3.750 \(\int \frac{1}{\cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx\)
Optimal. Leaf size=275 \[ \frac{5 \sqrt{\cot (c+d x)}}{8 d \left (a^3 \cot (c+d x)+i a^3\right )}+\frac{\left (\frac{5}{32}+\frac{7 i}{32}\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}-\frac{\left (\frac{5}{32}+\frac{7 i}{32}\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}-\frac{\left (\frac{5}{16}-\frac{7 i}{16}\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}+\frac{\left (\frac{5}{16}-\frac{7 i}{16}\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}+\frac{i \sqrt{\cot (c+d x)}}{3 a d (a \cot (c+d x)+i a)^2}-\frac{\sqrt{\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3} \]
[Out]
((-5/16 + (7*I)/16)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d) + ((5/16 - (7*I)/16)*ArcTan[1 + Sq
rt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d) - Sqrt[Cot[c + d*x]]/(6*d*(I*a + a*Cot[c + d*x])^3) + ((I/3)*Sqrt[C
ot[c + d*x]])/(a*d*(I*a + a*Cot[c + d*x])^2) + (5*Sqrt[Cot[c + d*x]])/(8*d*(I*a^3 + a^3*Cot[c + d*x])) + ((5/3
2 + (7*I)/32)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^3*d) - ((5/32 + (7*I)/32)*Log[1 +
Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^3*d)
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Rubi [A] time = 0.465134, antiderivative size = 275, normalized size of antiderivative = 1.,
number of steps used = 14, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used
= {3673, 3559, 3596, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac{5 \sqrt{\cot (c+d x)}}{8 d \left (a^3 \cot (c+d x)+i a^3\right )}+\frac{\left (\frac{5}{32}+\frac{7 i}{32}\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}-\frac{\left (\frac{5}{32}+\frac{7 i}{32}\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}-\frac{\left (\frac{5}{16}-\frac{7 i}{16}\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}+\frac{\left (\frac{5}{16}-\frac{7 i}{16}\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}+\frac{i \sqrt{\cot (c+d x)}}{3 a d (a \cot (c+d x)+i a)^2}-\frac{\sqrt{\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3} \]
Antiderivative was successfully verified.
[In]
Int[1/(Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^3),x]
[Out]
((-5/16 + (7*I)/16)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d) + ((5/16 - (7*I)/16)*ArcTan[1 + Sq
rt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d) - Sqrt[Cot[c + d*x]]/(6*d*(I*a + a*Cot[c + d*x])^3) + ((I/3)*Sqrt[C
ot[c + d*x]])/(a*d*(I*a + a*Cot[c + d*x])^2) + (5*Sqrt[Cot[c + d*x]])/(8*d*(I*a^3 + a^3*Cot[c + d*x])) + ((5/3
2 + (7*I)/32)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^3*d) - ((5/32 + (7*I)/32)*Log[1 +
Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^3*d)
Rule 3673
Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_.))^(p_.), x_Symbol] :> Dist
[d^(n*p), Int[(d*Cot[e + f*x])^(m - n*p)*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x
] && !IntegerQ[m] && IntegersQ[n, p]
Rule 3559
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(a*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(2*f*m*(b*c - a*d)), x] + Dist[1/(2*a*m*(b*c - a*d))
, Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*c*m - a*d*(2*m + n + 1) + b*d*(m + n + 1)*Tan
[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2
+ d^2, 0] && LtQ[m, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Rule 3596
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*A + b*B)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(2
*f*m*(b*c - a*d)), x] + Dist[1/(2*a*m*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Si
mp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*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] && LtQ[m, 0] && !GtQ[n,
0]
Rule 3534
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]
Rule 1168
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]
Rule 1162
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Rule 617
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 1165
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Rule 628
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rubi steps
