\(\int \frac {1}{(a+b \csc ^2(c+d x))^{7/2}} \, dx\) [15]
Optimal result
Integrand size = 16, antiderivative size = 180 \[
\int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{7/2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {a+b+b \cot ^2(c+d x)}}\right )}{a^{7/2} d}+\frac {b \cot (c+d x)}{5 a (a+b) d \left (a+b+b \cot ^2(c+d x)\right )^{5/2}}+\frac {b (9 a+5 b) \cot (c+d x)}{15 a^2 (a+b)^2 d \left (a+b+b \cot ^2(c+d x)\right )^{3/2}}+\frac {b \left (33 a^2+40 a b+15 b^2\right ) \cot (c+d x)}{15 a^3 (a+b)^3 d \sqrt {a+b+b \cot ^2(c+d x)}}
\]
[Out]
-arctan(cot(d*x+c)*a^(1/2)/(a+b+b*cot(d*x+c)^2)^(1/2))/a^(7/2)/d+1/5*b*cot(d*x+c)/a/(a+b)/d/(a+b+b*cot(d*x+c)^
2)^(5/2)+1/15*b*(9*a+5*b)*cot(d*x+c)/a^2/(a+b)^2/d/(a+b+b*cot(d*x+c)^2)^(3/2)+1/15*b*(33*a^2+40*a*b+15*b^2)*co
t(d*x+c)/a^3/(a+b)^3/d/(a+b+b*cot(d*x+c)^2)^(1/2)
Rubi [A] (verified)
Time = 0.21 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of
steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4213, 425, 541, 12, 385, 209}
\[
\int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{7/2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)+b}}\right )}{a^{7/2} d}+\frac {b (9 a+5 b) \cot (c+d x)}{15 a^2 d (a+b)^2 \left (a+b \cot ^2(c+d x)+b\right )^{3/2}}+\frac {b \left (33 a^2+40 a b+15 b^2\right ) \cot (c+d x)}{15 a^3 d (a+b)^3 \sqrt {a+b \cot ^2(c+d x)+b}}+\frac {b \cot (c+d x)}{5 a d (a+b) \left (a+b \cot ^2(c+d x)+b\right )^{5/2}}
\]
[In]
Int[(a + b*Csc[c + d*x]^2)^(-7/2),x]
[Out]
-(ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[a + b + b*Cot[c + d*x]^2]]/(a^(7/2)*d)) + (b*Cot[c + d*x])/(5*a*(a + b)*d
*(a + b + b*Cot[c + d*x]^2)^(5/2)) + (b*(9*a + 5*b)*Cot[c + d*x])/(15*a^2*(a + b)^2*d*(a + b + b*Cot[c + d*x]^
2)^(3/2)) + (b*(33*a^2 + 40*a*b + 15*b^2)*Cot[c + d*x])/(15*a^3*(a + b)^3*d*Sqrt[a + b + b*Cot[c + d*x]^2])
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 209
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 385
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 425
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]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]
Rule 541
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 4213
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*}
\text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+b+b x^2\right )^{7/2}} \, dx,x,\cot (c+d x)\right )}{d} \\ & = \frac {b \cot (c+d x)}{5 a (a+b) d \left (a+b+b \cot ^2(c+d x)\right )^{5/2}}-\frac {\text {Subst}\left (\int \frac {5 a+b-4 b x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^{5/2}} \, dx,x,\cot (c+d x)\right )}{5 a (a+b) d} \\ & = \frac {b \cot (c+d x)}{5 a (a+b) d \left (a+b+b \cot ^2(c+d x)\right )^{5/2}}+\frac {b (9 a+5 b) \cot (c+d x)}{15 a^2 (a+b)^2 d \left (a+b+b \cot ^2(c+d x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {15 a^2+12 a b+5 b^2-2 b (9 a+5 b) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^{3/2}} \, dx,x,\cot (c+d x)\right )}{15 a^2 (a+b)^2 d} \\ & = \frac {b \cot (c+d x)}{5 a (a+b) d \left (a+b+b \cot ^2(c+d x)\right )^{5/2}}+\frac {b (9 a+5 b) \cot (c+d x)}{15 a^2 (a+b)^2 d \left (a+b+b \cot ^2(c+d x)\right )^{3/2}}+\frac {b \left (33 a^2+40 a b+15 b^2\right ) \cot (c+d x)}{15 a^3 (a+b)^3 d \sqrt {a+b+b \cot ^2(c+d x)}}-\frac {\text {Subst}\left (\int \frac {15 (a+b)^3}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\cot (c+d x)\right )}{15 a^3 (a+b)^3 d} \\ & = \frac {b \cot (c+d x)}{5 a (a+b) d \left (a+b+b \cot ^2(c+d x)\right )^{5/2}}+\frac {b (9 a+5 b) \cot (c+d x)}{15 a^2 (a+b)^2 d \left (a+b+b \cot ^2(c+d x)\right )^{3/2}}+\frac {b \left (33 a^2+40 a b+15 b^2\right ) \cot (c+d x)}{15 a^3 (a+b)^3 d \sqrt {a+b+b \cot ^2(c+d x)}}-\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\cot (c+d x)\right )}{a^3 d} \\ & = \frac {b \cot (c+d x)}{5 a (a+b) d \left (a+b+b \cot ^2(c+d x)\right )^{5/2}}+\frac {b (9 a+5 b) \cot (c+d x)}{15 a^2 (a+b)^2 d \left (a+b+b \cot ^2(c+d x)\right )^{3/2}}+\frac {b \left (33 a^2+40 a b+15 b^2\right ) \cot (c+d x)}{15 a^3 (a+b)^3 d \sqrt {a+b+b \cot ^2(c+d x)}}-\frac {\text {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^3 d} \\ & = -\frac {\arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {a+b+b \cot ^2(c+d x)}}\right )}{a^{7/2} d}+\frac {b \cot (c+d x)}{5 a (a+b) d \left (a+b+b \cot ^2(c+d x)\right )^{5/2}}+\frac {b (9 a+5 b) \cot (c+d x)}{15 a^2 (a+b)^2 d \left (a+b+b \cot ^2(c+d x)\right )^{3/2}}+\frac {b \left (33 a^2+40 a b+15 b^2\right ) \cot (c+d x)}{15 a^3 (a+b)^3 d \sqrt {a+b+b \cot ^2(c+d x)}} \\
\end{align*}
Mathematica [A] (verified)
Time = 1.69 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.28
\[
\int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{7/2}} \, dx=\frac {\csc ^7(c+d x) \left (\frac {b \cos (c+d x) (a+2 b-a \cos (2 (c+d x))) \left (135 a^4+480 a^3 b+709 a^2 b^2+460 a b^3+120 b^4-4 a \left (45 a^3+135 a^2 b+117 a b^2+35 b^3\right ) \cos (2 (c+d x))+a^2 \left (45 a^2+60 a b+23 b^2\right ) \cos (4 (c+d x))\right )}{15 a^3 (a+b)^3}+\frac {\sqrt {2} (-a-2 b+a \cos (2 (c+d x)))^{7/2} \log \left (\sqrt {2} \sqrt {a} \cos (c+d x)+\sqrt {-a-2 b+a \cos (2 (c+d x))}\right )}{a^{7/2}}\right )}{16 d \left (a+b \csc ^2(c+d x)\right )^{7/2}}
\]
[In]
Integrate[(a + b*Csc[c + d*x]^2)^(-7/2),x]
[Out]
(Csc[c + d*x]^7*((b*Cos[c + d*x]*(a + 2*b - a*Cos[2*(c + d*x)])*(135*a^4 + 480*a^3*b + 709*a^2*b^2 + 460*a*b^3
+ 120*b^4 - 4*a*(45*a^3 + 135*a^2*b + 117*a*b^2 + 35*b^3)*Cos[2*(c + d*x)] + a^2*(45*a^2 + 60*a*b + 23*b^2)*C
os[4*(c + d*x)]))/(15*a^3*(a + b)^3) + (Sqrt[2]*(-a - 2*b + a*Cos[2*(c + d*x)])^(7/2)*Log[Sqrt[2]*Sqrt[a]*Cos[
c + d*x] + Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]])/a^(7/2)))/(16*d*(a + b*Csc[c + d*x]^2)^(7/2))
Maple [B] (warning: unable to verify)
Leaf count of result is larger than twice the leaf count of optimal. \(4845\) vs. \(2(162)=324\).
