3.2.82 \(\int \frac {\cot ^6(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx\) [182]

Optimal. Leaf size=335 \[ -\frac {2 \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{\sqrt {a} d}+\frac {835 \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{512 \sqrt {2} \sqrt {a} d}-\frac {189 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{512 a d}-\frac {323 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{768 a^2 d}+\frac {579 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{640 a^3 d}-\frac {101 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^3 d}-\frac {23 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{192 a^3 d}-\frac {\cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{48 a^3 d} \]

[Out]

-323/768*cot(d*x+c)^3*(a+a*sec(d*x+c))^(3/2)/a^2/d+579/640*cot(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)/a^3/d-101/128*c
os(d*x+c)*cot(d*x+c)^5*sec(1/2*d*x+1/2*c)^2*(a+a*sec(d*x+c))^(5/2)/a^3/d-23/192*cos(d*x+c)^2*cot(d*x+c)^5*sec(
1/2*d*x+1/2*c)^4*(a+a*sec(d*x+c))^(5/2)/a^3/d-1/48*cos(d*x+c)^3*cot(d*x+c)^5*sec(1/2*d*x+1/2*c)^6*(a+a*sec(d*x
+c))^(5/2)/a^3/d-2*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/d/a^(1/2)+835/1024*arctan(1/2*a^(1/2)*tan
(d*x+c)*2^(1/2)/(a+a*sec(d*x+c))^(1/2))*2^(1/2)/d/a^(1/2)-189/512*cot(d*x+c)*(a+a*sec(d*x+c))^(1/2)/a/d

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Rubi [A]
time = 0.21, antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3972, 483, 593, 597, 536, 209} \begin {gather*} \frac {579 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{640 a^3 d}-\frac {\cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{48 a^3 d}-\frac {23 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{192 a^3 d}-\frac {101 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{128 a^3 d}-\frac {323 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{768 a^2 d}-\frac {2 \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {835 \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{512 \sqrt {2} \sqrt {a} d}-\frac {189 \cot (c+d x) \sqrt {a \sec (c+d x)+a}}{512 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6/Sqrt[a + a*Sec[c + d*x]],x]

[Out]

(-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) + (835*ArcTan[(Sqrt[a]*Tan[c + d*x])/
(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(512*Sqrt[2]*Sqrt[a]*d) - (189*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(51
2*a*d) - (323*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(768*a^2*d) + (579*Cot[c + d*x]^5*(a + a*Sec[c + d*x]
)^(5/2))/(640*a^3*d) - (101*Cos[c + d*x]*Cot[c + d*x]^5*Sec[(c + d*x)/2]^2*(a + a*Sec[c + d*x])^(5/2))/(128*a^
3*d) - (23*Cos[c + d*x]^2*Cot[c + d*x]^5*Sec[(c + d*x)/2]^4*(a + a*Sec[c + d*x])^(5/2))/(192*a^3*d) - (Cos[c +
d*x]^3*Cot[c + d*x]^5*Sec[(c + d*x)/2]^6*(a + a*Sec[c + d*x])^(5/2))/(48*a^3*d)

