3.14 \(\int \frac{1}{(a+b \csc ^2(c+d x))^{5/2}} \, dx\)
Optimal. Leaf size=126 \[ \frac{b (5 a+3 b) \cot (c+d x)}{3 a^2 d (a+b)^2 \sqrt{a+b \cot ^2(c+d x)+b}}-\frac{\tan ^{-1}\left (\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)+b}}\right )}{a^{5/2} d}+\frac{b \cot (c+d x)}{3 a d (a+b) \left (a+b \cot ^2(c+d x)+b\right )^{3/2}} \]
[Out]
-(ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[a + b + b*Cot[c + d*x]^2]]/(a^(5/2)*d)) + (b*Cot[c + d*x])/(3*a*(a + b)*d
*(a + b + b*Cot[c + d*x]^2)^(3/2)) + (b*(5*a + 3*b)*Cot[c + d*x])/(3*a^2*(a + b)^2*d*Sqrt[a + b + b*Cot[c + d*
x]^2])
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Rubi [A] time = 0.100216, antiderivative size = 126, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used =
{4128, 414, 527, 12, 377, 203} \[ \frac{b (5 a+3 b) \cot (c+d x)}{3 a^2 d (a+b)^2 \sqrt{a+b \cot ^2(c+d x)+b}}-\frac{\tan ^{-1}\left (\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)+b}}\right )}{a^{5/2} d}+\frac{b \cot (c+d x)}{3 a d (a+b) \left (a+b \cot ^2(c+d x)+b\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Csc[c + d*x]^2)^(-5/2),x]
[Out]
-(ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[a + b + b*Cot[c + d*x]^2]]/(a^(5/2)*d)) + (b*Cot[c + d*x])/(3*a*(a + b)*d
*(a + b + b*Cot[c + d*x]^2)^(3/2)) + (b*(5*a + 3*b)*Cot[c + d*x])/(3*a^2*(a + b)^2*d*Sqrt[a + b + b*Cot[c + d*
x]^2])
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]
Rule 414
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[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]
Rule 527
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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 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])
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \csc ^2(c+d x)\right )^{5/2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \left (a+b+b x^2\right )^{5/2}} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{b \cot (c+d x)}{3 a (a+b) d \left (a+b+b \cot ^2(c+d x)\right )^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{3 a+b-2 b x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^{3/2}} \, dx,x,\cot (c+d x)\right )}{3 a (a+b) d}\\ &=\frac{b \cot (c+d x)}{3 a (a+b) d \left (a+b+b \cot ^2(c+d x)\right )^{3/2}}+\frac{b (5 a+3 b) \cot (c+d x)}{3 a^2 (a+b)^2 d \sqrt{a+b+b \cot ^2(c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{3 (a+b)^2}{\left (1+x^2\right ) \sqrt{a+b+b x^2}} \, dx,x,\cot (c+d x)\right )}{3 a^2 (a+b)^2 d}\\ &=\frac{b \cot (c+d x)}{3 a (a+b) d \left (a+b+b \cot ^2(c+d x)\right )^{3/2}}+\frac{b (5 a+3 b) \cot (c+d x)}{3 a^2 (a+b)^2 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^2 d}\\ &=\frac{b \cot (c+d x)}{3 a (a+b) d \left (a+b+b \cot ^2(c+d x)\right )^{3/2}}+\frac{b (5 a+3 b) \cot (c+d x)}{3 a^2 (a+b)^2 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^2 d}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b+b \cot ^2(c+d x)}}\right )}{a^{5/2} d}+\frac{b \cot (c+d x)}{3 a (a+b) d \left (a+b+b \cot ^2(c+d x)\right )^{3/2}}+\frac{b (5 a+3 b) \cot (c+d x)}{3 a^2 (a+b)^2 d \sqrt{a+b+b \cot ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.67913, size = 173, normalized size = 1.37 \[ \frac{\csc ^5(c+d x) \left (\frac{4 b \cos (c+d x) (a (-\cos (2 (c+d x)))+a+2 b) \left (3 a^2-a (3 a+2 b) \cos (2 (c+d x))+7 a b+3 b^2\right )}{3 a^2 (a+b)^2}-\frac{\sqrt{2} (a \cos (2 (c+d x))-a-2 b)^{5/2} \log \left (\sqrt{a \cos (2 (c+d x))-a-2 b}+\sqrt{2} \sqrt{a} \cos (c+d x)\right )}{a^{5/2}}\right )}{8 d \left (a+b \csc ^2(c+d x)\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Csc[c + d*x]^2)^(-5/2),x]
[Out]
(Csc[c + d*x]^5*((4*b*Cos[c + d*x]*(a + 2*b - a*Cos[2*(c + d*x)])*(3*a^2 + 7*a*b + 3*b^2 - a*(3*a + 2*b)*Cos[2
*(c + d*x)]))/(3*a^2*(a + b)^2) - (Sqrt[2]*(-a - 2*b + a*Cos[2*(c + d*x)])^(5/2)*Log[Sqrt[2]*Sqrt[a]*Cos[c + d
*x] + Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]])/a^(5/2)))/(8*d*(a + b*Csc[c + d*x]^2)^(5/2))
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Maple [B] time = 0.445, size = 2702, normalized size = 21.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b*csc(d*x+c)^2)^(5/2),x)
[Out]
-1/3/d*b^2/(((a+b)*a)^(1/2)+a)^2/(a+b)^2/(((a+b)*a)^(1/2)-a)^2/(-a)^(1/2)*sin(d*x+c)^5*(6*cos(d*x+c)*(-a)^(1/2
)*a^3*b+15*cos(d*x+c)*(-a)^(1/2)*a^2*b^2+12*cos(d*x+c)*(-a)^(1/2)*a*b^3-3*cos(d*x+c)^5*(-(a*cos(d*x+c)^2-a-b)/
(cos(d*x+c)+1)^2)^(5/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*a*cos(d*x+
c)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2))*a^2-3*cos(d*x+c)^5*(-(a*cos(d*x+c)^2-a-b)/(cos
(d*x+c)+1)^2)^(5/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*a*cos(d*x+c)+4
*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2))*b^2-15*cos(d*x+c)^4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*
x+c)+1)^2)^(5/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*a*cos(d*x+c)+4*(-
a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2))*a^2-15*cos(d*x+c)^4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c
)+1)^2)^(5/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*a*cos(d*x+c)+4*(-a)^
