3.95 \(\int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{\sqrt {a+i a \tan (c+d x)}} \, dx\)
Optimal. Leaf size=167 \[ \frac {(-2 B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {(B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}-\frac {(2 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{a d}+\frac {(A+i B) \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}} \]
[Out]
(I*A-2*B)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/d/a^(1/2)+1/2*(I*A+B)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)
*2^(1/2)/a^(1/2))/d*2^(1/2)/a^(1/2)+(A+I*B)*cot(d*x+c)/d/(a+I*a*tan(d*x+c))^(1/2)-(2*A+I*B)*cot(d*x+c)*(a+I*a*
tan(d*x+c))^(1/2)/a/d
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Rubi [A] time = 0.57, antiderivative size = 167, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 8, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used =
{3596, 3598, 3600, 3480, 206, 3599, 63, 208} \[ \frac {(-2 B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {(B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}-\frac {(2 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{a d}+\frac {(A+i B) \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
[In]
Int[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]],x]
[Out]
((I*A - 2*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + ((I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c
+ d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*Sqrt[a]*d) + ((A + I*B)*Cot[c + d*x])/(d*Sqrt[a + I*a*Tan[c + d*x]]) - ((
2*A + I*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(a*d)
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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 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]
Rule 3480
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 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 3598
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 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 3600
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 \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{\sqrt {a+i a \tan (c+d x)}} \, dx &=\frac {(A+i B) \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}+\frac {\int \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (a (2 A+i B)-\frac {3}{2} a (i A-B) \tan (c+d x)\right ) \, dx}{a^2}\\ &=\frac {(A+i B) \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {(2 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{a d}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {1}{2} a^2 (i A-2 B)-\frac {1}{2} a^2 (2 A+i B) \tan (c+d x)\right ) \, dx}{a^3}\\ &=\frac {(A+i B) \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {(2 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{a d}-\frac {(i A-2 B) \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \, dx}{2 a^2}-\frac {(A-i B) \int \sqrt {a+i a \tan (c+d x)} \, dx}{2 a}\\ &=\frac {(A+i B) \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {(2 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{a d}-\frac {(i A-2 B) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {(i A+B) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{d}\\ &=\frac {(i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}+\frac {(A+i B) \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {(2 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{a d}-\frac {(A+2 i B) \operatorname {Subst}\left (\int \frac {1}{i-\frac {i x^2}{a}} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{a d}\\ &=\frac {(i A-2 B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {(i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}+\frac {(A+i B) \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {(2 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{a d}\\ \end {align*}
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Mathematica [A] time = 4.50, size = 224, normalized size = 1.34 \[ \frac {(A \cot (c+d x)+B) \left (2 (B-2 i A) \sin (c+d x)+\frac {\left (-1+e^{2 i (c+d x)}\right ) \sqrt {\sec (c+d x)} \left ((A-i B) \sinh ^{-1}\left (e^{i (c+d x)}\right )+\sqrt {2} (A+2 i B) \tanh ^{-1}\left (\frac {\sqrt {2} e^{i (c+d x)}}{\sqrt {1+e^{2 i (c+d x)}}}\right )\right )}{\sqrt {2} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}}}-2 A \cos (c+d x)\right )}{2 d \sqrt {a+i a \tan (c+d x)} (A \cos (c+d x)+B \sin (c+d x))} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]],x]
[Out]
((B + A*Cot[c + d*x])*(-2*A*Cos[c + d*x] + ((-1 + E^((2*I)*(c + d*x)))*((A - I*B)*ArcSinh[E^(I*(c + d*x))] + S
qrt[2]*(A + (2*I)*B)*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[Sec[c + d*x]])/(Sq
rt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]) + 2*((-2*I)*A + B)*Sin[c
+ d*x]))/(2*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])
