3.64 \(\int \cos ^2(c+d x) (a+i a \tan (c+d x))^5 \, dx\)

Optimal. Leaf size=83 \[ \frac{i a^5 \tan ^2(c+d x)}{2 d}-\frac{8 i a^6}{d (a-i a \tan (c+d x))}+\frac{5 a^5 \tan (c+d x)}{d}+\frac{12 i a^5 \log (\cos (c+d x))}{d}-12 a^5 x \]

[Out]

-12*a^5*x + ((12*I)*a^5*Log[Cos[c + d*x]])/d + (5*a^5*Tan[c + d*x])/d + ((I/2)*a^5*Tan[c + d*x]^2)/d - ((8*I)*
a^6)/(d*(a - I*a*Tan[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.0585825, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3487, 43} \[ \frac{i a^5 \tan ^2(c+d x)}{2 d}-\frac{8 i a^6}{d (a-i a \tan (c+d x))}+\frac{5 a^5 \tan (c+d x)}{d}+\frac{12 i a^5 \log (\cos (c+d x))}{d}-12 a^5 x \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^5,x]

[Out]

-12*a^5*x + ((12*I)*a^5*Log[Cos[c + d*x]])/d + (5*a^5*Tan[c + d*x])/d + ((I/2)*a^5*Tan[c + d*x]^2)/d - ((8*I)*
a^6)/(d*(a - I*a*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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \cos ^2(c+d x) (a+i a \tan (c+d x))^5 \, dx &=-\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \frac{(a+x)^3}{(a-x)^2} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \left (5 a+\frac{8 a^3}{(a-x)^2}-\frac{12 a^2}{a-x}+x\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-12 a^5 x+\frac{12 i a^5 \log (\cos (c+d x))}{d}+\frac{5 a^5 \tan (c+d x)}{d}+\frac{i a^5 \tan ^2(c+d x)}{2 d}-\frac{8 i a^6}{d (a-i a \tan (c+d x))}\\ \end{align*}

Mathematica [B]  time = 6.55229, size = 649, normalized size = 7.82 \[ \frac{x \cos ^5(c+d x) \left (36 i \sin ^5(c)+24 i \sin ^3(c)-6 \cos ^5(c)+6 \cos ^3(c)+6 \sin ^5(c) \tan (c)+6 \sin ^3(c) \tan (c)+90 \sin ^2(c) \cos ^3(c)-120 i \sin ^3(c) \cos ^2(c)+36 i \sin (c) \cos ^4(c)-24 i \sin (c) \cos ^2(c)-90 \sin ^4(c) \cos (c)-36 \sin ^2(c) \cos (c)-i \tan (c) (12 \cos (5 c)-12 i \sin (5 c))\right ) (a+i a \tan (c+d x))^5}{(\cos (d x)+i \sin (d x))^5}+\frac{(5 \cos (5 c)-5 i \sin (5 c)) \sin (d x) \cos ^4(c+d x) (a+i a \tan (c+d x))^5}{d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) (\cos (d x)+i \sin (d x))^5}+\frac{(4 \cos (3 c)-4 i \sin (3 c)) \sin (2 d x) \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{d (\cos (d x)+i \sin (d x))^5}-\frac{12 x \cos (5 c) \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{(\cos (d x)+i \sin (d x))^5}+\frac{(-4 \sin (3 c)-4 i \cos (3 c)) \cos (2 d x) \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{d (\cos (d x)+i \sin (d x))^5}+\frac{\left (\frac{1}{2} \sin (5 c)+\frac{1}{2} i \cos (5 c)\right ) \cos ^3(c+d x) (a+i a \tan (c+d x))^5}{d (\cos (d x)+i \sin (d x))^5}+\frac{12 i x \sin (5 c) \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{(\cos (d x)+i \sin (d x))^5}+\frac{6 i \cos (5 c) \cos ^5(c+d x) (a+i a \tan (c+d x))^5 \log \left (\cos ^2(c+d x)\right )}{d (\cos (d x)+i \sin (d x))^5}+\frac{6 \sin (5 c) \cos ^5(c+d x) (a+i a \tan (c+d x))^5 \log \left (\cos ^2(c+d x)\right )}{d (\cos (d x)+i \sin (d x))^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^5,x]

