3.1.81 \(\int \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx\) [81]

Optimal. Leaf size=213 \[ \frac {a^{3/2} (23 i A+22 B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{8 d}-\frac {2 \sqrt {2} a^{3/2} (i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a (9 A-10 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a (7 i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d} \]

[Out]

1/8*a^(3/2)*(23*I*A+22*B)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/d-2*a^(3/2)*(I*A+B)*arctanh(1/2*(a+I*a*tan
(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)/d+1/8*a*(9*A-10*I*B)*cot(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/d-1/12*a*(7*I
*A+6*B)*cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(1/2)/d-1/3*a*A*cot(d*x+c)^3*(a+I*a*tan(d*x+c))^(1/2)/d

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Rubi [A]
time = 0.50, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3674, 3679, 3681, 3561, 212, 3680, 65, 214} \begin {gather*} \frac {a^{3/2} (22 B+23 i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{8 d}-\frac {2 \sqrt {2} a^{3/2} (B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {a (6 B+7 i A) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}+\frac {a (9 A-10 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]

[Out]

(a^(3/2)*((23*I)*A + 22*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(8*d) - (2*Sqrt[2]*a^(3/2)*(I*A + B)*A
rcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (a*(9*A - (10*I)*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c
+ d*x]])/(8*d) - (a*((7*I)*A + 6*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(12*d) - (a*A*Cot[c + d*x]^3*Sq
rt[a + I*a*Tan[c + d*x]])/(3*d)

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 214

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

Rule 3561

Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2*(b/d), Subst[Int[1/(2*a - x^2), x], x, Sq
rt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0]

Rule 3674

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

Rule 3679

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 3680

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 3681

Int[(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*tan[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(A*b + a*B)/(b*c + a*d), Int[(a + b*Tan[e + f*x])^m, x], x] - Dist[(B*c
- A*d)/(b*c + a*d), Int[(a + b*Tan[e + f*x])^m*((a - b*Tan[e + f*x])/(c + d*Tan[e + f*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] && NeQ[A*b + a*B, 0]

Rubi steps

\begin {align*} \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx &=-\frac {a A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}+\frac {1}{3} \int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \left (\frac {1}{2} a (7 i A+6 B)-\frac {1}{2} a (5 A-6 i B) \tan (c+d x)\right ) \, dx\\ &=-\frac {a (7 i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}+\frac {\int \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{4} a^2 (9 A-10 i B)-\frac {3}{4} a^2 (7 i A+6 B) \tan (c+d x)\right ) \, dx}{6 a}\\ &=\frac {a (9 A-10 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a (7 i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{8} a^3 (23 i A+22 B)+\frac {3}{8} a^3 (9 A-10 i B) \tan (c+d x)\right ) \, dx}{6 a^2}\\ &=\frac {a (9 A-10 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a (7 i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}+\frac {1}{16} (-23 i A-22 B) \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \, dx+(2 a (A-i B)) \int \sqrt {a+i a \tan (c+d x)} \, dx\\ &=\frac {a (9 A-10 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a (7 i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}-\frac {\left (4 a^2 (i A+B)\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{d}-\frac {\left (a^2 (23 i A+22 B)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{16 d}\\ &=-\frac {2 \sqrt {2} a^{3/2} (i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a (9 A-10 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a (7 i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}-\frac {(a (23 A-22 i B)) \text {Subst}\left (\int \frac {1}{i-\frac {i x^2}{a}} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{8 d}\\ &=\frac {a^{3/2} (23 i A+22 B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{8 d}-\frac {2 \sqrt {2} a^{3/2} (i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a (9 A-10 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a (7 i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(439\) vs. \(2(213)=426\).
time = 6.28, size = 439, normalized size = 2.06 \begin {gather*} \frac {\left (-\frac {2 i \left (64 \sqrt {2} (A-i B) \sinh ^{-1}\left (e^{i (c+d x)}\right )+(23 A-22 i B) \left (\log \left (\left (-1+e^{i (c+d x)}\right )^2\right )-\log \left (\left (1+e^{i (c+d x)}\right )^2\right )+\log \left (3+3 e^{2 i (c+d x)}+2 \sqrt {2} \sqrt {1+e^{2 i (c+d x)}}-2 e^{i (c+d x)} \left (1+\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}\right )\right )-\log \left (3+3 e^{2 i (c+d x)}+2 \sqrt {2} \sqrt {1+e^{2 i (c+d x)}}+2 e^{i (c+d x)} \left (1+\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}\right )\right )\right )\right )}{\left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{3/2} \left (1+e^{2 i (c+d x)}\right )^{3/2}}-\frac {4 \csc ^3(c+d x) (\cos (c)-i \sin (c)) (-19 A+30 i B+5 (7 A-6 i B) \cos (2 (c+d x))+2 (7 i A+6 B) \sin (2 (c+d x)))}{\sqrt {\sec (c+d x)} (3 \cos (d x)+3 i \sin (d x))}\right ) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{64 d \sec ^{\frac {5}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]

