∫ cot9 _ 2 (c + dx)(a + ia tan(c + dx))3∕2(A + B tan(c + dx)) dx

3.541 \(\int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx\)

Optimal. Leaf size=245 \[ \frac {(2-2 i) a^{3/2} (A-i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {2 a (7 B+8 i A) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}+\frac {4 a (19 A-21 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {4 a (63 B+67 i A) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d} \]

[Out]

(2-2*I)*a^(3/2)*(A-I*B)*arctanh((1+I)*a^(1/2)*tan(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(1/2))*cot(d*x+c)^(1/2)*tan(

d*x+c)^(1/2)/d+4/105*a*(19*A-21*I*B)*cot(d*x+c)^(3/2)*(a+I*a*tan(d*x+c))^(1/2)/d-2/35*a*(8*I*A+7*B)*cot(d*x+c)

^(5/2)*(a+I*a*tan(d*x+c))^(1/2)/d-2/7*a*A*cot(d*x+c)^(7/2)*(a+I*a*tan(d*x+c))^(1/2)/d+4/105*a*(67*I*A+63*B)*co

t(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^(1/2)/d

________________________________________________________________________________________

Rubi [A] time = 0.89, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4241, 3593, 3598, 12, 3544, 205} \[ \frac {(2-2 i) a^{3/2} (A-i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {2 a (7 B+8 i A) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}+\frac {4 a (19 A-21 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {4 a (63 B+67 i A) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d} \]

Antiderivative was successfully veriﬁed.

[In]

Int[Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]

[Out]

((2 - 2*I)*a^(3/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot

[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (4*a*((67*I)*A + 63*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(105*

d) + (4*a*(19*A - (21*I)*B)*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d) - (2*a*((8*I)*A + 7*B)*Cot[

c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(35*d) - (2*a*A*Cot[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]])/(7*d

)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]

&& PosQ[a/b]

Rule 3544

Int[Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[

(-2*a*b)/f, Subst[Int[1/(a*c - b*d - 2*a^2*x^2), x], x, Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]]], x]

/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]

Rule 3593

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_

.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(a^2*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^

(n + 1))/(d*f*(b*c + a*d)*(n + 1)), x] - Dist[a/(d*(b*c + a*d)*(n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +

d*Tan[e + f*x])^(n + 1)*Simp[A*b*d*(m - n - 2) - B*(b*c*(m - 1) + a*d*(n + 1)) + (a*A*d*(m + n) - B*(a*c*(m -

1) + b*d*(n + 1)))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ

[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]

Rule 3598

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_

.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((A*d - B*c)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(f

*(n + 1)*(c^2 + d^2)), x] - Dist[1/(a*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n

+ 1)*Simp[A*(b*d*m - a*c*(n + 1)) - B*(b*c*m + a*d*(n + 1)) - a*(B*c - A*d)*(m + n + 1)*Tan[e + f*x], x], x],

x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, -1]

Rule 4241

Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m, Int[A

ctivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownTangentIntegrandQ

[u, x]

