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

Optimal. Leaf size=244 \[ -\frac {(-1)^{3/4} a^{3/2} (12 A-11 i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tan ^{-1}\left (\frac {(-1)^{3/4} \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{4 d}-\frac {(2+2 i) a^{3/2} (A-i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}+\frac {a (5 B+4 i A) \sqrt {a+i a \tan (c+d x)}}{4 d \sqrt {\cot (c+d x)}}+\frac {i a B \sqrt {a+i a \tan (c+d x)}}{2 d \cot ^{\frac {3}{2}}(c+d x)} \]

[Out]

-1/4*(-1)^(3/4)*a^(3/2)*(12*A-11*I*B)*arctan((-1)^(3/4)*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-(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+1/2*I*a*B*(a+I*a*tan(d*x+c))^(1/2)/d/cot(d*x+c)^(3/2)+1/4*a*(4
*I*A+5*B)*(a+I*a*tan(d*x+c))^(1/2)/d/cot(d*x+c)^(1/2)

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Rubi [A]  time = 0.86, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {4241, 3594, 3597, 3601, 3544, 205, 3599, 63, 217, 203} \[ -\frac {(-1)^{3/4} a^{3/2} (12 A-11 i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tan ^{-1}\left (\frac {(-1)^{3/4} \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{4 d}-\frac {(2+2 i) a^{3/2} (A-i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}+\frac {a (5 B+4 i A) \sqrt {a+i a \tan (c+d x)}}{4 d \sqrt {\cot (c+d x)}}+\frac {i a B \sqrt {a+i a \tan (c+d x)}}{2 d \cot ^{\frac {3}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((-1)^(3/4)*a^(3/2)*(12*A - (11*I)*B)*ArcTan[((-1)^(3/4)*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]])/(4*d) - ((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 + ((I/2)*a*B*Sqrt[a + I*a*
Tan[c + d*x]])/(d*Cot[c + d*x]^(3/2)) + (a*((4*I)*A + 5*B)*Sqrt[a + I*a*Tan[c + d*x]])/(4*d*Sqrt[Cot[c + d*x]]
)

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 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])

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 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 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 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 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 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 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 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 \frac {(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx &=\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx\\ &=\frac {i a B \sqrt {a+i a \tan (c+d x)}}{2 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {1}{2} \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 {1}{2} a (4 A-3 i B)+\frac {1}{2} a (4 i A+5 B) \tan (c+d x)\right ) \, dx\\ &=\frac {i a B \sqrt {a+i a \tan (c+d x)}}{2 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {a (4 i A+5 B) \sqrt {a+i a \tan (c+d x)}}{4 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 {1}{4} a^2 (4 i A+5 B)+\frac {1}{4} a^2 (12 A-11 i B) \tan (c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx}{2 a}\\ &=\frac {i a B \sqrt {a+i a \tan (c+d x)}}{2 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {a (4 i A+5 B) \sqrt {a+i a \tan (c+d x)}}{4 d \sqrt {\cot (c+d x)}}-\left (2 a (i A+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 {1}{8} \left ((12 i A+11 B) \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\\ &=\frac {i a B \sqrt {a+i a \tan (c+d x)}}{2 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {a (4 i A+5 B) \sqrt {a+i a \tan (c+d x)}}{4 d \sqrt {\cot (c+d x)}}+\frac {\left (4 i a^3 (i A+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 {\left (a^2 (12 i A+11 B) \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 )}{8 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 {i a B \sqrt {a+i a \tan (c+d x)}}{2 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {a (4 i A+5 B) \sqrt {a+i a \tan (c+d x)}}{4 d \sqrt {\cot (c+d x)}}+\frac {\left (a^2 (12 i A+11 B) \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 )}{4 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 {i a B \sqrt {a+i a \tan (c+d x)}}{2 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {a (4 i A+5 B) \sqrt {a+i a \tan (c+d x)}}{4 d \sqrt {\cot (c+d x)}}+\frac {\left (a^2 (12 i A+11 B) \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 )}{4 d}\\ &=-\frac {\sqrt [4]{-1} a^{3/2} (12 i A+11 B) \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)}}{4 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 {i a B \sqrt {a+i a \tan (c+d x)}}{2 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {a (4 i A+5 B) \sqrt {a+i a \tan (c+d x)}}{4 d \sqrt {\cot (c+d x)}}\\ \end {align*}

