3.780 \(\int \frac{1}{\cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx\)
Optimal. Leaf size=258 \[ -\frac{3 \sqrt [4]{-1} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^{3/2} d}-\frac{7 \sqrt{a+i a \tan (c+d x)}}{2 a^2 d \sqrt{\cot (c+d x)}}+\frac{\left (\frac{1}{4}-\frac{i}{4}\right ) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^{3/2} d}+\frac{13 i}{6 a d \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}-\frac{1}{3 d \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \]
[Out]
(-3*(-1)^(1/4)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*S
qrt[Tan[c + d*x]])/(a^(3/2)*d) + ((1/4 - I/4)*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]])/(a^(3/2)*d) - 1/(3*d*Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])
^(3/2)) + ((13*I)/6)/(a*d*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (7*Sqrt[a + I*a*Tan[c + d*x]])/(2*a
^2*d*Sqrt[Cot[c + d*x]])
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Rubi [A] time = 0.748464, antiderivative size = 258, normalized size of antiderivative =
1., number of steps used = 11, number of rules used = 11, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) =
0.393, Rules used = {4241, 3558, 3595, 3597, 3601, 3544, 205, 3599, 63, 217, 203}
\[ -\frac{3 \sqrt [4]{-1} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^{3/2} d}-\frac{7 \sqrt{a+i a \tan (c+d x)}}{2 a^2 d \sqrt{\cot (c+d x)}}+\frac{\left (\frac{1}{4}-\frac{i}{4}\right ) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^{3/2} d}+\frac{13 i}{6 a d \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}-\frac{1}{3 d \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[1/(Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^(3/2)),x]
[Out]
(-3*(-1)^(1/4)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*S
qrt[Tan[c + d*x]])/(a^(3/2)*d) + ((1/4 - I/4)*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]])/(a^(3/2)*d) - 1/(3*d*Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])
^(3/2)) + ((13*I)/6)/(a*d*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (7*Sqrt[a + I*a*Tan[c + d*x]])/(2*a
^2*d*Sqrt[Cot[c + d*x]])
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]
Rule 3558
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[((b*c - a*d)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 1))/(2*a*f*m), x] + Dist[1/(2*a^2*m), Int[(a
+ b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)*Simp[c*(a*c*m + b*d*(n - 1)) - d*(b*c*m + a*d*(n - 1))
- d*(b*d*(m - n + 1) - a*c*(m + n - 1))*Tan[e + f*x], x], 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] && LtQ[m, 0] && GtQ[n, 1] && (IntegerQ[m] || IntegersQ[2*m,
2*n])
Rule 3595
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*b - a*B)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n)/(2*a*f*
m), x] + Dist[1/(2*a^2*m), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[A*(a*c*m + b*d*n
) - B*(b*c*m + a*d*n) - d*(b*B*(m - n) - a*A*(m + n))*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] && LtQ[m, 0] && GtQ[n, 0]
Rule 3597
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*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n)/(f*(m + n)), x] +
Dist[1/(a*(m + n)), Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 1)*Simp[a*A*c*(m + n) - B*(b*c*m + a*
d*n) + (a*A*d*(m + n) - B*(b*d*m - a*c*n))*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] && GtQ[n, 0]
Rule 3601
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(A*b + a*B)/b, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n, x]
, x] - Dist[B/b, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(a - b*Tan[e + f*x]), x], x] /; FreeQ[{a, b
, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]
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 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 3599
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*B)/f, Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x
]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && EqQ[A*b + a*B,
0]
Rule 63
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 217
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] && !GtQ[a, 0]
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rubi steps
