3.513 \(\int \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx\)
Optimal. Leaf size=171 \[ \frac{8 a^3 (23 A-21 i B) \cot ^{\frac{3}{2}}(c+d x)}{105 d}-\frac{2 (7 B+11 i A) \cot ^{\frac{3}{2}}(c+d x) \left (a^3 \cot (c+d x)+i a^3\right )}{35 d}+\frac{8 a^3 (B+i A) \sqrt{\cot (c+d x)}}{d}+\frac{8 \sqrt [4]{-1} a^3 (B+i A) \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\cot (c+d x)}\right )}{d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (a \cot (c+d x)+i a)^2}{7 d} \]
[Out]
(8*(-1)^(1/4)*a^3*(I*A + B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d + (8*a^3*(I*A + B)*Sqrt[Cot[c + d*x]])/d
+ (8*a^3*(23*A - (21*I)*B)*Cot[c + d*x]^(3/2))/(105*d) - (2*a*A*Cot[c + d*x]^(3/2)*(I*a + a*Cot[c + d*x])^2)/
(7*d) - (2*((11*I)*A + 7*B)*Cot[c + d*x]^(3/2)*(I*a^3 + a^3*Cot[c + d*x]))/(35*d)
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Rubi [A] time = 0.528106, antiderivative size = 171, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used =
{3581, 3594, 3592, 3528, 3533, 208} \[ \frac{8 a^3 (23 A-21 i B) \cot ^{\frac{3}{2}}(c+d x)}{105 d}-\frac{2 (7 B+11 i A) \cot ^{\frac{3}{2}}(c+d x) \left (a^3 \cot (c+d x)+i a^3\right )}{35 d}+\frac{8 a^3 (B+i A) \sqrt{\cot (c+d x)}}{d}+\frac{8 \sqrt [4]{-1} a^3 (B+i A) \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\cot (c+d x)}\right )}{d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (a \cot (c+d x)+i a)^2}{7 d} \]
Antiderivative was successfully verified.
[In]
Int[Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]
[Out]
(8*(-1)^(1/4)*a^3*(I*A + B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d + (8*a^3*(I*A + B)*Sqrt[Cot[c + d*x]])/d
+ (8*a^3*(23*A - (21*I)*B)*Cot[c + d*x]^(3/2))/(105*d) - (2*a*A*Cot[c + d*x]^(3/2)*(I*a + a*Cot[c + d*x])^2)/
(7*d) - (2*((11*I)*A + 7*B)*Cot[c + d*x]^(3/2)*(I*a^3 + a^3*Cot[c + d*x]))/(35*d)
Rule 3581
Int[(cot[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*tan[(e_.)
+ (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[g^(m + n), Int[(g*Cot[e + f*x])^(p - m - n)*(b + a*Cot[e + f*x])^m*(d
+ c*Cot[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && !IntegerQ[p] && IntegerQ[m] && IntegerQ
[n]
Rule 3594
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*B*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1))/(d*f
*(m + n)), x] + Dist[1/(d*(m + n)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n)
+ B*(a*c*(m - 1) - b*d*(n + 1)) - (B*(b*c - a*d)*(m - 1) - d*(A*b + a*B)*(m + n))*Tan[e + f*x], x], x], x] /;
FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[m, 1] && !LtQ[n, -1]
Rule 3592
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(B*d*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Int[(a + b*Tan[e
+ f*x])^m*Simp[A*c - B*d + (B*c + A*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b
*c - a*d, 0] && !LeQ[m, -1]
Rule 3528
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(d
*(a + b*Tan[e + f*x])^m)/(f*m), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
Rule 3533
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(2*c^2)/f, S
ubst[Int[1/(b*c - d*x^2), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 + d^2, 0]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx &=\int \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^3 (B+A \cot (c+d x)) \, dx\\ &=-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (i a+a \cot (c+d x))^2}{7 d}-\frac{2}{7} \int \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^2 \left (\frac{1}{2} a (3 A-7 i B)-\frac{1}{2} a (11 i A+7 B) \cot (c+d x)\right ) \, dx\\ &=-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (i a+a \cot (c+d x))^2}{7 d}-\frac{2 (11 i A+7 B) \cot ^{\frac{3}{2}}(c+d x) \left (i a^3+a^3 \cot (c+d x)\right )}{35 