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fricas [B] time = 0.53, size = 746, normalized size = 4.47 \[ \frac {{\left (a d e^{\left (3 i \, d x + 3 i \, c\right )} - a d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {-\frac {2 \, A^{2} - 4 i \, A B - 2 \, B^{2}}{a d^{2}}} \log \left (\frac {{\left ({\left (4 i \, A + 4 \, B\right )} a e^{\left (i \, d x + i \, c\right )} + 2 \, \sqrt {2} {\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {2 \, A^{2} - 4 i \, A B - 2 \, B^{2}}{a d^{2}}}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) - {\left (a d e^{\left (3 i \, d x + 3 i \, c\right )} - a d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {-\frac {2 \, A^{2} - 4 i \, A B - 2 \, B^{2}}{a d^{2}}} \log \left (\frac {{\left ({\left (4 i \, A + 4 \, B\right )} a e^{\left (i \, d x + i \, c\right )} - 2 \, \sqrt {2} {\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {2 \, A^{2} - 4 i \, A B - 2 \, B^{2}}{a d^{2}}}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) - {\left (a d e^{\left (3 i \, d x + 3 i \, c\right )} - a d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {-\frac {A^{2} + 4 i \, A B - 4 \, B^{2}}{a d^{2}}} \log \left (\frac {{\left ({\left (-48 i \, A + 96 \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-16 i \, A + 32 \, B\right )} a^{2} + 32 \, \sqrt {2} {\left (a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{2} d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {A^{2} + 4 i \, A B - 4 \, B^{2}}{a d^{2}}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{-i \, A + 2 \, B}\right ) + {\left (a d e^{\left (3 i \, d x + 3 i \, c\right )} - a d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {-\frac {A^{2} + 4 i \, A B - 4 \, B^{2}}{a d^{2}}} \log \left (\frac {{\left ({\left (-48 i \, A + 96 \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-16 i \, A + 32 \, B\right )} a^{2} - 32 \, \sqrt {2} {\left (a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{2} d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {A^{2} + 4 i \, A B - 4 \, B^{2}}{a d^{2}}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{-i \, A + 2 \, B}\right ) + \sqrt {2} {\left ({\left (-6 i \, A + 2 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - 4 i \, A e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i \, A - 2 \, B\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{4 \, {\left (a d e^{\left (3 i \, d x + 3 i \, c\right )} - a d e^{\left (i \, d x + i \, c\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="fricas")
[Out]
1/4*((a*d*e^(3*I*d*x + 3*I*c) - a*d*e^(I*d*x + I*c))*sqrt(-(2*A^2 - 4*I*A*B - 2*B^2)/(a*d^2))*log(((4*I*A + 4*
B)*a*e^(I*d*x + I*c) + 2*sqrt(2)*(a*d*e^(2*I*d*x + 2*I*c) + a*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-(2*A^
2 - 4*I*A*B - 2*B^2)/(a*d^2)))*e^(-I*d*x - I*c)/(I*A + B)) - (a*d*e^(3*I*d*x + 3*I*c) - a*d*e^(I*d*x + I*c))*s
qrt(-(2*A^2 - 4*I*A*B - 2*B^2)/(a*d^2))*log(((4*I*A + 4*B)*a*e^(I*d*x + I*c) - 2*sqrt(2)*(a*d*e^(2*I*d*x + 2*I
*c) + a*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-(2*A^2 - 4*I*A*B - 2*B^2)/(a*d^2)))*e^(-I*d*x - I*c)/(I*A +
B)) - (a*d*e^(3*I*d*x + 3*I*c) - a*d*e^(I*d*x + I*c))*sqrt(-(A^2 + 4*I*A*B - 4*B^2)/(a*d^2))*log(((-48*I*A +
96*B)*a^2*e^(2*I*d*x + 2*I*c) + (-16*I*A + 32*B)*a^2 + 32*sqrt(2)*(a^2*d*e^(3*I*d*x + 3*I*c) + a^2*d*e^(I*d*x
+ I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-(A^2 + 4*I*A*B - 4*B^2)/(a*d^2)))*e^(-2*I*d*x - 2*I*c)/(-I*A +
2*B)) + (a*d*e^(3*I*d*x + 3*I*c) - a*d*e^(I*d*x + I*c))*sqrt(-(A^2 + 4*I*A*B - 4*B^2)/(a*d^2))*log(((-48*I*A
+ 96*B)*a^2*e^(2*I*d*x + 2*I*c) + (-16*I*A + 32*B)*a^2 - 32*sqrt(2)*(a^2*d*e^(3*I*d*x + 3*I*c) + a^2*d*e^(I*d*
x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-(A^2 + 4*I*A*B - 4*B^2)/(a*d^2)))*e^(-2*I*d*x - 2*I*c)/(-I*A
+ 2*B)) + sqrt(2)*((-6*I*A + 2*B)*e^(4*I*d*x + 4*I*c) - 4*I*A*e^(2*I*d*x + 2*I*c) + 2*I*A - 2*B)*sqrt(a/(e^(2
*I*d*x + 2*I*c) + 1)))/(a*d*e^(3*I*d*x + 3*I*c) - a*d*e^(I*d*x + I*c))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{2}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="giac")
[Out]
integrate((B*tan(d*x + c) + A)*cot(d*x + c)^2/sqrt(I*a*tan(d*x + c) + a), x)
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maple [B] time = 3.93, size = 2727, normalized size = 16.33 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(1/2),x)
[Out]
1/4/d*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)*(A*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/
2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*sin(d*x+c)*cos(d*x+c)^2
+2*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+A*(-2*cos(d*x+c)/(1+c
os(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))*cos(d*x+c)+I
*A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/si
n(d*x+c))*sin(d*x+c)+2*I*B*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+