Rubi steps

\begin {align*} \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx &=\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac {9}{2}}(c+d x)} \, dx\\ &=-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d}+\frac {1}{7} \left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)} \left (\frac {1}{2} a (8 i A+7 B)-\frac {1}{2} a (6 A-7 i B) \tan (c+d x)\right )}{\tan ^{\frac {7}{2}}(c+d x)} \, dx\\ &=-\frac {2 a (8 i A+7 B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d}+\frac {\left (4 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)} \left (-\frac {1}{2} a^2 (19 A-21 i B)-a^2 (8 i A+7 B) \tan (c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)} \, dx}{35 a}\\ &=\frac {4 a (19 A-21 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a (8 i A+7 B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d}+\frac {\left (8 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)} \left (-\frac {1}{4} a^3 (67 i A+63 B)+\frac {1}{2} a^3 (19 A-21 i B) \tan (c+d x)\right )}{\tan ^{\frac {3}{2}}(c+d x)} \, dx}{105 a^2}\\ &=\frac {4 a (67 i A+63 B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {4 a (19 A-21 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a (8 i A+7 B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d}+\frac {\left (16 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {105 a^4 (A-i B) \sqrt {a+i a \tan (c+d x)}}{8 \sqrt {\tan (c+d x)}} \, dx}{105 a^3}\\ &=\frac {4 a (67 i A+63 B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {4 a (19 A-21 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a (8 i A+7 B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d}+\left (2 a (A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {\tan (c+d x)}} \, dx\\ &=\frac {4 a (67 i A+63 B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {4 a (19 A-21 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a (8 i A+7 B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d}-\frac {\left (4 i a^3 (A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{-i a-2 a^2 x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}\\ &=-\frac {(2+2 i) a^{3/2} (i A+B) \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {4 a (67 i A+63 B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {4 a (19 A-21 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a (8 i A+7 B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A] time = 8.15, size = 320, normalized size = 1.31 \[ \frac {(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \left (-2 i \sqrt {2} (A-i B) e^{-2 i (c+d x)} \sqrt {-1+e^{2 i (c+d x)}} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {\frac {i \left (1+e^{2 i (c+d x)}\right )}{-1+e^{2 i (c+d x)}}} \tanh ^{-1}\left (\frac {e^{i (c+d x)}}{\sqrt {-1+e^{2 i (c+d x)}}}\right )-\frac {1}{210} \sqrt {\cot (c+d x)} \csc ^3(c+d x) \sqrt {\sec (c+d x)} (\cos (c+d x)-i \sin (c+d x)) (7 (A+6 i B) \cos (c+d x)+(53 A-42 i B) \cos (3 (c+d x))+2 \sin (c+d x) ((147 B+158 i A) \cos (2 (c+d x))-110 i A-105 B))\right )}{d \sec ^{\frac {5}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]

[Out]

((((-2*I)*Sqrt[2]*(A - I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqr

t[(I*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*

x))]])/E^((2*I)*(c + d*x)) - (Sqrt[Cot[c + d*x]]*Csc[c + d*x]^3*Sqrt[Sec[c + d*x]]*(Cos[c + d*x] - I*Sin[c + d

*x])*(7*(A + (6*I)*B)*Cos[c + d*x] + (53*A - (42*I)*B)*Cos[3*(c + d*x)] + 2*((-110*I)*A - 105*B + ((158*I)*A +

147*B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/210)*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/(d*Sec[c + d*

x]^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))

________________________________________________________________________________________

fricas [B] time = 0.53, size = 564, normalized size = 2.30 \[ \frac {\sqrt {2} {\left ({\left (1688 i \, A + 1512 \, B\right )} a e^{\left (7 i \, d x + 7 i \, c\right )} + {\left (-2968 i \, A - 3192 \, B\right )} a e^{\left (5 i \, d x + 5 i \, c\right )} + {\left (3080 i \, A + 2520 \, B\right )} a e^{\left (3 i \, d x + 3 i \, c\right )} + {\left (-840 i \, A - 840 \, B\right )} a e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} + 105 \, \sqrt {\frac {{\left (-32 i \, A^{2} - 64 \, A B + 32 i \, B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (-\frac {{\left (8 \, {\left (A - i \, B\right )} a^{2} e^{\left (i \, d x + i \, c\right )} + \sqrt {2} \sqrt {\frac {{\left (-32 i \, A^{2} - 64 \, A B + 32 i \, B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{{\left (2 i \, A + 2 \, B\right )} a}\right ) - 105 \, \sqrt {\frac {{\left (-32 i \, A^{2} - 64 \, A B + 32 i \, B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (-\frac {{\left (8 \, {\left (A - i \, B\right )} a^{2} e^{\left (i \, d x + i \, c\right )} - \sqrt {2} \sqrt {\frac {{\left (-32 i \, A^{2} - 64 \, A B + 32 i \, B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{{\left (2 i \, A + 2 \, B\right )} a}\right )}{420 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^(9/2)*(a+I*a*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/420*(sqrt(2)*((1688*I*A + 1512*B)*a*e^(7*I*d*x + 7*I*c) + (-2968*I*A - 3192*B)*a*e^(5*I*d*x + 5*I*c) + (3080

*I*A + 2520*B)*a*e^(3*I*d*x + 3*I*c) + (-840*I*A - 840*B)*a*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))

*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)) + 105*sqrt((-32*I*A^2 - 64*A*B + 32*I*B^2)*a^3/d^

