3.639 \(\int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx\)
Optimal. Leaf size=163 \[ \frac {(A+i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}+\frac {(A-i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {b+i a}} \]
[Out]
(A+I*B)*arctan((I*a-b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/d/(I*a
-b)^(1/2)+(A-I*B)*arctanh((I*a+b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))*cot(d*x+c)^(1/2)*tan(d*x+c)^(
1/2)/d/(I*a+b)^(1/2)
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Rubi [A] time = 0.48, antiderivative size = 163, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used =
{4241, 3616, 3615, 93, 203, 206} \[ \frac {(A+i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}+\frac {(A-i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {b+i a}} \]
Antiderivative was successfully verified.
[In]
Int[(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]
[Out]
((A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c +
d*x]])/(Sqrt[I*a - b]*d) + ((A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sq
rt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a + b]*d)
Rule 93
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*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 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 3615
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^2/f, Subst[Int[((a + b*x)^m*(c + d*x)^n)/(A - B*x), 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] && NeQ[a^2 + b^2, 0] && EqQ[A^2 +
B^2, 0]
Rule 3616
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 + I*B)/2, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 -
I*Tan[e + f*x]), x], x] + Dist[(A - I*B)/2, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 + I*Tan[e +
f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[A^2
+ B^2, 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 {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx &=\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx\\ &=\frac {1}{2} \left ((A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1+i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{2} \left ((A+i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx\\ &=\frac {\left ((A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{(1-i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {\left ((A+i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{(1+i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac {\left ((A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1-(i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {\left ((A+i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1-(-i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}\\ &=\frac {(A+i B) \tan ^{-1}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {i a-b} d}+\frac {(A-i B) \tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {i a+b} d}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 157, normalized size = 0.96 \[ \frac {\sqrt [4]{-1} \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {(B-i A) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {a+i b}}-\frac {(B+i A) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {-a+i b}}\right )}{d} \]
Antiderivative was successfully verified.
[In]
Integrate[(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]
[Out]
((-1)^(1/4)*(-(((I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqr
t[-a + I*b]) + (((-I)*A + B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/S
qrt[a + I*b])*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^(1/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),x, algorithm="fricas")
[Out]
Timed out
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^(1/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),x, algorithm="giac")
[Out]
Timed out
________________________________________________________________________________________
maple [C] time = 3.60, size = 3474, normalized size = 21.31 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(d*x+c)^(1/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),x)
[Out]
1/d*2^(1/2)*(cos(d*x+c)/sin(d*x+c))^(1/2)*(1/cos(d*x+c)*(a*cos(d*x+c)+b*sin(d*x+c)))^(1/2)*(-2*I*A*EllipticPi(
(-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2),(-b+(a^2+b^
2)^(1/2))/(-I*a+(a^2+b^2)^(1/2)-b),1/2*2^(1/2)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*a*b^2+I*A*Ellipti
cPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2),(-b+(a^
2+b^2)^(1/2))/(I*a+(a^2+b^2)^(1/2)-b),1/2*2^(1/2)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*a^3-I*B*Ellipt
icPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2),(-b+(a
^2+b^2)^(1/2))/(I*a+(a^2+b^2)^(1/2)-b),1/2*2^(1/2)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*a^2*b+I*B*Ell
ipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2),(-b
+(a^2+b^2)^(1/2))/(-I*a+(a^2+b^2)^(1/2)-b),1/2*2^(1/2)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*a^2*b+2*I
*A*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/
2),(-b+(a^2+b^2)^(1/2))/(-I*a+(a^2+b^2)^(1/2)-b),1/2*2^(1/2)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*a*b
*(a^2+b^2)^(1/2)-I*B*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^