[Out]

((((-2*I)*(64*Sqrt[2]*(A - I*B)*ArcSinh[E^(I*(c + d*x))] + (23*A - (22*I)*B)*(Log[(-1 + E^(I*(c + d*x)))^2] -
Log[(1 + E^(I*(c + d*x)))^2] + Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] - 2*E^(
I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 +
E^((2*I)*(c + d*x))] + 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])))/((E^(I*(c + d*x))/(1
+ E^((2*I)*(c + d*x))))^(3/2)*(1 + E^((2*I)*(c + d*x)))^(3/2)) - (4*Csc[c + d*x]^3*(Cos[c] - I*Sin[c])*(-19*A
+ (30*I)*B + 5*(7*A - (6*I)*B)*Cos[2*(c + d*x)] + 2*((7*I)*A + 6*B)*Sin[2*(c + d*x)]))/(Sqrt[Sec[c + d*x]]*(3
*Cos[d*x] + (3*I)*Sin[d*x])))*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/(64*d*Sec[c + d*x]^(5/2)*(A*C
os[c + d*x] + B*Sin[c + d*x]))

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1803 vs. \(2 (175 ) = 350\).
time = 0.69, size = 1804, normalized size = 8.47

method	result	size
default	\(\text {Expression too large to display}\)	\(1804\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/48/d*a*((I*sin(d*x+c)+cos(d*x+c))*a/cos(d*x+c))^(1/2)*(54*A*cos(d*x+c)-60*B*cos(d*x+c)*sin(d*x+c)+98*A*cos(d
*x+c)^4+96*I*A*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*s
in(d*x+c)/cos(d*x+c))*2^(1/2)*cos(d*x+c)^4-82*A*cos(d*x+c)^2+69*I*A*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln((s
in(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-cos(d*x+c)+1)/sin(d*x+c))*cos(d*x+c)^4-66*I*B*(-2*cos(d*x+c)/(c
os(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^4-138*I*A*(-2*cos(d*x+c)/(cos(d*
x+c)+1))^(1/2)*ln((sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-cos(d*x+c)+1)/sin(d*x+c))*cos(d*x+c)^2+132*
I*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^2+96*I*A*(-
2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x
+c))*2^(1/2)-96*I*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/
2))*2^(1/2)+98*I*A*cos(d*x+c)^3*sin(d*x+c)-28*I*A*cos(d*x+c)^2*sin(d*x+c)+69*I*A*(-2*cos(d*x+c)/(cos(d*x+c)+1)
)^(1/2)*ln((sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-cos(d*x+c)+1)/sin(d*x+c))-54*I*A*cos(d*x+c)*sin(d*
x+c)-66*I*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+84*I*B*cos(d*x
+c)^2-60*I*B*cos(d*x+c)-84*I*B*cos(d*x+c)^4+60*I*B*cos(d*x+c)^3+84*B*cos(d*x+c)^3*sin(d*x+c)+69*A*(-2*cos(d*x+
c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+66*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1
/2)*ln((sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-cos(d*x+c)+1)/sin(d*x+c))-24*B*cos(d*x+c)^2*sin(d*x+c)
-70*A*cos(d*x+c)^3+96*B*2^(1/2)*cos(d*x+c)^4*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(
d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))-192*A*2^(1/2)*cos(d*x+c)^2*(-2*cos(d*x+c)/(cos(d*x+c)+1))^
(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-192*B*2^(1/2)*cos(d*x+c)^2*(-2*cos(d*x+c)/(cos(
d*x+c)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))+96*A*2^(1/2)*
cos(d*x+c)^4*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-138
*A*cos(d*x+c)^2*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-132*B*cos(
d*x+c)^2*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln((sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-cos(d*x+c)+1
)/sin(d*x+c))+96*A*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+
1))^(1/2))+96*B*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1)
)^(1/2)*sin(d*x+c)/cos(d*x+c))+69*A*cos(d*x+c)^4*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/
(cos(d*x+c)+1))^(1/2))+66*B*cos(d*x+c)^4*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln((sin(d*x+c)*(-2*cos(d*x+c)/(c
os(d*x+c)+1))^(1/2)-cos(d*x+c)+1)/sin(d*x+c))-96*I*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(
-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*2^(1/2)*cos(d*x+c)^4-192*I*A*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh
(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*2^(1/2)*cos(d*x+c)^2+192*I*B*(-2*cos(
d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*2^(1/2)*cos(d*x+c)^2)/(-
1+cos(d*x+c))/(I*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c)/(cos(d*x+c)+1)

