3.40 \(\int \frac{1}{(e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3} \, dx\)

Optimal. Leaf size=215 \[ -\frac{63}{8 a^3 d e^2 \sqrt{e \cot (c+d x)}}-\frac{59 \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{8 a^3 d e^{5/2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{2 \sqrt{2} a^3 d e^{5/2}}-\frac{11}{8 a^3 d e (\cot (c+d x)+1) (e \cot (c+d x))^{3/2}}+\frac{55}{24 a^3 d e (e \cot (c+d x))^{3/2}}-\frac{1}{4 a d e (a \cot (c+d x)+a)^2 (e \cot (c+d x))^{3/2}} \]

[Out]

(-59*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(8*a^3*d*e^(5/2)) + ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2
]*Sqrt[e*Cot[c + d*x]])]/(2*Sqrt[2]*a^3*d*e^(5/2)) + 55/(24*a^3*d*e*(e*Cot[c + d*x])^(3/2)) - 63/(8*a^3*d*e^2*
Sqrt[e*Cot[c + d*x]]) - 11/(8*a^3*d*e*(e*Cot[c + d*x])^(3/2)*(1 + Cot[c + d*x])) - 1/(4*a*d*e*(e*Cot[c + d*x])
^(3/2)*(a + a*Cot[c + d*x])^2)

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Rubi [A]  time = 1.10459, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {3569, 3649, 3650, 3653, 3532, 205, 3634, 63} \[ -\frac{63}{8 a^3 d e^2 \sqrt{e \cot (c+d x)}}-\frac{59 \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{8 a^3 d e^{5/2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{2 \sqrt{2} a^3 d e^{5/2}}-\frac{11}{8 a^3 d e (\cot (c+d x)+1) (e \cot (c+d x))^{3/2}}+\frac{55}{24 a^3 d e (e \cot (c+d x))^{3/2}}-\frac{1}{4 a d e (a \cot (c+d x)+a)^2 (e \cot (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3),x]

[Out]

(-59*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(8*a^3*d*e^(5/2)) + ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2
]*Sqrt[e*Cot[c + d*x]])]/(2*Sqrt[2]*a^3*d*e^(5/2)) + 55/(24*a^3*d*e*(e*Cot[c + d*x])^(3/2)) - 63/(8*a^3*d*e^2*
Sqrt[e*Cot[c + d*x]]) - 11/(8*a^3*d*e*(e*Cot[c + d*x])^(3/2)*(1 + Cot[c + d*x])) - 1/(4*a*d*e*(e*Cot[c + d*x])
^(3/2)*(a + a*Cot[c + d*x])^2)

Rule 3569

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 3649

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*T
an[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 3650

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

Rule 3653

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]

Rule 3532

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 205

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

Rule 3634

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 63

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]