[Out]

(-12*x*Cos[5*c]*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^5)/(Cos[d*x] + I*Sin[d*x])^5 + ((6*I)*Cos[5*c]*Cos[c + d
*x]^5*Log[Cos[c + d*x]^2]*(a + I*a*Tan[c + d*x])^5)/(d*(Cos[d*x] + I*Sin[d*x])^5) + (Cos[2*d*x]*Cos[c + d*x]^5
*((-4*I)*Cos[3*c] - 4*Sin[3*c])*(a + I*a*Tan[c + d*x])^5)/(d*(Cos[d*x] + I*Sin[d*x])^5) + (Cos[c + d*x]^3*((I/
2)*Cos[5*c] + Sin[5*c]/2)*(a + I*a*Tan[c + d*x])^5)/(d*(Cos[d*x] + I*Sin[d*x])^5) + ((12*I)*x*Cos[c + d*x]^5*S
in[5*c]*(a + I*a*Tan[c + d*x])^5)/(Cos[d*x] + I*Sin[d*x])^5 + (6*Cos[c + d*x]^5*Log[Cos[c + d*x]^2]*Sin[5*c]*(
a + I*a*Tan[c + d*x])^5)/(d*(Cos[d*x] + I*Sin[d*x])^5) + (Cos[c + d*x]^4*(5*Cos[5*c] - (5*I)*Sin[5*c])*Sin[d*x
]*(a + I*a*Tan[c + d*x])^5)/(d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[d*x] + I*Sin[d*x])^5) + (Cos[c
+ d*x]^5*(4*Cos[3*c] - (4*I)*Sin[3*c])*Sin[2*d*x]*(a + I*a*Tan[c + d*x])^5)/(d*(Cos[d*x] + I*Sin[d*x])^5) + (
x*Cos[c + d*x]^5*(6*Cos[c]^3 - 6*Cos[c]^5 - (24*I)*Cos[c]^2*Sin[c] + (36*I)*Cos[c]^4*Sin[c] - 36*Cos[c]*Sin[c]
^2 + 90*Cos[c]^3*Sin[c]^2 + (24*I)*Sin[c]^3 - (120*I)*Cos[c]^2*Sin[c]^3 - 90*Cos[c]*Sin[c]^4 + (36*I)*Sin[c]^5
+ 6*Sin[c]^3*Tan[c] + 6*Sin[c]^5*Tan[c] - I*(12*Cos[5*c] - (12*I)*Sin[5*c])*Tan[c])*(a + I*a*Tan[c + d*x])^5)
/(Cos[d*x] + I*Sin[d*x])^5

________________________________________________________________________________________

Maple [B]  time = 0.107, size = 175, normalized size = 2.1 \begin{align*}{\frac{6\,i{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{{\frac{i}{2}}{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d}}-{\frac{{\frac{5\,i}{2}}{a}^{5} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{{\frac{i}{2}}{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+5\,{\frac{{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{d\cos \left ( dx+c \right ) }}+5\,{\frac{{a}^{5}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{d}}+13\,{\frac{{a}^{5}\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }{d}}-12\,{a}^{5}x-12\,{\frac{{a}^{5}c}{d}}+{\frac{12\,i{a}^{5}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2*(a+I*a*tan(d*x+c))^5,x)

[Out]

6*I/d*a^5*sin(d*x+c)^2+1/2*I/d*a^5*sin(d*x+c)^4-5/2*I/d*a^5*cos(d*x+c)^2+1/2*I/d*a^5*sin(d*x+c)^6/cos(d*x+c)^2
+5/d*a^5*sin(d*x+c)^5/cos(d*x+c)+5/d*a^5*cos(d*x+c)*sin(d*x+c)^3+13/d*a^5*sin(d*x+c)*cos(d*x+c)-12*a^5*x-12/d*
a^5*c+12*I*a^5*ln(cos(d*x+c))/d

