3.50 \(\int \cos ^7(c+d x) (a+i a \tan (c+d x))^3 \, dx\)

Optimal. Leaf size=106 \[ \frac{3 a^3 \sin ^5(c+d x)}{35 d}-\frac{2 a^3 \sin ^3(c+d x)}{7 d}+\frac{3 a^3 \sin (c+d x)}{7 d}-\frac{3 i a^3 \cos ^5(c+d x)}{35 d}-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{7 d} \]

[Out]

(((-3*I)/35)*a^3*Cos[c + d*x]^5)/d + (3*a^3*Sin[c + d*x])/(7*d) - (2*a^3*Sin[c + d*x]^3)/(7*d) + (3*a^3*Sin[c
+ d*x]^5)/(35*d) - (((2*I)/7)*a*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^2)/d

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Rubi [A]  time = 0.0757118, antiderivative size = 106, 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 = {3496, 3486, 2633} \[ \frac{3 a^3 \sin ^5(c+d x)}{35 d}-\frac{2 a^3 \sin ^3(c+d x)}{7 d}+\frac{3 a^3 \sin (c+d x)}{7 d}-\frac{3 i a^3 \cos ^5(c+d x)}{35 d}-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^3,x]

[Out]

(((-3*I)/35)*a^3*Cos[c + d*x]^5)/d + (3*a^3*Sin[c + d*x])/(7*d) - (2*a^3*Sin[c + d*x]^3)/(7*d) + (3*a^3*Sin[c
+ d*x]^5)/(35*d) - (((2*I)/7)*a*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^2)/d

Rule 3496

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(2*b*(
d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1))/(f*m), x] - Dist[(b^2*(m + 2*n - 2))/(d^2*m), Int[(d*Sec[e + f
*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n,
1] && ((IGtQ[n/2, 0] && ILtQ[m - 1/2, 0]) || (EqQ[n, 2] && LtQ[m, 0]) || (LeQ[m, -1] && GtQ[m + n, 0]) || (ILt
Q[m, 0] && LtQ[m/2 + n - 1, 0] && IntegerQ[n]) || (EqQ[n, 3/2] && EqQ[m, -2^(-1)])) && IntegerQ[2*m]

Rule 3486

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*(d*Sec[
e + f*x])^m)/(f*m), x] + Dist[a, Int[(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2
*m] || NeQ[a^2 + b^2, 0])

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \cos ^7(c+d x) (a+i a \tan (c+d x))^3 \, dx &=-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{7 d}+\frac{1}{7} \left (3 a^2\right ) \int \cos ^5(c+d x) (a+i a \tan (c+d x)) \, dx\\ &=-\frac{3 i a^3 \cos ^5(c+d x)}{35 d}-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{7 d}+\frac{1}{7} \left (3 a^3\right ) \int \cos ^5(c+d x) \, dx\\ &=-\frac{3 i a^3 \cos ^5(c+d x)}{35 d}-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{7 d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{7 d}\\ &=-\frac{3 i a^3 \cos ^5(c+d x)}{35 d}+\frac{3 a^3 \sin (c+d x)}{7 d}-\frac{2 a^3 \sin ^3(c+d x)}{7 d}+\frac{3 a^3 \sin ^5(c+d x)}{35 d}-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{7 d}\\ \end{align*}

Mathematica [A]  time = 0.573259, size = 77, normalized size = 0.73 \[ \frac{a^3 (\sin (3 (c+d x))-i \cos (3 (c+d x))) (-56 i \sin (2 (c+d x))+20 i \sin (4 (c+d x))+84 \cos (2 (c+d x))-15 \cos (4 (c+d x))+35)}{280 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^3,x]

[Out]

(a^3*((-I)*Cos[3*(c + d*x)] + Sin[3*(c + d*x)])*(35 + 84*Cos[2*(c + d*x)] - 15*Cos[4*(c + d*x)] - (56*I)*Sin[2
*(c + d*x)] + (20*I)*Sin[4*(c + d*x)]))/(280*d)

