3.13 \(\int \frac {1}{(a+b \csc ^2(c+d x))^{3/2}} \, dx\)
Optimal. Leaf size=77 \[ \frac {b \cot (c+d x)}{a d (a+b) \sqrt {a+b \cot ^2(c+d x)+b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)+b}}\right )}{a^{3/2} d} \]
[Out]
-arctan(cot(d*x+c)*a^(1/2)/(a+b+b*cot(d*x+c)^2)^(1/2))/a^(3/2)/d+b*cot(d*x+c)/a/(a+b)/d/(a+b+b*cot(d*x+c)^2)^(
1/2)
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Rubi [A] time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used =
{4128, 382, 377, 203} \[ \frac {b \cot (c+d x)}{a d (a+b) \sqrt {a+b \cot ^2(c+d x)+b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)+b}}\right )}{a^{3/2} d} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Csc[c + d*x]^2)^(-3/2),x]
[Out]
-(ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[a + b + b*Cot[c + d*x]^2]]/(a^(3/2)*d)) + (b*Cot[c + d*x])/(a*(a + b)*d*S
qrt[a + b + b*Cot[c + d*x]^2])
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 377
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Rule 382
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d
)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] || !LtQ[q, -1]) && NeQ[p, -1]
Rule 4128
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{3/2}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+b+b x^2\right )^{3/2}} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {b \cot (c+d x)}{a (a+b) d \sqrt {a+b+b \cot ^2(c+d x)}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\cot (c+d x)\right )}{a d}\\ &=\frac {b \cot (c+d x)}{a (a+b) d \sqrt {a+b+b \cot ^2(c+d x)}}-\frac {\operatorname {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,\frac {\cot (c+d x)}{\sqrt {a+b+b \cot ^2(c+d x)}}\right )}{a d}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {a+b+b \cot ^2(c+d x)}}\right )}{a^{3/2} d}+\frac {b \cot (c+d x)}{a (a+b) d \sqrt {a+b+b \cot ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 146, normalized size = 1.90 \[ \frac {\csc ^2(c+d x) \left (\sqrt {2} \csc (c+d x) (a \cos (2 (c+d x))-a-2 b)^{3/2} \log \left (\sqrt {a \cos (2 (c+d x))-a-2 b}+\sqrt {2} \sqrt {a} \cos (c+d x)\right )-\frac {2 \sqrt {a} b \cot (c+d x) (a \cos (2 (c+d x))-a-2 b)}{a+b}\right )}{4 a^{3/2} d \left (a+b \csc ^2(c+d x)\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Csc[c + d*x]^2)^(-3/2),x]
[Out]
(Csc[c + d*x]^2*((-2*Sqrt[a]*b*(-a - 2*b + a*Cos[2*(c + d*x)])*Cot[c + d*x])/(a + b) + Sqrt[2]*(-a - 2*b + a*C
os[2*(c + d*x)])^(3/2)*Csc[c + d*x]*Log[Sqrt[2]*Sqrt[a]*Cos[c + d*x] + Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]]))/
(4*a^(3/2)*d*(a + b*Csc[c + d*x]^2)^(3/2))
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fricas [B] time = 0.72, size = 652, normalized size = 8.47 \[ \left [-\frac {8 \, a b \sqrt {\frac {a \cos \left (d x + c\right )^{2} - a - b}{\cos \left (d x + c\right )^{2} - 1}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left ({\left (a^{2} + a b\right )} \cos \left (d x + c\right )^{2} - a^{2} - 2 \, a b - b^{2}\right )} \sqrt {-a} \log \left (128 \, a^{4} \cos \left (d x + c\right )^{8} - 256 \, {\left (a^{4} + a^{3} b\right )} \cos \left (d x + c\right )^{6} + 160 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (16 \, a^{3} \cos \left (d x + c\right )^{7} - 24 \, {\left (a^{3} + a^{2} b\right )} \cos \left (d x + c\right )^{5} + 10 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right )^{2} - a - b}{\cos \left (d x + c\right )^{2} - 1}} \sin \left (d x + c\right )\right )}{8 \, {\left ({\left (a^{4} + a^{3} b\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} d\right )}}, -\frac {4 \, a b \sqrt {\frac {a \cos \left (d x + c\right )^{2} - a - b}{\cos \left (d x + c\right )^{2} - 1}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left ({\left (a^{2} + a b\right )} \cos \left (d x + c\right )^{2} - a^{2} - 2 \, a b - b^{2}\right )} \sqrt {a} \arctan \left (\frac {{\left (8 \, a^{2} \cos \left (d x + c\right )^{4} - 8 \, {\left (a^{2} + a b\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right )^{2} - a - b}{\cos \left (d x + c\right )^{2} - 1}} \sin \left (d x + c\right )}{4 \, {\left (2 \, a^{3} \cos \left (d x + c\right )^{5} - 3 \, {\left (a^{3} + a^{2} b\right )} \cos \left (d x + c\right )^{3} + {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \cos \left (d x + c\right )\right )}}\right )}{4 \, {\left ({\left (a^{4} + a^{3} b\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*csc(d*x+c)^2)^(3/2),x, algorithm="fricas")
[Out]
[-1/8*(8*a*b*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*cos(d*x + c)*sin(d*x + c) + ((a^2 + a*b)*co
s(d*x + c)^2 - a^2 - 2*a*b - b^2)*sqrt(-a)*log(128*a^4*cos(d*x + c)^8 - 256*(a^4 + a^3*b)*cos(d*x + c)^6 + 160
*(a^4 + 2*a^3*b + a^2*b^2)*cos(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^4 + 3*a^3*b + 3*
a^2*b^2 + a*b^3)*cos(d*x + c)^2 + 8*(16*a^3*cos(d*x + c)^7 - 24*(a^3 + a^2*b)*cos(d*x + c)^5 + 10*(a^3 + 2*a^2
*b + a*b^2)*cos(d*x + c)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c)^2 - a
- b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)))/((a^4 + a^3*b)*d*cos(d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d), -1/
4*(4*a*b*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*cos(d*x + c)*sin(d*x + c) - ((a^2 + a*b)*cos(d*
x + c)^2 - a^2 - 2*a*b - b^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(d*x + c)^4 - 8*(a^2 + a*b)*cos(d*x + c)^2 + a^2 +
2*a*b + b^2)*sqrt(a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(2*a^3*cos(d*x + c)^5
- 3*(a^3 + a^2*b)*cos(d*x + c)^3 + (a^3 + 2*a^2*b + a*b^2)*cos(d*x + c))))/((a^4 + a^3*b)*d*cos(d*x + c)^2 - (
a^4 + 2*a^3*b + a^2*b^2)*d)]
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*csc(d*x+c)^2)^(3/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(sin(d*t_nostep+c))]Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_n
ostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*
pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sig
n: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to che
ck sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Warning, replacing 0 by ` u`, a substitution variable should perh
aps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacin
g 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution var
iable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.
Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a
substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should p
erhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, integ
ration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(t_nostep
)]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.Non regul
ar value [0] was discarded and replaced randomly by 0=[15]Warning, need to choose a branch for the root of a p
olynomial with parameters. This might be wrong.Non regular value [0] was discarded and replaced randomly by 0=
[-24]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.Non re
gular value [0] was discarded and replaced randomly by 0=[16]Warning, need to choose a branch for the root of
a polynomial with parameters. This might be wrong.Non regular value [0] was discarded and replaced randomly by
0=[-59]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.Non
regular value [0] was discarded and replaced randomly by 0=[-92]Warning, need to choose a branch for the root
of a polynomial with parameters. This might be wrong.Non regular value [0] was discarded and replaced randoml
y by 0=[80]Evaluation time: 0.78index.cc index_m operator + Error: Bad Argument Value
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maple [B] time = 1.41, size = 652, normalized size = 8.47 \[ -\frac {\left (-1+\cos \left (d x +c \right )\right )^{2} \left (1+\cos \left (d x +c \right )\right )^{2} \left (a \left (\cos ^{2}\left (d x +c \right )\right )-a -b \right ) \left (-\cos \left (d x +c \right ) \sqrt {-\frac {a \left (\cos ^{2}\left (d x +c \right )\right )-a -b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \ln \left (4 \sqrt {-a}\, \cos \left (d x +c \right ) \sqrt {-\frac {a \left (\cos ^{2}\left (d x +c \right )\right )-a -b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}+4 \sqrt {-a}\, \sqrt {-\frac {a \left (\cos ^{2}\left (d x +c \right )\right )-a -b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}-4 a \cos \left (d x +c \right )\right ) a -\cos \left (d x +c \right ) \sqrt {-\frac {a \left (\cos ^{2}\left (d x +c \right )\right )-a -b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \ln \left (4 \sqrt {-a}\, \cos \left (d x +c \right ) \sqrt {-\frac {a \left (\cos ^{2}\left (d x +c \right )\right )-a -b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}+4 \sqrt {-a}\, \sqrt {-\frac {a \left (\cos ^{2}\left (d x +c \right )\right )-a -b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}-4 a \cos \left (d x +c \right )\right ) b +\sqrt {-a}\, \cos \left (d x +c \right ) b -\sqrt {-\frac {a \left (\cos ^{2}\left (d x +c \right )\right )-a -b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \ln \left (4 \sqrt {-a}\, \cos \left (d x +c \right ) \sqrt {-\frac {a \left (\cos ^{2}\left (d x +c \right )\right )-a -b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}+4 \sqrt {-a}\, \sqrt {-\frac {a \left (\cos ^{2}\left (d x +c \right )\right )-a -b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}-4 a \cos \left (d x +c \right )\right ) a -\sqrt {-\frac {a \left (\cos ^{2}\left (d x +c \right )\right )-a -b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \ln \left (4 \sqrt {-a}\, \cos \left (d x +c \right ) \sqrt {-\frac {a \left (\cos ^{2}\left (d x +c \right )\right )-a -b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}+4 \sqrt {-a}\, \sqrt {-\frac {a \left (\cos ^{2}\left (d x +c \right )\right )-a -b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}-4 a \cos \left (d x +c \right )\right ) b \right ) b}{d \left (\frac {a \left (\cos ^{2}\left (d x +c \right )\right )-a -b}{\cos ^{2}\left (d x +c \right )-1}\right )^{\frac {3}{2}} \sin \left (d x +c \right )^{7} \left (a +b \right ) \left (\sqrt {\left (a +b \right ) a}+a \right ) \left (\sqrt {\left (a +b \right ) a}-a \right ) \sqrt {-a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b*csc(d*x+c)^2)^(3/2),x)
[Out]
-1/d*(-1+cos(d*x+c))^2*(1+cos(d*x+c))^2*(a*cos(d*x+c)^2-a-b)*(-cos(d*x+c)*(-(a*cos(d*x+c)^2-a-b)/(1+cos(d*x+c)
)^2)^(1/2)*ln(4*(-a)^(1/2)*cos(d*x+c)*(-(a*cos(d*x+c)^2-a-b)/(1+cos(d*x+c))^2)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x
+c)^2-a-b)/(1+cos(d*x+c))^2)^(1/2)-4*a*cos(d*x+c))*a-cos(d*x+c)*(-(a*cos(d*x+c)^2-a-b)/(1+cos(d*x+c))^2)^(1/2)
*ln(4*(-a)^(1/2)*cos(d*x+c)*(-(a*cos(d*x+c)^2-a-b)/(1+cos(d*x+c))^2)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)
/(1+cos(d*x+c))^2)^(1/2)-4*a*cos(d*x+c))*b+(-a)^(1/2)*cos(d*x+c)*b-(-(a*cos(d*x+c)^2-a-b)/(1+cos(d*x+c))^2)^(1
/2)*ln(4*(-a)^(1/2)*cos(d*x+c)*(-(a*cos(d*x+c)^2-a-b)/(1+cos(d*x+c))^2)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a
-b)/(1+cos(d*x+c))^2)^(1/2)-4*a*cos(d*x+c))*a-(-(a*cos(d*x+c)^2-a-b)/(1+cos(d*x+c))^2)^(1/2)*ln(4*(-a)^(1/2)*c
os(d*x+c)*(-(a*cos(d*x+c)^2-a-b)/(1+cos(d*x+c))^2)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(1+cos(d*x+c))^2)
^(1/2)-4*a*cos(d*x+c))*b)*b/((a*cos(d*x+c)^2-a-b)/(cos(d*x+c)^2-1))^(3/2)/sin(d*x+c)^7/(a+b)/(((a+b)*a)^(1/2)+
a)/(((a+b)*a)^(1/2)-a)/(-a)^(1/2)
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maxima [B] time = 0.89, size = 2050, normalized size = 26.62 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*csc(d*x+c)^2)^(3/2),x, algorithm="maxima")
[Out]
-1/2*(2*a*b*cos(1/2*arctan2(a*sin(4*d*x + 4*c) - 2*(a + 2*b)*sin(2*d*x + 2*c), a*cos(4*d*x + 4*c) - 2*(a + 2*b
)*cos(2*d*x + 2*c) + a))*sin(2*d*x + 2*c) + 2*(a^2 + a*b)*sin(1/2*arctan2(a*sin(4*d*x + 4*c) - 2*(a + 2*b)*sin
(2*d*x + 2*c), a*cos(4*d*x + 4*c) - 2*(a + 2*b)*cos(2*d*x + 2*c) + a))^3 - 2*(a*b*cos(2*d*x + 2*c) - (a^2 + a*
b)*cos(1/2*arctan2(a*sin(4*d*x + 4*c) - 2*(a + 2*b)*sin(2*d*x + 2*c), a*cos(4*d*x + 4*c) - 2*(a + 2*b)*cos(2*d
*x + 2*c) + a))^2 + a^2 + 2*a*b)*sin(1/2*arctan2(a*sin(4*d*x + 4*c) - 2*(a + 2*b)*sin(2*d*x + 2*c), a*cos(4*d*
x + 4*c) - 2*(a + 2*b)*cos(2*d*x + 2*c) + a)) - (a^2*cos(4*d*x + 4*c)^2 + a^2*sin(4*d*x + 4*c)^2 + 4*(a^2 + 4*
a*b + 4*b^2)*cos(2*d*x + 2*c)^2 - 4*(a^2 + 2*a*b)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*(a^2 + 4*a*b + 4*b^2)*
sin(2*d*x + 2*c)^2 + a^2 + 2*(a^2 - 2*(a^2 + 2*a*b)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 4*(a^2 + 2*a*b)*cos(2
*d*x + 2*c))^(1/4)*(((a + b)*cos(1/2*arctan2(a*sin(4*d*x + 4*c) - 2*(a + 2*b)*sin(2*d*x + 2*c), a*cos(4*d*x +
4*c) - 2*(a + 2*b)*cos(2*d*x + 2*c) + a))^2 + (a + b)*sin(1/2*arctan2(a*sin(4*d*x + 4*c) - 2*(a + 2*b)*sin(2*d
*x + 2*c), a*cos(4*d*x + 4*c) - 2*(a + 2*b)*cos(2*d*x + 2*c) + a))^2)*arctan2(2*a*sin(2*d*x + 2*c) + 2*(a^2*co