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Mathematica [A]  time = 6.94, size = 441, normalized size = 1.81 \[ \frac {\cos ^2(c+d x) \sqrt {\cot (c+d x)} (\cos (d x)-i \sin (d x)) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \left (4 (\sin (c)+i \cos (c)) \tan (c+d x) (4 A+2 B \tan (c+d x)-5 i B)-\sqrt {2} (\cos (2 c+d x)-i \sin (2 c+d x)) \sqrt {i \sin ^2(c+d x) (\cot (c+d x)+i)} \left (\sqrt {2} (12 A-11 i B) \log \left (\frac {2 e^{\frac {5 i c}{2}} \left (2 i \sqrt {-1+e^{2 i (c+d x)}}-i \sqrt {2} e^{i (c+d x)}+\sqrt {2}\right )}{(12 A-11 i B) \left (e^{i (c+d x)}-i\right )}\right )+\sqrt {2} (-12 A+11 i B) \log \left (\frac {2 e^{\frac {5 i c}{2}} \left (2 \sqrt {-1+e^{2 i (c+d x)}}+\sqrt {2} e^{i (c+d x)}-i \sqrt {2}\right )}{(11 B+12 i A) \left (e^{i (c+d x)}+i\right )}\right )+32 (A-i B) \log \left ((\cos (c)-i \sin (c)) \left (i \sin (c+d x)+\cos (c+d x)+\sqrt {i \sin (2 (c+d x))+\cos (2 (c+d x))-1}\right )\right )\right )\right )}{16 d (A \cos (c+d x)+B \sin (c+d x))} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Cos[c + d*x]^2*Sqrt[Cot[c + d*x]]*(Cos[d*x] - I*Sin[d*x])*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x])*(
-(Sqrt[2]*(Sqrt[2]*(12*A - (11*I)*B)*Log[(2*E^(((5*I)/2)*c)*(Sqrt[2] - I*Sqrt[2]*E^(I*(c + d*x)) + (2*I)*Sqrt[
-1 + E^((2*I)*(c + d*x))]))/((12*A - (11*I)*B)*(-I + E^(I*(c + d*x))))] + Sqrt[2]*(-12*A + (11*I)*B)*Log[(2*E^
(((5*I)/2)*c)*((-I)*Sqrt[2] + Sqrt[2]*E^(I*(c + d*x)) + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]))/(((12*I)*A + 11*B)*
(I + E^(I*(c + d*x))))] + 32*(A - I*B)*Log[(Cos[c] - I*Sin[c])*(Cos[c + d*x] + I*Sin[c + d*x] + Sqrt[-1 + Cos[
2*(c + d*x)] + I*Sin[2*(c + d*x)]])])*Sqrt[I*(I + Cot[c + d*x])*Sin[c + d*x]^2]*(Cos[2*c + d*x] - I*Sin[2*c +
d*x])) + 4*(I*Cos[c] + Sin[c])*Tan[c + d*x]*(4*A - (5*I)*B + 2*B*Tan[c + d*x])))/(16*d*(A*Cos[c + d*x] + B*Sin
[c + d*x]))

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fricas [B]  time = 0.63, size = 907, normalized size = 3.72 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c))/cot(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