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Maple [A]  time = 0.062, size = 146, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ( -i{a}^{3} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{7}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35}} \right ) -3\,{a}^{3} \left ( -1/7\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}\sin \left ( dx+c \right ) +1/35\, \left ( 8/3+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+4/3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) -{\frac{3\,i}{7}}{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}+{\frac{{a}^{3}\sin \left ( dx+c \right ) }{7} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^3,x)

[Out]

1/d*(-I*a^3*(-1/7*sin(d*x+c)^2*cos(d*x+c)^5-2/35*cos(d*x+c)^5)-3*a^3*(-1/7*cos(d*x+c)^6*sin(d*x+c)+1/35*(8/3+c
os(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))-3/7*I*a^3*cos(d*x+c)^7+1/7*a^3*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8
/5*cos(d*x+c)^2)*sin(d*x+c))

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Maxima [A]  time = 1.15661, size = 166, normalized size = 1.57 \begin{align*} -\frac{15 i \, a^{3} \cos \left (d x + c\right )^{7} + i \,{\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{3} +{\left (15 \, \sin \left (d x + c\right )^{7} - 42 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3}\right )} a^{3} +{\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a^{3}}{35 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/35*(15*I*a^3*cos(d*x + c)^7 + I*(5*cos(d*x + c)^7 - 7*cos(d*x + c)^5)*a^3 + (15*sin(d*x + c)^7 - 42*sin(d*x
+ c)^5 + 35*sin(d*x + c)^3)*a^3 + (5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c)
)*a^3)/d

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Fricas [A]  time = 1.07833, size = 219, normalized size = 2.07 \begin{align*} \frac{{\left (-5 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 28 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 70 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 140 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 35 i \, a^{3}\right )} e^{\left (-i \, d x - i \, c\right )}}{560 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

1/560*(-5*I*a^3*e^(8*I*d*x + 8*I*c) - 28*I*a^3*e^(6*I*d*x + 6*I*c) - 70*I*a^3*e^(4*I*d*x + 4*I*c) - 140*I*a^3*
e^(2*I*d*x + 2*I*c) + 35*I*a^3)*e^(-I*d*x - I*c)/d

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Sympy [A]  time = 1.02439, size = 192, normalized size = 1.81 \begin{align*} \begin{cases} \frac{\left (- 10240 i a^{3} d^{4} e^{8 i c} e^{7 i d x} - 57344 i a^{3} d^{4} e^{6 i c} e^{5 i d x} - 143360 i a^{3} d^{4} e^{4 i c} e^{3 i d x} - 286720 i a^{3} d^{4} e^{2 i c} e^{i d x} + 71680 i a^{3} d^{4} e^{- i d x}\right ) e^{- i c}}{1146880 d^{5}} & \text{for}\: 1146880 d^{5} e^{i c} \neq 0 \\\frac{x \left (a^{3} e^{8 i c} + 4 a^{3} e^{6 i c} + 6 a^{3} e^{4 i c} + 4 a^{3} e^{2 i c} + a^{3}\right ) e^{- i c}}{16} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**7*(a+I*a*tan(d*x+c))**3,x)

[Out]

Piecewise(((-10240*I*a**3*d**4*exp(8*I*c)*exp(7*I*d*x) - 57344*I*a**3*d**4*exp(6*I*c)*exp(5*I*d*x) - 143360*I*
a**3*d**4*exp(4*I*c)*exp(3*I*d*x) - 286720*I*a**3*d**4*exp(2*I*c)*exp(I*d*x) + 71680*I*a**3*d**4*exp(-I*d*x))*
exp(-I*c)/(1146880*d**5), Ne(1146880*d**5*exp(I*c), 0)), (x*(a**3*exp(8*I*c) + 4*a**3*exp(6*I*c) + 6*a**3*exp(
4*I*c) + 4*a**3*exp(2*I*c) + a**3)*exp(-I*c)/16, True))