c)))^(1/2))+A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d
*x+c)-1)/sin(d*x+c))-4*B*cos(d*x+c)+4*A*cos(d*x+c)^2*sin(d*x+c)+4*B*cos(d*x+c)^3+I*B*2^(1/2)*(-2*cos(d*x+c)/(1
+cos(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(
1/2))*sin(d*x+c)+A*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/
2))-2*B*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+
cos(d*x+c)-1)/sin(d*x+c))+I*A*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+
c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*cos(d*x+c)-I*A*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+
c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*cos
(d*x+c)^3-I*A*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(
-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*cos(d*x+c)^2-I*A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*c
os(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))*sin(d*x+c)*cos(d*x+c)^2-2*I*B*(-2*cos(d*x
+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))*sin(d*x+c)*cos(d*x+c)^2-A*(-2*cos(d*x
+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))*cos(
d*x+c)^3-2*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))*cos(d*x+c)^3-
A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin
(d*x+c))*cos(d*x+c)^2-2*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))*
cos(d*x+c)^2+2*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))*cos(d*x+c
)+B*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x
+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))-I*A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x
+c)))^(1/2))+2*I*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+
cos(d*x+c)-1)/sin(d*x+c))-I*B*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+
c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*sin(d*x+c)*cos(d*x+c)^2-8*I*A*cos(d*x+c)^3+8*I*A*
cos(d*x+c)-2*I*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+co
s(d*x+c)-1)/sin(d*x+c))*cos(d*x+c)^3-2*I*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*
x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))*cos(d*x+c)^2-I*A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan
(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))*cos(d*x+c)+2*I*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d
*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))*cos(d*x+c)+I*A*(-2*cos(d*x+c)/(1+cos(d*x+c)))
^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))*cos(d*x+c)^3-B*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1
/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*cos(d*x+c)
^3+I*A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))*cos(d*x+c)^2-B*2^(1
/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+c
os(d*x+c)))^(1/2)*2^(1/2))*cos(d*x+c)^2+I*A*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x
+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))+B*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x
+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*co
s(d*x+c)-A*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*
cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*sin(d*x+c)-A*cos(d*x+c)^2*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^
(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+2*B*cos(d*x+c)^2*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c))
)^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c)))/(-1+cos(d*x+c))/(I*si
n(d*x+c)+cos(d*x+c))/(1+cos(d*x+c))/a
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maxima [A] time = 0.64, size = 184, normalized size = 1.10 \[ -\frac {i \, a {\left (\frac {\sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, {\left (A + 2 i \, B\right )} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {4 \, {\left ({\left (i \, a \tan \left (d x + c\right ) + a\right )} {\left (2 \, A + i \, B\right )} - {\left (A + i \, B\right )} a\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a - \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{2}}\right )}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="maxima")
[Out]
-1/4*I*a*(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*ta
n(d*x + c) + a)))/a^(3/2) + 2*(A + 2*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) + 4*((I*a*tan(d*x + c) + a)*(2*A + I*B) - (A + I*B)*a)/((I*a*tan(d*x + c) + a)^(3/2)*a
- sqrt(I*a*tan(d*x + c) + a)*a^2))/d
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mupad [B] time = 8.62, size = 2961, normalized size = 17.73 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((cot(c + d*x)^2*(A + B*tan(c + d*x)))/(a + a*tan(c + d*x)*1i)^(1/2),x)
[Out]
2*atanh((3*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*((9*B^2)/(16*a*d^2) - (3*A^2)/(16*a*d^2) - ((A^4*a^10)/d^4 + (49*
B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)/(16*a^6) - (A*B*3
i)/(8*a*d^2))^(1/2)*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^
