3.68 \(\int \cos ^{10}(c+d x) (a+i a \tan (c+d x))^5 \, dx\)

Optimal. Leaf size=144 \[ -\frac {i a^{10}}{10 d (a-i a \tan (c+d x))^5}-\frac {i a^9}{16 d (a-i a \tan (c+d x))^4}-\frac {i a^8}{24 d (a-i a \tan (c+d x))^3}-\frac {i a^7}{32 d (a-i a \tan (c+d x))^2}-\frac {i a^6}{32 d (a-i a \tan (c+d x))}+\frac {a^5 x}{32} \]

[Out]

1/32*a^5*x-1/10*I*a^10/d/(a-I*a*tan(d*x+c))^5-1/16*I*a^9/d/(a-I*a*tan(d*x+c))^4-1/24*I*a^8/d/(a-I*a*tan(d*x+c)
)^3-1/32*I*a^7/d/(a-I*a*tan(d*x+c))^2-1/32*I*a^6/d/(a-I*a*tan(d*x+c))

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Rubi [A]  time = 0.09, antiderivative size = 144, 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 = {3487, 44, 206} \[ -\frac {i a^{10}}{10 d (a-i a \tan (c+d x))^5}-\frac {i a^9}{16 d (a-i a \tan (c+d x))^4}-\frac {i a^8}{24 d (a-i a \tan (c+d x))^3}-\frac {i a^7}{32 d (a-i a \tan (c+d x))^2}-\frac {i a^6}{32 d (a-i a \tan (c+d x))}+\frac {a^5 x}{32} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^10*(a + I*a*Tan[c + d*x])^5,x]

[Out]

(a^5*x)/32 - ((I/10)*a^10)/(d*(a - I*a*Tan[c + d*x])^5) - ((I/16)*a^9)/(d*(a - I*a*Tan[c + d*x])^4) - ((I/24)*
a^8)/(d*(a - I*a*Tan[c + d*x])^3) - ((I/32)*a^7)/(d*(a - I*a*Tan[c + d*x])^2) - ((I/32)*a^6)/(d*(a - I*a*Tan[c
+ d*x]))

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])

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]

Rubi steps

\begin {align*} \int \cos ^{10}(c+d x) (a+i a \tan (c+d x))^5 \, dx &=-\frac {\left (i a^{11}\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x)^6 (a+x)} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {\left (i a^{11}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2 a (a-x)^6}+\frac {1}{4 a^2 (a-x)^5}+\frac {1}{8 a^3 (a-x)^4}+\frac {1}{16 a^4 (a-x)^3}+\frac {1}{32 a^5 (a-x)^2}+\frac {1}{32 a^5 \left (a^2-x^2\right )}\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {i a^{10}}{10 d (a-i a \tan (c+d x))^5}-\frac {i a^9}{16 d (a-i a \tan (c+d x))^4}-\frac {i a^8}{24 d (a-i a \tan (c+d x))^3}-\frac {i a^7}{32 d (a-i a \tan (c+d x))^2}-\frac {i a^6}{32 d (a-i a \tan (c+d x))}-\frac {\left (i a^6\right ) \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,i a \tan (c+d x)\right )}{32 d}\\ &=\frac {a^5 x}{32}-\frac {i a^{10}}{10 d (a-i a \tan (c+d x))^5}-\frac {i a^9}{16 d (a-i a \tan (c+d x))^4}-\frac {i a^8}{24 d (a-i a \tan (c+d x))^3}-\frac {i a^7}{32 d (a-i a \tan (c+d x))^2}-\frac {i a^6}{32 d (a-i a \tan (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 1.25, size = 137, normalized size = 0.95 \[ \frac {a^5 (-100 \sin (c+d x)-225 \sin (3 (c+d x))-120 i d x \sin (5 (c+d x))+12 \sin (5 (c+d x))-500 i \cos (c+d x)-375 i \cos (3 (c+d x))+120 d x \cos (5 (c+d x))-12 i \cos (5 (c+d x))) (\cos (5 (c+2 d x))+i \sin (5 (c+2 d x)))}{3840 d (\cos (d x)+i \sin (d x))^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^10*(a + I*a*Tan[c + d*x])^5,x]

