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

Optimal. Leaf size=325 \[ \frac {(a-b) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{3/2} \left (a^2+b^2\right )}-\frac {(a-b) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{3/2} \left (a^2+b^2\right )}-\frac {(a+b) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2} \left (a^2+b^2\right )}+\frac {(a+b) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d e^{3/2} \left (a^2+b^2\right )}+\frac {2 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{a^{3/2} d e^{3/2} \left (a^2+b^2\right )}+\frac {2}{a d e \sqrt {e \cot (c+d x)}} \]

[Out]

2*b^(5/2)*arctan(b^(1/2)*(e*cot(d*x+c))^(1/2)/a^(1/2)/e^(1/2))/a^(3/2)/(a^2+b^2)/d/e^(3/2)-1/2*(a+b)*arctan(1-
2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/(a^2+b^2)/d/e^(3/2)*2^(1/2)+1/2*(a+b)*arctan(1+2^(1/2)*(e*cot(d*x+c))^(1
/2)/e^(1/2))/(a^2+b^2)/d/e^(3/2)*2^(1/2)+1/4*(a-b)*ln(e^(1/2)+cot(d*x+c)*e^(1/2)-2^(1/2)*(e*cot(d*x+c))^(1/2))
/(a^2+b^2)/d/e^(3/2)*2^(1/2)-1/4*(a-b)*ln(e^(1/2)+cot(d*x+c)*e^(1/2)+2^(1/2)*(e*cot(d*x+c))^(1/2))/(a^2+b^2)/d
/e^(3/2)*2^(1/2)+2/a/d/e/(e*cot(d*x+c))^(1/2)

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Rubi [A]  time = 0.66, antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3569, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac {(a-b) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{3/2} \left (a^2+b^2\right )}-\frac {(a-b) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{3/2} \left (a^2+b^2\right )}+\frac {2 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{a^{3/2} d e^{3/2} \left (a^2+b^2\right )}-\frac {(a+b) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2} \left (a^2+b^2\right )}+\frac {(a+b) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d e^{3/2} \left (a^2+b^2\right )}+\frac {2}{a d e \sqrt {e \cot (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(a^(3/2)*(a^2 + b^2)*d*e^(3/2)) - ((a + b
)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d*e^(3/2)) + ((a + b)*ArcTan[1 + (S
qrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d*e^(3/2)) + 2/(a*d*e*Sqrt[e*Cot[c + d*x]]) + ((a
- b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d*e^(3/2)) - (
(a - b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d*e^(3/2))

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]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 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 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1168

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]

Rule 3534

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

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 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 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]

Rubi steps

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

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Mathematica [C]  time = 0.45, size = 198, normalized size = 0.61 \[ \frac {8 b^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {b \cot (c+d x)}{a}\right )+a \left (8 a \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\cot ^2(c+d x)\right )+\sqrt {2} b \sqrt {\cot (c+d x)} \left (-\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )+\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )-2 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )+2 \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )\right )\right )}{4 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*b^2*Hypergeometric2F1[-1/2, 1, 1/2, -((b*Cot[c + d*x])/a)] + a*(8*a*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c
+ d*x]^2] + Sqrt[2]*b*Sqrt[Cot[c + d*x]]*(-2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] + 2*ArcTan[1 + Sqrt[2]*Sqr
t[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]] + C
ot[c + d*x]])))/(4*a*(a^2 + b^2)*d*e*Sqrt[e*Cot[c + d*x]])