d}+\frac{4}{35} \int \sqrt{\cot (c+d x)} (i a+a \cot (c+d x)) \left (-2 a^2 (6 i A+7 B)-a^2 (23 A-21 i B) \cot (c+d x)\right ) \, dx\\ &=\frac{8 a^3 (23 A-21 i B) \cot ^{\frac{3}{2}}(c+d x)}{105 d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (i a+a \cot (c+d x))^2}{7 d}-\frac{2 (11 i A+7 B) \cot ^{\frac{3}{2}}(c+d x) \left (i a^3+a^3 \cot (c+d x)\right )}{35 d}+\frac{4}{35} \int \sqrt{\cot (c+d x)} \left (35 a^3 (A-i B)-35 a^3 (i A+B) \cot (c+d x)\right ) \, dx\\ &=\frac{8 a^3 (i A+B) \sqrt{\cot (c+d x)}}{d}+\frac{8 a^3 (23 A-21 i B) \cot ^{\frac{3}{2}}(c+d x)}{105 d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (i a+a \cot (c+d x))^2}{7 d}-\frac{2 (11 i A+7 B) \cot ^{\frac{3}{2}}(c+d x) \left (i a^3+a^3 \cot (c+d x)\right )}{35 d}+\frac{4}{35} \int \frac{35 a^3 (i A+B)+35 a^3 (A-i B) \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx\\ &=\frac{8 a^3 (i A+B) \sqrt{\cot (c+d x)}}{d}+\frac{8 a^3 (23 A-21 i B) \cot ^{\frac{3}{2}}(c+d x)}{105 d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (i a+a \cot (c+d x))^2}{7 d}-\frac{2 (11 i A+7 B) \cot ^{\frac{3}{2}}(c+d x) \left (i a^3+a^3 \cot (c+d x)\right )}{35 d}+\frac{\left (280 a^6 (i A+B)^2\right ) \operatorname{Subst}\left (\int \frac{1}{-35 a^3 (i A+B)+35 a^3 (A-i B) x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{d}\\ &=\frac{8 \sqrt [4]{-1} a^3 (i A+B) \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\cot (c+d x)}\right )}{d}+\frac{8 a^3 (i A+B) \sqrt{\cot (c+d x)}}{d}+\frac{8 a^3 (23 A-21 i B) \cot ^{\frac{3}{2}}(c+d x)}{105 d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (i a+a \cot (c+d x))^2}{7 d}-\frac{2 (11 i A+7 B) \cot ^{\frac{3}{2}}(c+d x) \left (i a^3+a^3 \cot (c+d x)\right )}{35 d}\\ \end{align*}
Mathematica [A] time = 9.16553, size = 161, normalized size = 0.94 \[ \frac{a^3 \sqrt{\cot (c+d x)} \left (\csc ^3(c+d x) (-((-95 A+105 i B) \cos (c+d x)+5 (31 A-21 i B) \cos (3 (c+d x))+42 \sin (c+d x) ((21 B+23 i A) \cos (2 (c+d x))-17 i A-19 B)))-1680 i (A-i B) \sqrt{i \tan (c+d x)} \tanh ^{-1}\left (\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right )\right )}{210 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]
[Out]
(a^3*Sqrt[Cot[c + d*x]]*(-(Csc[c + d*x]^3*((-95*A + (105*I)*B)*Cos[c + d*x] + 5*(31*A - (21*I)*B)*Cos[3*(c + d
*x)] + 42*((-17*I)*A - 19*B + ((23*I)*A + 21*B)*Cos[2*(c + d*x)])*Sin[c + d*x])) - (1680*I)*(A - I*B)*ArcTanh[
Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]]))/(210*d)
________________________________________________________________________________________
Maple [C] time = 0.568, size = 3132, normalized size = 18.3 \begin{align*} \text{output too large to display} \end{align*}
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*(A+B*tan(d*x+c)),x)
[Out]
-1/105*a^3/d*2^(1/2)*(-420*B*2^(1/2)*cos(d*x+c)*sin(d*x+c)+420*A*sin(d*x+c)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*
x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticF((-(cos(d*
x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))-105*I*B*cos(d*x+c)^4*2^(1/2)+105*I*B*cos(d*x+c)^2*2^(1/2)+44
1*B*cos(d*x+c)^3*sin(d*x+c)*2^(1/2)-140*A*2^(1/2)*cos(d*x+c)^2-420*A*sin(d*x+c)*(-(cos(d*x+c)-1-sin(d*x+c))/si
n(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticPi((-(c
os(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))+155*A*cos(d*x+c)^4*2^(1/2)-420*I*A*cos(d*x+c)
^2*sin(d*x+c)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticPi((-(cos
(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)-4
20*I*B*cos(d*x+c)^2*sin(d*x+c)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*
EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-(cos(d*x+c)-1-sin(d*x+c))/si