Rubi steps

\begin{align*} \int \frac{1}{(e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3} \, dx &=-\frac{1}{4 a d e (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2}-\frac{\int \frac{-\frac{11 a^2 e}{2}+2 a^2 e \cot (c+d x)-\frac{7}{2} a^2 e \cot ^2(c+d x)}{(e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^2} \, dx}{4 a^3 e}\\ &=-\frac{11}{8 a^3 d e (e \cot (c+d x))^{3/2} (1+\cot (c+d x))}-\frac{1}{4 a d e (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2}+\frac{\int \frac{\frac{55 a^4 e^2}{2}-4 a^4 e^2 \cot (c+d x)+\frac{55}{2} a^4 e^2 \cot ^2(c+d x)}{(e \cot (c+d x))^{5/2} (a+a \cot (c+d x))} \, dx}{8 a^6 e^2}\\ &=\frac{55}{24 a^3 d e (e \cot (c+d x))^{3/2}}-\frac{11}{8 a^3 d e (e \cot (c+d x))^{3/2} (1+\cot (c+d x))}-\frac{1}{4 a d e (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2}+\frac{\int \frac{-\frac{189}{4} a^5 e^4-\frac{165}{4} a^5 e^4 \cot ^2(c+d x)}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))} \, dx}{12 a^7 e^5}\\ &=\frac{55}{24 a^3 d e (e \cot (c+d x))^{3/2}}-\frac{63}{8 a^3 d e^2 \sqrt{e \cot (c+d x)}}-\frac{11}{8 a^3 d e (e \cot (c+d x))^{3/2} (1+\cot (c+d x))}-\frac{1}{4 a d e (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2}+\frac{\int \frac{\frac{189 a^6 e^6}{8}+3 a^6 e^6 \cot (c+d x)+\frac{189}{8} a^6 e^6 \cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{6 a^8 e^8}\\ &=\frac{55}{24 a^3 d e (e \cot (c+d x))^{3/2}}-\frac{63}{8 a^3 d e^2 \sqrt{e \cot (c+d x)}}-\frac{11}{8 a^3 d e (e \cot (c+d x))^{3/2} (1+\cot (c+d x))}-\frac{1}{4 a d e (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2}+\frac{\int \frac{3 a^7 e^6+3 a^7 e^6 \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{12 a^{10} e^8}+\frac{59 \int \frac{1+\cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{16 a^2 e^2}\\ &=\frac{55}{24 a^3 d e (e \cot (c+d x))^{3/2}}-\frac{63}{8 a^3 d e^2 \sqrt{e \cot (c+d x)}}-\frac{11}{8 a^3 d e (e \cot (c+d x))^{3/2} (1+\cot (c+d x))}-\frac{1}{4 a d e (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2}+\frac{59 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-e x} (a-a x)} \, dx,x,-\cot (c+d x)\right )}{16 a^2 d e^2}-\frac{\left (3 a^4 e^4\right ) \operatorname{Subst}\left (\int \frac{1}{-18 a^{14} e^{12}-e x^2} \, dx,x,\frac{3 a^7 e^6-3 a^7 e^6 \cot (c+d x)}{\sqrt{e \cot (c+d x)}}\right )}{2 d}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{2 \sqrt{2} a^3 d e^{5/2}}+\frac{55}{24 a^3 d e (e \cot (c+d x))^{3/2}}-\frac{63}{8 a^3 d e^2 \sqrt{e \cot (c+d x)}}-\frac{11}{8 a^3 d e (e \cot (c+d x))^{3/2} (1+\cot (c+d x))}-\frac{1}{4 a d e (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2}-\frac{59 \operatorname{Subst}\left (\int \frac{1}{a+\frac{a x^2}{e}} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{8 a^2 d e^3}\\ &=-\frac{59 \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{8 a^3 d e^{5/2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{2 \sqrt{2} a^3 d e^{5/2}}+\frac{55}{24 a^3 d e (e \cot (c+d x))^{3/2}}-\frac{63}{8 a^3 d e^2 \sqrt{e \cot (c+d x)}}-\frac{11}{8 a^3 d e (e \cot (c+d x))^{3/2} (1+\cot (c+d x))}-\frac{1}{4 a d e (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2}\\ \end{align*}

Mathematica [A]  time = 3.11719, size = 167, normalized size = 0.78 \[ \frac{\cot ^{\frac{5}{2}}(c+d x) \left (4 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-4 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-118 \tan ^{-1}\left (\sqrt{\cot (c+d x)}\right )-\frac{\sqrt{\cot (c+d x)} \sec ^2(c+d x) (678 \cos (2 (c+d x))+679 \cot (c+d x)+77 \cos (3 (c+d x)) \csc (c+d x)+614)}{6 (\cot (c+d x)+1)^2}\right )}{16 a^3 d (e \cot (c+d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3),x]

[Out]

(Cot[c + d*x]^(5/2)*(4*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 4*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[
c + d*x]]] - 118*ArcTan[Sqrt[Cot[c + d*x]]] - (Sqrt[Cot[c + d*x]]*(614 + 678*Cos[2*(c + d*x)] + 679*Cot[c + d*
x] + 77*Cos[3*(c + d*x)]*Csc[c + d*x])*Sec[c + d*x]^2)/(6*(1 + Cot[c + d*x])^2)))/(16*a^3*d*(e*Cot[c + d*x])^(
5/2))

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Maple [B]  time = 0.045, size = 482, normalized size = 2.2 \begin{align*} -{\frac{\sqrt{2}}{16\,d{a}^{3}{e}^{3}}\sqrt [4]{{e}^{2}}\ln \left ({ \left ( e\cot \left ( dx+c \right ) +\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) \left ( e\cot \left ( dx+c \right ) -\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}}{8\,d{a}^{3}{e}^{3}}\sqrt [4]{{e}^{2}}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }+{\frac{\sqrt{2}}{8\,d{a}^{3}{e}^{3}}\sqrt [4]{{e}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }-{\frac{\sqrt{2}}{16\,d{a}^{3}{e}^{2}}\ln \left ({ \left ( e\cot \left ( dx+c \right ) -\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) \left ( e\cot \left ( dx+c \right ) +\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}-{\frac{\sqrt{2}}{8\,d{a}^{3}{e}^{2}}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}+{\frac{\sqrt{2}}{8\,d{a}^{3}{e}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}+{\frac{2}{3\,d{a}^{3}e} \left ( e\cot \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}}-6\,{\frac{1}{d{a}^{3}{e}^{2}\sqrt{e\cot \left ( dx+c \right ) }}}-{\frac{15}{8\,d{a}^{3}{e}^{2} \left ( e\cot \left ( dx+c \right ) +e \right ) ^{2}} \left ( e\cot \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{17}{8\,d{a}^{3}e \left ( e\cot \left ( dx+c \right ) +e \right ) ^{2}}\sqrt{e\cot \left ( dx+c \right ) }}-{\frac{59}{8\,d{a}^{3}}\arctan \left ({\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt{e}}}} \right ){e}^{-{\frac{5}{2}}}} \end{align*}