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Giac [B]  time = 1.51986, size = 628, normalized size = 5.92 \begin{align*} \frac{19635 \, a^{3} e^{\left (5 i \, d x + 3 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 39270 \, a^{3} e^{\left (3 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 19635 \, a^{3} e^{\left (i \, d x - i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 19635 \, a^{3} e^{\left (5 i \, d x + 3 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) + 39270 \, a^{3} e^{\left (3 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) + 19635 \, a^{3} e^{\left (i \, d x - i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 19635 \, a^{3} e^{\left (5 i \, d x + 3 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 39270 \, a^{3} e^{\left (3 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 19635 \, a^{3} e^{\left (i \, d x - i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 19635 \, a^{3} e^{\left (5 i \, d x + 3 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 39270 \, a^{3} e^{\left (3 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 19635 \, a^{3} e^{\left (i \, d x - i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 640 i \, a^{3} e^{\left (12 i \, d x + 10 i \, c\right )} - 4864 i \, a^{3} e^{\left (10 i \, d x + 8 i \, c\right )} - 16768 i \, a^{3} e^{\left (8 i \, d x + 6 i \, c\right )} - 39424 i \, a^{3} e^{\left (6 i \, d x + 4 i \, c\right )} - 40320 i \, a^{3} e^{\left (4 i \, d x + 2 i \, c\right )} - 8960 i \, a^{3} e^{\left (2 i \, d x\right )} + 4480 i \, a^{3} e^{\left (-2 i \, c\right )}}{71680 \,{\left (d e^{\left (5 i \, d x + 3 i \, c\right )} + 2 \, d e^{\left (3 i \, d x + i \, c\right )} + d e^{\left (i \, d x - i \, c\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^3,x, algorithm="giac")

[Out]

1/71680*(19635*a^3*e^(5*I*d*x + 3*I*c)*log(I*e^(I*d*x + I*c) + 1) + 39270*a^3*e^(3*I*d*x + I*c)*log(I*e^(I*d*x
+ I*c) + 1) + 19635*a^3*e^(I*d*x - I*c)*log(I*e^(I*d*x + I*c) + 1) + 19635*a^3*e^(5*I*d*x + 3*I*c)*log(I*e^(I
*d*x + I*c) - 1) + 39270*a^3*e^(3*I*d*x + I*c)*log(I*e^(I*d*x + I*c) - 1) + 19635*a^3*e^(I*d*x - I*c)*log(I*e^
(I*d*x + I*c) - 1) - 19635*a^3*e^(5*I*d*x + 3*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 39270*a^3*e^(3*I*d*x + I*c)*l
og(-I*e^(I*d*x + I*c) + 1) - 19635*a^3*e^(I*d*x - I*c)*log(-I*e^(I*d*x + I*c) + 1) - 19635*a^3*e^(5*I*d*x + 3*
I*c)*log(-I*e^(I*d*x + I*c) - 1) - 39270*a^3*e^(3*I*d*x + I*c)*log(-I*e^(I*d*x + I*c) - 1) - 19635*a^3*e^(I*d*
x - I*c)*log(-I*e^(I*d*x + I*c) - 1) - 640*I*a^3*e^(12*I*d*x + 10*I*c) - 4864*I*a^3*e^(10*I*d*x + 8*I*c) - 167
68*I*a^3*e^(8*I*d*x + 6*I*c) - 39424*I*a^3*e^(6*I*d*x + 4*I*c) - 40320*I*a^3*e^(4*I*d*x + 2*I*c) - 8960*I*a^3*
e^(2*I*d*x) + 4480*I*a^3*e^(-2*I*c))/(d*e^(5*I*d*x + 3*I*c) + 2*d*e^(3*I*d*x + I*c) + d*e^(I*d*x - I*c))