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 483

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 536

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 593

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 597

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 3972

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 ^6(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx &=-\frac {2 \text {Subst}\left (\int \frac {1}{x^6 \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 ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{48 a^3 d}-\frac {\text {Subst}\left (\int \frac {a-11 a^2 x^2}{x^6 \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 {23 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{192 a^3 d}-\frac {\cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{48 a^3 d}-\frac {\text {Subst}\left (\int \frac {-111 a^2-207 a^3 x^2}{x^6 \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 {101 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^3 d}-\frac {23 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{192 a^3 d}-\frac {\cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{48 a^3 d}-\frac {\text {Subst}\left (\int \frac {-1737 a^3-2121 a^4 x^2}{x^6 \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 {579 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{640 a^3 d}-\frac {101 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^3 d}-\frac {23 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{192 a^3 d}-\frac {\cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{48 a^3 d}+\frac {\text {Subst}\left (\int \frac {-4845 a^4-8685 a^5 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 )}{1920 a^6 d}\\ &=-\frac {323 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{768 a^2 d}+\frac {579 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{640 a^3 d}-\frac {101 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^3 d}-\frac {23 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{192 a^3 d}-\frac {\cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{48 a^3 d}-\frac {\text {Subst}\left (\int \frac {8505 a^5-14535 a^6 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 )}{11520 a^6 d}\\ &=-\frac {189 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{512 a d}-\frac {323 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{768 a^2 d}+\frac {579 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{640 a^3 d}-\frac {101 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^3 d}-\frac {23 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{192 a^3 d}-\frac {\cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{48 a^3 d}+\frac {\text {Subst}\left (\int \frac {54585 a^6+8505 a^7 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 )}{23040 a^6 d}\\ &=-\frac {189 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{512 a d}-\frac {323 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{768 a^2 d}+\frac {579 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{640 a^3 d}-\frac {101 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^3 d}-\frac {23 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{192 a^3 d}-\frac {\cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{48 a^3 d}-\frac {835 \text {Subst}\left (\int \frac {1}{2+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{512 d}+\frac {2 \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}\\ &=-\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{\sqrt {a} d}+\frac {835 \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{512 \sqrt {2} \sqrt {a} d}-\frac {189 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{512 a d}-\frac {323 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{768 a^2 d}+\frac {579 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{640 a^3 d}-\frac {101 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^3 d}-\frac {23 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{192 a^3 d}-\frac {\cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{48 a^3 d}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 24.00, size = 5618, normalized size = 16.77 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[c + d*x]^6/Sqrt[a + a*Sec[c + d*x]],x]

[Out]

Result too large to show

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1067\) vs. \(2(291)=582\).
time = 0.20, size = 1068, normalized size = 3.19

method	result	size
default	\(\text {Expression too large to display}\)	\(1068\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^6/(a+a*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/15360/d*(a*(1+cos(d*x+c))/cos(d*x+c))^(1/2)*(1+cos(d*x+c))^2*(-1+cos(d*x+c))^3*(-15360*(-2*cos(d*x+c)/(1+cos
(d*x+c)))^(1/2)*cos(d*x+c)^5*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)/co
s(d*x+c)*2^(1/2))-12525*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^5*sin(d*x+c)*ln(-(-(-2*cos(d*x+c)/(1+c
os(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))-15360*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^4
*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^(1/2)-12525*(-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))+30720*cos(d*x+c)^3*sin(d*x+c)*(-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))*2^(1/2)+19474*cos(d*x+c)^6+25050*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))+30720*(-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))*cos(d*x+c)^2*sin(d*x+c)*2^(1/2)+6902*cos(d*x+c)^5+25050*sin(d*x+c)*cos(d*x+c)^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))*(-2*cos(d*x+c)/(1+cos(d*x+c))
)^(1/2)-15360*sin(d*x+c)*(-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))*2^(1/2)*cos(d*x+c)-28788*cos(d*x+c)^4-12525*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))*cos(d*x+c)-15360
*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*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))*sin(d*x+c)-12316*cos(d*x+c)^3-12525*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)*sin(d*x+c)+12130*cos(d*x+c)^2+5670*cos(d*x+c))/sin(
d*x+c)^11/a

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(cot(d*x + c)^6/sqrt(a*sec(d*x + c) + a), x)

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Fricas [A]
time = 3.18, size = 823, normalized size = 2.46 \begin {gather*} \left [-\frac {12525 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{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 ) + 15360 \, {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{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 (9737 \, \cos \left (d x + c\right )^{6} + 3451 \, \cos \left (d x + c\right )^{5} - 14394 \, \cos \left (d x + c\right )^{4} - 6158 \, \cos \left (d x + c\right )^{3} + 6065 \, \cos \left (d x + c\right )^{2} + 2835 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{30720 \, {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{3} - 2 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a d\right )} \sin \left (d x + c\right )}, -\frac {12525 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{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 ) + 15360 \, {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{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 (9737 \, \cos \left (d x + c\right )^{6} + 3451 \, \cos \left (d x + c\right )^{5} - 14394 \, \cos \left (d x + c\right )^{4} - 6158 \, \cos \left (d x + c\right )^{3} + 6065 \, \cos \left (d x + c\right )^{2} + 2835 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{15360 \, {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{3} - 2 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a d\right )} \sin \left (d x + c\right )}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