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))^3,x)

[Out]

-1/16/d/a^3/e^3*(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)))-1/8/d/a^3/e^3*(e^2)^(1/4)*2^(1/2)*arctan(2^(
1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)+1/8/d/a^3/e^3*(e^2)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(
d*x+c))^(1/2)+1)-1/16/d/a^3/e^2/(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)))-1/8/d/a^3/e^2/(e^2)^(1/4)*2^
(1/2)*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)+1/8/d/a^3/e^2/(e^2)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(e^
2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)+2/3/a^3/d/e/(e*cot(d*x+c))^(3/2)-6/a^3/d/e^2/(e*cot(d*x+c))^(1/2)-15/8/d/a^3/
e^2/(e*cot(d*x+c)+e)^2*(e*cot(d*x+c))^(3/2)-17/8/d/a^3/e/(e*cot(d*x+c)+e)^2*(e*cot(d*x+c))^(1/2)-59/8*arctan((
e*cot(d*x+c))^(1/2)/e^(1/2))/a^3/d/e^(5/2)

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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/(e*cot(d*x+c))^(5/2)/(a+a*cot(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.5437, size = 1850, normalized size = 8.6 \begin{align*} \text{result too large to display} \end{align*}

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))^3,x, algorithm="fricas")

[Out]

[-1/48*(6*sqrt(2)*((cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c) + cos(2*d*x + 2*c) + 1)*sqrt(-e)*log(sqrt(2)*sqrt(-
e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*(cos(2*d*x + 2*c) + sin(2*d*x + 2*c) - 1) - 2*e*sin(2*d*x +
2*c) + e) + 177*((cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c) + cos(2*d*x + 2*c) + 1)*sqrt(-e)*log((e*cos(2*d*x +
2*c) - e*sin(2*d*x + 2*c) + 2*sqrt(-e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*sin(2*d*x + 2*c) + e)/(
cos(2*d*x + 2*c) + sin(2*d*x + 2*c) + 1)) - (339*cos(2*d*x + 2*c)^2 - 7*(11*cos(2*d*x + 2*c) + 43)*sin(2*d*x +
2*c) - 32*cos(2*d*x + 2*c) - 307)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(a^3*d*e^3*cos(2*d*x + 2*c
) + a^3*d*e^3 + (a^3*d*e^3*cos(2*d*x + 2*c) + a^3*d*e^3)*sin(2*d*x + 2*c)), 1/48*(12*sqrt(2)*((cos(2*d*x + 2*c
) + 1)*sin(2*d*x + 2*c) + cos(2*d*x + 2*c) + 1)*sqrt(e)*arctan(-1/2*sqrt(2)*sqrt(e)*sqrt((e*cos(2*d*x + 2*c) +
e)/sin(2*d*x + 2*c))*(cos(2*d*x + 2*c) - sin(2*d*x + 2*c) + 1)/(e*cos(2*d*x + 2*c) + e)) - 354*((cos(2*d*x +
2*c) + 1)*sin(2*d*x + 2*c) + cos(2*d*x + 2*c) + 1)*sqrt(e)*arctan(sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*
c))/sqrt(e)) + (339*cos(2*d*x + 2*c)^2 - 7*(11*cos(2*d*x + 2*c) + 43)*sin(2*d*x + 2*c) - 32*cos(2*d*x + 2*c) -
307)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(a^3*d*e^3*cos(2*d*x + 2*c) + a^3*d*e^3 + (a^3*d*e^3*co
s(2*d*x + 2*c) + a^3*d*e^3)*sin(2*d*x + 2*c))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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))**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \cot \left (d x + c\right ) + a\right )}^{3} \left (e \cot \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}

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))^3,x, algorithm="giac")

[Out]

integrate(1/((a*cot(d*x + c) + a)^3*(e*cot(d*x + c))^(5/2)), x)