2+b^2)^(1/2)))^(1/2),(-b+(a^2+b^2)^(1/2))/(-I*a+(a^2+b^2)^(1/2)-b),1/2*2^(1/2)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)
^(1/2))^(1/2))*a^2*(a^2+b^2)^(1/2)-I*A*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a)/
sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2),(-b+(a^2+b^2)^(1/2))/(-I*a+(a^2+b^2)^(1/2)-b),1/2*2^(1/2)*((-b+(a^2+b^2
)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*a^3+I*B*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a
)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2),(-b+(a^2+b^2)^(1/2))/(I*a+(a^2+b^2)^(1/2)-b),1/2*2^(1/2)*((-b+(a^2+b^
2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*a^2*(a^2+b^2)^(1/2)+2*I*A*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*
x+c)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2),(-b+(a^2+b^2)^(1/2))/(I*a+(a^2+b^2)^(1/2)-b),1/2*2
^(1/2)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*a*b^2-2*I*A*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*c
os(d*x+c)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2),(-b+(a^2+b^2)^(1/2))/(I*a+(a^2+b^2)^(1/2)-b),
1/2*2^(1/2)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*a*b*(a^2+b^2)^(1/2)+A*EllipticPi((-(-(a^2+b^2)^(1/2)
*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2),(-b+(a^2+b^2)^(1/2))/(-I*a+(a^
2+b^2)^(1/2)-b),1/2*2^(1/2)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*a^2*(a^2+b^2)^(1/2)+A*EllipticPi((-(
-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2),(-b+(a^2+b^2)^
(1/2))/(I*a+(a^2+b^2)^(1/2)-b),1/2*2^(1/2)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*a^2*(a^2+b^2)^(1/2)-2
*A*EllipticF((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2
),1/2*2^(1/2)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*a^2*(a^2+b^2)^(1/2)-4*A*EllipticF((-(-(a^2+b^2)^(1
/2)*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((-b+(a^2+b^2)^
(1/2))/(a^2+b^2)^(1/2))^(1/2))*b^2*(a^2+b^2)^(1/2)-A*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+b*
sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2),(-b+(a^2+b^2)^(1/2))/(-I*a+(a^2+b^2)^(1/2)-b),1/2*2^(1/2)
*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*a^2*b-A*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+
b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2),(-b+(a^2+b^2)^(1/2))/(I*a+(a^2+b^2)^(1/2)-b),1/2*2^(1/2
)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*a^2*b+4*A*EllipticF((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c
)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1
/2))*a^2*b+4*A*EllipticF((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^
(1/2)))^(1/2),1/2*2^(1/2)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*b^3+2*B*EllipticPi((-(-(a^2+b^2)^(1/2)
*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2),(-b+(a^2+b^2)^(1/2))/(-I*a+(a^
2+b^2)^(1/2)-b),1/2*2^(1/2)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*a*b*(a^2+b^2)^(1/2)+2*B*EllipticPi((
-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2),(-b+(a^2+b^2
)^(1/2))/(I*a+(a^2+b^2)^(1/2)-b),1/2*2^(1/2)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*a*b*(a^2+b^2)^(1/2)
-B*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/
2),(-b+(a^2+b^2)^(1/2))/(-I*a+(a^2+b^2)^(1/2)-b),1/2*2^(1/2)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*a^3
-2*B*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(
1/2),(-b+(a^2+b^2)^(1/2))/(-I*a+(a^2+b^2)^(1/2)-b),1/2*2^(1/2)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*a
*b^2-B*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))
^(1/2),(-b+(a^2+b^2)^(1/2))/(I*a+(a^2+b^2)^(1/2)-b),1/2*2^(1/2)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))*
a^3-2*B*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2))
)^(1/2),(-b+(a^2+b^2)^(1/2))/(I*a+(a^2+b^2)^(1/2)-b),1/2*2^(1/2)*((-b+(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2))^(1/2))
*a*b^2)*(a*(-1+cos(d*x+c))/(-b+(a^2+b^2)^(1/2))/sin(d*x+c))^(1/2)*(((a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+b*
sin(d*x+c)-a)/(a^2+b^2)^(1/2)/sin(d*x+c))^(1/2)*(-(-(a^2+b^2)^(1/2)*sin(d*x+c)+a*cos(d*x+c)+b*sin(d*x+c)-a)/si
n(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2)*sin(d*x+c)^2/(-1+cos(d*x+c))/(a*cos(d*x+c)+b*sin(d*x+c))/a/(I*a+(a^2+b^2)
^(1/2)-b)/(I*a-(a^2+b^2)^(1/2)+b)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \sqrt {\cot \left (d x + c\right )}}{\sqrt {b \tan \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^(1/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),x, algorithm="maxima")
[Out]
integrate((B*tan(d*x + c) + A)*sqrt(cot(d*x + c))/sqrt(b*tan(d*x + c) + a), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\mathrm {cot}\left (c+d\,x\right )}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((cot(c + d*x)^(1/2)*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^(1/2),x)
[Out]
int((cot(c + d*x)^(1/2)*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^(1/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \sqrt {\cot {\left (c + d x \right )}}}{\sqrt {a + b \tan {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)**(1/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**(1/2),x)
[Out]
Integral((A + B*tan(c + d*x))*sqrt(cot(c + d*x))/sqrt(a + b*tan(c + d*x)), x)
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