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cot(d*x+c))^(3/2)/(a+b*cot(d*x+c)),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cot(d*x+c))^(3/2)/(a+b*cot(d*x+c)),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.61, size = 459, normalized size = 1.41 \[ \frac {2 b^{3} \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}\, b}{\sqrt {a e b}}\right )}{d e a \left (a^{2}+b^{2}\right ) \sqrt {a e b}}+\frac {b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{4 d \,e^{2} \left (a^{2}+b^{2}\right )}+\frac {b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d \,e^{2} \left (a^{2}+b^{2}\right )}-\frac {b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d \,e^{2} \left (a^{2}+b^{2}\right )}+\frac {a \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{4 d e \left (a^{2}+b^{2}\right ) \left (e^{2}\right )^{\frac {1}{4}}}+\frac {a \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d e \left (a^{2}+b^{2}\right ) \left (e^{2}\right )^{\frac {1}{4}}}-\frac {a \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d e \left (a^{2}+b^{2}\right ) \left (e^{2}\right )^{\frac {1}{4}}}+\frac {2}{a d e \sqrt {e \cot \left (d x +c \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.91, size = 261, normalized size = 0.80 \[ \frac {e {\left (\frac {8 \, b^{3} \arctan \left (\frac {b \sqrt {\frac {e}{\tan \left (d x + c\right )}}}{\sqrt {a b e}}\right )}{{\left (a^{3} + a b^{2}\right )} \sqrt {a b e} e^{2}} + \frac {\frac {2 \, \sqrt {2} {\left (a + b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {2 \, \sqrt {2} {\left (a + b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} - \frac {\sqrt {2} {\left (a - b\right )} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} + \frac {\sqrt {2} {\left (a - b\right )} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}}}{{\left (a^{2} + b^{2}\right )} e^{2}} + \frac {8}{a e^{2} \sqrt {\frac {e}{\tan \left (d x + c\right )}}}\right )}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cot(d*x+c))^(3/2)/(a+b*cot(d*x+c)),x, algorithm="maxima")

[Out]

1/4*e*(8*b^3*arctan(b*sqrt(e/tan(d*x + c))/sqrt(a*b*e))/((a^3 + a*b^2)*sqrt(a*b*e)*e^2) + (2*sqrt(2)*(a + b)*a
rctan(1/2*sqrt(2)*(sqrt(2)*sqrt(e) + 2*sqrt(e/tan(d*x + c)))/sqrt(e))/sqrt(e) + 2*sqrt(2)*(a + b)*arctan(-1/2*
sqrt(2)*(sqrt(2)*sqrt(e) - 2*sqrt(e/tan(d*x + c)))/sqrt(e))/sqrt(e) - sqrt(2)*(a - b)*log(sqrt(2)*sqrt(e)*sqrt
(e/tan(d*x + c)) + e + e/tan(d*x + c))/sqrt(e) + sqrt(2)*(a - b)*log(-sqrt(2)*sqrt(e)*sqrt(e/tan(d*x + c)) + e
+ e/tan(d*x + c))/sqrt(e))/((a^2 + b^2)*e^2) + 8/(a*e^2*sqrt(e/tan(d*x + c))))/d