Time = 7.53 (sec) , antiderivative size = 4846, normalized size of antiderivative =
26.92
| | |
| method | result | size |
| | |
| default |
\(\text {Expression too large to display}\) |
\(4846\) |
| | |
|
|
|
|
[In]
int(1/(a+b*csc(d*x+c)^2)^(7/2),x,method=_RETURNVERBOSE)
[Out]
1/30/d*csc(d*x+c)^3*(-45*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*sin(d*x+c)^2*cos(d*x+c)^5*ln(4*(-(a*co
s(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+
1)^2)^(1/2)-4*cos(d*x+c)*a)*a^4*b^2+45*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*sin(d*x+c)^4*cos(d*x+c)^
2*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b
)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^5*b-45*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*sin(d*x+c)^2
*cos(d*x+c)^4*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(
d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^4*b^2-90*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2
)*sin(d*x+c)^4*cos(d*x+c)*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/
2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^5*b+225*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)
+1)^2)^(1/2)*sin(d*x+c)^2*cos(d*x+c)^3*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/
2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^4*b^2+225*(-(a*cos(d*x+c)^2-a
-b)/(cos(d*x+c)+1)^2)^(1/2)*sin(d*x+c)^2*cos(d*x+c)^2*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(
d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^4*b^2-225*(-(a
*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*sin(d*x+c)^2*cos(d*x+c)*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^
2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^4
*b^2+45*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*sin(d*x+c)^4*cos(d*x+c)^3*ln(4*(-(a*cos(d*x+c)^2-a-b)/(
cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos
(d*x+c)*a)*a^5*b-45*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)^5*ln(4*(-(a*cos(d*x+c)^2-a-b)/(c
os(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(
d*x+c)*a)*a^2*b^4+180*(-a)^(1/2)*sin(d*x+c)^4*cos(d*x+c)*a^4*b^2-200*(-a)^(1/2)*sin(d*x+c)^2*cos(d*x+c)^3*a^3*
b^3-90*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*sin(d*x+c)^4*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^
2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^5
*b-180*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)^4*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^
2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^3
*b^3-45*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)^4*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)
^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^
2*b^4+450*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)^3*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+
1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*
a^3*b^3+225*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)^3*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c
)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a
)*a^2*b^4+45*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)^3*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+
c)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*
a)*a*b^5+285*(-a)^(1/2)*sin(d*x+c)^2*cos(d*x+c)*a^3*b^3-225*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*sin
(d*x+c)^2*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+
c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^4*b^2-15*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*si
n(d*x+c)^6*cos(d*x+c)*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(
-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^6+15*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^
(1/2)*cos(d*x+c)^7*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a
*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^3*b^3+45*(-a)^(1/2)*sin(d*x+c)^6*cos(d*x+c)*a^5*b
-60*(-a)^(1/2)*sin(d*x+c)^4*cos(d*x+c)^3*a^4*b^2+23*(-a)^(1/2)*sin(d*x+c)^2*cos(d*x+c)^5*a^3*b^3+15*(-(a*cos(d
*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)^6*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+
c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^3*b^3+450*(-(a*cos
(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)^2*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*
x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^3*b^3+225*(-(a*c
os(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)^2*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(
d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^2*b^4+45*(-(a*
cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)^2*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos
(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a*b^5-300*(-(a*
cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d
*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^3*b^3-225*(-(a*
cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d
*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^2*b^4-90*(-(a*c
os(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*
x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a*b^5-180*(-(a*cos
(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)^5*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*
x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^3*b^3-15*(-(a*co
s(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1
/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*b^6+15*(-a)^(1/2)*cos(d*x+c)*b
^6+58*(-a)^(1/2)*cos(d*x+c)^5*a^2*b^4-250*(-a)^(1/2)*cos(d*x+c)^3*a^2*b^4-50*(-a)^(1/2)*cos(d*x+c)^3*a*b^5+225
*(-a)^(1/2)*cos(d*x+c)*a^2*b^4+90*(-a)^(1/2)*cos(d*x+c)*a*b^5-90*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2
)*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b
)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a*b^5-15*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*sin(d*x+c)^6
*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)
/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^6-15*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*ln(4
*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos
(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*b^6-300*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*ln(4*(-(a*cos(d*x+c
)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(
1/2)-4*cos(d*x+c)*a)*a^3*b^3-225*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*ln(4*(-(a*cos(d*x+c)^2-a-b)/(c
os(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(
d*x+c)*a)*a^2*b^4)*b^3/(a+b*csc(d*x+c)^2)^(7/2)/(cos(d*x+c)-1)^2/(cos(d*x+c)+1)^2*4^(1/2)/((a*(a+b))^(1/2)-a)^
3/((a*(a+b))^(1/2)+a)^3/(-a)^(1/2)/(a+b)^3
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 667 vs. \(2 (162) = 324\).