(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2))*b^2-30*cos(d*x+c)^3*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1
)^2)^(5/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*a*cos(d*x+c)+4*(-a)^(1/
2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2))*a^2-30*cos(d*x+c)^3*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2
)^(5/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*a*cos(d*x+c)+4*(-a)^(1/2)*
(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2))*b^2+6*cos(d*x+c)^5*(-a)^(1/2)*a^3*b+4*cos(d*x+c)^5*(-a)^(1/2)*
a^2*b^2-30*cos(d*x+c)^2*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(5/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d*x
+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*a*cos(d*x+c)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2))
*a^2-30*cos(d*x+c)^2*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(5/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d*x+c)
^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*a*cos(d*x+c)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2))*b^
2-15*cos(d*x+c)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(5/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-
b)/(cos(d*x+c)+1)^2)^(1/2)-4*a*cos(d*x+c)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2))*a^2-15*
cos(d*x+c)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(5/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(c
os(d*x+c)+1)^2)^(1/2)-4*a*cos(d*x+c)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2))*b^2-12*cos(d
*x+c)^3*(-a)^(1/2)*a^3*b-19*cos(d*x+c)^3*(-a)^(1/2)*a^2*b^2-7*cos(d*x+c)^3*(-a)^(1/2)*a*b^3-6*cos(d*x+c)^5*(-(
a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(5/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2
)^(1/2)-4*a*cos(d*x+c)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2))*a*b-30*cos(d*x+c)^4*(-(a*c
os(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(5/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(
1/2)-4*a*cos(d*x+c)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2))*a*b-60*cos(d*x+c)^3*(-(a*cos(
d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(5/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2
)-4*a*cos(d*x+c)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2))*a*b-60*cos(d*x+c)^2*(-(a*cos(d*x
+c)^2-a-b)/(cos(d*x+c)+1)^2)^(5/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4
*a*cos(d*x+c)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2))*a*b-30*cos(d*x+c)*(-(a*cos(d*x+c)^2
-a-b)/(cos(d*x+c)+1)^2)^(5/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*a*co
s(d*x+c)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2))*a*b-3*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)
+1)^2)^(5/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*a*cos(d*x+c)+4*(-a)^(
1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2))*b^2+3*cos(d*x+c)*(-a)^(1/2)*b^4-3*(-(a*cos(d*x+c)^2-a-b)/
(cos(d*x+c)+1)^2)^(5/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*a*cos(d*x+
c)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2))*a^2-6*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)
^(5/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*a*cos(d*x+c)+4*(-a)^(1/2)*(
-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2))*a*b)/(-1+cos(d*x+c))^5/((a*cos(d*x+c)^2-a-b)/(cos(d*x+c)^2-1))^
(5/2)/(cos(d*x+c)+1)^5
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*csc(d*x+c)^2)^(5/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.15094, size = 2234, normalized size = 17.73 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*csc(d*x+c)^2)^(5/2),x, algorithm="fricas")
[Out]
[-1/24*(3*((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 - 2*(a^4 + 3*a
^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c)^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)) + 8*(2*(3*a^3*b + 2*a^2*b^2)*cos(d*x + c)^3 - 3*(2*a^3*b + 3
*a^2*b^2 + a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c))/((a^7 + 2*
a^6*b + a^5*b^2)*d*cos(d*x + c)^4 - 2*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cos(d*x + c)^2 + (a^7 + 4*a^6*b
+ 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d), 1/12*(3*((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 - 2*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c)^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)*co
s(d*x + c))) - 4*(2*(3*a^3*b + 2*a^2*b^2)*cos(d*x + c)^3 - 3*(2*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c))*sqrt(
(a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c))/((a^7 + 2*a^6*b + a^5*b^2)*d*cos(d*x + c)^4 - 2
*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cos(d*x + c)^2 + (a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d)
]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \csc ^{2}{\left (c + d x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*csc(d*x+c)**2)**(5/2),x)
[Out]
Integral((a + b*csc(c + d*x)**2)**(-5/2), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \csc \left (d x + c\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*csc(d*x+c)^2)^(5/2),x, algorithm="giac")
[Out]
integrate((b*csc(d*x + c)^2 + a)^(-5/2), x)