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mupad [B]  time = 1.86, size = 4899, normalized size = 15.07 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log((e*cot(c + d*x))^(1/2)*(64*a^7*b^7*d^5*e^13 - 32*a^9*b^5*d^5*e^13) - ((-1/(b^2*d^2*e^3*1i - a^2*d^2*e^3*1
i + 2*a*b*d^2*e^3))^(1/2)*(((((-1/(b^2*d^2*e^3*1i - a^2*d^2*e^3*1i + 2*a*b*d^2*e^3))^(1/2)*(((e*cot(c + d*x))^
(1/2)*(-1/(b^2*d^2*e^3*1i - a^2*d^2*e^3*1i + 2*a*b*d^2*e^3))^(1/2)*(512*a^9*b^9*d^9*e^19 + 512*a^11*b^7*d^9*e^
19 - 512*a^13*b^5*d^9*e^19 - 512*a^15*b^3*d^9*e^19))/2 - 512*a^8*b^9*d^8*e^18 - 640*a^10*b^7*d^8*e^18 + 256*a^
12*b^5*d^8*e^18 + 384*a^14*b^3*d^8*e^18))/2 - (e*cot(c + d*x))^(1/2)*(512*a^8*b^8*d^7*e^16 - 448*a^10*b^6*d^7*
e^16 + 128*a^12*b^4*d^7*e^16 + 64*a^14*b^2*d^7*e^16))*(-1/(b^2*d^2*e^3*1i - a^2*d^2*e^3*1i + 2*a*b*d^2*e^3))^(
1/2))/2 - 128*a^7*b^8*d^6*e^15 + 32*a^11*b^4*d^6*e^15 + 32*a^13*b^2*d^6*e^15))/2)*(-1/(b^2*d^2*e^3*1i - a^2*d^
2*e^3*1i + 2*a*b*d^2*e^3))^(1/2))/2 - atan(((-1i/(4*(b^2*d^2*e^3 - a^2*d^2*e^3 + a*b*d^2*e^3*2i)))^(1/2)*((e*c
ot(c + d*x))^(1/2)*(64*a^7*b^7*d^5*e^13 - 32*a^9*b^5*d^5*e^13) - (-1i/(4*(b^2*d^2*e^3 - a^2*d^2*e^3 + a*b*d^2*
e^3*2i)))^(1/2)*((-1i/(4*(b^2*d^2*e^3 - a^2*d^2*e^3 + a*b*d^2*e^3*2i)))^(1/2)*((-1i/(4*(b^2*d^2*e^3 - a^2*d^2*
e^3 + a*b*d^2*e^3*2i)))^(1/2)*((-1i/(4*(b^2*d^2*e^3 - a^2*d^2*e^3 + a*b*d^2*e^3*2i)))^(1/2)*(e*cot(c + d*x))^(
1/2)*(512*a^9*b^9*d^9*e^19 + 512*a^11*b^7*d^9*e^19 - 512*a^13*b^5*d^9*e^19 - 512*a^15*b^3*d^9*e^19) - 512*a^8*
b^9*d^8*e^18 - 640*a^10*b^7*d^8*e^18 + 256*a^12*b^5*d^8*e^18 + 384*a^14*b^3*d^8*e^18) - (e*cot(c + d*x))^(1/2)
*(512*a^8*b^8*d^7*e^16 - 448*a^10*b^6*d^7*e^16 + 128*a^12*b^4*d^7*e^16 + 64*a^14*b^2*d^7*e^16)) - 128*a^7*b^8*
d^6*e^15 + 32*a^11*b^4*d^6*e^15 + 32*a^13*b^2*d^6*e^15))*1i + (-1i/(4*(b^2*d^2*e^3 - a^2*d^2*e^3 + a*b*d^2*e^3
*2i)))^(1/2)*((e*cot(c + d*x))^(1/2)*(64*a^7*b^7*d^5*e^13 - 32*a^9*b^5*d^5*e^13) - (-1i/(4*(b^2*d^2*e^3 - a^2*
d^2*e^3 + a*b*d^2*e^3*2i)))^(1/2)*((-1i/(4*(b^2*d^2*e^3 - a^2*d^2*e^3 + a*b*d^2*e^3*2i)))^(1/2)*((-1i/(4*(b^2*
