3.139 \(\int \frac{\cos ^4(c+d x)}{(a+i a \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=195 \[ \frac{i a^2}{40 d (a+i a \tan (c+d x))^5}-\frac{3 i}{64 d \left (a^3-i a^3 \tan (c+d x)\right )}+\frac{15 i}{128 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{21 x}{128 a^3}+\frac{3 i a}{64 d (a+i a \tan (c+d x))^4}+\frac{i}{16 d (a+i a \tan (c+d x))^3}-\frac{i}{128 a d (a-i a \tan (c+d x))^2}+\frac{5 i}{64 a d (a+i a \tan (c+d x))^2} \]

[Out]

(21*x)/(128*a^3) - (I/128)/(a*d*(a - I*a*Tan[c + d*x])^2) + ((I/40)*a^2)/(d*(a + I*a*Tan[c + d*x])^5) + (((3*I
)/64)*a)/(d*(a + I*a*Tan[c + d*x])^4) + (I/16)/(d*(a + I*a*Tan[c + d*x])^3) + ((5*I)/64)/(a*d*(a + I*a*Tan[c +
d*x])^2) - ((3*I)/64)/(d*(a^3 - I*a^3*Tan[c + d*x])) + ((15*I)/128)/(d*(a^3 + I*a^3*Tan[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.116134, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3487, 44, 206} \[ \frac{i a^2}{40 d (a+i a \tan (c+d x))^5}-\frac{3 i}{64 d \left (a^3-i a^3 \tan (c+d x)\right )}+\frac{15 i}{128 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{21 x}{128 a^3}+\frac{3 i a}{64 d (a+i a \tan (c+d x))^4}+\frac{i}{16 d (a+i a \tan (c+d x))^3}-\frac{i}{128 a d (a-i a \tan (c+d x))^2}+\frac{5 i}{64 a d (a+i a \tan (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^3,x]

[Out]

(21*x)/(128*a^3) - (I/128)/(a*d*(a - I*a*Tan[c + d*x])^2) + ((I/40)*a^2)/(d*(a + I*a*Tan[c + d*x])^5) + (((3*I
)/64)*a)/(d*(a + I*a*Tan[c + d*x])^4) + (I/16)/(d*(a + I*a*Tan[c + d*x])^3) + ((5*I)/64)/(a*d*(a + I*a*Tan[c +
d*x])^2) - ((3*I)/64)/(d*(a^3 - I*a^3*Tan[c + d*x])) + ((15*I)/128)/(d*(a^3 + I*a^3*Tan[c + d*x]))

Rule 3487

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(a^(m - 2)*b
*f), Subst[Int[(a - x)^(m/2 - 1)*(a + x)^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x
] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\cos ^4(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=-\frac{\left (i a^5\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x)^3 (a+x)^6} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{\left (i a^5\right ) \operatorname{Subst}\left (\int \left (\frac{1}{64 a^6 (a-x)^3}+\frac{3}{64 a^7 (a-x)^2}+\frac{1}{8 a^3 (a+x)^6}+\frac{3}{16 a^4 (a+x)^5}+\frac{3}{16 a^5 (a+x)^4}+\frac{5}{32 a^6 (a+x)^3}+\frac{15}{128 a^7 (a+x)^2}+\frac{21}{128 a^7 \left (a^2-x^2\right )}\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{i}{128 a d (a-i a \tan (c+d x))^2}+\frac{i a^2}{40 d (a+i a \tan (c+d x))^5}+\frac{3 i a}{64 d (a+i a \tan (c+d x))^4}+\frac{i}{16 d (a+i a \tan (c+d x))^3}+\frac{5 i}{64 a d (a+i a \tan (c+d x))^2}-\frac{3 i}{64 d \left (a^3-i a^3 \tan (c+d x)\right )}+\frac{15 i}{128 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{(21 i) \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,i a \tan (c+d x)\right )}{128 a^2 d}\\ &=\frac{21 x}{128 a^3}-\frac{i}{128 a d (a-i a \tan (c+d x))^2}+\frac{i a^2}{40 d (a+i a \tan (c+d x))^5}+\frac{3 i a}{64 d (a+i a \tan (c+d x))^4}+\frac{i}{16 d (a+i a \tan (c+d x))^3}+\frac{5 i}{64 a d (a+i a \tan (c+d x))^2}-\frac{3 i}{64 d \left (a^3-i a^3 \tan (c+d x)\right )}+\frac{15 i}{128 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.373805, size = 137, normalized size = 0.7 \[ \frac{\sec ^3(c+d x) (-350 i \sin (c+d x)-840 d x \sin (3 (c+d x))+140 i \sin (3 (c+d x))+175 i \sin (5 (c+d x))+14 i \sin (7 (c+d x))-1050 \cos (c+d x)+140 i (6 d x+i) \cos (3 (c+d x))+105 \cos (5 (c+d x))+6 \cos (7 (c+d x)))}{5120 a^3 d (\tan (c+d x)-i)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^3,x]