\begin{align*} \int \frac{1}{\cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx &=\int \frac{1}{\sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^3} \, dx\\ &=-\frac{\sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{\int \frac{-\frac{11 i a}{2}+\frac{5}{2} a \cot (c+d x)}{\sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^2} \, dx}{6 a^2}\\ &=-\frac{\sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{i \sqrt{\cot (c+d x)}}{3 a d (i a+a \cot (c+d x))^2}+\frac{\int \frac{-18 a^2-12 i a^2 \cot (c+d x)}{\sqrt{\cot (c+d x)} (i a+a \cot (c+d x))} \, dx}{24 a^4}\\ &=-\frac{\sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{i \sqrt{\cot (c+d x)}}{3 a d (i a+a \cot (c+d x))^2}+\frac{5 \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{\int \frac{21 i a^3-15 a^3 \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx}{48 a^6}\\ &=-\frac{\sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{i \sqrt{\cot (c+d x)}}{3 a d (i a+a \cot (c+d x))^2}+\frac{5 \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{-21 i a^3+15 a^3 x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{24 a^6 d}\\ &=-\frac{\sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{i \sqrt{\cot (c+d x)}}{3 a d (i a+a \cot (c+d x))^2}+\frac{5 \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+-\frac{\left (\frac{5}{16}+\frac{7 i}{16}\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a^3 d}+\frac{\left (\frac{5}{16}-\frac{7 i}{16}\right ) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a^3 d}\\ &=-\frac{\sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{i \sqrt{\cot (c+d x)}}{3 a d (i a+a \cot (c+d x))^2}+\frac{5 \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{\left (\frac{5}{32}-\frac{7 i}{32}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a^3 d}+\frac{\left (\frac{5}{32}-\frac{7 i}{32}\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a^3 d}+\frac{\left (\frac{5}{32}+\frac{7 i}{32}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}+\frac{\left (\frac{5}{32}+\frac{7 i}{32}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}\\ &=-\frac{\sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{i \sqrt{\cot (c+d x)}}{3 a d (i a+a \cot (c+d x))^2}+\frac{5 \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{\left (\frac{5}{32}+\frac{7 i}{32}\right ) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt{2} a^3 d}-\frac{\left (\frac{5}{32}+\frac{7 i}{32}\right ) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt{2} a^3 d}+-\frac{\left (\frac{5}{16}-\frac{7 i}{16}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}+\frac{\left (\frac{5}{16}-\frac{7 i}{16}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}\\ &=-\frac{\left (\frac{5}{16}-\frac{7 i}{16}\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}+\frac{\left (\frac{5}{16}-\frac{7 i}{16}\right ) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}-\frac{\sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{i \sqrt{\cot (c+d x)}}{3 a d (i a+a \cot (c+d x))^2}+\frac{5 \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{\left (\frac{5}{32}+\frac{7 i}{32}\right ) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt{2} a^3 d}-\frac{\left (\frac{5}{32}+\frac{7 i}{32}\right ) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt{2} a^3 d}\\ \end{align*}
Mathematica [A] time = 1.83931, size = 235, normalized size = 0.85 \[ \frac{\cot ^{\frac{5}{2}}(c+d x) \csc (c+d x) \sec ^3(c+d x) \left (-12 \sin (2 (c+d x))+21 \sin (4 (c+d x))-19 i \cos (4 (c+d x))+(21+15 i) \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\sin (3 (c+d x))-i \cos (3 (c+d x)))+(21-15 i) \sqrt{\sin (2 (c+d x))} \sin (3 (c+d x)) \log \left (\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right )-(15+21 i) \sqrt{\sin (2 (c+d x))} \cos (3 (c+d x)) \log \left (\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right )+19 i\right )}{96 a^3 d (\cot (c+d x)+i)^3} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^3),x]
[Out]
(Cot[c + d*x]^(5/2)*Csc[c + d*x]*Sec[c + d*x]^3*(19*I - (19*I)*Cos[4*(c + d*x)] - (15 + 21*I)*Cos[3*(c + d*x)]
*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]] - 12*Sin[2*(c + d*x)] + (21
- 15*I)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]]*Sin[3*(c + d*x)] + (2
1 + 15*I)*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Sqrt[Sin[2*(c + d*x)]]*((-I)*Cos[3*(c + d*x)] + Sin[3*(c + d*x)]
) + 21*Sin[4*(c + d*x)]))/(96*a^3*d*(I + Cot[c + d*x])^3)
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Maple [C] time = 0.326, size = 3184, normalized size = 11.6 \begin{align*} \text{output 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,x)
[Out]
-1/48/a^3/d*2^(1/2)*(cos(d*x+c)-1)*(75*I*(-(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))*cos(d*x+c)^2+72*I*((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)^4*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))-72*I*((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)^3*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)+54*I*((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)*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)*sin(d*x+c)-60*I*((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)^4*EllipticF((-(cos(d*x