d^2*e^3 - a^2*d^2*e^3 + a*b*d^2*e^3*2i)))^(1/2)*((-1i/(4*(b^2*d^2*e^3 - a^2*d^2*e^3 + a*b*d^2*e^3*2i)))^(1/2)*
(e*cot(c + d*x))^(1/2)*(512*a^9*b^9*d^9*e^19 + 512*a^11*b^7*d^9*e^19 - 512*a^13*b^5*d^9*e^19 - 512*a^15*b^3*d^
9*e^19) + 512*a^8*b^9*d^8*e^18 + 640*a^10*b^7*d^8*e^18 - 256*a^12*b^5*d^8*e^18 - 384*a^14*b^3*d^8*e^18) - (e*c
ot(c + d*x))^(1/2)*(512*a^8*b^8*d^7*e^16 - 448*a^10*b^6*d^7*e^16 + 128*a^12*b^4*d^7*e^16 + 64*a^14*b^2*d^7*e^1
6)) + 128*a^7*b^8*d^6*e^15 - 32*a^11*b^4*d^6*e^15 - 32*a^13*b^2*d^6*e^15))*1i)/((-1i/(4*(b^2*d^2*e^3 - a^2*d^2
*e^3 + a*b*d^2*e^3*2i)))^(1/2)*((e*cot(c + d*x))^(1/2)*(64*a^7*b^7*d^5*e^13 - 32*a^9*b^5*d^5*e^13) - (-1i/(4*(
b^2*d^2*e^3 - a^2*d^2*e^3 + a*b*d^2*e^3*2i)))^(1/2)*((-1i/(4*(b^2*d^2*e^3 - a^2*d^2*e^3 + a*b*d^2*e^3*2i)))^(1
/2)*((-1i/(4*(b^2*d^2*e^3 - a^2*d^2*e^3 + a*b*d^2*e^3*2i)))^(1/2)*((-1i/(4*(b^2*d^2*e^3 - a^2*d^2*e^3 + a*b*d^
2*e^3*2i)))^(1/2)*(e*cot(c + d*x))^(1/2)*(512*a^9*b^9*d^9*e^19 + 512*a^11*b^7*d^9*e^19 - 512*a^13*b^5*d^9*e^19
- 512*a^15*b^3*d^9*e^19) - 512*a^8*b^9*d^8*e^18 - 640*a^10*b^7*d^8*e^18 + 256*a^12*b^5*d^8*e^18 + 384*a^14*b^
3*d^8*e^18) - (e*cot(c + d*x))^(1/2)*(512*a^8*b^8*d^7*e^16 - 448*a^10*b^6*d^7*e^16 + 128*a^12*b^4*d^7*e^16 + 6
4*a^14*b^2*d^7*e^16)) - 128*a^7*b^8*d^6*e^15 + 32*a^11*b^4*d^6*e^15 + 32*a^13*b^2*d^6*e^15)) - (-1i/(4*(b^2*d^
2*e^3 - a^2*d^2*e^3 + a*b*d^2*e^3*2i)))^(1/2)*((e*cot(c + d*x))^(1/2)*(64*a^7*b^7*d^5*e^13 - 32*a^9*b^5*d^5*e^
13) - (-1i/(4*(b^2*d^2*e^3 - a^2*d^2*e^3 + a*b*d^2*e^3*2i)))^(1/2)*((-1i/(4*(b^2*d^2*e^3 - a^2*d^2*e^3 + a*b*d
^2*e^3*2i)))^(1/2)*((-1i/(4*(b^2*d^2*e^3 - a^2*d^2*e^3 + a*b*d^2*e^3*2i)))^(1/2)*((-1i/(4*(b^2*d^2*e^3 - a^2*d
^2*e^3 + a*b*d^2*e^3*2i)))^(1/2)*(e*cot(c + d*x))^(1/2)*(512*a^9*b^9*d^9*e^19 + 512*a^11*b^7*d^9*e^19 - 512*a^
13*b^5*d^9*e^19 - 512*a^15*b^3*d^9*e^19) + 512*a^8*b^9*d^8*e^18 + 640*a^10*b^7*d^8*e^18 - 256*a^12*b^5*d^8*e^1
8 - 384*a^14*b^3*d^8*e^18) - (e*cot(c + d*x))^(1/2)*(512*a^8*b^8*d^7*e^16 - 448*a^10*b^6*d^7*e^16 + 128*a^12*b
^4*d^7*e^16 + 64*a^14*b^2*d^7*e^16)) + 128*a^7*b^8*d^6*e^15 - 32*a^11*b^4*d^6*e^15 - 32*a^13*b^2*d^6*e^15))))*
