3.193 \(\int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=303 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{a^{3/2} d}-\frac {533 \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{256 \sqrt {2} a^{3/2} d}+\frac {277 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{384 a^3 d}-\frac {\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{48 a^3 d}-\frac {7 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{64 a^3 d}-\frac {81 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{128 a^3 d}-\frac {21 \cot (c+d x) \sqrt {a \sec (c+d x)+a}}{256 a^2 d} \]

[Out]

2*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/a^(3/2)/d+277/384*cot(d*x+c)^3*(a+a*sec(d*x+c))^(3/2)/a^3/
d-81/128*cos(d*x+c)*cot(d*x+c)^3*sec(1/2*d*x+1/2*c)^2*(a+a*sec(d*x+c))^(3/2)/a^3/d-7/64*cos(d*x+c)^2*cot(d*x+c
)^3*sec(1/2*d*x+1/2*c)^4*(a+a*sec(d*x+c))^(3/2)/a^3/d-1/48*cos(d*x+c)^3*cot(d*x+c)^3*sec(1/2*d*x+1/2*c)^6*(a+a
*sec(d*x+c))^(3/2)/a^3/d-533/512*arctan(1/2*a^(1/2)*tan(d*x+c)*2^(1/2)/(a+a*sec(d*x+c))^(1/2))*2^(1/2)/a^(3/2)
/d-21/256*cot(d*x+c)*(a+a*sec(d*x+c))^(1/2)/a^2/d

________________________________________________________________________________________

Rubi [A]  time = 0.28, antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3887, 472, 579, 583, 522, 203} \[ \frac {277 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{384 a^3 d}-\frac {21 \cot (c+d x) \sqrt {a \sec (c+d x)+a}}{256 a^2 d}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{a^{3/2} d}-\frac {533 \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{256 \sqrt {2} a^{3/2} d}-\frac {\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{48 a^3 d}-\frac {7 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{64 a^3 d}-\frac {81 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{128 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4/(a + a*Sec[c + d*x])^(3/2),x]

[Out]

(2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) - (533*ArcTan[(Sqrt[a]*Tan[c + d*x])/(
Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(256*Sqrt[2]*a^(3/2)*d) - (21*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(256*
a^2*d) + (277*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(384*a^3*d) - (81*Cos[c + d*x]*Cot[c + d*x]^3*Sec[(c
+ d*x)/2]^2*(a + a*Sec[c + d*x])^(3/2))/(128*a^3*d) - (7*Cos[c + d*x]^2*Cot[c + d*x]^3*Sec[(c + d*x)/2]^4*(a +
a*Sec[c + d*x])^(3/2))/(64*a^3*d) - (Cos[c + d*x]^3*Cot[c + d*x]^3*Sec[(c + d*x)/2]^6*(a + a*Sec[c + d*x])^(3
/2))/(48*a^3*d)

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 472

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*(e*x
)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*e*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d)*(
p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n*(
p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p
, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 579

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*g*n*(b*c - a*d)*(p +
1)), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)*(
m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 3887

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[(-2*a^(m/2 +
n + 1/2))/d, Subst[Int[(x^m*(2 + a*x^2)^(m/2 + n - 1/2))/(1 + a*x^2), x], x, Cot[c + d*x]/Sqrt[a + b*Csc[c +
d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m/2] && IntegerQ[n - 1/2]