+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))+12*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))^(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)^3*sin(d*x+c)+15*(-(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))*cos(d*x+c)^2-9*(-(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))*cos(d*x+c)*sin(d*x+c)-72*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))^(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)^3*sin(d*x+c)-90*I*((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))+60*sin(d*x+c)*((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)^3*EllipticF((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))+27*2^(1/2)*cos(d*x+c)^2
+15*I*((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)*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
)^2-12*I*((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)^4*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))-27*2^(1/2)*cos(d*x+c)-38*I*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)+38*I*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)-3*(-(co
s(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))+42*2^(1/2)*cos(d*x+c
)^3-18*((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)*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))-3*I*((co
s(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)*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))-45*(-(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)*E
llipticF((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))*cos(d*x+c)*sin(d*x+c)+90*(-(cos(d*x+c)-1-s
in(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)*El
lipticPi((-(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)^2+54*EllipticPi((-(co
s(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)*sin(d*x+c)*((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)+18*I*((co
s(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)*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))-15*I*((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)
*EllipticF((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))-12*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))^(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)^4-72*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))^(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)^4-42*cos(d*x+c)^4*2^(1/2)-12*I*EllipticP
i((-(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))^(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)^3*sin(d*x+c)+
9*I*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)*sin(d*x+c)*((co
s(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)^3*(cos(d*x+c)+1)^2/(4*I*sin(d*x+c)*cos(d*x+c)^2+4*cos(d*x+c)^3-I*sin(d*x+c)-3*cos(d*x+c
))/(cos(d*x+c)/sin(d*x+c))^(7/2)/sin(d*x+c)^7
________________________________________________________________________________________
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,x, algorithm="maxima")
[Out]
Exception raised: RuntimeError
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Fricas [B] time = 1.45689, size = 1485, normalized size = 5.4 \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,x, algorithm="fricas")
[Out]
1/48*(12*a^3*d*sqrt(-1/64*I/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(((16*I*a^3*d*e^(2*I*d*x + 2*I*c) - 16*I*a^3*d)*
sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(-1/64*I/(a^6*d^2)) + 2*I*e^(2*I*d*x + 2*I*c))
*e^(-2*I*d*x - 2*I*c)) - 12*a^3*d*sqrt(-1/64*I/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(((-16*I*a^3*d*e^(2*I*d*x + 2
*I*c) + 16*I*a^3*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(-1/64*I/(a^6*d^2)) + 2*I*
e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)) + 12*a^3*d*sqrt(9/16*I/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(1/4*(4*(a
^3*d*e^(2*I*d*x + 2*I*c) - a^3*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(9/16*I/(a^6
*d^2)) + 3)*e^(-2*I*d*x - 2*I*c)/(a^3*d)) - 12*a^3*d*sqrt(9/16*I/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(-1/4*(4*(a
^3*d*e^(2*I*d*x + 2*I*c) - a^3*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(9/16*I/(a^6
*d^2)) - 3)*e^(-2*I*d*x - 2*I*c)/(a^3*d)) + sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*(-20*I
*e^(6*I*d*x + 6*I*c) + 26*I*e^(4*I*d*x + 4*I*c) - 7*I*e^(2*I*d*x + 2*I*c) + I))*e^(-6*I*d*x - 6*I*c)/(a^3*d)
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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,x)
[Out]
Exception raised: AttributeError
________________________________________________________________________________________
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 )}^{3} \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,x, algorithm="giac")
[Out]
integrate(1/((I*a*tan(d*x + c) + a)^3*cot(d*x + c)^(7/2)), x)