2)*(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) - d)*log(-(8*(A - I*B)*a^2*e^(I*

d*x + I*c) + sqrt(2)*sqrt((-32*I*A^2 - 64*A*B + 32*I*B^2)*a^3/d^2)*(d*e^(2*I*d*x + 2*I*c) - d)*sqrt(a/(e^(2*I*

d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))*e^(-I*d*x - I*c)/((2*I*A + 2*B

)*a)) - 105*sqrt((-32*I*A^2 - 64*A*B + 32*I*B^2)*a^3/d^2)*(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*x + 4*I*c) + 3

*d*e^(2*I*d*x + 2*I*c) - d)*log(-(8*(A - I*B)*a^2*e^(I*d*x + I*c) - sqrt(2)*sqrt((-32*I*A^2 - 64*A*B + 32*I*B^

2)*a^3/d^2)*(d*e^(2*I*d*x + 2*I*c) - d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^

(2*I*d*x + 2*I*c) - 1)))*e^(-I*d*x - I*c)/((2*I*A + 2*B)*a)))/(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*x + 4*I*c)

+ 3*d*e^(2*I*d*x + 2*I*c) - d)

________________________________________________________________________________________

giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^(9/2)*(a+I*a*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [B] time = 4.09, size = 3124, normalized size = 12.75 \[ \text {output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^(9/2)*(a+I*a*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x)

[Out]

-1/105/d*(211*A*2^(1/2)*cos(d*x+c)^4-53*A*2^(1/2)*cos(d*x+c)^3-330*A*2^(1/2)*cos(d*x+c)^2+38*A*2^(1/2)*cos(d*x

+c)+210*I*A*cos(d*x+c)^4*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/

2)+210*I*A*cos(d*x+c)^4*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1

)+105*I*A*cos(d*x+c)^4*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(

d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1

))-210*I*B*cos(d*x+c)^4*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2

)-210*I*B*cos(d*x+c)^4*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)

-105*I*B*cos(d*x+c)^4*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d

*x+c)-cos(d*x+c)-sin(d*x+c)+1)/(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)

)+211*I*A*cos(d*x+c)^3*sin(d*x+c)*2^(1/2)+126*B*sin(d*x+c)*2^(1/2)+134*A*2^(1/2)-420*A*((-1+cos(d*x+c))/sin(d*

x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*cos(d*x+c)^2-420*B*arctan(2^(1/2)*((-1+cos(d*

x+c))/sin(d*x+c))^(1/2)+1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^2-420*B*((-1+cos(d*x+c))/sin(d*x+c))^

(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*cos(d*x+c)^2-210*B*((-1+cos(d*x+c))/sin(d*x+c))^(1/

2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(2^(1/2)*((-1+cos(d*x+c

))/sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))*cos(d*x+c)^2+134*I*A*sin(d*x+c)*2^(1/2)+210*I*A*arct

an(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+210*I*A*((-1+cos(d*x+c))/s

in(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)+105*I*A*((-1+cos(d*x+c))/sin(d*x+c))^(1/

2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(2^(1/2)*((-1+cos(d*x+c

))/sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))-210*I*B*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^

(1/2)+1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-210*I*B*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos

(d*x+c))/sin(d*x+c))^(1/2)-1)-105*I*B*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x

+c))^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1)/(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+

c)+sin(d*x+c)-1))-189*I*B*cos(d*x+c)^4*2^(1/2)+42*I*B*cos(d*x+c)^3*2^(1/2)+315*I*B*cos(d*x+c)^2*2^(1/2)+210*A*

cos(d*x+c)^4*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+210*A*cos

(d*x+c)^4*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)+105*A*cos(d*

x+c)^4*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*x+c

)-sin(d*x+c)+1)/(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))+210*B*cos(d*x

+c)^4*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+210*B*cos(d*x+c)

^4*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)+105*B*cos(d*x+c)^4*

((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d

*x+c)-1)/(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))-42*I*B*cos(d*x+c)*2^

(1/2)-126*I*B*2^(1/2)-420*A*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*((-1+cos(d*x+c))/sin(d*x+c))^

(1/2)*cos(d*x+c)^2-210*A*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*si

n(d*x+c)-cos(d*x+c)-sin(d*x+c)+1)/(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)

