\(\int \cos ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx\) [299]

Optimal result

Integrand size = 26, antiderivative size = 239 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=-\frac {105 i a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} d}+\frac {35 i a^3}{128 d (a+i a \tan (c+d x))^{3/2}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{3/2}}-\frac {3 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{3/2}}-\frac {21 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{3/2}}+\frac {105 i a^2}{256 d \sqrt {a+i a \tan (c+d x)}} \]

[Out]

-105/512*I*a^(3/2)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2))/d*2^(1/2)+105/256*I*a^2/d/(a+I*a*tan(
d*x+c))^(1/2)+35/128*I*a^3/d/(a+I*a*tan(d*x+c))^(3/2)-1/6*I*a^6/d/(a-I*a*tan(d*x+c))^3/(a+I*a*tan(d*x+c))^(3/2
)-3/16*I*a^5/d/(a-I*a*tan(d*x+c))^2/(a+I*a*tan(d*x+c))^(3/2)-21/64*I*a^4/d/(a-I*a*tan(d*x+c))/(a+I*a*tan(d*x+c
))^(3/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3568, 44, 53, 65, 212} \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=-\frac {105 i a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} d}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{3/2}}-\frac {3 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{3/2}}-\frac {21 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{3/2}}+\frac {35 i a^3}{128 d (a+i a \tan (c+d x))^{3/2}}+\frac {105 i a^2}{256 d \sqrt {a+i a \tan (c+d x)}} \]

[In]

Int[Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^(3/2),x]

[Out]

(((-105*I)/256)*a^(3/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*d) + (((35*I)/128)*a^3
)/(d*(a + I*a*Tan[c + d*x])^(3/2)) - ((I/6)*a^6)/(d*(a - I*a*Tan[c + d*x])^3*(a + I*a*Tan[c + d*x])^(3/2)) - (
((3*I)/16)*a^5)/(d*(a - I*a*Tan[c + d*x])^2*(a + I*a*Tan[c + d*x])^(3/2)) - (((21*I)/64)*a^4)/(d*(a - I*a*Tan[
c + d*x])*(a + I*a*Tan[c + d*x])^(3/2)) + (((105*I)/256)*a^2)/(d*Sqrt[a + I*a*Tan[c + d*x]])

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 212

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

Rule 3568

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*} \text {integral}& = -\frac {\left (i a^7\right ) \text {Subst}\left (\int \frac {1}{(a-x)^4 (a+x)^{5/2}} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{3/2}}-\frac {\left (3 i a^6\right ) \text {Subst}\left (\int \frac {1}{(a-x)^3 (a+x)^{5/2}} \, dx,x,i a \tan (c+d x)\right )}{4 d} \\ & = -\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{3/2}}-\frac {3 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{3/2}}-\frac {\left (21 i a^5\right ) \text {Subst}\left (\int \frac {1}{(a-x)^2 (a+x)^{5/2}} \, dx,x,i a \tan (c+d x)\right )}{32 d} \\ & = -\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{3/2}}-\frac {3 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{3/2}}-\frac {21 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{3/2}}-\frac {\left (105 i a^4\right ) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{5/2}} \, dx,x,i a \tan (c+d x)\right )}{128 d} \\ & = \frac {35 i a^3}{128 d (a+i a \tan (c+d x))^{3/2}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{3/2}}-\frac {3 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{3/2}}-\frac {21 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{3/2}}-\frac {\left (105 i a^3\right ) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,i a \tan (c+d x)\right )}{256 d} \\ & = \frac {35 i a^3}{128 d (a+i a \tan (c+d x))^{3/2}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{3/2}}-\frac {3 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{3/2}}-\frac {21 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{3/2}}+\frac {105 i a^2}{256 d \sqrt {a+i a \tan (c+d x)}}-\frac {\left (105 i a^2\right ) \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{512 d} \\ & = \frac {35 i a^3}{128 d (a+i a \tan (c+d x))^{3/2}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{3/2}}-\frac {3 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{3/2}}-\frac {21 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{3/2}}+\frac {105 i a^2}{256 d \sqrt {a+i a \tan (c+d x)}}-\frac {\left (105 i a^2\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{256 d} \\ & = -\frac {105 i a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} d}+\frac {35 i a^3}{128 d (a+i a \tan (c+d x))^{3/2}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{3/2}}-\frac {3 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{3/2}}-\frac {21 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{3/2}}+\frac {105 i a^2}{256 d \sqrt {a+i a \tan (c+d x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.15 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.22 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\frac {i a^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},4,-\frac {1}{2},\frac {1}{2} (1+i \tan (c+d x))\right )}{24 d (a+i a \tan (c+d x))^{3/2}} \]

[In]

Integrate[Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^(3/2),x]

[Out]

((I/24)*a^3*Hypergeometric2F1[-3/2, 4, -1/2, (1 + I*Tan[c + d*x])/2])/(d*(a + I*a*Tan[c + d*x])^(3/2))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 685 vs. \(2 (193 ) = 386\).