(-1i/(4*(b^2*d^2*e^3 - a^2*d^2*e^3 + a*b*d^2*e^3*2i)))^(1/2)*2i - log((e*cot(c + d*x))^(1/2)*(64*a^7*b^7*d^5*e
^13 - 32*a^9*b^5*d^5*e^13) - (-1/(4*(b^2*d^2*e^3*1i - a^2*d^2*e^3*1i + 2*a*b*d^2*e^3)))^(1/2)*((-1/(4*(b^2*d^2
*e^3*1i - a^2*d^2*e^3*1i + 2*a*b*d^2*e^3)))^(1/2)*((-1/(4*(b^2*d^2*e^3*1i - a^2*d^2*e^3*1i + 2*a*b*d^2*e^3)))^
(1/2)*((-1/(4*(b^2*d^2*e^3*1i - a^2*d^2*e^3*1i + 2*a*b*d^2*e^3)))^(1/2)*(e*cot(c + d*x))^(1/2)*(512*a^9*b^9*d^
9*e^19 + 512*a^11*b^7*d^9*e^19 - 512*a^13*b^5*d^9*e^19 - 512*a^15*b^3*d^9*e^19) + 512*a^8*b^9*d^8*e^18 + 640*a
^10*b^7*d^8*e^18 - 256*a^12*b^5*d^8*e^18 - 384*a^14*b^3*d^8*e^18) - (e*cot(c + d*x))^(1/2)*(512*a^8*b^8*d^7*e^
16 - 448*a^10*b^6*d^7*e^16 + 128*a^12*b^4*d^7*e^16 + 64*a^14*b^2*d^7*e^16)) + 128*a^7*b^8*d^6*e^15 - 32*a^11*b
^4*d^6*e^15 - 32*a^13*b^2*d^6*e^15))*(-1/(4*(b^2*d^2*e^3*1i - a^2*d^2*e^3*1i + 2*a*b*d^2*e^3)))^(1/2) - (atan(
((((e*cot(c + d*x))^(1/2)*(64*a^7*b^7*d^5*e^13 - 32*a^9*b^5*d^5*e^13) + ((-a^3*b^5*e^3)^(1/2)*((((e*cot(c + d*
x))^(1/2)*(512*a^8*b^8*d^7*e^16 - 448*a^10*b^6*d^7*e^16 + 128*a^12*b^4*d^7*e^16 + 64*a^14*b^2*d^7*e^16) - ((-a
^3*b^5*e^3)^(1/2)*(256*a^12*b^5*d^8*e^18 - 640*a^10*b^7*d^8*e^18 - 512*a^8*b^9*d^8*e^18 + 384*a^14*b^3*d^8*e^1
8 + ((e*cot(c + d*x))^(1/2)*(-a^3*b^5*e^3)^(1/2)*(512*a^9*b^9*d^9*e^19 + 512*a^11*b^7*d^9*e^19 - 512*a^13*b^5*
d^9*e^19 - 512*a^15*b^3*d^9*e^19))/(a^5*d*e^3 + a^3*b^2*d*e^3)))/(a^5*d*e^3 + a^3*b^2*d*e^3))*(-a^3*b^5*e^3)^(
1/2))/(a^5*d*e^3 + a^3*b^2*d*e^3) + 128*a^7*b^8*d^6*e^15 - 32*a^11*b^4*d^6*e^15 - 32*a^13*b^2*d^6*e^15))/(a^5*
d*e^3 + a^3*b^2*d*e^3))*(-a^3*b^5*e^3)^(1/2)*1i)/(a^5*d*e^3 + a^3*b^2*d*e^3) + (((e*cot(c + d*x))^(1/2)*(64*a^
7*b^7*d^5*e^13 - 32*a^9*b^5*d^5*e^13) + ((-a^3*b^5*e^3)^(1/2)*((((e*cot(c + d*x))^(1/2)*(512*a^8*b^8*d^7*e^16
- 448*a^10*b^6*d^7*e^16 + 128*a^12*b^4*d^7*e^16 + 64*a^14*b^2*d^7*e^16) - ((-a^3*b^5*e^3)^(1/2)*(512*a^8*b^9*d
^8*e^18 + 640*a^10*b^7*d^8*e^18 - 256*a^12*b^5*d^8*e^18 - 384*a^14*b^3*d^8*e^18 + ((e*cot(c + d*x))^(1/2)*(-a^
