3.870 \(\int \frac{1}{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx\)
Optimal. Leaf size=189 \[ -\frac{2 b}{d \left (a^2+b^2\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (-b+i a)^{3/2}}+\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (b+i a)^{3/2}} \]
[Out]
-((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]])/
((I*a - b)^(3/2)*d)) + (ArcTanh[(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]])/((I*a + b)^(3/2)*d) - (2*b)/((a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])
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Rubi [A] time = 0.504191, antiderivative size = 189, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used =
{4241, 3568, 3616, 3615, 93, 203, 206} \[ -\frac{2 b}{d \left (a^2+b^2\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (-b+i a)^{3/2}}+\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (b+i a)^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[1/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)),x]
[Out]
-((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]])/
((I*a - b)^(3/2)*d)) + (ArcTanh[(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]])/((I*a + b)^(3/2)*d) - (2*b)/((a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])
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]
Rule 3568
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n)/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(a^2
+ b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[a*c*(m + 1) - b*d*n - (b*c - a*d)*
(m + 1)*Tan[e + f*x] - b*d*(m + n + 1)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[2*m]
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 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 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])
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx &=\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{\tan (c+d x)}}{(a+b \tan (c+d x))^{3/2}} \, dx\\ &=-\frac{2 b}{\left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{\left (2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{-\frac{b}{2}-\frac{1}{2} a \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{a^2+b^2}\\ &=-\frac{2 b}{\left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{\left ((-i a-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}{2 \left (a^2+b^2\right )}-\frac{\left ((i a-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}{2 \left (a^2+b^2\right )}\\ &=-\frac{2 b}{\left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{\left ((-i a-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 \left (a^2+b^2\right ) d}-\frac{\left ((i a-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 \left (a^2+b^2\right ) d}\\ &=-\frac{2 b}{\left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{\left ((-i a-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 )}{\left (a^2+b^2\right ) d}-\frac{\left ((i a-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 )}{\left (a^2+b^2\right ) d}\\ &=-\frac{\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)}}{(i a-b)^{3/2} d}+\frac{\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)}}{(i a+b)^{3/2} d}-\frac{2 b}{\left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 2.13645, size = 184, normalized size = 0.97 \[ -\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left (\frac{2 b \sqrt{\tan (c+d x)}}{\left (a^2+b^2\right ) \sqrt{a+b \tan (c+d x)}}+\frac{\sqrt [4]{-1} \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{(-a-i b)^{3/2}}+\frac{\sqrt [4]{-1} \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{(a-i b)^{3/2}}\right )}{d} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)),x]
[Out]
-((Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((-1)^(1/4)*ArcTanh[((-1)^(1/4)*Sqrt[-a - I*b]*Sqrt[Tan[c + d*x]])/S
qrt[a + b*Tan[c + d*x]]])/(-a - I*b)^(3/2) + ((-1)^(1/4)*ArcTanh[((-1)^(1/4)*Sqrt[a - I*b]*Sqrt[Tan[c + d*x]])
/Sqrt[a + b*Tan[c + d*x]]])/(a - I*b)^(3/2) + (2*b*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]]))
)/d)
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Maple [C] time = 0.413, size = 4901, normalized size = 25.9 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(3/2),x)
[Out]
1/d*2^(1/2)/(a^2+b^2)/(I*a+(a^2+b^2)^(1/2)-b)/(I*a-(a^2+b^2)^(1/2)+b)/a*(2*I*sin(d*x+c)*(-(-(a^2+b^2)^(1/2)*si
n(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2)*(((a^2+b^2)^(1/2)*sin(d*x+c)+b*si
n(d*x+c)+a*cos(d*x+c)-a)/(a^2+b^2)^(1/2)/sin(d*x+c))^(1/2)*(a*(cos(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))/sin(d*x+c))^
(1/2)*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(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^3-2*I*sin(d*x+c)*(a^2+b^2)^(1/2)*(-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/sin(d*x+c)/(-b
+(a^2+b^2)^(1/2)))^(1/2)*(((a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/(a^2+b^2)^(1/2)/sin(d*x+c))
^(1/2)*(a*(cos(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))/sin(d*x+c))^(1/2)*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*si
n(d*x+c)+a*cos(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*sin(d*x+c)*(a^2+b^2)^(1/2)*(-(-(a^2+b^2)^(1/2
)*sin(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2)*(((a^2+b^2)^(1/2)*sin(d*x+c)+
b*sin(d*x+c)+a*cos(d*x+c)-a)/(a^2+b^2)^(1/2)/sin(d*x+c))^(1/2)*(a*(cos(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))/sin(d*x+
c))^(1/2)*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(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*I*sin(d*x+c)*(a^2+b^2)^(1/2)*(-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/sin(d*x+c)/
(-b+(a^2+b^2)^(1/2)))^(1/2)*(((a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/(a^2+b^2)^(1/2)/sin(d*x+
c))^(1/2)*(a*(cos(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))/sin(d*x+c))^(1/2)*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b
*sin(d*x+c)+a*cos(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*sin(d*x+c)*(-(-(a^2+b^2)^(1/2)*sin(d*x+