[Out]

(a^5*((-500*I)*Cos[c + d*x] - (375*I)*Cos[3*(c + d*x)] - (12*I)*Cos[5*(c + d*x)] + 120*d*x*Cos[5*(c + d*x)] -
100*Sin[c + d*x] - 225*Sin[3*(c + d*x)] + 12*Sin[5*(c + d*x)] - (120*I)*d*x*Sin[5*(c + d*x)])*(Cos[5*(c + 2*d*
x)] + I*Sin[5*(c + 2*d*x)]))/(3840*d*(Cos[d*x] + I*Sin[d*x])^5)

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fricas [A]  time = 0.74, size = 83, normalized size = 0.58 \[ \frac {120 \, a^{5} d x - 12 i \, a^{5} e^{\left (10 i \, d x + 10 i \, c\right )} - 75 i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} - 200 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 300 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} - 300 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )}}{3840 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^10*(a+I*a*tan(d*x+c))^5,x, algorithm="fricas")

[Out]

1/3840*(120*a^5*d*x - 12*I*a^5*e^(10*I*d*x + 10*I*c) - 75*I*a^5*e^(8*I*d*x + 8*I*c) - 200*I*a^5*e^(6*I*d*x + 6
*I*c) - 300*I*a^5*e^(4*I*d*x + 4*I*c) - 300*I*a^5*e^(2*I*d*x + 2*I*c))/d

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giac [B]  time = 6.40, size = 857, normalized size = 5.95 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^10*(a+I*a*tan(d*x+c))^5,x, algorithm="giac")

[Out]

1/30720*(960*a^5*d*x*e^(16*I*d*x + 8*I*c) + 7680*a^5*d*x*e^(14*I*d*x + 6*I*c) + 26880*a^5*d*x*e^(12*I*d*x + 4*
I*c) + 53760*a^5*d*x*e^(10*I*d*x + 2*I*c) + 53760*a^5*d*x*e^(6*I*d*x - 2*I*c) + 26880*a^5*d*x*e^(4*I*d*x - 4*I
*c) + 7680*a^5*d*x*e^(2*I*d*x - 6*I*c) + 67200*a^5*d*x*e^(8*I*d*x) + 960*a^5*d*x*e^(-8*I*c) - 390*I*a^5*e^(16*
I*d*x + 8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 3120*I*a^5*e^(14*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 1
0920*I*a^5*e^(12*I*d*x + 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 21840*I*a^5*e^(10*I*d*x + 2*I*c)*log(e^(2*I*d*x
+ 2*I*c) + 1) - 21840*I*a^5*e^(6*I*d*x - 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 10920*I*a^5*e^(4*I*d*x - 4*I*c
)*log(e^(2*I*d*x + 2*I*c) + 1) - 3120*I*a^5*e^(2*I*d*x - 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 27300*I*a^5*e^(
8*I*d*x)*log(e^(2*I*d*x + 2*I*c) + 1) - 390*I*a^5*e^(-8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 390*I*a^5*e^(16*I*
d*x + 8*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 3120*I*a^5*e^(14*I*d*x + 6*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 1
0920*I*a^5*e^(12*I*d*x + 4*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 21840*I*a^5*e^(10*I*d*x + 2*I*c)*log(e^(2*I*d*
x) + e^(-2*I*c)) + 21840*I*a^5*e^(6*I*d*x - 2*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 10920*I*a^5*e^(4*I*d*x - 4*
I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 3120*I*a^5*e^(2*I*d*x - 6*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 27300*I*a^
5*e^(8*I*d*x)*log(e^(2*I*d*x) + e^(-2*I*c)) + 390*I*a^5*e^(-8*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) - 96*I*a^5*e^
(26*I*d*x + 18*I*c) - 1368*I*a^5*e^(24*I*d*x + 16*I*c) - 9088*I*a^5*e^(22*I*d*x + 14*I*c) - 37376*I*a^5*e^(20*
I*d*x + 12*I*c) - 106720*I*a^5*e^(18*I*d*x + 10*I*c) - 223376*I*a^5*e^(16*I*d*x + 8*I*c) - 349888*I*a^5*e^(14*
I*d*x + 6*I*c) - 409568*I*a^5*e^(12*I*d*x + 4*I*c) - 352096*I*a^5*e^(10*I*d*x + 2*I*c) - 88000*I*a^5*e^(6*I*d*
x - 2*I*c) - 21600*I*a^5*e^(4*I*d*x - 4*I*c) - 2400*I*a^5*e^(2*I*d*x - 6*I*c) - 215000*I*a^5*e^(8*I*d*x))/(d*e
^(16*I*d*x + 8*I*c) + 8*d*e^(14*I*d*x + 6*I*c) + 28*d*e^(12*I*d*x + 4*I*c) + 56*d*e^(10*I*d*x + 2*I*c) + 56*d*
e^(6*I*d*x - 2*I*c) + 28*d*e^(4*I*d*x - 4*I*c) + 8*d*e^(2*I*d*x - 6*I*c) + 70*d*e^(8*I*d*x) + d*e^(-8*I*c))