3*B*a^10*20i)/d^4)^(1/2))/((A^3*a^5*d*1i)/2 + (35*B^3*a^5*d)/2 + (A*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 -
(114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)*3i)/2 - (3*B*d^3*((A^4*a^10)/d^4
+ (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2))/2 - (A*B^2
*a^5*d*57i)/2 - (15*A^2*B*a^5*d)/2) + (A^2*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((9*B^2)/(16*a*d^2) - (3*A^2)
/(16*a*d^2) - ((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^
10*20i)/d^4)^(1/2)/(16*a^6) - (A*B*3i)/(8*a*d^2))^(1/2))/((A^3*a^2*d*1i)/2 + (35*B^3*a^2*d)/2 - (A*B^2*a^2*d*5
7i)/2 - (15*A^2*B*a^2*d)/2 + (A*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10
*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)*3i)/(2*a^3) - (3*B*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2
*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2))/(2*a^3)) - (7*B^2*a^2*d^2*(a + a*tan(c +
d*x)*1i)^(1/2)*((9*B^2)/(16*a*d^2) - (3*A^2)/(16*a*d^2) - ((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*
a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)/(16*a^6) - (A*B*3i)/(8*a*d^2))^(1/2))/((A^3*a^
2*d*1i)/2 + (35*B^3*a^2*d)/2 - (A*B^2*a^2*d*57i)/2 - (15*A^2*B*a^2*d)/2 + (A*d^3*((A^4*a^10)/d^4 + (49*B^4*a^1
0)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)*3i)/(2*a^3) - (3*B*d^3*(
(A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1
/2))/(2*a^3)) + (A*B*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((9*B^2)/(16*a*d^2) - (3*A^2)/(16*a*d^2) - ((A^4*a^
10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)/(16
*a^6) - (A*B*3i)/(8*a*d^2))^(1/2)*10i)/((A^3*a^2*d*1i)/2 + (35*B^3*a^2*d)/2 - (A*B^2*a^2*d*57i)/2 - (15*A^2*B*
a^2*d)/2 + (A*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*
B*a^10*20i)/d^4)^(1/2)*3i)/(2*a^3) - (3*B*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (
A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2))/(2*a^3)))*((9*B^2)/(16*a*d^2) - (3*A^2)/(16*a*d^2) - ((A^4
*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)/
(16*a^6) - (A*B*3i)/(8*a*d^2))^(1/2) - 2*atanh((3*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*(((A^4*a^10)/d^4 + (49*B^4
*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)/(16*a^6) - (3*A^2)/(
16*a*d^2) + (9*B^2)/(16*a*d^2) - (A*B*3i)/(8*a*d^2))^(1/2)*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*
a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2))/((A^3*a^5*d*1i)/2 + (35*B^3*a^5*d)/2 - (A*d^3
*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^
(1/2)*3i)/2 + (3*B*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 +
(A^3*B*a^10*20i)/d^4)^(1/2))/2 - (A*B^2*a^5*d*57i)/2 - (15*A^2*B*a^5*d)/2) - (A^2*a^2*d^2*(a + a*tan(c + d*x)*
1i)^(1/2)*(((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*
20i)/d^4)^(1/2)/(16*a^6) - (3*A^2)/(16*a*d^2) + (9*B^2)/(16*a*d^2) - (A*B*3i)/(8*a*d^2))^(1/2))/((A^3*a^2*d*1i
)/2 + (35*B^3*a^2*d)/2 - (A*B^2*a^2*d*57i)/2 - (15*A^2*B*a^2*d)/2 - (A*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4
- (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)*3i)/(2*a^3) + (3*B*d^3*((A^4*a
^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2))/(
2*a^3)) + (7*B^2*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^1
0)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)/(16*a^6) - (3*A^2)/(16*a*d^2) + (9*B^2)/(16*a*d^2
) - (A*B*3i)/(8*a*d^2))^(1/2))/((A^3*a^2*d*1i)/2 + (35*B^3*a^2*d)/2 - (A*B^2*a^2*d*57i)/2 - (15*A^2*B*a^2*d)/2
- (A*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*2
0i)/d^4)^(1/2)*3i)/(2*a^3) + (3*B*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^
10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2))/(2*a^3)) - (A*B*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((A^4*a^10)
/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)/(16*a^
6) - (3*A^2)/(16*a*d^2) + (9*B^2)/(16*a*d^2) - (A*B*3i)/(8*a*d^2))^(1/2)*10i)/((A^3*a^2*d*1i)/2 + (35*B^3*a^2*
d)/2 - (A*B^2*a^2*d*57i)/2 - (15*A^2*B*a^2*d)/2 - (A*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^
10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)*3i)/(2*a^3) + (3*B*d^3*((A^4*a^10)/d^4 + (49*B^4
*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2))/(2*a^3)))*(((A^4*a^
10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)/(16
*a^6) - (3*A^2)/(16*a*d^2) + (9*B^2)/(16*a*d^2) - (A*B*3i)/(8*a*d^2))^(1/2) - (((A*a + B*a*1i)*1i)/d - ((2*A +
B*1i)*(a + a*tan(c + d*x)*1i)*1i)/d)/(a*(a + a*tan(c + d*x)*1i)^(1/2) - (a + a*tan(c + d*x)*1i)^(3/2))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{2}{\left (c + d x \right )}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)**2*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))**(1/2),x)
[Out]
Integral((A + B*tan(c + d*x))*cot(c + d*x)**2/sqrt(I*a*(tan(c + d*x) - I)), x)
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