Time = 14.05 (sec) , antiderivative size = 686, normalized size of antiderivative = 2.87

method	result	size
default	\(-\frac {\left (-\tan \left (d x +c \right )+i\right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, a \cos \left (d x +c \right ) \left (384 i \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+630 i \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \left (\cos ^{2}\left (d x +c \right )\right )+630 i \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right )-128 \left (\cos ^{5}\left (d x +c \right )\right )+315 i \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+315 i \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sin \left (d x +c \right )+840 i \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+630 \,\operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right ) \sin \left (d x +c \right )-630 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \left (\cos ^{2}\left (d x +c \right )\right )-315 i \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+315 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sin \left (d x +c \right )-315 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \cos \left (d x +c \right )-504 \left (\cos ^{3}\left (d x +c \right )\right )+315 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )+630 \cos \left (d x +c \right )\right )}{1536 d}\)	\(686\)

[In]

int(cos(d*x+c)^6*(a+I*a*tan(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/1536/d*(-tan(d*x+c)+I)*(a*(1+I*tan(d*x+c)))^(1/2)*a*cos(d*x+c)*(384*I*cos(d*x+c)^4*sin(d*x+c)+630*I*(-cos(d
*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^2
+630*I*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)*sin(d*x+c)-128
*cos(d*x+c)^5+315*I*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x
+c)+1))^(1/2))*cos(d*x+c)+315*I*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*
sin(d*x+c)+840*I*cos(d*x+c)^2*sin(d*x+c)+630*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1
/2))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)*sin(d*x+c)-630*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan((-
cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^2-315*I*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+c)/(co
s(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+315*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+c)/(cos
(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*sin(d*x+c)-315*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan((-cos(
d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)-504*cos(d*x+c)^3+315*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan((-cos(
d*x+c)/(cos(d*x+c)+1))^(1/2))+630*cos(d*x+c))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.30 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=-\frac {{\left (315 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{3}}{d^{2}}} d e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (-\frac {4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {-\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} - a^{2} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{a}\right ) - 315 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{3}}{d^{2}}} d e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (-\frac {4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {-\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} - a^{2} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{a}\right ) - \sqrt {2} {\left (-8 i \, a e^{\left (10 i \, d x + 10 i \, c\right )} - 58 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} - 215 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 43 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 224 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 16 i \, a\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{1536 \, d} \]

[In]

integrate(cos(d*x+c)^6*(a+I*a*tan(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

-1/1536*(315*sqrt(1/2)*sqrt(-a^3/d^2)*d*e^(3*I*d*x + 3*I*c)*log(-4*(sqrt(2)*sqrt(1/2)*(I*d*e^(2*I*d*x + 2*I*c)
+ I*d)*sqrt(-a^3/d^2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)) - a^2*e^(I*d*x + I*c))*e^(-I*d*x - I*c)/a) - 315*sqrt
(1/2)*sqrt(-a^3/d^2)*d*e^(3*I*d*x + 3*I*c)*log(-4*(sqrt(2)*sqrt(1/2)*(-I*d*e^(2*I*d*x + 2*I*c) - I*d)*sqrt(-a^
3/d^2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)) - a^2*e^(I*d*x + I*c))*e^(-I*d*x - I*c)/a) - sqrt(2)*(-8*I*a*e^(10*I*
d*x + 10*I*c) - 58*I*a*e^(8*I*d*x + 8*I*c) - 215*I*a*e^(6*I*d*x + 6*I*c) + 43*I*a*e^(4*I*d*x + 4*I*c) + 224*I*
a*e^(2*I*d*x + 2*I*c) + 16*I*a)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-3*I*d*x - 3*I*c)/d

Sympy [F(-1)]

Timed out. \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*(a+I*a*tan(d*x+c))**(3/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.89 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\frac {i \, {\left (315 \, \sqrt {2} a^{\frac {5}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right ) + \frac {4 \, {\left (315 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4} a^{3} - 1680 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} a^{4} + 2772 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a^{5} - 1152 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{6} - 256 \, a^{7}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} - 6 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a + 12 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - 8 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3}}\right )}}{3072 \, a d} \]

[In]

integrate(cos(d*x+c)^6*(a+I*a*tan(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

1/3072*I*(315*sqrt(2)*a^(5/2)*log(-(sqrt(2)*sqrt(a) - sqrt(I*a*tan(d*x + c) + a))/(sqrt(2)*sqrt(a) + sqrt(I*a*
tan(d*x + c) + a))) + 4*(315*(I*a*tan(d*x + c) + a)^4*a^3 - 1680*(I*a*tan(d*x + c) + a)^3*a^4 + 2772*(I*a*tan(
d*x + c) + a)^2*a^5 - 1152*(I*a*tan(d*x + c) + a)*a^6 - 256*a^7)/((I*a*tan(d*x + c) + a)^(9/2) - 6*(I*a*tan(d*
x + c) + a)^(7/2)*a + 12*(I*a*tan(d*x + c) + a)^(5/2)*a^2 - 8*(I*a*tan(d*x + c) + a)^(3/2)*a^3))/(a*d)

Giac [F(-1)]

Timed out. \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)^6*(a+I*a*tan(d*x+c))^(3/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\int {\cos \left (c+d\,x\right )}^6\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2} \,d x \]

[In]

int(cos(c + d*x)^6*(a + a*tan(c + d*x)*1i)^(3/2),x)

[Out]

int(cos(c + d*x)^6*(a + a*tan(c + d*x)*1i)^(3/2), x)