s(4*d*x + 4*c)^2 + a^2*sin(4*d*x + 4*c)^2 + 4*(a^2 + 4*a*b + 4*b^2)*cos(2*d*x + 2*c)^2 - 4*(a^2 + 2*a*b)*sin(4
*d*x + 4*c)*sin(2*d*x + 2*c) + 4*(a^2 + 4*a*b + 4*b^2)*sin(2*d*x + 2*c)^2 + a^2 + 2*(a^2 - 2*(a^2 + 2*a*b)*cos
(2*d*x + 2*c))*cos(4*d*x + 4*c) - 4*(a^2 + 2*a*b)*cos(2*d*x + 2*c))^(1/4)*sqrt(a)*sin(1/2*arctan2(a*sin(4*d*x
+ 4*c) - 2*(a + 2*b)*sin(2*d*x + 2*c), a*cos(4*d*x + 4*c) - 2*(a + 2*b)*cos(2*d*x + 2*c) + a)), 2*a*cos(2*d*x
+ 2*c) + 2*(a^2*cos(4*d*x + 4*c)^2 + a^2*sin(4*d*x + 4*c)^2 + 4*(a^2 + 4*a*b + 4*b^2)*cos(2*d*x + 2*c)^2 - 4*(
a^2 + 2*a*b)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*(a^2 + 4*a*b + 4*b^2)*sin(2*d*x + 2*c)^2 + a^2 + 2*(a^2 - 2
*(a^2 + 2*a*b)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 4*(a^2 + 2*a*b)*cos(2*d*x + 2*c))^(1/4)*sqrt(a)*cos(1/2*ar
ctan2(a*sin(4*d*x + 4*c) - 2*(a + 2*b)*sin(2*d*x + 2*c), a*cos(4*d*x + 4*c) - 2*(a + 2*b)*cos(2*d*x + 2*c) + a
)) - 2*a - 4*b) + ((a + b)*cos(1/2*arctan2(a*sin(4*d*x + 4*c) - 2*(a + 2*b)*sin(2*d*x + 2*c), a*cos(4*d*x + 4*
c) - 2*(a + 2*b)*cos(2*d*x + 2*c) + a))^2 + (a + b)*sin(1/2*arctan2(a*sin(4*d*x + 4*c) - 2*(a + 2*b)*sin(2*d*x
+ 2*c), a*cos(4*d*x + 4*c) - 2*(a + 2*b)*cos(2*d*x + 2*c) + a))^2)*arctan2(2*(a^2*cos(4*d*x + 4*c)^2 + a^2*si
n(4*d*x + 4*c)^2 + 4*(a^2 + 4*a*b + 4*b^2)*cos(2*d*x + 2*c)^2 - 4*(a^2 + 2*a*b)*sin(4*d*x + 4*c)*sin(2*d*x + 2
*c) + 4*(a^2 + 4*a*b + 4*b^2)*sin(2*d*x + 2*c)^2 + a^2 + 2*(a^2 - 2*(a^2 + 2*a*b)*cos(2*d*x + 2*c))*cos(4*d*x
+ 4*c) - 4*(a^2 + 2*a*b)*cos(2*d*x + 2*c))^(1/4)*sqrt(a)*sin(1/2*arctan2(a*sin(4*d*x + 4*c) - 2*(a + 2*b)*sin(
2*d*x + 2*c), a*cos(4*d*x + 4*c) - 2*(a + 2*b)*cos(2*d*x + 2*c) + a)), 2*(a^2*cos(4*d*x + 4*c)^2 + a^2*sin(4*d
*x + 4*c)^2 + 4*(a^2 + 4*a*b + 4*b^2)*cos(2*d*x + 2*c)^2 - 4*(a^2 + 2*a*b)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) +
4*(a^2 + 4*a*b + 4*b^2)*sin(2*d*x + 2*c)^2 + a^2 + 2*(a^2 - 2*(a^2 + 2*a*b)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c
) - 4*(a^2 + 2*a*b)*cos(2*d*x + 2*c))^(1/4)*sqrt(a)*cos(1/2*arctan2(a*sin(4*d*x + 4*c) - 2*(a + 2*b)*sin(2*d*x
+ 2*c), a*cos(4*d*x + 4*c) - 2*(a + 2*b)*cos(2*d*x + 2*c) + a)) - 4*b))*sqrt(a))/((a^2*cos(4*d*x + 4*c)^2 + a
^2*sin(4*d*x + 4*c)^2 + 4*(a^2 + 4*a*b + 4*b^2)*cos(2*d*x + 2*c)^2 - 4*(a^2 + 2*a*b)*sin(4*d*x + 4*c)*sin(2*d*
x + 2*c) + 4*(a^2 + 4*a*b + 4*b^2)*sin(2*d*x + 2*c)^2 + a^2 + 2*(a^2 - 2*(a^2 + 2*a*b)*cos(2*d*x + 2*c))*cos(4
*d*x + 4*c) - 4*(a^2 + 2*a*b)*cos(2*d*x + 2*c))^(1/4)*((a^3 + a^2*b)*cos(1/2*arctan2(a*sin(4*d*x + 4*c) - 2*(a
+ 2*b)*sin(2*d*x + 2*c), a*cos(4*d*x + 4*c) - 2*(a + 2*b)*cos(2*d*x + 2*c) + a))^2 + (a^3 + a^2*b)*sin(1/2*ar
ctan2(a*sin(4*d*x + 4*c) - 2*(a + 2*b)*sin(2*d*x + 2*c), a*cos(4*d*x + 4*c) - 2*(a + 2*b)*cos(2*d*x + 2*c) + a
))^2)*d)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a+\frac {b}{{\sin \left (c+d\,x\right )}^2}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a + b/sin(c + d*x)^2)^(3/2),x)
[Out]
int(1/(a + b/sin(c + d*x)^2)^(3/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \csc ^{2}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*csc(d*x+c)**2)**(3/2),x)
[Out]
Integral((a + b*csc(c + d*x)**2)**(-3/2), x)
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