3*I*a^4*e^(4*I*d*x + 4*I*c) + 3*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^(6*I*d*x
+ 6*I*c) + 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) + d)

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Sympy [A]
time = 0.21, size = 138, normalized size = 1.55 \begin {gather*} - \frac {8 i a^{4} \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {- 72 i a^{4} e^{4 i c} e^{4 i d x} - 108 i a^{4} e^{2 i c} e^{2 i d x} - 44 i a^{4}}{3 d e^{6 i c} e^{6 i d x} + 9 d e^{4 i c} e^{4 i d x} + 9 d e^{2 i c} e^{2 i d x} + 3 d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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 + (-72*I*a**4*exp(4*I*c)*exp(4*I*d*x) - 108*I*a**4*exp(2*I*c)*exp(
2*I*d*x) - 44*I*a**4)/(3*d*exp(6*I*c)*exp(6*I*d*x) + 9*d*exp(4*I*c)*exp(4*I*d*x) + 9*d*exp(2*I*c)*exp(2*I*d*x)
+ 3*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. 170 vs. \(2 (77) = 154\).
time = 0.51, size = 170, normalized size = 1.91 \begin {gather*} -\frac {4 \, {\left (6 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 ) + 18 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 ) + 18 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 ) + 18 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 27 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 6 i \, a^{4} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 11 i \, a^{4}\right )}}{3 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, 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((a+I*a*tan(d*x+c))^4,x, algorithm="giac")

[Out]

-4/3*(6*I*a^4*e^(6*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 18*I*a^4*e^(4*I*d*x + 4*I*c)*log(e^(2*I*d*x +
2*I*c) + 1) + 18*I*a^4*e^(2*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 18*I*a^4*e^(4*I*d*x + 4*I*c) + 27*I
*a^4*e^(2*I*d*x + 2*I*c) + 6*I*a^4*log(e^(2*I*d*x + 2*I*c) + 1) + 11*I*a^4)/(d*e^(6*I*d*x + 6*I*c) + 3*d*e^(4*
I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) + d)

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Mupad [B]
time = 3.69, size = 59, normalized size = 0.66 \begin {gather*} \frac {\frac {a^4\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}-7\,a^4\,\mathrm {tan}\left (c+d\,x\right )+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\,2{}\mathrm {i}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*tan(c + d*x)*1i)^4,x)

[Out]

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

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