Time = 1.99 (sec) , antiderivative size = 1445, normalized size of antiderivative = 8.03
\[
\int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{7/2}} \, dx=\text {Too large to display}
\]
[In]
integrate(1/(a+b*csc(d*x+c)^2)^(7/2),x, algorithm="fricas")
[Out]
[-1/120*(15*((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cos(d*x + c)^6 - a^6 - 6*a^5*b - 15*a^4*b^2 - 20*a^3*b^3 -
15*a^2*b^4 - 6*a*b^5 - b^6 - 3*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cos(d*x + c)^4 + 3*(a^6 + 5*a
^5*b + 10*a^4*b^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5)*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))*s
qrt(-a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)) + 8*((45*a^5*b + 60*a^4*b^2 + 23*a
^3*b^3)*cos(d*x + c)^5 - 5*(18*a^5*b + 39*a^4*b^2 + 28*a^3*b^3 + 7*a^2*b^4)*cos(d*x + c)^3 + 15*(3*a^5*b + 9*a
^4*b^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5)*cos(d*x + c))*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*s
in(d*x + c))/((a^10 + 3*a^9*b + 3*a^8*b^2 + a^7*b^3)*d*cos(d*x + c)^6 - 3*(a^10 + 4*a^9*b + 6*a^8*b^2 + 4*a^7*
b^3 + a^6*b^4)*d*cos(d*x + c)^4 + 3*(a^10 + 5*a^9*b + 10*a^8*b^2 + 10*a^7*b^3 + 5*a^6*b^4 + a^5*b^5)*d*cos(d*x
+ c)^2 - (a^10 + 6*a^9*b + 15*a^8*b^2 + 20*a^7*b^3 + 15*a^6*b^4 + 6*a^5*b^5 + a^4*b^6)*d), 1/60*(15*((a^6 + 3
*a^5*b + 3*a^4*b^2 + a^3*b^3)*cos(d*x + c)^6 - a^6 - 6*a^5*b - 15*a^4*b^2 - 20*a^3*b^3 - 15*a^2*b^4 - 6*a*b^5
- b^6 - 3*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cos(d*x + c)^4 + 3*(a^6 + 5*a^5*b + 10*a^4*b^2 + 1
0*a^3*b^3 + 5*a^2*b^4 + a*b^5)*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)*cos(d*x + c))) - 4*((45*a^5*b + 60
*a^4*b^2 + 23*a^3*b^3)*cos(d*x + c)^5 - 5*(18*a^5*b + 39*a^4*b^2 + 28*a^3*b^3 + 7*a^2*b^4)*cos(d*x + c)^3 + 15
*(3*a^5*b + 9*a^4*b^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5)*cos(d*x + c))*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x
+ c)^2 - 1))*sin(d*x + c))/((a^10 + 3*a^9*b + 3*a^8*b^2 + a^7*b^3)*d*cos(d*x + c)^6 - 3*(a^10 + 4*a^9*b + 6*a
^8*b^2 + 4*a^7*b^3 + a^6*b^4)*d*cos(d*x + c)^4 + 3*(a^10 + 5*a^9*b + 10*a^8*b^2 + 10*a^7*b^3 + 5*a^6*b^4 + a^5
*b^5)*d*cos(d*x + c)^2 - (a^10 + 6*a^9*b + 15*a^8*b^2 + 20*a^7*b^3 + 15*a^6*b^4 + 6*a^5*b^5 + a^4*b^6)*d)]
Sympy [F]
\[
\int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{7/2}} \, dx=\int \frac {1}{\left (a + b \csc ^{2}{\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx
\]
[In]
integrate(1/(a+b*csc(d*x+c)**2)**(7/2),x)
[Out]
Integral((a + b*csc(c + d*x)**2)**(-7/2), x)
Maxima [F(-1)]
Timed out. \[
\int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{7/2}} \, dx=\text {Timed out}
\]
[In]
integrate(1/(a+b*csc(d*x+c)^2)^(7/2),x, algorithm="maxima")
[Out]
Timed out
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 737 vs. \(2 (162) = 324\).
