3.1.28 \(\int \frac {1}{(e \cot (c+d x))^{5/2} (a+a \cot (c+d x))} \, dx\) [28]

Optimal. Leaf size=135 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{a d e^{5/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d e^{5/2}}+\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {2}{a d e^2 \sqrt {e \cot (c+d x)}} \]

[Out]

-arctan((e*cot(d*x+c))^(1/2)/e^(1/2))/a/d/e^(5/2)+2/3/a/d/e/(e*cot(d*x+c))^(3/2)+1/2*arctanh(1/2*(e^(1/2)+cot(
d*x+c)*e^(1/2))*2^(1/2)/(e*cot(d*x+c))^(1/2))/a/d/e^(5/2)*2^(1/2)-2/a/d/e^2/(e*cot(d*x+c))^(1/2)

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Rubi [A]
time = 0.36, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3650, 3730, 12, 16, 3654, 3613, 214, 3715, 65, 211} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{a d e^{5/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {e} \cot (c+d x)+\sqrt {e}}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d e^{5/2}}-\frac {2}{a d e^2 \sqrt {e \cot (c+d x)}}+\frac {2}{3 a d e (e \cot (c+d x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])),x]

[Out]

-(ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]]/(a*d*e^(5/2))) + ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt
[e*Cot[c + d*x]])]/(Sqrt[2]*a*d*e^(5/2)) + 2/(3*a*d*e*(e*Cot[c + d*x])^(3/2)) - 2/(a*d*e^2*Sqrt[e*Cot[c + d*x]
])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

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 211

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

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 3613

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[-2*(d^2/f),
Subst[Int[1/(2*c*d + b*x^2), x], x, (c - d*Tan[e + f*x])/Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x
] && EqQ[c^2 - d^2, 0]

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 3654

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(3/2)/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1
/(c^2 + d^2), Int[Simp[a^2*c - b^2*c + 2*a*b*d + (2*a*b*c - a^2*d + b^2*d)*Tan[e + f*x], x]/Sqrt[a + b*Tan[e +
f*x]], x], x] + Dist[(b*c - a*d)^2/(c^2 + d^2), Int[(1 + Tan[e + f*x]^2)/(Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan
[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2
, 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])))

Rubi steps

\begin {align*} \int \frac {1}{(e \cot (c+d x))^{5/2} (a+a \cot (c+d x))} \, dx &=\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}+\frac {2 \int \frac {-\frac {3 a e^2}{2}-\frac {3}{2} a e^2 \cot (c+d x)-\frac {3}{2} a e^2 \cot ^2(c+d x)}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))} \, dx}{3 a e^3}\\ &=\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {2}{a d e^2 \sqrt {e \cot (c+d x)}}+\frac {4 \int \frac {3 a^2 e^4 \cot ^2(c+d x)}{4 \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{3 a^2 e^6}\\ &=\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {2}{a d e^2 \sqrt {e \cot (c+d x)}}+\frac {\int \frac {\cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{e^2}\\ &=\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {2}{a d e^2 \sqrt {e \cot (c+d x)}}+\frac {\int \frac {(e \cot (c+d x))^{3/2}}{a+a \cot (c+d x)} \, dx}{e^4}\\ &=\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {2}{a d e^2 \sqrt {e \cot (c+d x)}}+\frac {\int \frac {-a e^2+a e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{2 a^2 e^4}+\frac {\int \frac {1+\cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{2 e^2}\\ &=\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {2}{a d e^2 \sqrt {e \cot (c+d x)}}-\frac {\text {Subst}\left (\int \frac {1}{2 a^2 e^4-e x^2} \, dx,x,\frac {-a e^2-a e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{d}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-a x)} \, dx,x,-\cot (c+d x)\right )}{2 d e^2}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d e^{5/2}}+\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {2}{a d e^2 \sqrt {e \cot (c+d x)}}-\frac {\text {Subst}\left (\int \frac {1}{a+\frac {a x^2}{e}} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e^3}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{a d e^{5/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d e^{5/2}}+\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {2}{a d e^2 \sqrt {e \cot (c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 1.41, size = 131, normalized size = 0.97 \begin {gather*} \frac {-12 \text {ArcTan}\left (\sqrt {\cot (c+d x)}\right ) \sqrt {\cot (c+d x)}-3 \sqrt {2} \sqrt {\cot (c+d x)} \left (\log \left (-1+\sqrt {2} \sqrt {\cot (c+d x)}-\cot (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )+8 (-3+\tan (c+d x))}{12 a d e^2 \sqrt {e \cot (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])),x]

[Out]

(-12*ArcTan[Sqrt[Cot[c + d*x]]]*Sqrt[Cot[c + d*x]] - 3*Sqrt[2]*Sqrt[Cot[c + d*x]]*(Log[-1 + Sqrt[2]*Sqrt[Cot[c
+ d*x]] - Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]) + 8*(-3 + Tan[c + d*x]))/(12*a*
d*e^2*Sqrt[e*Cot[c + d*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(332\) vs. \(2(113)=226\).
time = 0.48, size = 333, normalized size = 2.47

method	result	size
derivativedivides	\(-\frac {2 e^{2} \left (-\frac {1}{3 e^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {1}{e^{4} \sqrt {e \cot \left (d x +c \right )}}+\frac {\arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2 e^{\frac {9}{2}}}+\frac {-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{2 e^{4}}\right )}{d a}\)	\(333\)
default	\(-\frac {2 e^{2} \left (-\frac {1}{3 e^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {1}{e^{4} \sqrt {e \cot \left (d x +c \right )}}+\frac {\arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2 e^{\frac {9}{2}}}+\frac {-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{2 e^{4}}\right )}{d a}\)	\(333\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*cot(d*x+c))^(5/2)/(a+a*cot(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-2/d/a*e^2*(-1/3/e^3/(e*cot(d*x+c))^(3/2)+1/e^4/(e*cot(d*x+c))^(1/2)+1/2/e^(9/2)*arctan((e*cot(d*x+c))^(1/2)/e
^(1/2))+1/2/e^4*(-1/8/e*(e^2)^(1/4)*2^(1/2)*(ln((e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(
1/2))/(e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot
(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1))+1/8/(e^2)^(1/4)*2^(1/2)*(ln((e*cot(d*
x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1
/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*
x+c))^(1/2)+1))))