\begin{align*} \int \frac{1}{\cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx &=\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\tan ^{\frac{7}{2}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx\\ &=-\frac{1}{3 d \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}-\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\tan ^{\frac{3}{2}}(c+d x) \left (-\frac{5 a}{2}+4 i a \tan (c+d x)\right )}{\sqrt{a+i a \tan (c+d x)}} \, dx}{3 a^2}\\ &=-\frac{1}{3 d \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{13 i}{6 a d \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}+\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)} \left (-\frac{39 i a^2}{4}-\frac{21}{2} a^2 \tan (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac{1}{3 d \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{13 i}{6 a d \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}-\frac{7 \sqrt{a+i a \tan (c+d x)}}{2 a^2 d \sqrt{\cot (c+d x)}}+\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)} \left (\frac{21 a^3}{4}-\frac{9}{2} i a^3 \tan (c+d x)\right )}{\sqrt{\tan (c+d x)}} \, dx}{3 a^5}\\ &=-\frac{1}{3 d \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{13 i}{6 a d \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}-\frac{7 \sqrt{a+i a \tan (c+d x)}}{2 a^2 d \sqrt{\cot (c+d x)}}+\frac{\left (3 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{(a-i a \tan (c+d x)) \sqrt{a+i a \tan (c+d x)}}{\sqrt{\tan (c+d x)}} \, dx}{2 a^3}+\frac{\left (\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}{4 a^2}\\ &=-\frac{1}{3 d \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{13 i}{6 a d \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}-\frac{7 \sqrt{a+i a \tan (c+d x)}}{2 a^2 d \sqrt{\cot (c+d x)}}-\frac{\left (i \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 )}{2 d}+\frac{\left (3 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{a+i a x}} \, dx,x,\tan (c+d x)\right )}{2 a d}\\ &=\frac{\left (\frac{1}{4}-\frac{i}{4}\right ) \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)}}{a^{3/2} d}-\frac{1}{3 d \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{13 i}{6 a d \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}-\frac{7 \sqrt{a+i a \tan (c+d x)}}{2 a^2 d \sqrt{\cot (c+d x)}}+\frac{\left (3 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+i a x^2}} \, dx,x,\sqrt{\tan (c+d x)}\right )}{a d}\\ &=\frac{\left (\frac{1}{4}-\frac{i}{4}\right ) \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)}}{a^{3/2} d}-\frac{1}{3 d \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{13 i}{6 a d \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}-\frac{7 \sqrt{a+i a \tan (c+d x)}}{2 a^2 d \sqrt{\cot (c+d x)}}+\frac{\left (3 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-i a x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a d}\\ &=-\frac{3 \sqrt [4]{-1} \tan ^{-1}\left (\frac{(-1)^{3/4} \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)}}{a^{3/2} d}+\frac{\left (\frac{1}{4}-\frac{i}{4}\right ) \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)}}{a^{3/2} d}-\frac{1}{3 d \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{13 i}{6 a d \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}-\frac{7 \sqrt{a+i a \tan (c+d x)}}{2 a^2 d \sqrt{\cot (c+d x)}}\\ \end{align*}
Mathematica [A] time = 2.64651, size = 268, normalized size = 1.04 \[ -\frac{i \sqrt{\cot (c+d x)} \left (16 e^{2 i (c+d x)}+13 e^{4 i (c+d x)}-28 e^{6 i (c+d x)}+3 e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \left (1+e^{2 i (c+d x)}\right ) \tanh ^{-1}\left (\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right )+18 \sqrt{2} e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \left (1+e^{2 i (c+d x)}\right ) \tanh ^{-1}\left (\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right )-1\right )}{6 \sqrt{2} d \left (1+e^{2 i (c+d x)}\right )^3 \left (\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^(3/2)),x]
[Out]
((-I/6)*(-1 + 16*E^((2*I)*(c + d*x)) + 13*E^((4*I)*(c + d*x)) - 28*E^((6*I)*(c + d*x)) + 3*E^((3*I)*(c + d*x))
*Sqrt[-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)
)]] + 18*Sqrt[2]*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*(1 + E^((2*I)*(c + d*x)))*ArcTanh[(Sqrt[2]
*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(Sqrt[2]*d*((a*E^((2*I)*(c + d*x)))/(1
+ E^((2*I)*(c + d*x))))^(3/2)*(1 + E^((2*I)*(c + d*x)))^3)
________________________________________________________________________________________
Maple [B] time = 0.384, size = 1265, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/cot(d*x+c)^(7/2)/(a+I*a*tan(d*x+c))^(3/2),x)
[Out]
(-1/12-1/12*I)/d/a^2*(18*sin(d*x+c)*cos(d*x+c)^2*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-I)-18*sin(d*x+c)*cos(d*x
+c)^2*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+I)+18*sin(d*x+c)*cos(d*x+c)^2*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-
1)+27*sin(d*x+c)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2+9*cos(d*x+c)*sin(d*x+c)*ln(((cos(d*x+c)-1)/sin
(d*x+c))^(1/2)+1)-9*cos(d*x+c)*sin(d*x+c)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-I)+9*cos(d*x+c)*sin(d*x+c)*ln((