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Maxima [A]
time = 0.51, size = 253, normalized size = 1.19 \begin {gather*} \frac {i \, a^{3} {\left (\frac {48 \, \sqrt {2} {\left (A - i \, B\right )} \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 )}{a^{\frac {3}{2}}} - \frac {3 \, {\left (23 \, A - 22 i \, B\right )} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} {\left (9 \, A - 10 i \, B\right )} - 8 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} {\left (5 \, A - 6 i \, B\right )} a + 3 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} {\left (7 \, A - 6 i \, B\right )} a^{2}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} a - 3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a^{2} + 3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{3} - a^{4}}\right )}}{48 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/48*I*a^3*(48*sqrt(2)*(A - I*B)*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)))/a^(3/2) - 3*(23*A - 22*I*B)*log((sqrt(I*a*tan(d*x + c) + a) - sqrt(a))/(sqrt(I*a*tan(d*
x + c) + a) + sqrt(a)))/a^(3/2) + 2*(3*(I*a*tan(d*x + c) + a)^(5/2)*(9*A - 10*I*B) - 8*(I*a*tan(d*x + c) + a)^
(3/2)*(5*A - 6*I*B)*a + 3*sqrt(I*a*tan(d*x + c) + a)*(7*A - 6*I*B)*a^2)/((I*a*tan(d*x + c) + a)^3*a - 3*(I*a*t
an(d*x + c) + a)^2*a^2 + 3*(I*a*tan(d*x + c) + a)*a^3 - a^4))/d