Rubi steps

\begin {align*} \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )^4} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^3 d}\\ &=-\frac {\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{48 a^3 d}-\frac {\operatorname {Subst}\left (\int \frac {3 a-9 a^2 x^2}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )^3} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{6 a^4 d}\\ &=-\frac {7 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{64 a^3 d}-\frac {\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{48 a^3 d}-\frac {\operatorname {Subst}\left (\int \frac {-51 a^2-147 a^3 x^2}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{48 a^5 d}\\ &=-\frac {81 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^3 d}-\frac {7 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{64 a^3 d}-\frac {\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{48 a^3 d}-\frac {\operatorname {Subst}\left (\int \frac {-831 a^3-1215 a^4 x^2}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{192 a^6 d}\\ &=\frac {277 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{384 a^3 d}-\frac {81 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^3 d}-\frac {7 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{64 a^3 d}-\frac {\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{48 a^3 d}+\frac {\operatorname {Subst}\left (\int \frac {-189 a^4-2493 a^5 x^2}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{1152 a^6 d}\\ &=-\frac {21 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{256 a^2 d}+\frac {277 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{384 a^3 d}-\frac {81 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^3 d}-\frac {7 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{64 a^3 d}-\frac {\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{48 a^3 d}-\frac {\operatorname {Subst}\left (\int \frac {4419 a^5-189 a^6 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{2304 a^6 d}\\ &=-\frac {21 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{256 a^2 d}+\frac {277 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{384 a^3 d}-\frac {81 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^3 d}-\frac {7 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{64 a^3 d}-\frac {\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{48 a^3 d}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a d}+\frac {533 \operatorname {Subst}\left (\int \frac {1}{2+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{256 a d}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^{3/2} d}-\frac {533 \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{256 \sqrt {2} a^{3/2} d}-\frac {21 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{256 a^2 d}+\frac {277 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{384 a^3 d}-\frac {81 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^3 d}-\frac {7 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{64 a^3 d}-\frac {\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{48 a^3 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 24.01, size = 5620, normalized size = 18.55 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[c + d*x]^4/(a + a*Sec[c + d*x])^(3/2),x]

[Out]

Result too large to show

________________________________________________________________________________________

fricas [A]  time = 0.90, size = 712, normalized size = 2.35 \[ \left [-\frac {1599 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right ) - 1\right )} \sqrt {-a} \log \left (-\frac {2 \, \sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 3 \, a \cos \left (d x + c\right )^{2} - 2 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) \sin \left (d x + c\right ) + 1536 \, {\left (\cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right ) - 1\right )} \sqrt {-a} \log \left (-\frac {8 \, a \cos \left (d x + c\right )^{3} + 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right ) + 1}\right ) \sin \left (d x + c\right ) - 4 \, {\left (819 \, \cos \left (d x + c\right )^{5} + 492 \, \cos \left (d x + c\right )^{4} - 690 \, \cos \left (d x + c\right )^{3} - 428 \, \cos \left (d x + c\right )^{2} + 63 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{3072 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )}, \frac {1599 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right ) - 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 1536 \, {\left (\cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right ) - 1\right )} \sqrt {a} \arctan \left (\frac {2 \, \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a}\right ) \sin \left (d x + c\right ) + 2 \, {\left (819 \, \cos \left (d x + c\right )^{5} + 492 \, \cos \left (d x + c\right )^{4} - 690 \, \cos \left (d x + c\right )^{3} - 428 \, \cos \left (d x + c\right )^{2} + 63 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{1536 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

[-1/3072*(1599*sqrt(2)*(cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c) - 1)*sqrt(-a)*log(-(2*sqrt(2)*sqrt(
-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) - 3*a*cos(d*x + c)^2 - 2*a*cos(d*x + c)
+ a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 1536*(cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x
+ c) - 1)*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 + 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x +
c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) - 4*(819*cos(d*x +
c)^5 + 492*cos(d*x + c)^4 - 690*cos(d*x + c)^3 - 428*cos(d*x + c)^2 + 63*cos(d*x + c))*sqrt((a*cos(d*x + c) +
a)/cos(d*x + c)))/((a^2*d*cos(d*x + c)^4 + 2*a^2*d*cos(d*x + c)^3 - 2*a^2*d*cos(d*x + c) - a^2*d)*sin(d*x + c
)), 1/1536*(1599*sqrt(2)*(cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c) - 1)*sqrt(a)*arctan(sqrt(2)*sqrt(
(a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 1536*(cos(d*x + c)^4 +
2*cos(d*x + c)^3 - 2*cos(d*x + c) - 1)*sqrt(a)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*
x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 2*(819*cos(d*x + c)^5 + 492*cos(
d*x + c)^4 - 690*cos(d*x + c)^3 - 428*cos(d*x + c)^2 + 63*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)
))/((a^2*d*cos(d*x + c)^4 + 2*a^2*d*cos(d*x + c)^3 - 2*a^2*d*cos(d*x + c) - a^2*d)*sin(d*x + c))]

