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

Optimal. Leaf size=190 \[ \frac{b \left (33 a^2-26 a b+8 b^2\right ) \cot (c+d x)}{15 a^3 d (a-b)^3 \sqrt{a+b \cot ^2(c+d x)}}+\frac{b (9 a-4 b) \cot (c+d x)}{15 a^2 d (a-b)^2 \left (a+b \cot ^2(c+d x)\right )^{3/2}}+\frac{b \cot (c+d x)}{5 a d (a-b) \left (a+b \cot ^2(c+d x)\right )^{5/2}}-\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right )}{d (a-b)^{7/2}} \]

[Out]

-(ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]]/((a - b)^(7/2)*d)) + (b*Cot[c + d*x])/(5*a*(a
- b)*d*(a + b*Cot[c + d*x]^2)^(5/2)) + ((9*a - 4*b)*b*Cot[c + d*x])/(15*a^2*(a - b)^2*d*(a + b*Cot[c + d*x]^2)
^(3/2)) + (b*(33*a^2 - 26*a*b + 8*b^2)*Cot[c + d*x])/(15*a^3*(a - b)^3*d*Sqrt[a + b*Cot[c + d*x]^2])

________________________________________________________________________________________

Rubi [A]  time = 0.194083, antiderivative size = 190, 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 = {3661, 414, 527, 12, 377, 203} \[ \frac{b \left (33 a^2-26 a b+8 b^2\right ) \cot (c+d x)}{15 a^3 d (a-b)^3 \sqrt{a+b \cot ^2(c+d x)}}+\frac{b (9 a-4 b) \cot (c+d x)}{15 a^2 d (a-b)^2 \left (a+b \cot ^2(c+d x)\right )^{3/2}}+\frac{b \cot (c+d x)}{5 a d (a-b) \left (a+b \cot ^2(c+d x)\right )^{5/2}}-\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right )}{d (a-b)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cot[c + d*x]^2)^(-7/2),x]

[Out]

-(ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]]/((a - b)^(7/2)*d)) + (b*Cot[c + d*x])/(5*a*(a
- b)*d*(a + b*Cot[c + d*x]^2)^(5/2)) + ((9*a - 4*b)*b*Cot[c + d*x])/(15*a^2*(a - b)^2*d*(a + b*Cot[c + d*x]^2)
^(3/2)) + (b*(33*a^2 - 26*a*b + 8*b^2)*Cot[c + d*x])/(15*a^3*(a - b)^3*d*Sqrt[a + b*Cot[c + d*x]^2])

Rule 3661

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[(c*ff)/f, Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ
[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

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 \cot ^2(c+d x)\right )^{7/2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \left (a+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 \cot ^2(c+d x)\right )^{5/2}}-\frac{\operatorname{Subst}\left (\int \frac{5 a-4 b-4 b x^2}{\left (1+x^2\right ) \left (a+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 \cot ^2(c+d x)\right )^{5/2}}+\frac{(9 a-4 b) b \cot (c+d x)}{15 a^2 (a-b)^2 d \left (a+b \cot ^2(c+d x)\right )^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{15 a^2-18 a b+8 b^2-2 (9 a-4 b) b x^2}{\left (1+x^2\right ) \left (a+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 \cot ^2(c+d x)\right )^{5/2}}+\frac{(9 a-4 b) b \cot (c+d x)}{15 a^2 (a-b)^2 d \left (a+b \cot ^2(c+d x)\right )^{3/2}}+\frac{b \left (33 a^2-26 a b+8 b^2\right ) \cot (c+d x)}{15 a^3 (a-b)^3 d \sqrt{a+b \cot ^2(c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{15 a^3}{\left (1+x^2\right ) \sqrt{a+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 \cot ^2(c+d x)\right )^{5/2}}+\frac{(9 a-4 b) b \cot (c+d x)}{15 a^2 (a-b)^2 d \left (a+b \cot ^2(c+d x)\right )^{3/2}}+\frac{b \left (33 a^2-26 a b+8 b^2\right ) \cot (c+d x)}{15 a^3 (a-b)^3 d \sqrt{a+b \cot ^2(c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\cot (c+d x)\right )}{(a-b)^3 d}\\ &=\frac{b \cot (c+d x)}{5 a (a-b) d \left (a+b \cot ^2(c+d x)\right )^{5/2}}+\frac{(9 a-4 b) b \cot (c+d x)}{15 a^2 (a-b)^2 d \left (a+b \cot ^2(c+d x)\right )^{3/2}}+\frac{b \left (33 a^2-26 a b+8 b^2\right ) \cot (c+d x)}{15 a^3 (a-b)^3 d \sqrt{a+b \cot ^2(c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{1-(-a+b) x^2} \, dx,x,\frac{\cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right )}{(a-b)^3 d}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right )}{(a-b)^{7/2} d}+\frac{b \cot (c+d x)}{5 a (a-b) d \left (a+b \cot ^2(c+d x)\right )^{5/2}}+\frac{(9 a-4 b) b \cot (c+d x)}{15 a^2 (a-b)^2 d \left (a+b \cot ^2(c+d x)\right )^{3/2}}+\frac{b \left (33 a^2-26 a b+8 b^2\right ) \cot (c+d x)}{15 a^3 (a-b)^3 d \sqrt{a+b \cot ^2(c+d x)}}\\ \end{align*}

