3.6.53 \(\int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx\) [553]

Optimal. Leaf size=245 \[ \frac {5 b \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{7/2} d}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}-\frac {b \left (3 a^2+5 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {b \left (a^4+10 a^2 b^2+5 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}} \]

[Out]

5*b*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/a^(7/2)/d+I*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^
(5/2)/d-I*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/(a+I*b)^(5/2)/d-b*(a^4+10*a^2*b^2+5*b^4)/a^3/(a^2+b^2)
^2/d/(a+b*tan(d*x+c))^(1/2)-1/3*b*(3*a^2+5*b^2)/a^2/(a^2+b^2)/d/(a+b*tan(d*x+c))^(3/2)-cot(d*x+c)/a/d/(a+b*tan
(d*x+c))^(3/2)

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Rubi [A]
time = 0.65, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3650, 3730, 3734, 3620, 3618, 65, 214, 3715} \begin {gather*} \frac {5 b \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{7/2} d}-\frac {b \left (3 a^2+5 b^2\right )}{3 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {b \left (a^4+10 a^2 b^2+5 b^4\right )}{a^3 d \left (a^2+b^2\right )^2 \sqrt {a+b \tan (c+d x)}}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{5/2}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{5/2}}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^2/(a + b*Tan[c + d*x])^(5/2),x]

[Out]

(5*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(a^(7/2)*d) + (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b
]])/((a - I*b)^(5/2)*d) - (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) - (b*(3*a^2
+ 5*b^2))/(3*a^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - Cot[c + d*x]/(a*d*(a + b*Tan[c + d*x])^(3/2)) - (
b*(a^4 + 10*a^2*b^2 + 5*b^4))/(a^3*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 3620

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]

Rule 3650

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 0])))

Rule 3715

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rule 3730

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3734

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +
b^2), Int[(c + d*Tan[e + f*x])^n*((1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e,
f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0] &&  !LeQ[n, -
1]

Rubi steps

\begin {align*} \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx &=-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {\int \frac {\cot (c+d x) \left (\frac {5 b}{2}+a \tan (c+d x)+\frac {5}{2} b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^{5/2}} \, dx}{a}\\ &=-\frac {b \left (3 a^2+5 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {2 \int \frac {\cot (c+d x) \left (\frac {15}{4} b \left (a^2+b^2\right )+\frac {3}{2} a^3 \tan (c+d x)+\frac {3}{4} b \left (3 a^2+5 b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^{3/2}} \, dx}{3 a^2 \left (a^2+b^2\right )}\\ &=-\frac {b \left (3 a^2+5 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {b \left (a^4+10 a^2 b^2+5 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {4 \int \frac {\cot (c+d x) \left (\frac {15}{8} b \left (a^2+b^2\right )^2+\frac {3}{4} a^3 \left (a^2-b^2\right ) \tan (c+d x)+\frac {3}{8} b \left (a^4+10 a^2 b^2+5 b^4\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{3 a^3 \left (a^2+b^2\right )^2}\\ &=-\frac {b \left (3 a^2+5 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {b \left (a^4+10 a^2 b^2+5 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {(5 b) \int \frac {\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 a^3}-\frac {4 \int \frac {\frac {3}{4} a^3 \left (a^2-b^2\right )-\frac {3}{2} a^4 b \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{3 a^3 \left (a^2+b^2\right )^2}\\ &=-\frac {b \left (3 a^2+5 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {b \left (a^4+10 a^2 b^2+5 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {\int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (a-i b)^2}-\frac {\int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (a+i b)^2}-\frac {(5 b) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 a^3 d}\\ &=-\frac {b \left (3 a^2+5 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {b \left (a^4+10 a^2 b^2+5 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {5 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{a^3 d}-\frac {i \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 (a-i b)^2 d}+\frac {i \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 (a+i b)^2 d}\\ &=\frac {5 b \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{7/2} d}-\frac {b \left (3 a^2+5 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {b \left (a^4+10 a^2 b^2+5 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{-1-\frac {i a}{b}+\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{(a-i b)^2 b d}+\frac {\text {Subst}\left (\int \frac {1}{-1+\frac {i a}{b}-\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{(a+i b)^2 b d}\\ &=\frac {5 b \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{7/2} d}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}-\frac {b \left (3 a^2+5 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {b \left (a^4+10 a^2 b^2+5 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 4.88, size = 232, normalized size = 0.95 \begin {gather*} -\frac {-\frac {15 b \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2}}-3 i a^2 \left (\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2}}\right )+\frac {b \left (3 a^2+5 b^2\right )}{\left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {3 a \cot (c+d x)}{(a+b \tan (c+d x))^{3/2}}+\frac {3 b \left (a^4+10 a^2 b^2+5 b^4\right )}{a \left (a^2+b^2\right )^2 \sqrt {a+b \tan (c+d x)}}}{3 a^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^2/(a + b*Tan[c + d*x])^(5/2),x]