________________________________________________________________________________________

giac [A]  time = 1.95, size = 289, normalized size = 0.95 \[ \frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} {\left (2 \, {\left (\frac {4 \, \sqrt {2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} - \frac {37 \, \sqrt {2}}{a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {417 \, \sqrt {2}}{a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {32 \, \sqrt {2} {\left (21 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{4} - 36 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} a + 19 \, a^{2}\right )}}{{\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - a\right )}^{3} \sqrt {-a} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}}{1536 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^(3/2),x, algorithm="giac")

[Out]

1/1536*(sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*(2*(4*sqrt(2)*tan(1/2*d*x + 1/2*c)^2/(a^2*sgn(tan(1/2*d*x + 1/2*c)
^2 - 1)) - 37*sqrt(2)/(a^2*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)))*tan(1/2*d*x + 1/2*c)^2 + 417*sqrt(2)/(a^2*sgn(tan
(1/2*d*x + 1/2*c)^2 - 1)))*tan(1/2*d*x + 1/2*c) - 32*sqrt(2)*(21*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(
1/2*d*x + 1/2*c)^2 + a))^4 - 36*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2*a + 19
*a^2)/(((sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 - a)^3*sqrt(-a)*sgn(tan(1/2*d*
x + 1/2*c)^2 - 1)))/d

________________________________________________________________________________________

maple [B]  time = 1.47, size = 732, normalized size = 2.42 \[ \frac {\sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (1+\cos \left (d x +c \right )\right ) \left (-1+\cos \left (d x +c \right )\right )^{3} \left (-1536 \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right )-1599 \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \ln \left (-\frac {-\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+\cos \left (d x +c \right )-1}{\sin \left (d x +c \right )}\right )-3072 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right )-3198 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \ln \left (-\frac {-\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+\cos \left (d x +c \right )-1}{\sin \left (d x +c \right )}\right )+1638 \left (\cos ^{5}\left (d x +c \right )\right )+3072 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right )+984 \left (\cos ^{4}\left (d x +c \right )\right )+3198 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \ln \left (-\frac {-\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+\cos \left (d x +c \right )-1}{\sin \left (d x +c \right )}\right )+1536 \sqrt {2}\, \sin \left (d x +c \right ) \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-1380 \left (\cos ^{3}\left (d x +c \right )\right )+1599 \sin \left (d x +c \right ) \ln \left (-\frac {-\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+\cos \left (d x +c \right )-1}{\sin \left (d x +c \right )}\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-856 \left (\cos ^{2}\left (d x +c \right )\right )+126 \cos \left (d x +c \right )\right )}{1536 d \sin \left (d x +c \right )^{9} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^4/(a+a*sec(d*x+c))^(3/2),x)

[Out]

1/1536/d*(a*(1+cos(d*x+c))/cos(d*x+c))^(1/2)*(1+cos(d*x+c))*(-1+cos(d*x+c))^3*(-1536*(-2*cos(d*x+c)/(1+cos(d*x
+c)))^(1/2)*cos(d*x+c)^4*sin(d*x+c)*2^(1/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*
x+c)*2^(1/2))-1599*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^4*sin(d*x+c)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*
x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))-3072*cos(d*x+c)^3*sin(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d
*x+c)))^(1/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))-3198*cos(d*x+c)^
3*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*
x+c)-1)/sin(d*x+c))+1638*cos(d*x+c)^5+3072*cos(d*x+c)*sin(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*
arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))+984*cos(d*x+c)^4+3198*cos(d*x+
c)*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d
*x+c)-1)/sin(d*x+c))+1536*2^(1/2)*sin(d*x+c)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d
*x+c)*2^(1/2))*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)-1380*cos(d*x+c)^3+1599*sin(d*x+c)*ln(-(-(-2*cos(d*x+c)/(1+
cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)-856*cos(d*x+c)^2+
126*cos(d*x+c))/sin(d*x+c)^9/a^2

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \left (d x + c\right )^{4}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate(cot(d*x + c)^4/(a*sec(d*x + c) + a)^(3/2), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^4}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^4/(a + a/cos(c + d*x))^(3/2),x)

[Out]

int(cot(c + d*x)^4/(a + a/cos(c + d*x))^(3/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{4}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**4/(a+a*sec(d*x+c))**(3/2),x)

[Out]

Integral(cot(c + d*x)**4/(a*(sec(c + d*x) + 1))**(3/2), x)

________________________________________________________________________________________