3*b^5*e^3)^(1/2)*(512*a^9*b^9*d^9*e^19 + 512*a^11*b^7*d^9*e^19 - 512*a^13*b^5*d^9*e^19 - 512*a^15*b^3*d^9*e^19
))/(a^5*d*e^3 + a^3*b^2*d*e^3)))/(a^5*d*e^3 + a^3*b^2*d*e^3))*(-a^3*b^5*e^3)^(1/2))/(a^5*d*e^3 + a^3*b^2*d*e^3
) - 128*a^7*b^8*d^6*e^15 + 32*a^11*b^4*d^6*e^15 + 32*a^13*b^2*d^6*e^15))/(a^5*d*e^3 + a^3*b^2*d*e^3))*(-a^3*b^
5*e^3)^(1/2)*1i)/(a^5*d*e^3 + a^3*b^2*d*e^3))/((((e*cot(c + d*x))^(1/2)*(64*a^7*b^7*d^5*e^13 - 32*a^9*b^5*d^5*
e^13) + ((-a^3*b^5*e^3)^(1/2)*((((e*cot(c + d*x))^(1/2)*(512*a^8*b^8*d^7*e^16 - 448*a^10*b^6*d^7*e^16 + 128*a^
12*b^4*d^7*e^16 + 64*a^14*b^2*d^7*e^16) - ((-a^3*b^5*e^3)^(1/2)*(256*a^12*b^5*d^8*e^18 - 640*a^10*b^7*d^8*e^18
- 512*a^8*b^9*d^8*e^18 + 384*a^14*b^3*d^8*e^18 + ((e*cot(c + d*x))^(1/2)*(-a^3*b^5*e^3)^(1/2)*(512*a^9*b^9*d^
9*e^19 + 512*a^11*b^7*d^9*e^19 - 512*a^13*b^5*d^9*e^19 - 512*a^15*b^3*d^9*e^19))/(a^5*d*e^3 + a^3*b^2*d*e^3)))
/(a^5*d*e^3 + a^3*b^2*d*e^3))*(-a^3*b^5*e^3)^(1/2))/(a^5*d*e^3 + a^3*b^2*d*e^3) + 128*a^7*b^8*d^6*e^15 - 32*a^
11*b^4*d^6*e^15 - 32*a^13*b^2*d^6*e^15))/(a^5*d*e^3 + a^3*b^2*d*e^3))*(-a^3*b^5*e^3)^(1/2))/(a^5*d*e^3 + a^3*b
^2*d*e^3) - (((e*cot(c + d*x))^(1/2)*(64*a^7*b^7*d^5*e^13 - 32*a^9*b^5*d^5*e^13) + ((-a^3*b^5*e^3)^(1/2)*((((e
*cot(c + d*x))^(1/2)*(512*a^8*b^8*d^7*e^16 - 448*a^10*b^6*d^7*e^16 + 128*a^12*b^4*d^7*e^16 + 64*a^14*b^2*d^7*e
^16) - ((-a^3*b^5*e^3)^(1/2)*(512*a^8*b^9*d^8*e^18 + 640*a^10*b^7*d^8*e^18 - 256*a^12*b^5*d^8*e^18 - 384*a^14*
b^3*d^8*e^18 + ((e*cot(c + d*x))^(1/2)*(-a^3*b^5*e^3)^(1/2)*(512*a^9*b^9*d^9*e^19 + 512*a^11*b^7*d^9*e^19 - 51
2*a^13*b^5*d^9*e^19 - 512*a^15*b^3*d^9*e^19))/(a^5*d*e^3 + a^3*b^2*d*e^3)))/(a^5*d*e^3 + a^3*b^2*d*e^3))*(-a^3
*b^5*e^3)^(1/2))/(a^5*d*e^3 + a^3*b^2*d*e^3) - 128*a^7*b^8*d^6*e^15 + 32*a^11*b^4*d^6*e^15 + 32*a^13*b^2*d^6*e
^15))/(a^5*d*e^3 + a^3*b^2*d*e^3))*(-a^3*b^5*e^3)^(1/2))/(a^5*d*e^3 + a^3*b^2*d*e^3)))*(-a^3*b^5*e^3)^(1/2)*2i
)/(a^5*d*e^3 + a^3*b^2*d*e^3) + 2/(a*d*e*(e*cot(c + d*x))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((e*cot(c + d*x))**(3/2)*(a + b*cot(c + d*x))), x)

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