3.15 \(\int \frac{1}{(a+b \csc ^2(c+d x))^{7/2}} \, dx\)

Optimal. Leaf size=180 \[ \frac{b \left (33 a^2+40 a b+15 b^2\right ) \cot (c+d x)}{15 a^3 d (a+b)^3 \sqrt{a+b \cot ^2(c+d x)+b}}+\frac{b (9 a+5 b) \cot (c+d x)}{15 a^2 d (a+b)^2 \left (a+b \cot ^2(c+d x)+b\right )^{3/2}}-\frac{\tan ^{-1}\left (\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)+b}}\right )}{a^{7/2} d}+\frac{b \cot (c+d x)}{5 a d (a+b) \left (a+b \cot ^2(c+d x)+b\right )^{5/2}} \]

[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])

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Rubi [A]  time = 0.175658, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{b \left (33 a^2+40 a b+15 b^2\right ) \cot (c+d x)}{15 a^3 d (a+b)^3 \sqrt{a+b \cot ^2(c+d x)+b}}+\frac{b (9 a+5 b) \cot (c+d x)}{15 a^2 d (a+b)^2 \left (a+b \cot ^2(c+d x)+b\right )^{3/2}}-\frac{\tan ^{-1}\left (\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)+b}}\right )}{a^{7/2} d}+\frac{b \cot (c+d x)}{5 a d (a+b) \left (a+b \cot ^2(c+d x)+b\right )^{5/2}} \]

Antiderivative was successfully verified.

[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 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 )^{7/2}} \, dx &=-\frac{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\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^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{\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^3 d}\\ &=-\frac{\tan ^{-1}\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]  time = 1.42582, size = 231, normalized size = 1.28 \[ \frac{\csc ^7(c+d x) \left (\frac{b \cos (c+d x) (a (-\cos (2 (c+d x)))+a+2 b) \left (a^2 \left (45 a^2+60 a b+23 b^2\right ) \cos (4 (c+d x))-4 a \left (135 a^2 b+45 a^3+117 a b^2+35 b^3\right ) \cos (2 (c+d x))+709 a^2 b^2+480 a^3 b+135 a^4+460 a b^3+120 b^4\right )}{15 a^3 (a+b)^3}+\frac{\sqrt{2} (a \cos (2 (c+d x))-a-2 b)^{7/2} \log \left (\sqrt{a \cos (2 (c+d x))-a-2 b}+\sqrt{2} \sqrt{a} \cos (c+d x)\right )}{a^{7/2}}\right )}{16 d \left (a+b \csc ^2(c+d x)\right )^{7/2}} \]

Antiderivative was successfully verified.

[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))

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Maple [B]  time = 0.561, size = 4815, normalized size = 26.8 \begin{align*} \text{output 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)^(7/2),x)

[Out]

1/15/d*b^3/(((a+b)*a)^(1/2)-a)^3/(a+b)^3/(((a+b)*a)^(1/2)+a)^3/(-a)^(1/2)*sin(d*x+c)^7*(23*cos(d*x+c)^7*(-a)^(
1/2)*a^3*b^3+525*cos(d*x+c)^3*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(7/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*c
os(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^3+525*cos(d*x+c)^3*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(7/2)*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^3+315*cos(d*x+c)^2*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(7/2)*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^3+315*cos(d*x+c)^2*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(7/2)*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^3-135*cos(d*x+c)^5*(-a)^(1/2)*a^5*b-300*cos(d*x+c)^5*(-a)^(1/2)*a^4*b^2-223*cos(d*x+c)^5*(-a)^(1/2)*a^3*b^3
-58*cos(d*x+c)^5*(-a)^(1/2)*a^2*b^4+105*cos(d*x+c)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(7/2)*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^3+105*cos(d*x+c)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(7/2)*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^3+45*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(7/2)*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*b+45*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(7/2)*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^2+
135*cos(d*x+c)^3*(-a)^(1/2)*a^5*b+420*cos(d*x+c)^3*(-a)^(1/2)*a^4*b^2+485*cos(d*x+c)^3*(-a)^(1/2)*a^3*b^3+250*
cos(d*x+c)^3*(-a)^(1/2)*a^2*b^4+50*cos(d*x+c)^3*(-a)^(1/2)*a*b^5-45*cos(d*x+c)*(-a)^(1/2)*a^5*b-180*cos(d*x+c)
*(-a)^(1/2)*a^4*b^2-285*cos(d*x+c)*(-a)^(1/2)*a^3*b^3-225*cos(d*x+c)*(-a)^(1/2)*a^2*b^4-90*cos(d*x+c)*(-a)^(1/
2)*a*b^5+15*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(7/2)*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^3-15*cos(
d*x+c)*(-a)^(1/2)*b^6+15*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(7/2)*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^3+15*cos(d*x+c)^7*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(7/2)*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
^3+15*cos(d*x+c)^7*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(7/2)*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^3+
105*cos(d*x+c)^6*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(7/2)*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^3+10
5*cos(d*x+c)^6*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(7/2)*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^3+315*
cos(d*x+c)^5*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(7/2)*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^3+315*co
s(d*x+c)^5*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(7/2)*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^3+525*cos(
d*x+c)^4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(7/2)*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^3+525*cos(d*
x+c)^4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(7/2)*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^3+45*cos(d*x+c
)^7*(-a)^(1/2)*a^5*b+60*cos(d*x+c)^7*(-a)^(1/2)*a^4*b^2+45*cos(d*x+c)^7*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^
2)^(7/2)*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*b+45*cos(d*x+c)^7*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2
)^(7/2)*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^2+315*cos(d*x+c)^6*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2
)^(7/2)*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*b+315*cos(d*x+c)^6*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2
)^(7/2)*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^2+945*cos(d*x+c)^5*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2
)^(7/2)*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*b+945*cos(d*x+c)^5*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2
)^(7/2)*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^2+1575*cos(d*x+c)^4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^
2)^(7/2)*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*b+1575*cos(d*x+c)^4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)
^2)^(7/2)*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^2+1575*cos(d*x+c)^3*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1
)^2)^(7/2)*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*b+1575*cos(d*x+c)^3*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+
1)^2)^(7/2)*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^2+945*cos(d*x+c)^2*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+
1)^2)^(7/2)*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*b+945*cos(d*x+c)^2*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+
1)^2)^(7/2)*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^2+315*cos(d*x+c)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)
^2)^(7/2)*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*b+315*cos(d*x+c)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2
)^(7/2)*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^2)/(-1+cos(d*x+c))^7/((a*cos(d*x+c)^2-a-b)/(cos(d*x+c)^2-1
))^(7/2)/(cos(d*x+c)+1)^7

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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)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 6.96993, size = 3267, normalized size = 18.15 \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)^(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)]

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Sympy [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/(a+b*csc(d*x+c)**2)**(7/2),x)

[Out]

Timed out

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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{7}{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)^(7/2),x, algorithm="giac")

[Out]

integrate((b*csc(d*x + c)^2 + a)^(-7/2), x)