Time = 0.85 (sec) , antiderivative size = 737, normalized size of antiderivative = 4.09
\[
\int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{7/2}} \, dx=\text {Too large to display}
\]
[In]
integrate(1/(a+b*csc(d*x+c)^2)^(7/2),x, algorithm="giac")
[Out]
-1/15*(((((((33*a^20*b^3*sgn(sin(d*x + c)) + 40*a^19*b^4*sgn(sin(d*x + c)) + 15*a^18*b^5*sgn(sin(d*x + c)))*ta
n(1/2*d*x + 1/2*c)^2/(a^24 + 3*a^23*b + 3*a^22*b^2 + a^21*b^3) + 5*(60*a^21*b^2*sgn(sin(d*x + c)) + 95*a^20*b^
3*sgn(sin(d*x + c)) + 52*a^19*b^4*sgn(sin(d*x + c)) + 9*a^18*b^5*sgn(sin(d*x + c)))/(a^24 + 3*a^23*b + 3*a^22*
b^2 + a^21*b^3))*tan(1/2*d*x + 1/2*c)^2 + 10*(72*a^22*b*sgn(sin(d*x + c)) + 126*a^21*b^2*sgn(sin(d*x + c)) + 8
1*a^20*b^3*sgn(sin(d*x + c)) + 22*a^19*b^4*sgn(sin(d*x + c)) + 3*a^18*b^5*sgn(sin(d*x + c)))/(a^24 + 3*a^23*b
+ 3*a^22*b^2 + a^21*b^3))*tan(1/2*d*x + 1/2*c)^2 - 10*(72*a^22*b*sgn(sin(d*x + c)) + 126*a^21*b^2*sgn(sin(d*x
+ c)) + 81*a^20*b^3*sgn(sin(d*x + c)) + 22*a^19*b^4*sgn(sin(d*x + c)) + 3*a^18*b^5*sgn(sin(d*x + c)))/(a^24 +
3*a^23*b + 3*a^22*b^2 + a^21*b^3))*tan(1/2*d*x + 1/2*c)^2 - 5*(60*a^21*b^2*sgn(sin(d*x + c)) + 95*a^20*b^3*sgn
(sin(d*x + c)) + 52*a^19*b^4*sgn(sin(d*x + c)) + 9*a^18*b^5*sgn(sin(d*x + c)))/(a^24 + 3*a^23*b + 3*a^22*b^2 +
a^21*b^3))*tan(1/2*d*x + 1/2*c)^2 - (33*a^20*b^3*sgn(sin(d*x + c)) + 40*a^19*b^4*sgn(sin(d*x + c)) + 15*a^18*
b^5*sgn(sin(d*x + c)))/(a^24 + 3*a^23*b + 3*a^22*b^2 + a^21*b^3))/(b*tan(1/2*d*x + 1/2*c)^4 + 4*a*tan(1/2*d*x
+ 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c)^2 + b)^(5/2) - 30*arctan(-1/2*(sqrt(b)*tan(1/2*d*x + 1/2*c)^2 - sqrt(b*t
an(1/2*d*x + 1/2*c)^4 + 4*a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c)^2 + b) + sqrt(b))/sqrt(a))/(a^(7
/2)*sgn(sin(d*x + c))))/d
Mupad [F(-1)]
Timed out. \[
\int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{7/2}} \, dx=\int \frac {1}{{\left (a+\frac {b}{{\sin \left (c+d\,x\right )}^2}\right )}^{7/2}} \,d x
\]
[In]
int(1/(a + b/sin(c + d*x)^2)^(7/2),x)
[Out]
int(1/(a + b/sin(c + d*x)^2)^(7/2), x)