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

Optimal. Leaf size=116 \[ -\frac {\sqrt {2} a e^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}+\frac {2 a e^2 \sqrt {e \cot (c+d x)}}{d}-\frac {2 a e (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 a (e \cot (c+d x))^{5/2}}{5 d} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3609, 3613, 214} \begin {gather*} -\frac {\sqrt {2} a e^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} \cot (c+d x)+\sqrt {e}}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}+\frac {2 a e^2 \sqrt {e \cot (c+d x)}}{d}-\frac {2 a e (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 a (e \cot (c+d x))^{5/2}}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 3609

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*
((a + b*Tan[e + f*x])^m/(f*m)), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*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] && GtQ[m, 0]

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]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.21, size = 68, normalized size = 0.59 \begin {gather*} -\frac {2 a e (e \cot (c+d x))^{3/2} \left (3 \cot (c+d x) \, _2F_1\left (-\frac {5}{4},1;-\frac {1}{4};-\tan ^2(c+d x)\right )+5 \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\tan ^2(c+d x)\right )\right )}{15 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*e*(e*Cot[c + d*x])^(3/2)*(3*Cot[c + d*x]*Hypergeometric2F1[-5/4, 1, -1/4, -Tan[c + d*x]^2] + 5*Hypergeom
etric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2]))/(15*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(318\) vs. \(2(95)=190\).
time = 0.42, size = 319, normalized size = 2.75

method	result	size
derivativedivides	\(-\frac {a \left (\frac {2 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-2 e^{2} \sqrt {e \cot \left (d x +c \right )}+2 e^{3} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d}\)	\(319\)
default	\(-\frac {a \left (\frac {2 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-2 e^{2} \sqrt {e \cot \left (d x +c \right )}+2 e^{3} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d}\)	\(319\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d*a*(2/5*(e*cot(d*x+c))^(5/2)+2/3*e*(e*cot(d*x+c))^(3/2)-2*e^2*(e*cot(d*x+c))^(1/2)+2*e^3*(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 = 0.53, size = 101, normalized size = 0.87 \begin {gather*} -\frac {{\left (15 \, {\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 {60 \, a}{\sqrt {\tan \left (d x + c\right )}} + \frac {20 \, a}{\tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {12 \, a}{\tan \left (d x + c\right )^{\frac {5}{2}}}\right )} e^{\frac {5}{2}}}{30 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(15*(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 - 60*a/sqrt(tan(d*x + c)) + 20*a/tan(d*x + c)^(3/2) + 12*a/tan(d*x + c)^(5/2))*e^
(5/2)/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (88) = 176\).
time = 3.29, size = 181, normalized size = 1.56 \begin {gather*} \frac {15 \, {\left (\sqrt {2} a \cos \left (2 \, d x + 2 \, c\right ) e^{\frac {5}{2}} - \sqrt {2} a e^{\frac {5}{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 ) + 4 \, {\left (18 \, a \cos \left (2 \, d x + 2 \, c\right ) e^{\frac {5}{2}} + 5 \, a e^{\frac {5}{2}} \sin \left (2 \, d x + 2 \, c\right ) - 12 \, a e^{\frac {5}{2}}\right )} \sqrt {\frac {\cos \left (2 \, d x + 2 \, c\right ) + 1}{\sin \left (2 \, d x + 2 \, c\right )}}}{30 \, {\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(15*(sqrt(2)*a*cos(2*d*x + 2*c)*e^(5/2) - sqrt(2)*a*e^(5/2))*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) + 4*(18*a*cos(
2*d*x + 2*c)*e^(5/2) + 5*a*e^(5/2)*sin(2*d*x + 2*c) - 12*a*e^(5/2))*sqrt((cos(2*d*x + 2*c) + 1)/sin(2*d*x + 2*
c)))/(d*cos(2*d*x + 2*c) - d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Mupad [B]
time = 1.98, size = 144, normalized size = 1.24 \begin {gather*} \frac {2\,a\,e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d}-\frac {2\,a\,e\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{3\,d}-\frac {2\,a\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}}{5\,d}+\frac {{\left (-1\right )}^{1/4}\,a\,e^{5/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,1{}\mathrm {i}}{\sqrt {e}}\right )}{d}-\frac {{\left (-1\right )}^{1/4}\,a\,e^{5/2}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{d}+\frac {{\left (-1\right )}^{1/4}\,a\,e^{5/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,\left (1+1{}\mathrm {i}\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a*e^2*(e*cot(c + d*x))^(1/2))/d - (2*a*e*(e*cot(c + d*x))^(3/2))/(3*d) - (2*a*(e*cot(c + d*x))^(5/2))/(5*d)
+ ((-1)^(1/4)*a*e^(5/2)*atan(((-1)^(1/4)*(e*cot(c + d*x))^(1/2))/e^(1/2))*(1 + 1i))/d + ((-1)^(1/4)*a*e^(5/2)
*atan(((-1)^(1/4)*(e*cot(c + d*x))^(1/2)*1i)/e^(1/2)))/d - ((-1)^(1/4)*a*e^(5/2)*atanh(((-1)^(1/4)*(e*cot(c +
d*x))^(1/2))/e^(1/2)))/d

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