[Out]

(Sec[c + d*x]^3*(-1050*Cos[c + d*x] + (140*I)*(I + 6*d*x)*Cos[3*(c + d*x)] + 105*Cos[5*(c + d*x)] + 6*Cos[7*(c
+ d*x)] - (350*I)*Sin[c + d*x] + (140*I)*Sin[3*(c + d*x)] - 840*d*x*Sin[3*(c + d*x)] + (175*I)*Sin[5*(c + d*x
)] + (14*I)*Sin[7*(c + d*x)]))/(5120*a^3*d*(-I + Tan[c + d*x])^3)

________________________________________________________________________________________

Maple [A]  time = 0.102, size = 176, normalized size = 0.9 \begin{align*}{\frac{-{\frac{21\,i}{256}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) }{{a}^{3}d}}+{\frac{{\frac{3\,i}{64}}}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{4}}}-{\frac{{\frac{5\,i}{64}}}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}+{\frac{1}{40\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{5}}}-{\frac{1}{16\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}+{\frac{15}{128\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{{\frac{i}{128}}}{{a}^{3}d \left ( \tan \left ( dx+c \right ) +i \right ) ^{2}}}+{\frac{{\frac{21\,i}{256}}\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{{a}^{3}d}}+{\frac{3}{64\,{a}^{3}d \left ( \tan \left ( dx+c \right ) +i \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4/(a+I*a*tan(d*x+c))^3,x)

[Out]

-21/256*I/a^3/d*ln(tan(d*x+c)-I)+3/64*I/a^3/d/(tan(d*x+c)-I)^4-5/64*I/a^3/d/(tan(d*x+c)-I)^2+1/40/a^3/d/(tan(d
*x+c)-I)^5-1/16/a^3/d/(tan(d*x+c)-I)^3+15/128/a^3/d/(tan(d*x+c)-I)+1/128*I/a^3/d/(tan(d*x+c)+I)^2+21/256*I/a^3
/d*ln(tan(d*x+c)+I)+3/64/a^3/d/(tan(d*x+c)+I)

________________________________________________________________________________________

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(cos(d*x+c)^4/(a+I*a*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [A]  time = 2.41382, size = 333, normalized size = 1.71 \begin{align*} \frac{{\left (840 \, d x e^{\left (10 i \, d x + 10 i \, c\right )} - 10 i \, e^{\left (14 i \, d x + 14 i \, c\right )} - 140 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 700 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 350 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 140 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 35 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 4 i\right )} e^{\left (-10 i \, d x - 10 i \, c\right )}}{5120 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4/(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

1/5120*(840*d*x*e^(10*I*d*x + 10*I*c) - 10*I*e^(14*I*d*x + 14*I*c) - 140*I*e^(12*I*d*x + 12*I*c) + 700*I*e^(8*
I*d*x + 8*I*c) + 350*I*e^(6*I*d*x + 6*I*c) + 140*I*e^(4*I*d*x + 4*I*c) + 35*I*e^(2*I*d*x + 2*I*c) + 4*I)*e^(-1
0*I*d*x - 10*I*c)/(a^3*d)