1/16*(4*sqrt(2)*((4*A - 7*I*B)*a*e^(5*I*d*x + 5*I*c) + 4*I*B*a*e^(3*I*d*x + 3*I*c) - (4*A - 3*I*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)) - sqrt((
144*I*A^2 + 264*A*B - 121*I*B^2)*a^3/d^2)*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*log(((-576*I*A
- 528*B)*a^2*e^(2*I*d*x + 2*I*c) + (192*I*A + 176*B)*a^2 + 32*sqrt(2)*sqrt((144*I*A^2 + 264*A*B - 121*I*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))*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)/(12*I*A + 11*B)) + sqrt((144*I*A^2 + 264*A*B - 121*
I*B^2)*a^3/d^2)*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*log(((-576*I*A - 528*B)*a^2*e^(2*I*d*x +
2*I*c) + (192*I*A + 176*B)*a^2 - 32*sqrt(2)*sqrt((144*I*A^2 + 264*A*B - 121*I*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))*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)/(12*I*A + 11*B)) - 4*sqrt((32*I*A^2 + 64*A*B - 32*I*B^2)*a^3/d^2)*(d*e^(4*I*d
*x + 4*I*c) + 2*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)*(I*d*e^(2*I*d*x + 2*I*c) - I*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)) + 4*sqrt((32*I*A^2 + 64*A*B - 3
2*I*B^2)*a^3/d^2)*(d*e^(4*I*d*x + 4*I*c) + 2*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)*(-I*d*e^(2*I*d*x + 2*I*c) + I*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^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\sqrt {\cot \left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c))/cot(d*x+c)^(1/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 4.01, size = 4490, normalized size = 18.40 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16/d*(8*I*A*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)-32*A*arctan(2^(1/2)*((-1+cos(
d*x+c))/sin(d*x+c))^(1/2)+1)*cos(d*x+c)^3-8*A*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)*cos(d*x+c)-32*I*A*arc
tan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*cos(d*x+c)^2-32*I*A*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+
c))^(1/2)-1)*cos(d*x+c)^2-16*I*A*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)^2-32*I*B*ar
ctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*cos(d*x+c)^2-32*I*B*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x
+c))^(1/2)-1)*cos(d*x+c)^2-16*I*B*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)^2-4*I*B*((
-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)+14*B*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)*cos(d*x+c)^3+8*A*((-1
+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)*cos(d*x+c)^3+32*I*A*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1
)*cos(d*x+c)^3+32*I*A*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*cos(d*x+c)^3+16*I*A*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)^3+32*I*B*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+
1)*cos(d*x+c)^3+32*I*B*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*cos(d*x+c)^3+16*I*B*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)^3+24*A*arctan(((-1+cos(d*x+c))/sin(d*x+c))^(1/2))*2^(1/2)*
cos(d*x+c)^3-12*A*ln(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*2^(1/2)*cos(d*x+c)^3+12*A*ln(((-1+cos(d*x+c))/sin(d
*x+c))^(1/2)+1)*2^(1/2)*cos(d*x+c)^3+16*A*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*
sin(d*x+c)-32*B*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*cos(d*x+c)^2*sin(d*x+c)-32*B*arctan(2^(1/
2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*cos(d*x+c)^2*sin(d*x+c)-16*B*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*sin(d*x+c)-22*B*arctan(((-1+cos(d*x+c))/sin(d*x+c))^(1/2))*2^(1/2)*cos(d*x+c)^3-11
*B*ln(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*2^(1/2)*cos(d*x+c)^3+11*B*ln(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)
*2^(1/2)*cos(d*x+c)^3+32*A*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*cos(d*x+c)^2*sin(d*x+c)+32*A*a
rctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*cos(d*x+c)^2*sin(d*x+c)+12*A*ln(((-1+cos(d*x+c))/sin(d*x+c
))^(1/2)-1)*2^(1/2)*cos(d*x+c)^2-12*A*ln(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*2^(1/2)*cos(d*x+c)^2-24*A*arcta
n(((-1+cos(d*x+c))/sin(d*x+c))^(1/2))*2^(1/2)*cos(d*x+c)^2-4*B*2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+14*I
*B*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)-12*I*A*ln(((-1+cos(d*x+c))/sin(d*x+c))^(
1/2)+1)*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)+22*I*B*arctan(((-1+cos(d*x+c))/sin(d*x+c))^(1/2))*2^(1/2)*cos(d*x+c)^2
*sin(d*x+c)-11*I*B*ln(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)+11*I*B*ln(((-1+cos
(d*x+c))/sin(d*x+c))^(1/2)+1)*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)-8*I*A*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)
*cos(d*x+c)*sin(d*x+c)-14*B*cos(d*x+c)*2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+10*I*B*((-1+cos(d*x+c))/sin(
d*x+c))^(1/2)*2^(1/2)*cos(d*x+c)*sin(d*x+c)+24*I*A*arctan(((-1+cos(d*x+c))/sin(d*x+c))^(1/2))*2^(1/2)*cos(d*x+