n(d*x+c))^(1/2)+420*I*B*cos(d*x+c)^2*sin(d*x+c)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/s
in(d*x+c))^(1/2)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*EllipticF((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c
))^(1/2),1/2*2^(1/2))+420*I*A*cos(d*x+c)*sin(d*x+c)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-
1)/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-(cos(d*
x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)+420*I*B*cos(d*x+c)*sin(d*x+c)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2
)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(
1/2))*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)-420*I*B*cos(d*x+c)*sin(d*x+c)*((cos(d*x+c)-1+sin(d*x+c))/s
in(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*EllipticF((-(
cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))-420*I*A*cos(d*x+c)^3*sin(d*x+c)*((cos(d*x+c)-1+sin(d*x
+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1
/2),1/2+1/2*I,1/2*2^(1/2))*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)-420*I*B*cos(d*x+c)^3*sin(d*x+c)*((cos
(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c
))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)+420*I*B*cos(d*x+c)^3
*sin(d*x+c)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*(-(cos(d*x+c)-1-sin
(d*x+c))/sin(d*x+c))^(1/2)*EllipticF((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))+420*A*cos(d*x+
c)*sin(d*x+c)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((cos
(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticF((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))-420*A*cos(d*
x+c)*sin(d*x+c)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((c
os(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))
+420*B*cos(d*x+c)*sin(d*x+c)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+
c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I
,1/2*2^(1/2))+483*I*A*cos(d*x+c)^3*sin(d*x+c)*2^(1/2)-420*I*A*cos(d*x+c)*sin(d*x+c)*2^(1/2)+420*B*sin(d*x+c)*(
-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(
d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))-420*A*cos(d*x+c)
^3*sin(d*x+c)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*(-(cos(d*x+c)-1-s
in(d*x+c))/sin(d*x+c))^(1/2)*EllipticF((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))-420*B*cos(d*
x+c)^3*sin(d*x+c)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticPi((-
(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/
2)+420*I*A*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(cos(d*x+c)-1-sin(
d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^
(1/2))+420*I*B*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(cos(d*x+c)-1-
sin(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/
2*2^(1/2))-420*I*B*sin(d*x+c)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*(
-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*EllipticF((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/
2))+420*A*cos(d*x+c)^3*sin(d*x+c)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/
2)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-(cos(d*x+c)-1-sin(d*x+c))
/sin(d*x+c))^(1/2)-420*A*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(cos
(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^2*EllipticF((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1
/2*2^(1/2))*sin(d*x+c)+420*A*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*(-
(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^2*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1
/2),1/2+1/2*I,1/2*2^(1/2))*sin(d*x+c)-420*B*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d
*x+c))^(1/2)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^2*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))