c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2)*(((a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+
c)+a*cos(d*x+c)-a)/(a^2+b^2)^(1/2)/sin(d*x+c))^(1/2)*(a*(cos(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))/sin(d*x+c))^(1/2)*
EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(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^3+
I*sin(d*x+c)*(a^2+b^2)^(1/2)*(-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b
^2)^(1/2)))^(1/2)*(((a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/(a^2+b^2)^(1/2)/sin(d*x+c))^(1/2)*
(a*(cos(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))/sin(d*x+c))^(1/2)*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c
)+a*cos(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+3*sin(d*x+c)*(a^2+b^2)^(1/2)*(-(-(a^2+b^2)^(1/2)*sin(d*x
+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2)*(((a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x
+c)+a*cos(d*x+c)-a)/(a^2+b^2)^(1/2)/sin(d*x+c))^(1/2)*(a*(cos(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))/sin(d*x+c))^(1/2)
*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(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+
3*sin(d*x+c)*(a^2+b^2)^(1/2)*(-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b
^2)^(1/2)))^(1/2)*(((a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/(a^2+b^2)^(1/2)/sin(d*x+c))^(1/2)*
(a*(cos(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))/sin(d*x+c))^(1/2)*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c
)+a*cos(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*sin(d*x+c)*(a^2+b^2)^(1/2)*(-(-(a^2+b^2)^(1/2)*sin(
d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2)*(((a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(
d*x+c)+a*cos(d*x+c)-a)/(a^2+b^2)^(1/2)/sin(d*x+c))^(1/2)*(a*(cos(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))/sin(d*x+c))^(1
/2)*EllipticF((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(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*sin(d*x+c)*(a^2+b^2)^(1/2)*(-(-(a^2+b^2)^
(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2)*(((a^2+b^2)^(1/2)*sin(d*x
+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/(a^2+b^2)^(1/2)/sin(d*x+c))^(1/2)*(a*(cos(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))/sin(
d*x+c))^(1/2)*EllipticF((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(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-sin(d*x+c)*(-(-(a^2+b^2)^(1/2)*sin(
d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2)*(((a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(
d*x+c)+a*cos(d*x+c)-a)/(a^2+b^2)^(1/2)/sin(d*x+c))^(1/2)*(a*(cos(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))/sin(d*x+c))^(1
/2)*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(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^4
-3*sin(d*x+c)*(-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/
2)*(((a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/(a^2+b^2)^(1/2)/sin(d*x+c))^(1/2)*(a*(cos(d*x+c)-
1)/(-b+(a^2+b^2)^(1/2))/sin(d*x+c))^(1/2)*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(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-sin(d*x+c)*(-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)
-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2)*(((a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/(a^2+b^2)
^(1/2)/sin(d*x+c))^(1/2)*(a*(cos(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))/sin(d*x+c))^(1/2)*EllipticPi((-(-(a^2+b^2)^(1/
2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(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^4-3*sin(d*x+c)*(-(-(a^2+b^2)^(1/
2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/2)))^(1/2)*(((a^2+b^2)^(1/2)*sin(d*x+c)
+b*sin(d*x+c)+a*cos(d*x+c)-a)/(a^2+b^2)^(1/2)/sin(d*x+c))^(1/2)*(a*(cos(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))/sin(d*x
+c))^(1/2)*EllipticPi((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(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+4*sin(d*x+c)*(-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)
^(1/2)))^(1/2)*(((a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/(a^2+b^2)^(1/2)/sin(d*x+c))^(1/2)*(a*
(cos(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))/sin(d*x+c))^(1/2)*EllipticF((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*
cos(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^2+4*sin(d*x+c)*(-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/sin(d*x+c)/(-b+(a^2+b^2)^(1/
2)))^(1/2)*(((a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(d*x+c)-a)/(a^2+b^2)^(1/2)/sin(d*x+c))^(1/2)*(a*(cos
(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))/sin(d*x+c))^(1/2)*EllipticF((-(-(a^2+b^2)^(1/2)*sin(d*x+c)+b*sin(d*x+c)+a*cos(
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^4
-2*2^(1/2)*cos(d*x+c)*(a^2+b^2)^(1/2)*a*b^2+2*2^(1/2)*cos(d*x+c)*a^3*b+2*2^(1/2)*cos(d*x+c)*a*b^3+2*2^(1/2)*(a
^2+b^2)^(1/2)*a*b^2-2*2^(1/2)*a^3*b-2*2^(1/2)*a*b^3)*cos(d*x+c)*(1/cos(d*x+c)*(a*cos(d*x+c)+b*sin(d*x+c)))^(1/
2)/(a*cos(d*x+c)+b*sin(d*x+c))/(cos(d*x+c)-1)/(cos(d*x+c)/sin(d*x+c))^(1/2)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \sqrt{\cot \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(3/2),x, algorithm="maxima")
[Out]
integrate(1/((b*tan(d*x + c) + a)^(3/2)*sqrt(cot(d*x + c))), x)
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(3/2),x, algorithm="fricas")
[Out]
Timed out
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \tan{\left (c + d x \right )}\right )^{\frac{3}{2}} \sqrt{\cot{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/cot(d*x+c)**(1/2)/(a+b*tan(d*x+c))**(3/2),x)
[Out]
Integral(1/((a + b*tan(c + d*x))**(3/2)*sqrt(cot(c + d*x))), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \sqrt{\cot \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(3/2),x, algorithm="giac")
[Out]
integrate(1/((b*tan(d*x + c) + a)^(3/2)*sqrt(cot(d*x + c))), x)