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maple [B]  time = 0.62, size = 331, normalized size = 2.30 \[ \frac {i a^{5} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{20}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{60}\right )+5 a^{5} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )-10 i a^{5} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{8}\left (d x +c \right )\right )}{10}-\frac {\left (\cos ^{8}\left (d x +c \right )\right )}{40}\right )-10 a^{5} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{9}\left (d x +c \right )\right )}{10}+\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{80}+\frac {7 d x}{256}+\frac {7 c}{256}\right )-\frac {i a^{5} \left (\cos ^{10}\left (d x +c \right )\right )}{2}+a^{5} \left (\frac {\left (\cos ^{9}\left (d x +c \right )+\frac {9 \left (\cos ^{7}\left (d x +c \right )\right )}{8}+\frac {21 \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {105 \left (\cos ^{3}\left (d x +c \right )\right )}{64}+\frac {315 \cos \left (d x +c \right )}{128}\right ) \sin \left (d x +c \right )}{10}+\frac {63 d x}{256}+\frac {63 c}{256}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^10*(a+I*a*tan(d*x+c))^5,x)

[Out]

1/d*(I*a^5*(-1/10*sin(d*x+c)^4*cos(d*x+c)^6-1/20*sin(d*x+c)^2*cos(d*x+c)^6-1/60*cos(d*x+c)^6)+5*a^5*(-1/10*sin
(d*x+c)^3*cos(d*x+c)^7-3/80*sin(d*x+c)*cos(d*x+c)^7+1/160*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(
d*x+c)+3/256*d*x+3/256*c)-10*I*a^5*(-1/10*sin(d*x+c)^2*cos(d*x+c)^8-1/40*cos(d*x+c)^8)-10*a^5*(-1/10*sin(d*x+c
)*cos(d*x+c)^9+1/80*(cos(d*x+c)^7+7/6*cos(d*x+c)^5+35/24*cos(d*x+c)^3+35/16*cos(d*x+c))*sin(d*x+c)+7/256*d*x+7
/256*c)-1/2*I*a^5*cos(d*x+c)^10+a^5*(1/10*(cos(d*x+c)^9+9/8*cos(d*x+c)^7+21/16*cos(d*x+c)^5+105/64*cos(d*x+c)^
3+315/128*cos(d*x+c))*sin(d*x+c)+63/256*d*x+63/256*c))

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maxima [A]  time = 0.65, size = 164, normalized size = 1.14 \[ \frac {120 \, {\left (d x + c\right )} a^{5} + \frac {120 \, a^{5} \tan \left (d x + c\right )^{9} + 560 \, a^{5} \tan \left (d x + c\right )^{7} + 1024 \, a^{5} \tan \left (d x + c\right )^{5} - 640 i \, a^{5} \tan \left (d x + c\right )^{4} - 1840 \, a^{5} \tan \left (d x + c\right )^{3} + 4480 i \, a^{5} \tan \left (d x + c\right )^{2} + 3720 \, a^{5} \tan \left (d x + c\right ) - 1024 i \, a^{5}}{\tan \left (d x + c\right )^{10} + 5 \, \tan \left (d x + c\right )^{8} + 10 \, \tan \left (d x + c\right )^{6} + 10 \, \tan \left (d x + c\right )^{4} + 5 \, \tan \left (d x + c\right )^{2} + 1}}{3840 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^10*(a+I*a*tan(d*x+c))^5,x, algorithm="maxima")