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 856 vs. \(2 (166) = 332\).
time = 3.28, size = 856, normalized size = 4.02 \begin {gather*} \frac {96 \, \sqrt {2} \sqrt {-\frac {{\left (A^{2} - 2 i \, A B - 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 {4 \, {\left ({\left (-i \, A - B\right )} a^{2} e^{\left (i \, d x + i \, c\right )} + \sqrt {-\frac {{\left (A^{2} - 2 i \, A B - 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}}\right )} e^{\left (-i \, d x - i \, c\right )}}{{\left (-i \, A - B\right )} a}\right ) - 96 \, \sqrt {2} \sqrt {-\frac {{\left (A^{2} - 2 i \, A B - 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 {4 \, {\left ({\left (-i \, A - B\right )} a^{2} e^{\left (i \, d x + i \, c\right )} - \sqrt {-\frac {{\left (A^{2} - 2 i \, A B - 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}}\right )} e^{\left (-i \, d x - i \, c\right )}}{{\left (-i \, A - B\right )} a}\right ) - 3 \, \sqrt {-\frac {{\left (529 \, A^{2} - 1012 i \, A B - 484 \, 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 {16 \, {\left (3 \, {\left (-23 i \, A - 22 \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-23 i \, A - 22 \, B\right )} a^{2} + 2 \, \sqrt {2} \sqrt {-\frac {{\left (529 \, A^{2} - 1012 i \, A B - 484 \, B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (3 i \, d x + 3 i \, c\right )} + d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{23 i \, A + 22 \, B}\right ) + 3 \, \sqrt {-\frac {{\left (529 \, A^{2} - 1012 i \, A B - 484 \, 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 {16 \, {\left (3 \, {\left (-23 i \, A - 22 \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-23 i \, A - 22 \, B\right )} a^{2} - 2 \, \sqrt {2} \sqrt {-\frac {{\left (529 \, A^{2} - 1012 i \, A B - 484 \, B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (3 i \, d x + 3 i \, c\right )} + d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{23 i \, A + 22 \, B}\right ) - 4 \, \sqrt {2} {\left (7 \, {\left (-7 i \, A - 6 \, B\right )} a e^{\left (7 i \, d x + 7 i \, c\right )} - {\left (11 i \, A - 18 \, B\right )} a e^{\left (5 i \, d x + 5 i \, c\right )} - {\left (-17 i \, A - 42 \, B\right )} a e^{\left (3 i \, d x + 3 i \, c\right )} + 3 \, {\left (-7 i \, A - 6 \, 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}}}{96 \, {\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/96*(96*sqrt(2)*sqrt(-(A^2 - 2*I*A*B - 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(4*((-I*A - B)*a^2*e^(I*d*x + I*c) + sqrt(-(A^2 - 2*I*A*B - 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)))*e^(-I*d*x - I*c)/((-I*A - B)*a)) - 96*sqrt(2)*sqrt(-(A^2
- 2*I*A*B - 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
(4*((-I*A - B)*a^2*e^(I*d*x + I*c) - sqrt(-(A^2 - 2*I*A*B - 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)))*e^(-I*d*x - I*c)/((-I*A - B)*a)) - 3*sqrt(-(529*A^2 - 1012*I*A*B - 484*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(-16*(3*(-23*I*A - 22*B)*
a^2*e^(2*I*d*x + 2*I*c) + (-23*I*A - 22*B)*a^2 + 2*sqrt(2)*sqrt(-(529*A^2 - 1012*I*A*B - 484*B^2)*a^3/d^2)*(d*
e^(3*I*d*x + 3*I*c) + d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-2*I*d*x - 2*I*c)/(23*I*A + 22*
B)) + 3*sqrt(-(529*A^2 - 1012*I*A*B - 484*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(-16*(3*(-23*I*A - 22*B)*a^2*e^(2*I*d*x + 2*I*c) + (-23*I*A - 22*B)*a^2 - 2*sqrt(
2)*sqrt(-(529*A^2 - 1012*I*A*B - 484*B^2)*a^3/d^2)*(d*e^(3*I*d*x + 3*I*c) + d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*
d*x + 2*I*c) + 1)))*e^(-2*I*d*x - 2*I*c)/(23*I*A + 22*B)) - 4*sqrt(2)*(7*(-7*I*A - 6*B)*a*e^(7*I*d*x + 7*I*c)
- (11*I*A - 18*B)*a*e^(5*I*d*x + 5*I*c) - (-17*I*A - 42*B)*a*e^(3*I*d*x + 3*I*c) + 3*(-7*I*A - 6*B)*a*e^(I*d*x
+ I*c))*sqrt(a/(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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}} \left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{4}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**4*(a+I*a*tan(d*x+c))**(3/2)*(A+B*tan(d*x+c)),x)

[Out]

Integral((I*a*(tan(c + d*x) - I))**(3/2)*(A + B*tan(c + d*x))*cot(c + d*x)**4, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)*(I*a*tan(d*x + c) + a)^(3/2)*cot(d*x + c)^4, x)

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Mupad [B]
time = 7.86, size = 2500, normalized size = 11.74 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^4*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^(3/2),x)

[Out]