Mathematica [C]  time = 14.7461, size = 2553, normalized size = 13.44 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Cot[c + d*x]^2)^(-7/2),x]

[Out]

-(Cot[c + d*x]*(-33075*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]] + (99225*(a - b)*ArcSin[Sqrt[((a - b)*Cos[c +
d*x]^2)/a]]*Cos[c + d*x]^2)/a - (99225*(a - b)^2*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^4)/a^2
+ (33075*(a - b)^3*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^6)/a^3 - (66150*b*ArcSin[Sqrt[((a - b
)*Cos[c + d*x]^2)/a]]*Cot[c + d*x]^2)/a + (198450*(a - b)*b*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d
*x]^2*Cot[c + d*x]^2)/a^2 + (66150*(a - b)^3*b*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^6*Cot[c +
d*x]^2)/a^4 - (52920*b^2*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cot[c + d*x]^4)/a^2 + (158760*(a - b)*b^2*A
rcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^2*Cot[c + d*x]^4)/a^3 - (158760*(a - b)^2*b^2*ArcSin[Sqrt
[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^4*Cot[c + d*x]^4)/a^4 + (52920*(a - b)^3*b^2*ArcSin[Sqrt[((a - b)*C
os[c + d*x]^2)/a]]*Cos[c + d*x]^6*Cot[c + d*x]^4)/a^5 - (15120*b^3*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Co
t[c + d*x]^6)/a^3 + (45360*(a - b)*b^3*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^2*Cot[c + d*x]^6)
/a^4 - (45360*(a - b)^2*b^3*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^4*Cot[c + d*x]^6)/a^5 + (151
20*(a - b)^3*b^3*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^6*Cot[c + d*x]^6)/a^6 - 77175*(((a - b)
*Cos[c + d*x]^2)/a)^(3/2)*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a] + 50715*(((a - b)*Cos[c + d*x]^2)/a)
^(5/2)*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a] - (154350*b*(((a - b)*Cos[c + d*x]^2)/a)^(3/2)*Cot[c +
d*x]^2*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a + (101430*b*(((a - b)*Cos[c + d*x]^2)/a)^(5/2)*Cot[c
+ d*x]^2*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a - (123480*b^2*(((a - b)*Cos[c + d*x]^2)/a)^(3/2)*
Cot[c + d*x]^4*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^2 + (81144*b^2*(((a - b)*Cos[c + d*x]^2)/a)^
(5/2)*Cot[c + d*x]^4*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^2 - (35280*b^3*(((a - b)*Cos[c + d*x]^
2)/a)^(3/2)*Cot[c + d*x]^6*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^3 + (23184*b^3*(((a - b)*Cos[c +
d*x]^2)/a)^(5/2)*Cot[c + d*x]^6*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^3 + 1420*(((a - b)*Cos[c +
d*x]^2)/a)^(9/2)*Hypergeometric2F1[2, 2, 11/2, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c
+ d*x]^2))/a] + (3540*b*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^2*Hypergeometric2F1[2, 2, 11/2, ((a -
b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a + (3000*b^2*(((a - b)*Cos[c + d*x]^2)
/a)^(9/2)*Cot[c + d*x]^4*Hypergeometric2F1[2, 2, 11/2, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a
*Tan[c + d*x]^2))/a])/a^2 + (880*b^3*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^6*Hypergeometric2F1[2, 2,
11/2, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^3 + 600*(((a - b)*Cos[c
+ d*x]^2)/a)^(9/2)*HypergeometricPFQ[{2, 2, 2}, {1, 11/2}, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b
+ a*Tan[c + d*x]^2))/a] + (1680*b*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^2*HypergeometricPFQ[{2, 2,
2}, {1, 11/2}, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a + (1560*b^2*(((a
- b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^4*HypergeometricPFQ[{2, 2, 2}, {1, 11/2}, ((a - b)*Cos[c + d*x]^2)
/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^2 + (480*b^3*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c +
d*x]^6*HypergeometricPFQ[{2, 2, 2}, {1, 11/2}, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c
+ d*x]^2))/a])/a^3 + 80*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 11/2}, ((a -
b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a] + (240*b*(((a - b)*Cos[c + d*x]^2)/a)^(
9/2)*Cot[c + d*x]^2*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 11/2}, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*
x]^2*(b + a*Tan[c + d*x]^2))/a])/a + (240*b^2*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^4*Hypergeometric
PFQ[{2, 2, 2, 2}, {1, 1, 11/2}, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a
^2 + (80*b^3*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^6*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 11/2}, (
(a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^3 + 33075*Sqrt[((a - b)*Cos[c +
d*x]^4*(b + a*Tan[c + d*x]^2))/a^2] + (66150*b*Cot[c + d*x]^2*Sqrt[((a - b)*Cos[c + d*x]^4*(b + a*Tan[c + d*x]
^2))/a^2])/a + (52920*b^2*Cot[c + d*x]^4*Sqrt[((a - b)*Cos[c + d*x]^4*(b + a*Tan[c + d*x]^2))/a^2])/a^2 + (151
20*b^3*Cot[c + d*x]^6*Sqrt[((a - b)*Cos[c + d*x]^4*(b + a*Tan[c + d*x]^2))/a^2])/a^3 - (198450*(a - b)^2*b*Arc
Sin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]])/(a^3*(Tan[c + d*x] + Tan[c + d*x]^3)^2)))/(4725*a^3*d*(((a - b)*Cos[c +
d*x]^2)/a)^(7/2)*(1 + Cot[c + d*x]^2)*Sqrt[a + b*Cot[c + d*x]^2]*(1 + (b*Cot[c + d*x]^2)/a)^2*Sqrt[(Cos[c + d
*x]^2*(b + a*Tan[c + d*x]^2))/a])