/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*sin(d*x+c))*(cos(d*x+c)/sin(d*x+c))^(9/2)*sin(d*x+c)/cos(d*x+c)^5
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Maxima [A] time = 1.52589, size = 292, normalized size = 1.71 \begin{align*} -\frac{105 \,{\left (\sqrt{2}{\left (\left (2 i + 2\right ) \, A - \left (2 i - 2\right ) \, B\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + \frac{2}{\sqrt{\tan \left (d x + c\right )}}\right )}\right ) + \sqrt{2}{\left (\left (2 i + 2\right ) \, A - \left (2 i - 2\right ) \, B\right )} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - \frac{2}{\sqrt{\tan \left (d x + c\right )}}\right )}\right ) + \sqrt{2}{\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \log \left (\frac{\sqrt{2}}{\sqrt{\tan \left (d x + c\right )}} + \frac{1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt{2}{\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \log \left (-\frac{\sqrt{2}}{\sqrt{\tan \left (d x + c\right )}} + \frac{1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{3} - \frac{840 \,{\left (i \, A + B\right )} a^{3}}{\sqrt{\tan \left (d x + c\right )}} - \frac{2 \,{\left (140 \, A - 105 i \, B\right )} a^{3}}{\tan \left (d x + c\right )^{\frac{3}{2}}} - \frac{42 \,{\left (-3 i \, A - B\right )} a^{3}}{\tan \left (d x + c\right )^{\frac{5}{2}}} + \frac{30 \, A a^{3}}{\tan \left (d x + c\right )^{\frac{7}{2}}}}{105 \, d} \end{align*}
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*(A+B*tan(d*x+c)),x, algorithm="maxima")
[Out]
-1/105*(105*(sqrt(2)*((2*I + 2)*A - (2*I - 2)*B)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + sqrt(2
)*((2*I + 2)*A - (2*I - 2)*B)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) + sqrt(2)*((I - 1)*A + (I
+ 1)*B)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - sqrt(2)*((I - 1)*A + (I + 1)*B)*log(-sqrt(2)/sq
rt(tan(d*x + c)) + 1/tan(d*x + c) + 1))*a^3 - 840*(I*A + B)*a^3/sqrt(tan(d*x + c)) - 2*(140*A - 105*I*B)*a^3/t
an(d*x + c)^(3/2) - 42*(-3*I*A - B)*a^3/tan(d*x + c)^(5/2) + 30*A*a^3/tan(d*x + c)^(7/2))/d
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Fricas [B] time = 1.75402, size = 1386, normalized size = 8.11 \begin{align*} \text{result too large to display} \end{align*}
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*(A+B*tan(d*x+c)),x, algorithm="fricas")
[Out]
1/420*(105*sqrt((-64*I*A^2 - 128*A*B + 64*I*B^2)*a^6/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^3*e^(2*I*d*x + 2*I*c) + sqrt((-64*I*A^2 - 128*A*B + 64*I*B^2)*
a^6/d^2)*(d*e^(2*I*d*x + 2*I*c) - d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))*e^(-2*I*d*x
- 2*I*c)/((4*I*A + 4*B)*a^3)) - 105*sqrt((-64*I*A^2 - 128*A*B + 64*I*B^2)*a^6/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^3*e^(2*I*d*x + 2*I*c) - sqrt((-64*I*A
^2 - 128*A*B + 64*I*B^2)*a^6/d^2)*(d*e^(2*I*d*x + 2*I*c) - d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2
*I*c) - 1)))*e^(-2*I*d*x - 2*I*c)/((4*I*A + 4*B)*a^3)) + ((5104*I*A + 4368*B)*a^3*e^(6*I*d*x + 6*I*c) + (-1033
6*I*A - 10752*B)*a^3*e^(4*I*d*x + 4*I*c) + (8816*I*A + 9072*B)*a^3*e^(2*I*d*x + 2*I*c) + (-2624*I*A - 2688*B)*
a^3)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))/(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)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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*(A+B*tan(d*x+c)),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \tan \left (d x + c\right ) + A\right )}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac{9}{2}}\,{d x} \end{align*}
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*(A+B*tan(d*x+c)),x, algorithm="giac")
[Out]
integrate((B*tan(d*x + c) + A)*(I*a*tan(d*x + c) + a)^3*cot(d*x + c)^(9/2), x)