-1))*cos(d*x+c)^2+189*B*cos(d*x+c)^3*sin(d*x+c)*2^(1/2)-168*B*2^(1/2)*cos(d*x+c)*sin(d*x+c)-147*B*2^(1/2)*cos(

d*x+c)^2*sin(d*x+c)+210*A*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*((-1+cos(d*x+c))/sin(d*x+c))^(1

/2)+105*A*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*

x+c)-sin(d*x+c)+1)/(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))+210*A*((-1

+cos(d*x+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)+210*B*arctan(2^(1/2)*((-1+

cos(d*x+c))/sin(d*x+c))^(1/2)+1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+210*B*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*a

rctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)+105*B*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1

+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)

*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))-158*I*A*cos(d*x+c)^2*sin(d*x+c)*2^(1/2)-420*I*A*cos(d*x+c)^2*arctan(2^(1

/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-420*I*A*cos(d*x+c)^2*((-1+cos(d*x

+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)-210*I*A*cos(d*x+c)^2*((-1+cos(d*x+

c))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(2^(

1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))+420*I*B*cos(d*x+c)^2*arctan(2^(1/

2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+420*I*B*cos(d*x+c)^2*((-1+cos(d*x+

c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)+210*I*B*cos(d*x+c)^2*((-1+cos(d*x+c

))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1)/(2^(1

/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))-172*I*A*cos(d*x+c)*sin(d*x+c)*2^(1

/2))*(cos(d*x+c)/sin(d*x+c))^(9/2)*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)*sin(d*x+c)/(I*sin(d*x+c)+cos

(d*x+c)-1)/cos(d*x+c)^4*a*2^(1/2)

________________________________________________________________________________________

maxima [B] time = 7.07, size = 3719, normalized size = 15.18 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^(9/2)*(a+I*a*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/176400*(sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*((((352800*I - 352800)*A + (3

52800*I + 352800)*B)*a*cos(7*d*x + 7*c) + (-(176400*I - 176400)*A - (529200*I + 529200)*B)*a*cos(5*d*x + 5*c)

+ ((167580*I - 167580)*A + (255780*I + 255780)*B)*a*cos(3*d*x + 3*c) + ((59220*I - 59220)*A - (79380*I + 79380

)*B)*a*cos(d*x + c) + (-(352800*I + 352800)*A + (352800*I - 352800)*B)*a*sin(7*d*x + 7*c) + ((176400*I + 17640

0)*A - (529200*I - 529200)*B)*a*sin(5*d*x + 5*c) + (-(167580*I + 167580)*A + (255780*I - 255780)*B)*a*sin(3*d*

x + 3*c) + (-(59220*I + 59220)*A - (79380*I - 79380)*B)*a*sin(d*x + c))*cos(7/2*arctan2(sin(2*d*x + 2*c), cos(

2*d*x + 2*c) - 1)) + (((-(236880*I - 236880)*A - (199920*I + 199920)*B)*a*cos(d*x + c) + ((236880*I + 236880)*

A - (199920*I - 199920)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^2 + (-(236880*I - 236880)*A - (199920*I + 199920)*

B)*a*cos(d*x + c) + ((-(236880*I - 236880)*A - (199920*I + 199920)*B)*a*cos(d*x + c) + ((236880*I + 236880)*A

- (199920*I - 199920)*B)*a*sin(d*x + c))*sin(2*d*x + 2*c)^2 + ((236880*I + 236880)*A - (199920*I - 199920)*B)*

a*sin(d*x + c) + (((352800*I - 352800)*A + (352800*I + 352800)*B)*a*cos(2*d*x + 2*c)^2 + ((352800*I - 352800)*

A + (352800*I + 352800)*B)*a*sin(2*d*x + 2*c)^2 + (-(705600*I - 705600)*A - (705600*I + 705600)*B)*a*cos(2*d*x

+ 2*c) + ((352800*I - 352800)*A + (352800*I + 352800)*B)*a)*cos(3*d*x + 3*c) + (((473760*I - 473760)*A + (399

840*I + 399840)*B)*a*cos(d*x + c) + (-(473760*I + 473760)*A + (399840*I - 399840)*B)*a*sin(d*x + c))*cos(2*d*x