________________________________________________________________________________________

Maple [A]  time = 0.027, size = 284, normalized size = 1.5 \begin{align*} -{\frac{1}{d \left ( a-b \right ) ^{4}{b}^{2}}\sqrt{{b}^{4} \left ( a-b \right ) }\arctan \left ({ \left ( a-b \right ){b}^{2}\cot \left ( dx+c \right ){\frac{1}{\sqrt{{b}^{4} \left ( a-b \right ) }}}{\frac{1}{\sqrt{a+b \left ( \cot \left ( dx+c \right ) \right ) ^{2}}}}} \right ) }+{\frac{b\cot \left ( dx+c \right ) }{d \left ( a-b \right ) ^{3}a}{\frac{1}{\sqrt{a+b \left ( \cot \left ( dx+c \right ) \right ) ^{2}}}}}+{\frac{b\cot \left ( dx+c \right ) }{5\,a \left ( a-b \right ) d} \left ( a+b \left ( \cot \left ( dx+c \right ) \right ) ^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{4\,b\cot \left ( dx+c \right ) }{15\,d \left ( a-b \right ){a}^{2}} \left ( a+b \left ( \cot \left ( dx+c \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,b\cot \left ( dx+c \right ) }{15\,d \left ( a-b \right ){a}^{3}}{\frac{1}{\sqrt{a+b \left ( \cot \left ( dx+c \right ) \right ) ^{2}}}}}+{\frac{b\cot \left ( dx+c \right ) }{3\,d \left ( a-b \right ) ^{2}a} \left ( a+b \left ( \cot \left ( dx+c \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,b\cot \left ( dx+c \right ) }{3\,d \left ( a-b \right ) ^{2}{a}^{2}}{\frac{1}{\sqrt{a+b \left ( \cot \left ( dx+c \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*cot(d*x+c)^2)^(7/2),x)

[Out]

-1/d/(a-b)^4*(b^4*(a-b))^(1/2)/b^2*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))+1/d
*b/(a-b)^3*cot(d*x+c)/a/(a+b*cot(d*x+c)^2)^(1/2)+1/5*b*cot(d*x+c)/a/(a-b)/d/(a+b*cot(d*x+c)^2)^(5/2)+4/15/d/(a
-b)*b/a^2*cot(d*x+c)/(a+b*cot(d*x+c)^2)^(3/2)+8/15/d/(a-b)*b/a^3*cot(d*x+c)/(a+b*cot(d*x+c)^2)^(1/2)+1/3/d/(a-
b)^2*b*cot(d*x+c)/a/(a+b*cot(d*x+c)^2)^(3/2)+2/3/d/(a-b)^2*b/a^2*cot(d*x+c)/(a+b*cot(d*x+c)^2)^(1/2)

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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*cot(d*x+c)^2)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.4875, size = 3156, normalized size = 16.61 \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*cot(d*x+c)^2)^(7/2),x, algorithm="fricas")