[Out]

-1/3*((-15*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/a^(3/2) - (3*I)*a^2*(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/
Sqrt[a - I*b]]/(a - I*b)^(5/2) - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/(a + I*b)^(5/2)) + (b*(3*a^2
+ 5*b^2))/((a^2 + b^2)*(a + b*Tan[c + d*x])^(3/2)) + (3*a*Cot[c + d*x])/(a + b*Tan[c + d*x])^(3/2) + (3*b*(a^4
+ 10*a^2*b^2 + 5*b^4))/(a*(a^2 + b^2)^2*Sqrt[a + b*Tan[c + d*x]]))/(a^2*d)

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 4.66, size = 175534, normalized size = 716.47

method	result	size
default	\(\text {Expression too large to display}\)	\(175534\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^2/(a+b*tan(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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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)^2/(a+b*tan(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate(cot(d*x + c)^2/(b*tan(d*x + c) + a)^(5/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 11132 vs. \(2 (207) = 414\).
time = 4.46, size = 22339, normalized size = 91.18 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2/(a+b*tan(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

[-1/12*(12*sqrt(2)*((a^22 + a^20*b^2 - 20*a^18*b^4 - 84*a^16*b^6 - 154*a^14*b^8 - 154*a^12*b^10 - 84*a^10*b^12
- 20*a^8*b^14 + a^6*b^16 + a^4*b^18)*d^5*cos(d*x + c)^6 - (a^22 - 5*a^20*b^2 - 60*a^18*b^4 - 196*a^16*b^6 - 3
22*a^14*b^8 - 294*a^12*b^10 - 140*a^10*b^12 - 20*a^8*b^14 + 9*a^6*b^16 + 3*a^4*b^18)*d^5*cos(d*x + c)^4 - 3*(2
*a^20*b^2 + 13*a^18*b^4 + 35*a^16*b^6 + 49*a^14*b^8 + 35*a^12*b^10 + 7*a^10*b^12 - 7*a^8*b^14 - 5*a^6*b^16 - a
^4*b^18)*d^5*cos(d*x + c)^2 - (a^18*b^4 + 7*a^16*b^6 + 21*a^14*b^8 + 35*a^12*b^10 + 35*a^10*b^12 + 21*a^8*b^14
+ 7*a^6*b^16 + a^4*b^18)*d^5 + 4*((a^21*b + 6*a^19*b^3 + 14*a^17*b^5 + 14*a^15*b^7 - 14*a^11*b^11 - 14*a^9*b^
13 - 6*a^7*b^15 - a^5*b^17)*d^5*cos(d*x + c)^5 - (a^21*b + 5*a^19*b^3 + 7*a^17*b^5 - 7*a^15*b^7 - 35*a^13*b^9
- 49*a^11*b^11 - 35*a^9*b^13 - 13*a^7*b^15 - 2*a^5*b^17)*d^5*cos(d*x + c)^3 - (a^19*b^3 + 7*a^17*b^5 + 21*a^15
*b^7 + 35*a^13*b^9 + 35*a^11*b^11 + 21*a^9*b^13 + 7*a^7*b^15 + a^5*b^17)*d^5*cos(d*x + c))*sin(d*x + c))*sqrt(
(a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 + (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6
- 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5
*a^2*b^8 + b^10)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((25*a^8*b^2 - 100*a
^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 2
52*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*(1/((a^10 + 5*a^8*b^2 + 1
0*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))^(3/4)*arctan(((5*a^20 + 30*a^18*b^2 + 61*a^16*b^4 + 8*a^14*b^
6 - 182*a^12*b^8 - 364*a^10*b^10 - 350*a^8*b^12 - 184*a^6*b^14 - 47*a^4*b^16 - 2*a^2*b^18 + b^20)*d^4*sqrt((25
*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 +
210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqrt(1/((
a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)) + (5*a^15 + 15*a^13*b^2 + a^11*b^4 - 45*a
^9*b^6 - 65*a^7*b^8 - 35*a^5*b^10 - 5*a^3*b^12 + a*b^14)*d^2*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20
*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^
12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)) - sqrt(2)*((3*a^22 + 29*a^20*b^2 + 125*a^18*b^4 +
315*a^16*b^6 + 510*a^14*b^8 + 546*a^12*b^10 + 378*a^10*b^12 + 150*a^8*b^14 + 15*a^6*b^16 - 15*a^4*b^18 - 7*a^2
*b^20 - b^22)*d^7*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*
a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^
18 + b^20)*d^4))*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)) + 2*(a^17 + 8*a
^15*b^2 + 28*a^13*b^4 + 56*a^11*b^6 + 70*a^9*b^8 + 56*a^7*b^10 + 28*a^5*b^12 + 8*a^3*b^14 + a*b^16)*d^5*sqrt((
25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6
+ 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)))*sqrt((
a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 + (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6
- 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*
a^2*b^8 + b^10)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt(((25*a^14*b^2 - 25*a
^12*b^4 - 115*a^10*b^6 + 35*a^8*b^8 + 171*a^6*b^10 + 53*a^4*b^12 - 17*a^2*b^14 + b^16)*d^2*sqrt(1/((a^10 + 5*a
^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))*cos(d*x + c) + sqrt(2)*(2*(25*a^15*b^3 - 25*a^13*b^
5 - 115*a^11*b^7 + 35*a^9*b^9 + 171*a^7*b^11 + 53*a^5*b^13 - 17*a^3*b^15 + a*b^17)*d^3*sqrt(1/((a^10 + 5*a^8*b
^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))*cos(d*x + c) + (75*a^10*b^3 - 325*a^8*b^5 + 430*a^6*b^7
- 170*a^4*b^9 + 23*a^2*b^11 - b^13)*d*cos(d*x + c))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*
b^8 + b^10 + (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d
^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 +
110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^10 + 5*a^8*b^2 +
10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))^(1/4) + (25*a^9*b^2 - 100*a^7*b^4 + 110*a^5*b^6 - 20*a^3*b^
8 + a*b^10)*cos(d*x + c) + (25*a^8*b^3 - 100*a^6*b^5 + 110*a^4*b^7 - 20*a^2*b^9 + b^11)*sin(d*x + c))/cos(d*x
+ c))*(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))^(3/4) + sqrt(2)*((15*a^26*b +
115*a^24*b^3 + 338*a^22*b^5 + 354*a^20*b^7 - 475*a^18*b^9 - 2055*a^16*b^11 - 3060*a^14*b^13 - 2484*a^12*b^15 -
1047*a^10*b^17 - 75*a^8*b^19 + 130*a^6*b^21 + 50*a^4*b^23 + 3*a^2*b^25 - b^27)*d^7*sqrt((25*a^8*b^2 - 100*a^6
*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot ^{2}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**2/(a+b*tan(d*x+c))**(5/2),x)