2*atanh((6*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*((249*B^2*a^3)/(128*d^2) - (1041*A^2*a^3)/(512*d^2) - ((289*A^4*a
^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51
i)/(8*d^4))^(1/2)/(64*a^6) + (A*B*a^3*509i)/(128*d^2))^(1/2)*((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4)
+ (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2))/((A^3*a^11*d*663i)/
32 + (133*B^3*a^11*d)/4 + (A*B^2*a^11*d*387i)/8 + (89*A^2*B*a^11*d)/16 + (A*a^2*d^3*((289*A^4*a^18)/(64*d^4) +
(49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/
2)*7i)/4 + (3*B*a^2*d^3*((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3
*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2))/2) + (17*A^2*a^6*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((249
*B^2*a^3)/(128*d^2) - (1041*A^2*a^3)/(512*d^2) - ((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B
^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2)/(64*a^6) + (A*B*a^3*509i)/(128*d
^2))^(1/2))/(4*((A^3*a^8*d*663i)/32 + (133*B^3*a^8*d)/4 + (A*B^2*a^8*d*387i)/8 + (89*A^2*B*a^8*d)/16 + (A*d^3*
((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^
3*B*a^18*51i)/(8*d^4))^(1/2)*7i)/(4*a) + (3*B*d^3*((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*
B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2))/(2*a))) + (7*B^2*a^6*d^2*(a +
a*tan(c + d*x)*1i)^(1/2)*((249*B^2*a^3)/(128*d^2) - (1041*A^2*a^3)/(512*d^2) - ((289*A^4*a^18)/(64*d^4) + (49*
B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2)/(6
4*a^6) + (A*B*a^3*509i)/(128*d^2))^(1/2))/((A^3*a^8*d*663i)/32 + (133*B^3*a^8*d)/4 + (A*B^2*a^8*d*387i)/8 + (8
9*A^2*B*a^8*d)/16 + (A*d^3*((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*
B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2)*7i)/(4*a) + (3*B*d^3*((289*A^4*a^18)/(64*d^4) + (49*B^
4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2))/(2*
a)) + (A*B*a^6*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((249*B^2*a^3)/(128*d^2) - (1041*A^2*a^3)/(512*d^2) - ((289*A
^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^1
8*51i)/(8*d^4))^(1/2)/(64*a^6) + (A*B*a^3*509i)/(128*d^2))^(1/2)*3i)/((A^3*a^8*d*663i)/32 + (133*B^3*a^8*d)/4
+ (A*B^2*a^8*d*387i)/8 + (89*A^2*B*a^8*d)/16 + (A*d^3*((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*
A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2)*7i)/(4*a) + (3*B*d^3*((289*
A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^
18*51i)/(8*d^4))^(1/2))/(2*a)))*((249*B^2*a^3)/(128*d^2) - (1041*A^2*a^3)/(512*d^2) - ((289*A^4*a^18)/(64*d^4)
+ (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(
1/2)/(64*a^6) + (A*B*a^3*509i)/(128*d^2))^(1/2) + 2*atanh((17*A^2*a^6*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((289
*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a
^18*51i)/(8*d^4))^(1/2)/(64*a^6) - (1041*A^2*a^3)/(512*d^2) + (249*B^2*a^3)/(128*d^2) + (A*B*a^3*509i)/(128*d^
2))^(1/2))/(4*((A^3*a^8*d*663i)/32 + (133*B^3*a^8*d)/4 + (A*B^2*a^8*d*387i)/8 + (89*A^2*B*a^8*d)/16 - (A*d^3*(
(289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3
*B*a^18*51i)/(8*d^4))^(1/2)*7i)/(4*a) - (3*B*d^3*((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B
^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2))/(2*a))) - (6*d^4*(a + a*tan(c +
d*x)*1i)^(1/2)*(((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*2
1i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2)/(64*a^6) - (1041*A^2*a^3)/(512*d^2) + (249*B^2*a^3)/(128*d^2) +
(A*B*a^3*509i)/(128*d^2))^(1/2)*((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4)
+ (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2))/((A^3*a^11*d*663i)/32 + (133*B^3*a^11*d)/4 + (A*
B^2*a^11*d*387i)/8 + (89*A^2*B*a^11*d)/16 - (A*a^2*d^3*((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101
*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2)*7i)/4 - (3*B*a^2*d^3*((289
*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a
^18*51i)/(8*d^4))^(1/2))/2) + (7*B^2*a^6*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((289*A^4*a^18)/(64*d^4) + (49*B^4
*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2)/(64*a
^6) - (1041*A^2*a^3)/(512*d^2) + (249*B^2*a^3)/(128*d^2) + (A*B*a^3*509i)/(128*d^2))^(1/2))/((A^3*a^8*d*663i)/
32 + (133*B^3*a^8*d)/4 + (A*B^2*a^8*d*387i)/8 +...

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