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Maxima [A]
time = 0.51, size = 109, normalized size = 0.81 \begin {gather*} -\frac {{\left (\frac {8 \, {\left (\frac {3}{\tan \left (d x + c\right )} - 1\right )} \tan \left (d x + c\right )^{\frac {3}{2}}}{a} - \frac {3 \, {\left (\sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )}}{a} + \frac {12 \, \arctan \left (\frac {1}{\sqrt {\tan \left (d x + c\right )}}\right )}{a}\right )} e^{\left (-\frac {5}{2}\right )}}{12 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cot(d*x+c))^(5/2)/(a+a*cot(d*x+c)),x, algorithm="maxima")

[Out]

-1/12*(8*(3/tan(d*x + c) - 1)*tan(d*x + c)^(3/2)/a - 3*(sqrt(2)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c
) + 1) - sqrt(2)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1))/a + 12*arctan(1/sqrt(tan(d*x + c)))/a)
*e^(-5/2)/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (97) = 194\).
time = 2.48, size = 229, normalized size = 1.70 \begin {gather*} \frac {12 \, {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \arctan \left (\frac {\sqrt {\frac {\cos \left (2 \, d x + 2 \, c\right ) + 1}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right )}{\cos \left (2 \, d x + 2 \, c\right ) + 1}\right ) + 3 \, {\left (\sqrt {2} \cos \left (2 \, d x + 2 \, c\right ) + \sqrt {2}\right )} \log \left (-{\left (\sqrt {2} \cos \left (2 \, d x + 2 \, c\right ) - \sqrt {2} \sin \left (2 \, d x + 2 \, c\right ) - \sqrt {2}\right )} \sqrt {\frac {\cos \left (2 \, d x + 2 \, c\right ) + 1}{\sin \left (2 \, d x + 2 \, c\right )}} + 2 \, \sin \left (2 \, d x + 2 \, c\right ) + 1\right ) - 8 \, \sqrt {\frac {\cos \left (2 \, d x + 2 \, c\right ) + 1}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + 3 \, \sin \left (2 \, d x + 2 \, c\right ) - 1\right )}}{12 \, {\left (a d \cos \left (2 \, d x + 2 \, c\right ) e^{\frac {5}{2}} + a d e^{\frac {5}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cot(d*x+c))^(5/2)/(a+a*cot(d*x+c)),x, algorithm="fricas")

[Out]

1/12*(12*(cos(2*d*x + 2*c) + 1)*arctan(sqrt((cos(2*d*x + 2*c) + 1)/sin(2*d*x + 2*c))*sin(2*d*x + 2*c)/(cos(2*d
*x + 2*c) + 1)) + 3*(sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*log(-(sqrt(2)*cos(2*d*x + 2*c) - sqrt(2)*sin(2*d*x +
2*c) - sqrt(2))*sqrt((cos(2*d*x + 2*c) + 1)/sin(2*d*x + 2*c)) + 2*sin(2*d*x + 2*c) + 1) - 8*sqrt((cos(2*d*x +
2*c) + 1)/sin(2*d*x + 2*c))*(cos(2*d*x + 2*c) + 3*sin(2*d*x + 2*c) - 1))/(a*d*cos(2*d*x + 2*c)*e^(5/2) + a*d*e
^(5/2))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}} \cot {\left (c + d x \right )} + \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cot(d*x+c))**(5/2)/(a+a*cot(d*x+c)),x)

[Out]

Integral(1/((e*cot(c + d*x))**(5/2)*cot(c + d*x) + (e*cot(c + d*x))**(5/2)), x)/a

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cot(d*x+c))^(5/2)/(a+a*cot(d*x+c)),x, algorithm="giac")

[Out]

integrate(1/((a*cot(d*x + c) + a)*(e*cot(d*x + c))^(5/2)), x)

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Mupad [B]
time = 0.93, size = 132, normalized size = 0.98 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {12\,\sqrt {2}\,a^3\,d^3\,e^{21/2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{12\,a^3\,d^3\,e^{11}+12\,a^3\,d^3\,e^{11}\,\mathrm {cot}\left (c+d\,x\right )}\right )}{2\,a\,d\,e^{5/2}}-\frac {\mathrm {atan}\left (\frac {\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{a\,d\,e^{5/2}}-\frac {\frac {2\,\mathrm {cot}\left (c+d\,x\right )}{e}-\frac {2}{3\,e}}{a\,d\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((e*cot(c + d*x))^(5/2)*(a + a*cot(c + d*x))),x)

[Out]

(2^(1/2)*atanh((12*2^(1/2)*a^3*d^3*e^(21/2)*(e*cot(c + d*x))^(1/2))/(12*a^3*d^3*e^11 + 12*a^3*d^3*e^11*cot(c +
d*x))))/(2*a*d*e^(5/2)) - atan((e*cot(c + d*x))^(1/2)/e^(1/2))/(a*d*e^(5/2)) - ((2*cot(c + d*x))/e - 2/(3*e))
/(a*d*(e*cot(c + d*x))^(3/2))

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