+ 2*c) + ((-(352800*I + 352800)*A + (352800*I - 352800)*B)*a*cos(2*d*x + 2*c)^2 + (-(352800*I + 352800)*A + (

352800*I - 352800)*B)*a*sin(2*d*x + 2*c)^2 + ((705600*I + 705600)*A - (705600*I - 705600)*B)*a*cos(2*d*x + 2*c

) + (-(352800*I + 352800)*A + (352800*I - 352800)*B)*a)*sin(3*d*x + 3*c))*cos(3/2*arctan2(sin(2*d*x + 2*c), co

s(2*d*x + 2*c) - 1)) + (((352800*I + 352800)*A - (352800*I - 352800)*B)*a*cos(7*d*x + 7*c) + (-(176400*I + 176

400)*A + (529200*I - 529200)*B)*a*cos(5*d*x + 5*c) + ((167580*I + 167580)*A - (255780*I - 255780)*B)*a*cos(3*d

*x + 3*c) + ((59220*I + 59220)*A + (79380*I - 79380)*B)*a*cos(d*x + c) + ((352800*I - 352800)*A + (352800*I +

352800)*B)*a*sin(7*d*x + 7*c) + (-(176400*I - 176400)*A - (529200*I + 529200)*B)*a*sin(5*d*x + 5*c) + ((167580

*I - 167580)*A + (255780*I + 255780)*B)*a*sin(3*d*x + 3*c) + ((59220*I - 59220)*A - (79380*I + 79380)*B)*a*sin

(d*x + c))*sin(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + (((-(236880*I + 236880)*A + (199920*I -

199920)*B)*a*cos(d*x + c) + (-(236880*I - 236880)*A - (199920*I + 199920)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^

2 + (-(236880*I + 236880)*A + (199920*I - 199920)*B)*a*cos(d*x + c) + ((-(236880*I + 236880)*A + (199920*I - 1

99920)*B)*a*cos(d*x + c) + (-(236880*I - 236880)*A - (199920*I + 199920)*B)*a*sin(d*x + c))*sin(2*d*x + 2*c)^2

+ (-(236880*I - 236880)*A - (199920*I + 199920)*B)*a*sin(d*x + c) + (((352800*I + 352800)*A - (352800*I - 352

800)*B)*a*cos(2*d*x + 2*c)^2 + ((352800*I + 352800)*A - (352800*I - 352800)*B)*a*sin(2*d*x + 2*c)^2 + (-(70560

0*I + 705600)*A + (705600*I - 705600)*B)*a*cos(2*d*x + 2*c) + ((352800*I + 352800)*A - (352800*I - 352800)*B)*

a)*cos(3*d*x + 3*c) + (((473760*I + 473760)*A - (399840*I - 399840)*B)*a*cos(d*x + c) + ((473760*I - 473760)*A

+ (399840*I + 399840)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c) + (((352800*I - 352800)*A + (352800*I + 352800)*B)*

a*cos(2*d*x + 2*c)^2 + ((352800*I - 352800)*A + (352800*I + 352800)*B)*a*sin(2*d*x + 2*c)^2 + (-(705600*I - 70

5600)*A - (705600*I + 705600)*B)*a*cos(2*d*x + 2*c) + ((352800*I - 352800)*A + (352800*I + 352800)*B)*a)*sin(3

*d*x + 3*c))*sin(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)))*sqrt(a) + ((((352800*I + 352800)*A - (3

52800*I - 352800)*B)*a*cos(2*d*x + 2*c)^4 + ((352800*I + 352800)*A - (352800*I - 352800)*B)*a*sin(2*d*x + 2*c)

^4 + (-(1411200*I + 1411200)*A + (1411200*I - 1411200)*B)*a*cos(2*d*x + 2*c)^3 + ((2116800*I + 2116800)*A - (2

116800*I - 2116800)*B)*a*cos(2*d*x + 2*c)^2 + (-(1411200*I + 1411200)*A + (1411200*I - 1411200)*B)*a*cos(2*d*x

+ 2*c) + (((705600*I + 705600)*A - (705600*I - 705600)*B)*a*cos(2*d*x + 2*c)^2 + (-(1411200*I + 1411200)*A +