________________________________________________________________________________________

Sympy [A]  time = 1.33778, size = 294, normalized size = 1.51 \begin{align*} \begin{cases} \frac{\left (- 11258999068426240 i a^{18} d^{6} e^{34 i c} e^{4 i d x} - 157625986957967360 i a^{18} d^{6} e^{32 i c} e^{2 i d x} + 788129934789836800 i a^{18} d^{6} e^{28 i c} e^{- 2 i d x} + 394064967394918400 i a^{18} d^{6} e^{26 i c} e^{- 4 i d x} + 157625986957967360 i a^{18} d^{6} e^{24 i c} e^{- 6 i d x} + 39406496739491840 i a^{18} d^{6} e^{22 i c} e^{- 8 i d x} + 4503599627370496 i a^{18} d^{6} e^{20 i c} e^{- 10 i d x}\right ) e^{- 30 i c}}{5764607523034234880 a^{21} d^{7}} & \text{for}\: 5764607523034234880 a^{21} d^{7} e^{30 i c} \neq 0 \\x \left (\frac{\left (e^{14 i c} + 7 e^{12 i c} + 21 e^{10 i c} + 35 e^{8 i c} + 35 e^{6 i c} + 21 e^{4 i c} + 7 e^{2 i c} + 1\right ) e^{- 10 i c}}{128 a^{3}} - \frac{21}{128 a^{3}}\right ) & \text{otherwise} \end{cases} + \frac{21 x}{128 a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**4/(a+I*a*tan(d*x+c))**3,x)

[Out]

Piecewise(((-11258999068426240*I*a**18*d**6*exp(34*I*c)*exp(4*I*d*x) - 157625986957967360*I*a**18*d**6*exp(32*
I*c)*exp(2*I*d*x) + 788129934789836800*I*a**18*d**6*exp(28*I*c)*exp(-2*I*d*x) + 394064967394918400*I*a**18*d**
6*exp(26*I*c)*exp(-4*I*d*x) + 157625986957967360*I*a**18*d**6*exp(24*I*c)*exp(-6*I*d*x) + 39406496739491840*I*
a**18*d**6*exp(22*I*c)*exp(-8*I*d*x) + 4503599627370496*I*a**18*d**6*exp(20*I*c)*exp(-10*I*d*x))*exp(-30*I*c)/
(5764607523034234880*a**21*d**7), Ne(5764607523034234880*a**21*d**7*exp(30*I*c), 0)), (x*((exp(14*I*c) + 7*exp
(12*I*c) + 21*exp(10*I*c) + 35*exp(8*I*c) + 35*exp(6*I*c) + 21*exp(4*I*c) + 7*exp(2*I*c) + 1)*exp(-10*I*c)/(12
8*a**3) - 21/(128*a**3)), True)) + 21*x/(128*a**3)

________________________________________________________________________________________

Giac [A]  time = 1.18832, size = 184, normalized size = 0.94 \begin{align*} -\frac{-\frac{420 i \, \log \left (\tan \left (d x + c\right ) + i\right )}{a^{3}} + \frac{420 i \, \log \left (\tan \left (d x + c\right ) - i\right )}{a^{3}} + \frac{10 \,{\left (-63 i \, \tan \left (d x + c\right )^{2} + 150 \, \tan \left (d x + c\right ) + 91 i\right )}}{a^{3}{\left (i \, \tan \left (d x + c\right ) - 1\right )}^{2}} - \frac{959 i \, \tan \left (d x + c\right )^{5} + 5395 \, \tan \left (d x + c\right )^{4} - 12390 i \, \tan \left (d x + c\right )^{3} - 14710 \, \tan \left (d x + c\right )^{2} + 9275 i \, \tan \left (d x + c\right ) + 2647}{a^{3}{\left (\tan \left (d x + c\right ) - i\right )}^{5}}}{5120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4/(a+I*a*tan(d*x+c))^3,x, algorithm="giac")

[Out]

-1/5120*(-420*I*log(tan(d*x + c) + I)/a^3 + 420*I*log(tan(d*x + c) - I)/a^3 + 10*(-63*I*tan(d*x + c)^2 + 150*t
an(d*x + c) + 91*I)/(a^3*(I*tan(d*x + c) - 1)^2) - (959*I*tan(d*x + c)^5 + 5395*tan(d*x + c)^4 - 12390*I*tan(d
*x + c)^3 - 14710*tan(d*x + c)^2 + 9275*I*tan(d*x + c) + 2647)/(a^3*(tan(d*x + c) - I)^5))/d