[Out]

1/3840*(120*(d*x + c)*a^5 + (120*a^5*tan(d*x + c)^9 + 560*a^5*tan(d*x + c)^7 + 1024*a^5*tan(d*x + c)^5 - 640*I
*a^5*tan(d*x + c)^4 - 1840*a^5*tan(d*x + c)^3 + 4480*I*a^5*tan(d*x + c)^2 + 3720*a^5*tan(d*x + c) - 1024*I*a^5
)/(tan(d*x + c)^10 + 5*tan(d*x + c)^8 + 10*tan(d*x + c)^6 + 10*tan(d*x + c)^4 + 5*tan(d*x + c)^2 + 1))/d

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mupad [B]  time = 3.64, size = 122, normalized size = 0.85 \[ \frac {a^5\,x}{32}+\frac {\frac {a^5\,{\mathrm {tan}\left (c+d\,x\right )}^4}{32}+\frac {a^5\,{\mathrm {tan}\left (c+d\,x\right )}^3\,5{}\mathrm {i}}{32}-\frac {31\,a^5\,{\mathrm {tan}\left (c+d\,x\right )}^2}{96}-\frac {a^5\,\mathrm {tan}\left (c+d\,x\right )\,35{}\mathrm {i}}{96}+\frac {4\,a^5}{15}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^5+{\mathrm {tan}\left (c+d\,x\right )}^4\,5{}\mathrm {i}-10\,{\mathrm {tan}\left (c+d\,x\right )}^3-{\mathrm {tan}\left (c+d\,x\right )}^2\,10{}\mathrm {i}+5\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^10*(a + a*tan(c + d*x)*1i)^5,x)

[Out]

(a^5*x)/32 + ((4*a^5)/15 - (a^5*tan(c + d*x)*35i)/96 - (31*a^5*tan(c + d*x)^2)/96 + (a^5*tan(c + d*x)^3*5i)/32
+ (a^5*tan(c + d*x)^4)/32)/(d*(5*tan(c + d*x) - tan(c + d*x)^2*10i - 10*tan(c + d*x)^3 + tan(c + d*x)^4*5i +
tan(c + d*x)^5 + 1i))

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sympy [A]  time = 0.73, size = 211, normalized size = 1.47 \[ \frac {a^{5} x}{32} + \begin {cases} - \frac {100663296 i a^{5} d^{4} e^{10 i c} e^{10 i d x} + 629145600 i a^{5} d^{4} e^{8 i c} e^{8 i d x} + 1677721600 i a^{5} d^{4} e^{6 i c} e^{6 i d x} + 2516582400 i a^{5} d^{4} e^{4 i c} e^{4 i d x} + 2516582400 i a^{5} d^{4} e^{2 i c} e^{2 i d x}}{32212254720 d^{5}} & \text {for}\: 32212254720 d^{5} \neq 0 \\x \left (\frac {a^{5} e^{10 i c}}{32} + \frac {5 a^{5} e^{8 i c}}{32} + \frac {5 a^{5} e^{6 i c}}{16} + \frac {5 a^{5} e^{4 i c}}{16} + \frac {5 a^{5} e^{2 i c}}{32}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**10*(a+I*a*tan(d*x+c))**5,x)

[Out]

a**5*x/32 + Piecewise((-(100663296*I*a**5*d**4*exp(10*I*c)*exp(10*I*d*x) + 629145600*I*a**5*d**4*exp(8*I*c)*ex
p(8*I*d*x) + 1677721600*I*a**5*d**4*exp(6*I*c)*exp(6*I*d*x) + 2516582400*I*a**5*d**4*exp(4*I*c)*exp(4*I*d*x) +
2516582400*I*a**5*d**4*exp(2*I*c)*exp(2*I*d*x))/(32212254720*d**5), Ne(32212254720*d**5, 0)), (x*(a**5*exp(10
*I*c)/32 + 5*a**5*exp(8*I*c)/32 + 5*a**5*exp(6*I*c)/16 + 5*a**5*exp(4*I*c)/16 + 5*a**5*exp(2*I*c)/32), True))

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