(1411200*I - 1411200)*B)*a*cos(2*d*x + 2*c) + ((705600*I + 705600)*A - (705600*I - 705600)*B)*a)*sin(2*d*x + 2

*c)^2 + ((352800*I + 352800)*A - (352800*I - 352800)*B)*a)*arctan2(2*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2

- 2*cos(2*d*x + 2*c) + 1)^(1/4)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + 2*sin(d*x + c), 2*(

cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(

2*d*x + 2*c) - 1)) + 2*cos(d*x + c)) + ((-(176400*I - 176400)*A - (176400*I + 176400)*B)*a*cos(2*d*x + 2*c)^4

+ (-(176400*I - 176400)*A - (176400*I + 176400)*B)*a*sin(2*d*x + 2*c)^4 + ((705600*I - 705600)*A + (705600*I +

705600)*B)*a*cos(2*d*x + 2*c)^3 + (-(1058400*I - 1058400)*A - (1058400*I + 1058400)*B)*a*cos(2*d*x + 2*c)^2 +

((705600*I - 705600)*A + (705600*I + 705600)*B)*a*cos(2*d*x + 2*c) + ((-(352800*I - 352800)*A - (352800*I + 3

52800)*B)*a*cos(2*d*x + 2*c)^2 + ((705600*I - 705600)*A + (705600*I + 705600)*B)*a*cos(2*d*x + 2*c) + (-(35280

0*I - 352800)*A - (352800*I + 352800)*B)*a)*sin(2*d*x + 2*c)^2 + (-(176400*I - 176400)*A - (176400*I + 176400)

*B)*a)*log(4*cos(d*x + c)^2 + 4*sin(d*x + c)^2 + 4*sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x

+ 2*c) + 1)*(cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1))^2 + sin(1/2*arctan2(sin(2*d*x + 2*c), co

s(2*d*x + 2*c) - 1))^2) + 8*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*(cos(d*x

+ c)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + sin(d*x + c)*sin(1/2*arctan2(sin(2*d*x + 2*c),

cos(2*d*x + 2*c) - 1)))))*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*sqrt(a) +

(((((524580*I - 524580)*A + (496860*I + 496860)*B)*a*cos(d*x + c) + (-(524580*I + 524580)*A + (496860*I - 4968

60)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^2 + ((524580*I - 524580)*A + (496860*I + 496860)*B)*a*cos(d*x + c) + (

((524580*I - 524580)*A + (496860*I + 496860)*B)*a*cos(d*x + c) + (-(524580*I + 524580)*A + (496860*I - 496860)

*B)*a*sin(d*x + c))*sin(2*d*x + 2*c)^2 + (-(524580*I + 524580)*A + (496860*I - 496860)*B)*a*sin(d*x + c) + (((

352800*I - 352800)*A + (352800*I + 352800)*B)*a*cos(2*d*x + 2*c)^2 + ((352800*I - 352800)*A + (352800*I + 3528

00)*B)*a*sin(2*d*x + 2*c)^2 + (-(705600*I - 705600)*A - (705600*I + 705600)*B)*a*cos(2*d*x + 2*c) + ((352800*I

- 352800)*A + (352800*I + 352800)*B)*a)*cos(5*d*x + 5*c) + ((-(823200*I - 823200)*A - (823200*I + 823200)*B)*

a*cos(2*d*x + 2*c)^2 + (-(823200*I - 823200)*A - (823200*I + 823200)*B)*a*sin(2*d*x + 2*c)^2 + ((1646400*I - 1

646400)*A + (1646400*I + 1646400)*B)*a*cos(2*d*x + 2*c) + (-(823200*I - 823200)*A - (823200*I + 823200)*B)*a)*

cos(3*d*x + 3*c) + ((-(1049160*I - 1049160)*A - (993720*I + 993720)*B)*a*cos(d*x + c) + ((1049160*I + 1049160)

*A - (993720*I - 993720)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c) + ((-(352800*I + 352800)*A + (352800*I - 352800)*

B)*a*cos(2*d*x + 2*c)^2 + (-(352800*I + 352800)*A + (352800*I - 352800)*B)*a*sin(2*d*x + 2*c)^2 + ((705600*I +