c)^2*sin(d*x+c)+12*I*A*ln(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)+4*B*2^(1/2)*co
s(d*x+c)^2*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-4*B*2^(1/2)*sin(d*x+c)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+11*B*l
n(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*2^(1/2)*cos(d*x+c)^2-11*B*ln(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*2^(
1/2)*cos(d*x+c)^2+22*B*arctan(((-1+cos(d*x+c))/sin(d*x+c))^(1/2))*2^(1/2)*cos(d*x+c)^2-32*A*arctan(2^(1/2)*((-
1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*cos(d*x+c)^3-16*A*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))*co
s(d*x+c)^3+32*B*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*cos(d*x+c)^3+32*B*arctan(2^(1/2)*((-1+cos
(d*x+c))/sin(d*x+c))^(1/2)-1)*cos(d*x+c)^3+16*B*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)^3-32*B*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*cos(d*x+c)^2-16*B*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+32*A*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*cos(d*x+c)^2
+32*A*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*cos(d*x+c)^2+16*A*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)^2-32*B*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*cos(d*x+c)^2-24*A
*arctan(((-1+cos(d*x+c))/sin(d*x+c))^(1/2))*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)+12*A*ln(((-1+cos(d*x+c))/sin(d*x+c
))^(1/2)-1)*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)+4*I*B*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)*cos(d*x+c)^2+14*I
*B*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)*cos(d*x+c)-8*A*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)*cos(d*
x+c)^2*sin(d*x+c)-8*A*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)*cos(d*x+c)*sin(d*x+c)-24*I*A*arctan(((-1+cos(
d*x+c))/sin(d*x+c))^(1/2))*2^(1/2)*cos(d*x+c)^3-12*I*A*ln(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*2^(1/2)*cos(d*
x+c)^3+12*I*A*ln(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*2^(1/2)*cos(d*x+c)^3+8*I*A*((-1+cos(d*x+c))/sin(d*x+c))
^(1/2)*2^(1/2)*cos(d*x+c)^3-22*I*B*arctan(((-1+cos(d*x+c))/sin(d*x+c))^(1/2))*2^(1/2)*cos(d*x+c)^3+11*I*B*ln((
(-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*2^(1/2)*cos(d*x+c)^3-11*I*B*ln(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*2^(1
/2)*cos(d*x+c)^3-14*I*B*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)*cos(d*x+c)^3-12*A*ln(((-1+cos(d*x+c))/sin(d
*x+c))^(1/2)+1)*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)+22*B*arctan(((-1+cos(d*x+c))/sin(d*x+c))^(1/2))*2^(1/2)*cos(d*
x+c)^2*sin(d*x+c)+11*B*ln(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)-11*B*ln(((-1+c
os(d*x+c))/sin(d*x+c))^(1/2)+1)*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)+14*B*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2
)*cos(d*x+c)^2*sin(d*x+c)-32*I*A*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*cos(d*x+c)^2*sin(d*x+c)-
32*I*A*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*cos(d*x+c)^2*sin(d*x+c)-16*I*A*ln(-(2^(1/2)*((-1+c
os(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)*s
in(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))*cos(d*x+c)^2*sin(d*x+c)+24*I*A*arctan(((-1+cos(d*x+c))/sin(d*x+c))^(1/2))*
2^(1/2)*cos(d*x+c)^2+12*I*A*ln(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*2^(1/2)*cos(d*x+c)^2-12*I*A*ln(((-1+cos(d
*x+c))/sin(d*x+c))^(1/2)+1)*2^(1/2)*cos(d*x+c)^2-32*I*B*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*c
os(d*x+c)^2*sin(d*x+c)-32*I*B*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*cos(d*x+c)^2*sin(d*x+c)+22*
I*B*arctan(((-1+cos(d*x+c))/sin(d*x+c))^(1/2))*2^(1/2)*cos(d*x+c)^2-11*I*B*ln(((-1+cos(d*x+c))/sin(d*x+c))^(1/
2)-1)*2^(1/2)*cos(d*x+c)^2+11*I*B*ln(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)*2^(1/2)*cos(d*x+c)^2-16*I*B*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*sin(d*x+c)-8*I*A*((-1+cos(d*x+c))/sin(d*x+c))^(1/
2)*2^(1/2)*cos(d*x+c)-4*I*B*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-10*B*2^(1/2)*cos(d*x+c)*sin(
d*x+c)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2))*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)/(I*cos(d*x+c)+I*sin(
d*x+c)-1+I+cos(d*x+c)-sin(d*x+c))/cos(d*x+c)/(cos(d*x+c)/sin(d*x+c))^(1/2)/((-1+cos(d*x+c))/sin(d*x+c))^(1/2)/
sin(d*x+c)*2^(1/2)*a

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\sqrt {\cot \left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c))/cot(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}} \left (A + B \tan {\left (c + d x \right )}\right )}{\sqrt {\cot {\left (c + d x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(d*x+c))**(3/2)*(A+B*tan(d*x+c))/cot(d*x+c)**(1/2),x)

[Out]

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

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