[Out]

[-1/60*(15*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3 - (a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*cos(2*d*x + 2*c)^3 + 3
*(a^6 - a^5*b - a^4*b^2 + a^3*b^3)*cos(2*d*x + 2*c)^2 - 3*(a^6 + a^5*b - a^4*b^2 - a^3*b^3)*cos(2*d*x + 2*c))*
sqrt(-a + b)*log(-2*(a^2 - 2*a*b + b^2)*cos(2*d*x + 2*c)^2 + 2*((a - b)*cos(2*d*x + 2*c) - b)*sqrt(-a + b)*sqr
t(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + a^2 - 2*b^2 + 4*(a*b - b^2)*co
s(2*d*x + 2*c)) + 4*(45*a^5*b - 15*a^4*b^2 - 47*a^3*b^3 + 11*a^2*b^4 + 14*a*b^5 - 8*b^6 + (45*a^5*b - 165*a^4*
b^2 + 233*a^3*b^3 - 159*a^2*b^4 + 54*a*b^5 - 8*b^6)*cos(2*d*x + 2*c)^2 - 2*(45*a^5*b - 90*a^4*b^2 + 27*a^3*b^3
+ 44*a^2*b^4 - 34*a*b^5 + 8*b^6)*cos(2*d*x + 2*c))*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c)
- 1))*sin(2*d*x + 2*c))/((a^10 - 7*a^9*b + 21*a^8*b^2 - 35*a^7*b^3 + 35*a^6*b^4 - 21*a^5*b^5 + 7*a^4*b^6 - a^3
*b^7)*d*cos(2*d*x + 2*c)^3 - 3*(a^10 - 5*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 - 5*a^6*b^4 + 9*a^5*b^5 - 5*a^4*b^6 + a
^3*b^7)*d*cos(2*d*x + 2*c)^2 + 3*(a^10 - 3*a^9*b + a^8*b^2 + 5*a^7*b^3 - 5*a^6*b^4 - a^5*b^5 + 3*a^4*b^6 - a^3
*b^7)*d*cos(2*d*x + 2*c) - (a^10 - a^9*b - 3*a^8*b^2 + 3*a^7*b^3 + 3*a^6*b^4 - 3*a^5*b^5 - a^4*b^6 + a^3*b^7)*
d), 1/30*(15*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3 - (a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*cos(2*d*x + 2*c)^3 +
3*(a^6 - a^5*b - a^4*b^2 + a^3*b^3)*cos(2*d*x + 2*c)^2 - 3*(a^6 + a^5*b - a^4*b^2 - a^3*b^3)*cos(2*d*x + 2*c)
)*sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x +
2*c)/((a - b)*cos(2*d*x + 2*c) - b)) - 2*(45*a^5*b - 15*a^4*b^2 - 47*a^3*b^3 + 11*a^2*b^4 + 14*a*b^5 - 8*b^6 +
(45*a^5*b - 165*a^4*b^2 + 233*a^3*b^3 - 159*a^2*b^4 + 54*a*b^5 - 8*b^6)*cos(2*d*x + 2*c)^2 - 2*(45*a^5*b - 90
*a^4*b^2 + 27*a^3*b^3 + 44*a^2*b^4 - 34*a*b^5 + 8*b^6)*cos(2*d*x + 2*c))*sqrt(((a - b)*cos(2*d*x + 2*c) - a -
b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c))/((a^10 - 7*a^9*b + 21*a^8*b^2 - 35*a^7*b^3 + 35*a^6*b^4 - 21*a^5*
b^5 + 7*a^4*b^6 - a^3*b^7)*d*cos(2*d*x + 2*c)^3 - 3*(a^10 - 5*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 - 5*a^6*b^4 + 9*a^
5*b^5 - 5*a^4*b^6 + a^3*b^7)*d*cos(2*d*x + 2*c)^2 + 3*(a^10 - 3*a^9*b + a^8*b^2 + 5*a^7*b^3 - 5*a^6*b^4 - a^5*
b^5 + 3*a^4*b^6 - a^3*b^7)*d*cos(2*d*x + 2*c) - (a^10 - a^9*b - 3*a^8*b^2 + 3*a^7*b^3 + 3*a^6*b^4 - 3*a^5*b^5
- a^4*b^6 + a^3*b^7)*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \cot ^{2}{\left (c + d x \right )}\right )^{\frac{7}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cot(d*x+c)**2)**(7/2),x)

[Out]

Integral((a + b*cot(c + d*x)**2)**(-7/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \cot \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*cot(d*x+c)^2)^(7/2),x, algorithm="giac")

[Out]

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