________________________________________________________________________________________

Maxima [A]  time = 1.70172, size = 116, normalized size = 1.4 \begin{align*} -\frac{-i \, a^{5} \tan \left (d x + c\right )^{2} + 24 \,{\left (d x + c\right )} a^{5} + 12 i \, a^{5} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 10 \, a^{5} \tan \left (d x + c\right ) - \frac{16 \,{\left (a^{5} \tan \left (d x + c\right ) - i \, a^{5}\right )}}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+I*a*tan(d*x+c))^5,x, algorithm="maxima")

[Out]

-1/2*(-I*a^5*tan(d*x + c)^2 + 24*(d*x + c)*a^5 + 12*I*a^5*log(tan(d*x + c)^2 + 1) - 10*a^5*tan(d*x + c) - 16*(
a^5*tan(d*x + c) - I*a^5)/(tan(d*x + c)^2 + 1))/d

________________________________________________________________________________________

Fricas [A]  time = 1.08967, size = 352, normalized size = 4.24 \begin{align*} \frac{-4 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 8 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 8 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + 10 i \, a^{5} +{\left (12 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 24 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + 12 i \, a^{5}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}{d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+I*a*tan(d*x+c))^5,x, algorithm="fricas")

[Out]

(-4*I*a^5*e^(6*I*d*x + 6*I*c) - 8*I*a^5*e^(4*I*d*x + 4*I*c) + 8*I*a^5*e^(2*I*d*x + 2*I*c) + 10*I*a^5 + (12*I*a
^5*e^(4*I*d*x + 4*I*c) + 24*I*a^5*e^(2*I*d*x + 2*I*c) + 12*I*a^5)*log(e^(2*I*d*x + 2*I*c) + 1))/(d*e^(4*I*d*x
+ 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)

________________________________________________________________________________________

Sympy [A]  time = 1.38713, size = 128, normalized size = 1.54 \begin{align*} 8 a^{5} \left (\begin{cases} - \frac{i e^{2 i d x}}{2 d} & \text{for}\: d \neq 0 \\x & \text{otherwise} \end{cases}\right ) e^{2 i c} + \frac{12 i a^{5} \log{\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac{\frac{12 i a^{5} e^{- 2 i c} e^{2 i d x}}{d} + \frac{10 i a^{5} e^{- 4 i c}}{d}}{e^{4 i d x} + 2 e^{- 2 i c} e^{2 i d x} + e^{- 4 i c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**2*(a+I*a*tan(d*x+c))**5,x)

[Out]

8*a**5*Piecewise((-I*exp(2*I*d*x)/(2*d), Ne(d, 0)), (x, True))*exp(2*I*c) + 12*I*a**5*log(exp(2*I*d*x) + exp(-
2*I*c))/d + (12*I*a**5*exp(-2*I*c)*exp(2*I*d*x)/d + 10*I*a**5*exp(-4*I*c)/d)/(exp(4*I*d*x) + 2*exp(-2*I*c)*exp
(2*I*d*x) + exp(-4*I*c))

________________________________________________________________________________________

Giac [A]  time = 1.43947, size = 196, normalized size = 2.36 \begin{align*} \frac{12 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 24 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 4 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 8 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 8 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + 12 i \, a^{5} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 10 i \, a^{5}}{d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+I*a*tan(d*x+c))^5,x, algorithm="giac")

[Out]

(12*I*a^5*e^(4*I*d*x + 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 24*I*a^5*e^(2*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I
*c) + 1) - 4*I*a^5*e^(6*I*d*x + 6*I*c) - 8*I*a^5*e^(4*I*d*x + 4*I*c) + 8*I*a^5*e^(2*I*d*x + 2*I*c) + 12*I*a^5*
log(e^(2*I*d*x + 2*I*c) + 1) + 10*I*a^5)/(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)