[Out]

Integral(cot(c + d*x)**2/(a + b*tan(c + d*x))**(5/2), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2/(a+b*tan(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 4.95, size = 2500, normalized size = 10.20 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^2/(a + b*tan(c + d*x))^(5/2),x)

[Out]

((2*b^3)/(3*a*(a^2 + b^2)) + (2*b^3*(11*a^2 + 5*b^2)*(a + b*tan(c + d*x)))/(3*(a*b^2 + a^3)^2) - (b*(a + b*tan
(c + d*x))^2*(a^4 + 5*b^4 + 10*a^2*b^2))/(a^3*(a^2 + b^2)^2))/(d*(a + b*tan(c + d*x))^(5/2) - a*d*(a + b*tan(c
+ d*x))^(3/2)) + (log(400*a^22*b^39*d^4 - ((((((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4
- 25*a^8*b^2*d^4)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a
^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(90304*a^29*b^37*d^6 - 800*a^21*b^45*d^6 - 10400*a^23*b^43
*d^6 - 54400*a^25*b^41*d^6 - 121600*a^27*b^39*d^6 - (((a + b*tan(c + d*x))^(1/2)*(1600*a^22*b^46*d^7 + 28800*a
^24*b^44*d^7 + 244800*a^26*b^42*d^7 + 1304256*a^28*b^40*d^7 + 4880128*a^30*b^38*d^7 + 13627392*a^32*b^36*d^7 +
29476608*a^34*b^34*d^7 + 50615552*a^36*b^32*d^7 + 70152576*a^38*b^30*d^7 + 79329536*a^40*b^28*d^7 + 73600384*
a^42*b^26*d^7 + 56025216*a^44*b^24*d^7 + 34754304*a^46*b^22*d^7 + 17296384*a^48*b^20*d^7 + 6713088*a^50*b^18*d
^7 + 1934592*a^52*b^16*d^7 + 377408*a^54*b^14*d^7 + 39552*a^56*b^12*d^7 + 64*a^58*b^10*d^7 - 320*a^60*b^8*d^7)
- ((((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) - a^5*d^2 - 5*a*b
^4*d^2 + 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^
4))^(1/2)*(((((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) - a^5*d^2
- 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^
8*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(512*a^27*b^46*d^9 + 9984*a^29*b^44*d^9 + 92160*a^31*b^42*d^9 + 5
35296*a^33*b^40*d^9 + 2193408*a^35*b^38*d^9 + 6736896*a^37*b^36*d^9 + 16084992*a^39*b^34*d^9 + 30551040*a^41*b
^32*d^9 + 46844928*a^43*b^30*d^9 + 58499584*a^45*b^28*d^9 + 59744256*a^47*b^26*d^9 + 49900032*a^49*b^24*d^9 +
33945600*a^51*b^22*d^9 + 18643968*a^53*b^20*d^9 + 8146944*a^55*b^18*d^9 + 2767872*a^57*b^16*d^9 + 705024*a^59*
b^14*d^9 + 126720*a^61*b^12*d^9 + 14336*a^63*b^10*d^9 + 768*a^65*b^8*d^9))/2 + 1280*a^24*b^47*d^8 + 24320*a^26
*b^45*d^8 + 219008*a^28*b^43*d^8 + 1241984*a^30*b^41*d^8 + 4970496*a^32*b^39*d^8 + 14909440*a^34*b^37*d^8 + 34
746880*a^36*b^35*d^8 + 64356864*a^38*b^33*d^8 + 96092672*a^40*b^31*d^8 + 116633088*a^42*b^29*d^8 + 115498240*a