705600)*A - (705600*I - 705600)*B)*a*cos(2*d*x + 2*c) + (-(352800*I + 352800)*A + (352800*I - 352800)*B)*a)*s

in(5*d*x + 5*c) + (((823200*I + 823200)*A - (823200*I - 823200)*B)*a*cos(2*d*x + 2*c)^2 + ((823200*I + 823200)

*A - (823200*I - 823200)*B)*a*sin(2*d*x + 2*c)^2 + (-(1646400*I + 1646400)*A + (1646400*I - 1646400)*B)*a*cos(

2*d*x + 2*c) + ((823200*I + 823200)*A - (823200*I - 823200)*B)*a)*sin(3*d*x + 3*c))*cos(5/2*arctan2(sin(2*d*x

+ 2*c), cos(2*d*x + 2*c) - 1)) + (((-(349440*I - 349440)*A - (423360*I + 423360)*B)*a*cos(d*x + c) + ((349440*

I + 349440)*A - (423360*I - 423360)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^4 + ((-(349440*I - 349440)*A - (423360

*I + 423360)*B)*a*cos(d*x + c) + ((349440*I + 349440)*A - (423360*I - 423360)*B)*a*sin(d*x + c))*sin(2*d*x + 2

*c)^4 + (((1397760*I - 1397760)*A + (1693440*I + 1693440)*B)*a*cos(d*x + c) + (-(1397760*I + 1397760)*A + (169

3440*I - 1693440)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^3 + ((-(2096640*I - 2096640)*A - (2540160*I + 2540160)*B

)*a*cos(d*x + c) + ((2096640*I + 2096640)*A - (2540160*I - 2540160)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^2 + (-

(349440*I - 349440)*A - (423360*I + 423360)*B)*a*cos(d*x + c) + (((-(698880*I - 698880)*A - (846720*I + 846720

)*B)*a*cos(d*x + c) + ((698880*I + 698880)*A - (846720*I - 846720)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^2 + (-(

698880*I - 698880)*A - (846720*I + 846720)*B)*a*cos(d*x + c) + ((698880*I + 698880)*A - (846720*I - 846720)*B)

*a*sin(d*x + c) + (((1397760*I - 1397760)*A + (1693440*I + 1693440)*B)*a*cos(d*x + c) + (-(1397760*I + 1397760

)*A + (1693440*I - 1693440)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + ((349440*I + 349440)*A -

(423360*I - 423360)*B)*a*sin(d*x + c) + (((1397760*I - 1397760)*A + (1693440*I + 1693440)*B)*a*cos(d*x + c) +

(-(1397760*I + 1397760)*A + (1693440*I - 1693440)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c))*cos(1/2*arctan2(sin(2*

d*x + 2*c), cos(2*d*x + 2*c) - 1)) + ((((524580*I + 524580)*A - (496860*I - 496860)*B)*a*cos(d*x + c) + ((5245

80*I - 524580)*A + (496860*I + 496860)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^2 + ((524580*I + 524580)*A - (49686

0*I - 496860)*B)*a*cos(d*x + c) + (((524580*I + 524580)*A - (496860*I - 496860)*B)*a*cos(d*x + c) + ((524580*I

- 524580)*A + (496860*I + 496860)*B)*a*sin(d*x + c))*sin(2*d*x + 2*c)^2 + ((524580*I - 524580)*A + (496860*I

+ 496860)*B)*a*sin(d*x + c) + (((352800*I + 352800)*A - (352800*I - 352800)*B)*a*cos(2*d*x + 2*c)^2 + ((352800

*I + 352800)*A - (352800*I - 352800)*B)*a*sin(2*d*x + 2*c)^2 + (-(705600*I + 705600)*A + (705600*I - 705600)*B

)*a*cos(2*d*x + 2*c) + ((352800*I + 352800)*A - (352800*I - 352800)*B)*a)*cos(5*d*x + 5*c) + ((-(823200*I + 82

3200)*A + (823200*I - 823200)*B)*a*cos(2*d*x + 2*c)^2 + (-(823200*I + 823200)*A + (823200*I - 823200)*B)*a*sin

(2*d*x + 2*c)^2 + ((1646400*I + 1646400)*A - (1646400*I - 1646400)*B)*a*cos(2*d*x + 2*c) + (-(823200*I + 82320