(cos(d*x+c)-1)/sin(d*x+c))^(1/2)+I)-9*cos(d*x+c)*sin(d*x+c)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)+18*I*cos(d
*x+c)^3*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)-18*I*cos(d*x+c)^3*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-I)+18*I
*cos(d*x+c)^3*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+I)-18*I*cos(d*x+c)^3*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1
)-29*I*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^3-9*I*cos(d*x+c)^2*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)
+9*I*cos(d*x+c)^2*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-I)-9*I*cos(d*x+c)^2*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2
)+I)+9*I*cos(d*x+c)^2*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)-9*I*cos(d*x+c)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1
/2)+1)+9*I*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-I)*cos(d*x+c)-9*I*cos(d*x+c)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1
/2)+I)+9*I*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)*cos(d*x+c)+29*I*cos(d*x+c)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2
)+6*2^(1/2)*cos(d*x+c)^3*arctan((1/2+1/2*I)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2))+6*I*((cos(d*x+c)-1)/sin
(d*x+c))^(1/2)*sin(d*x+c)-27*I*sin(d*x+c)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2-3*2^(1/2)*cos(d*x+c)^
2*arctan((1/2+1/2*I)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2))-3*2^(1/2)*cos(d*x+c)*arctan((1/2+1/2*I)*((cos(
d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2))-18*sin(d*x+c)*cos(d*x+c)^2*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)+29*cos
(d*x+c)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-29*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^3-6*((cos(d*x+c)-1)/
sin(d*x+c))^(1/2)*sin(d*x+c)+6*I*sin(d*x+c)*2^(1/2)*cos(d*x+c)^2*arctan((1/2+1/2*I)*((cos(d*x+c)-1)/sin(d*x+c)
)^(1/2)*2^(1/2))-3*I*cos(d*x+c)*sin(d*x+c)*2^(1/2)*arctan((1/2+1/2*I)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2
)))*cos(d*x+c)^3*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)/(2*I*cos(d*x+c)*sin(d*x+c)+2*cos(d*x+c)^2-1)/(
(cos(d*x+c)-1)/sin(d*x+c))^(1/2)/(cos(d*x+c)/sin(d*x+c))^(7/2)/sin(d*x+c)^4
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/cot(d*x+c)^(7/2)/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="maxima")
[Out]
Exception raised: RuntimeError
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Fricas [B] time = 2.13656, size = 2118, normalized size = 8.21 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/cot(d*x+c)^(7/2)/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="fricas")
[Out]
1/12*(sqrt(2)*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))*(2
8*I*e^(6*I*d*x + 6*I*c) - 13*I*e^(4*I*d*x + 4*I*c) - 16*I*e^(2*I*d*x + 2*I*c) + I)*e^(I*d*x + I*c) + 3*(a^2*d*
e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))*sqrt(-1/2*I/(a^3*d^2))*log(1/4*(2*I*a^2*d*sqrt(-1/2*I/(a^3*d^
2))*e^(2*I*d*x + 2*I*c) + sqrt(2)*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^(2*I*d*x + 2*I*c) - 1)*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) - 3*(a^2*d*e^(6*I*d*x + 6*I*c)
+ a^2*d*e^(4*I*d*x + 4*I*c))*sqrt(-1/2*I/(a^3*d^2))*log(1/4*(-2*I*a^2*d*sqrt(-1/2*I/(a^3*d^2))*e^(2*I*d*x + 2*
I*c) + sqrt(2)*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^(2*I*d*x + 2*I*c) - 1)*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) + 3*(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x
+ 4*I*c))*sqrt(-9*I/(a^3*d^2))*log(1/1815*(624*sqrt(2)*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^(2*I*d*x + 2*I*c) - 1)*e^(I*d*x + I*c) + (312*I*a^2*d*e^(2*I*d*x + 2
*I*c) - 104*I*a^2*d)*sqrt(-9*I/(a^3*d^2)))/(e^(2*I*d*x + 2*I*c) + 1)) - 3*(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e
^(4*I*d*x + 4*I*c))*sqrt(-9*I/(a^3*d^2))*log(1/1815*(624*sqrt(2)*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^(2*I*d*x + 2*I*c) - 1)*e^(I*d*x + I*c) + (-312*I*a^2*d*e^(
2*I*d*x + 2*I*c) + 104*I*a^2*d)*sqrt(-9*I/(a^3*d^2)))/(e^(2*I*d*x + 2*I*c) + 1)))/(a^2*d*e^(6*I*d*x + 6*I*c) +
a^2*d*e^(4*I*d*x + 4*I*c))
________________________________________________________________________________________
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(1/cot(d*x+c)**(7/2)/(a+I*a*tan(d*x+c))**(3/2),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cot \left (d x + c\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/cot(d*x+c)^(7/2)/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="giac")
[Out]
integrate(1/((I*a*tan(d*x + c) + a)^(3/2)*cot(d*x + c)^(7/2)), x)