^44*b^27*d^8 + 93267200*a^46*b^25*d^8 + 61128704*a^48*b^23*d^8 + 32212992*a^50*b^21*d^8 + 13439488*a^52*b^19*d
^8 + 4334080*a^54*b^17*d^8 + 1040640*a^56*b^15*d^8 + 174848*a^58*b^13*d^8 + 18304*a^60*b^11*d^8 + 896*a^62*b^9
*d^8))/2)*(((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) - a^5*d^2 -
5*a*b^4*d^2 + 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*
b^2*d^4))^(1/2))/2 + 1465856*a^31*b^35*d^6 + 5014464*a^33*b^33*d^6 + 10323456*a^35*b^31*d^6 + 14661504*a^37*b^
29*d^6 + 14908608*a^39*b^27*d^6 + 10808512*a^41*b^25*d^6 + 5328128*a^43*b^23*d^6 + 1531712*a^45*b^21*d^6 + 878
08*a^47*b^19*d^6 - 85696*a^49*b^17*d^6 - 6144*a^51*b^15*d^6 + 15264*a^53*b^13*d^6 + 5856*a^55*b^11*d^6 + 704*a
^57*b^9*d^6))/2 + (a + b*tan(c + d*x))^(1/2)*(67232*a^27*b^36*d^5 - 3200*a^23*b^40*d^5 - 3200*a^25*b^38*d^5 -
400*a^21*b^42*d^5 + 437248*a^29*b^34*d^5 + 1458912*a^31*b^32*d^5 + 3214848*a^33*b^30*d^5 + 5065632*a^35*b^28*d
^5 + 5898464*a^37*b^26*d^5 + 5129696*a^39*b^24*d^5 + 3313024*a^41*b^22*d^5 + 1552096*a^43*b^20*d^5 + 500864*a^
45*b^18*d^5 + 99232*a^47*b^16*d^5 + 8448*a^49*b^14*d^5 - 288*a^51*b^12*d^5 + 48*a^53*b^10*d^5 + 32*a^55*b^8*d^
5))*(((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) - a^5*d^2 - 5*a*b
^4*d^2 + 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^
4))^(1/2))/2 + 5520*a^24*b^37*d^4 + 35280*a^26*b^35*d^4 + 138320*a^28*b^33*d^4 + 371280*a^30*b^31*d^4 + 720720
*a^32*b^29*d^4 + 1041040*a^34*b^27*d^4 + 1132560*a^36*b^25*d^4 + 926640*a^38*b^23*d^4 + 560560*a^40*b^21*d^4 +
240240*a^42*b^19*d^4 + 65520*a^44*b^17*d^4 + 7280*a^46*b^15*d^4 - 1680*a^48*b^13*d^4 - 720*a^50*b^11*d^4 - 80
*a^52*b^9*d^4)*(((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) - a^5*
d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5
*a^8*b^2*d^4))^(1/2))/2 + (log(400*a^22*b^39*d^4 - ((((-((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^
6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d
^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(90304*a^29*b^37*d^6 - 800*a^21*b^45*d^6 - 10400*
a^23*b^43*d^6 - 54400*a^25*b^41*d^6 - 121600*a^27*b^39*d^6 - (((a + b*tan(c + d*x))^(1/2)*(1600*a^22*b^46*d^7
+ 28800*a^24*b^44*d^7 + 244800*a^26*b^42*d^7 + 1304256*a^28*b^40*d^7 + 4880128*a^30*b^38*d^7 + 13627392*a^32*b
^36*d^7 + 29476608*a^34*b^34*d^7 + 50615552*a^36*b^32*d^7 + 70152576*a^38*b^30*d^7 + 79329536*a^40*b^28*d^7 +
73600384*a^42*b^26*d^7 + 56025216*a^44*b^24*d^7...

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