0)*A + (823200*I - 823200)*B)*a)*cos(3*d*x + 3*c) + ((-(1049160*I + 1049160)*A + (993720*I - 993720)*B)*a*cos(

d*x + c) + (-(1049160*I - 1049160)*A - (993720*I + 993720)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c) + (((352800*I -

352800)*A + (352800*I + 352800)*B)*a*cos(2*d*x + 2*c)^2 + ((352800*I - 352800)*A + (352800*I + 352800)*B)*a*s

in(2*d*x + 2*c)^2 + (-(705600*I - 705600)*A - (705600*I + 705600)*B)*a*cos(2*d*x + 2*c) + ((352800*I - 352800)

*A + (352800*I + 352800)*B)*a)*sin(5*d*x + 5*c) + ((-(823200*I - 823200)*A - (823200*I + 823200)*B)*a*cos(2*d*

x + 2*c)^2 + (-(823200*I - 823200)*A - (823200*I + 823200)*B)*a*sin(2*d*x + 2*c)^2 + ((1646400*I - 1646400)*A

+ (1646400*I + 1646400)*B)*a*cos(2*d*x + 2*c) + (-(823200*I - 823200)*A - (823200*I + 823200)*B)*a)*sin(3*d*x

+ 3*c))*sin(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + (((-(349440*I + 349440)*A + (423360*I - 423

360)*B)*a*cos(d*x + c) + (-(349440*I - 349440)*A - (423360*I + 423360)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^4 +

((-(349440*I + 349440)*A + (423360*I - 423360)*B)*a*cos(d*x + c) + (-(349440*I - 349440)*A - (423360*I + 4233

60)*B)*a*sin(d*x + c))*sin(2*d*x + 2*c)^4 + (((1397760*I + 1397760)*A - (1693440*I - 1693440)*B)*a*cos(d*x + c

) + ((1397760*I - 1397760)*A + (1693440*I + 1693440)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^3 + ((-(2096640*I + 2

096640)*A + (2540160*I - 2540160)*B)*a*cos(d*x + c) + (-(2096640*I - 2096640)*A - (2540160*I + 2540160)*B)*a*s

in(d*x + c))*cos(2*d*x + 2*c)^2 + (-(349440*I + 349440)*A + (423360*I - 423360)*B)*a*cos(d*x + c) + (((-(69888

0*I + 698880)*A + (846720*I - 846720)*B)*a*cos(d*x + c) + (-(698880*I - 698880)*A - (846720*I + 846720)*B)*a*s

in(d*x + c))*cos(2*d*x + 2*c)^2 + (-(698880*I + 698880)*A + (846720*I - 846720)*B)*a*cos(d*x + c) + (-(698880*

I - 698880)*A - (846720*I + 846720)*B)*a*sin(d*x + c) + (((1397760*I + 1397760)*A - (1693440*I - 1693440)*B)*a

*cos(d*x + c) + ((1397760*I - 1397760)*A + (1693440*I + 1693440)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c))*sin(2*d*

x + 2*c)^2 + (-(349440*I - 349440)*A - (423360*I + 423360)*B)*a*sin(d*x + c) + (((1397760*I + 1397760)*A - (16

93440*I - 1693440)*B)*a*cos(d*x + c) + ((1397760*I - 1397760)*A + (1693440*I + 1693440)*B)*a*sin(d*x + c))*cos

(2*d*x + 2*c))*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)))*sqrt(a))/((cos(2*d*x + 2*c)^4 + sin(2

*d*x + 2*c)^4 - 4*cos(2*d*x + 2*c)^3 + 2*(cos(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c)^2 + 6*

cos(2*d*x + 2*c)^2 - 4*cos(2*d*x + 2*c) + 1)*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1

)^(1/4)*d)

________________________________________________________________________________________

mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {cot}\left (c+d\,x\right )}^{9/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2} \,d x \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^(9/2)*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^(3/2),x)

[Out]

int(cot(c + d*x)^(9/2)*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^(3/2), x)

________________________________________________________________________________________

sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**(9/2)*(a+I*a*tan(d*x+c))**(3/2)*(A+B*tan(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]