[-1/30720*(12525*sqrt(2)*(cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 2*cos(d*x + c)^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) + 15360*(cos(d*x +
c)^5 + cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 2*cos(d*x + c)^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*(9737*cos(d*x + c)^6 + 3451*cos(d*x + c)^5 - 14394*cos
(d*x + c)^4 - 6158*cos(d*x + c)^3 + 6065*cos(d*x + c)^2 + 2835*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x
+ c)))/((a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^3 - 2*a*d*cos(d*x + c)^2 + a*d*cos(d*x
+ c) + a*d)*sin(d*x + c)), -1/15360*(12525*sqrt(2)*(cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 2*cos
(d*x + c)^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)/(s
qrt(a)*sin(d*x + c)))*sin(d*x + c) + 15360*(cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 2*cos(d*x + c
)^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*(9737*cos(d*x + c)^6 + 3451*cos(d*x + c)^5 -
14394*cos(d*x + c)^4 - 6158*cos(d*x + c)^3 + 6065*cos(d*x + c)^2 + 2835*cos(d*x + c))*sqrt((a*cos(d*x + c) + a
)/cos(d*x + c)))/((a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^3 - 2*a*d*cos(d*x + c)^2 + a*d
*cos(d*x + c) + a*d)*sin(d*x + c))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot ^{6}{\left (c + d x \right )}}{\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**6/(a+a*sec(d*x+c))**(1/2),x)

[Out]

Integral(cot(c + d*x)**6/sqrt(a*(sec(c + d*x) + 1)), x)

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Giac [A]
time = 1.32, size = 500, normalized size = 1.49 \begin {gather*} \frac {\sqrt {2} {\left (10 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} {\left (2 \, {\left (\frac {4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a} - \frac {43}{a}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {567}{a}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {15360 \, \sqrt {2} \sqrt {-a} \log \left (\frac {{\left | 2 \, {\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} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\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} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right )}{{\left | a \right |}} - \frac {12525 \, \log \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}\right )}{\sqrt {-a}} + \frac {192 \, {\left (145 \, {\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 )}^{8} \sqrt {-a} - 500 \, {\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 )}^{6} \sqrt {-a} a + 710 \, {\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} \sqrt {-a} a^{2} - 460 \, {\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} \sqrt {-a} a^{3} + 121 \, \sqrt {-a} a^{4}\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 )}^{5}}\right )}}{30720 \, d \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

1/30720*sqrt(2)*(10*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*(2*(4*tan(1/2*d*x + 1/2*c)^2/a - 43/a)*tan(1/2*d*x + 1
/2*c)^2 + 567/a)*tan(1/2*d*x + 1/2*c) + 15360*sqrt(2)*sqrt(-a)*log(abs(2*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt
(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 - 4*sqrt(2)*abs(a) - 6*a)/abs(2*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*ta
n(1/2*d*x + 1/2*c)^2 + a))^2 + 4*sqrt(2)*abs(a) - 6*a))/abs(a) - 12525*log((sqrt(-a)*tan(1/2*d*x + 1/2*c) - sq
rt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2)/sqrt(-a) + 192*(145*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x
+ 1/2*c)^2 + a))^8*sqrt(-a) - 500*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^6*sqr
t(-a)*a + 710*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^4*sqrt(-a)*a^2 - 460*(sqrt
(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2*sqrt(-a)*a^3 + 121*sqrt(-a)*a^4)/((sqrt(-a)
*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 - a)^5)/(d*sgn(cos(d*x + c)))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^6}{\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^6/(a + a/cos(c + d*x))^(1/2),x)

[Out]

int(cot(c + d*x)^6/(a + a/cos(c + d*x))^(1/2), x)

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