3.1.50 \(\int \cos ^7(c+d x) (a+i a \tan (c+d x))^3 \, dx\) [50]

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

[Out]

-3/35*I*a^3*cos(d*x+c)^5/d+3/7*a^3*sin(d*x+c)/d-2/7*a^3*sin(d*x+c)^3/d+3/35*a^3*sin(d*x+c)^5/d-2/7*I*a*cos(d*x
+c)^7*(a+I*a*tan(d*x+c))^2/d

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Rubi [A]
time = 0.06, antiderivative size = 106, normalized size of antiderivative = 1.00, 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 = {3577, 3567, 2713} \begin {gather*} \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} \end {gather*}

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 2713

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]

Rule 3567

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 3577

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]

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 ) \text {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*}

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Mathematica [A]
time = 0.55, size = 77, normalized size = 0.73 \begin {gather*} \frac {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} \end {gather*}

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 = 0.22, size = 146, normalized size = 1.38

method	result	size
risch	\(-\frac {i a^{3} {\mathrm e}^{7 i \left (d x +c \right )}}{112 d}-\frac {i a^{3} {\mathrm e}^{5 i \left (d x +c \right )}}{20 d}-\frac {i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{8 d}-\frac {3 i a^{3} \cos \left (d x +c \right )}{16 d}+\frac {5 a^{3} \sin \left (d x +c \right )}{16 d}\)	\(85\)
derivativedivides	\(\frac {-i a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )-3 a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {3 i a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7}+\frac {a^{3} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\)	\(146\)
default	\(\frac {-i a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )-3 a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {3 i a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7}+\frac {a^{3} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\)	\(146\)

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,method=_RETURNVERBOSE)

[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*sin(d*x+c)*cos(d*x+c)^6+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 = 0.27, size = 123, normalized size = 1.16 \begin {gather*} -\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 {gather*}

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 = 0.37, size = 76, normalized size = 0.72 \begin {gather*} \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 {gather*}

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 = 0.35, size = 190, normalized size = 1.79 \begin {gather*} \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}\: 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 {gather*}

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(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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (90) = 180\).
time = 0.99, size = 465, normalized size = 4.39 \begin {gather*} \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 {gather*}

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))

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Mupad [B]
time = 4.54, size = 134, normalized size = 1.26 \begin {gather*} -\frac {2\,a^3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {17\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}-\frac {17\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{2}+\frac {31\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{2}-\frac {5\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{2}+\frac {\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,35{}\mathrm {i}}{8}-\frac {\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,35{}\mathrm {i}}{8}+\frac {\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )\,119{}\mathrm {i}}{8}-\frac {\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,15{}\mathrm {i}}{8}\right )}{35\,d\,\left (\cos \left (3\,c+3\,d\,x\right )-\sin \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^7*(a + a*tan(c + d*x)*1i)^3,x)

[Out]

-(2*a^3*cos(c/2 + (d*x)/2)*((cos(c/2 + (d*x)/2)*35i)/8 - (cos((3*c)/2 + (3*d*x)/2)*35i)/8 + (cos((5*c)/2 + (5*
d*x)/2)*119i)/8 - (cos((7*c)/2 + (7*d*x)/2)*15i)/8 + (17*sin(c/2 + (d*x)/2))/2 - (17*sin((3*c)/2 + (3*d*x)/2))
/2 + (31*sin((5*c)/2 + (5*d*x)/2))/2 - (5*sin((7*c)/2 + (7*d*x)/2))/2))/(35*d*(cos(3*c + 3*d*x) - sin(3*c + 3*
d*x)*1i))

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