3.4.36 \(\int \frac {\cot (e+f x)}{(a+b \tan ^2(e+f x))^{3/2}} \, dx\) [336]

Optimal. Leaf size=106 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2} f}-\frac {b}{a (a-b) f \sqrt {a+b \tan ^2(e+f x)}} \]

[Out]

-arctanh((a+b*tan(f*x+e)^2)^(1/2)/a^(1/2))/a^(3/2)/f+arctanh((a+b*tan(f*x+e)^2)^(1/2)/(a-b)^(1/2))/(a-b)^(3/2)
/f-b/a/(a-b)/f/(a+b*tan(f*x+e)^2)^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3751, 457, 87, 162, 65, 214} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f}-\frac {b}{a f (a-b) \sqrt {a+b \tan ^2(e+f x)}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f (a-b)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]/(a + b*Tan[e + f*x]^2)^(3/2),x]

[Out]

-(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f)) + ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/(
(a - b)^(3/2)*f) - b/(a*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2])

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 87

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[f*((e + f*x)^(p +
1)/((p + 1)*(b*e - a*f)*(d*e - c*f))), x] + Dist[1/((b*e - a*f)*(d*e - c*f)), Int[(b*d*e - b*c*f - a*d*f - b*
d*f*x)*((e + f*x)^(p + 1)/((a + b*x)*(c + d*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[p, -1]

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 3751

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2
+ ff^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.09, size = 91, normalized size = 0.86 \begin {gather*} \frac {-a \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {a+b \tan ^2(e+f x)}{a-b}\right )+(a-b) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1+\frac {b \tan ^2(e+f x)}{a}\right )}{a (a-b) f \sqrt {a+b \tan ^2(e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]/(a + b*Tan[e + f*x]^2)^(3/2),x]

[Out]

(-(a*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[e + f*x]^2)/(a - b)]) + (a - b)*Hypergeometric2F1[-1/2, 1, 1/2
, 1 + (b*Tan[e + f*x]^2)/a])/(a*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(32887\) vs. \(2(92)=184\).
time = 0.97, size = 32888, normalized size = 310.26

method	result	size
default	\(\text {Expression too large to display}\)	\(32888\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)/(a+b*tan(f*x+e)^2)^(3/2),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(f*x+e)/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(cot(f*x + e)/(b*tan(f*x + e)^2 + a)^(3/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (95) = 190\).
time = 3.46, size = 952, normalized size = 8.98 \begin {gather*} \left [-\frac {{\left (a^{2} b \tan \left (f x + e\right )^{2} + a^{3}\right )} \sqrt {a - b} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 2 \, a - b}{\tan \left (f x + e\right )^{2} + 1}\right ) - {\left (a^{3} - 2 \, a^{2} b + a b^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {a} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a} + 2 \, a}{\tan \left (f x + e\right )^{2}}\right ) + 2 \, {\left (a^{2} b - a b^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{2 \, {\left ({\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} f\right )}}, \frac {2 \, {\left (a^{2} b \tan \left (f x + e\right )^{2} + a^{3}\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{a - b}\right ) + {\left (a^{3} - 2 \, a^{2} b + a b^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {a} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a} + 2 \, a}{\tan \left (f x + e\right )^{2}}\right ) - 2 \, {\left (a^{2} b - a b^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{2 \, {\left ({\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} f\right )}}, \frac {2 \, {\left (a^{3} - 2 \, a^{2} b + a b^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a}}{a}\right ) - {\left (a^{2} b \tan \left (f x + e\right )^{2} + a^{3}\right )} \sqrt {a - b} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 2 \, a - b}{\tan \left (f x + e\right )^{2} + 1}\right ) - 2 \, {\left (a^{2} b - a b^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{2 \, {\left ({\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} f\right )}}, \frac {{\left (a^{3} - 2 \, a^{2} b + a b^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a}}{a}\right ) + {\left (a^{2} b \tan \left (f x + e\right )^{2} + a^{3}\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{a - b}\right ) - {\left (a^{2} b - a b^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{{\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} f}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

[-1/2*((a^2*b*tan(f*x + e)^2 + a^3)*sqrt(a - b)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a -
b) + 2*a - b)/(tan(f*x + e)^2 + 1)) - (a^3 - 2*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3)*tan(f*x + e)^2)*sqrt(a)
*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) + 2*(a^2*b - a*b^2)*sqrt(
b*tan(f*x + e)^2 + a))/((a^4*b - 2*a^3*b^2 + a^2*b^3)*f*tan(f*x + e)^2 + (a^5 - 2*a^4*b + a^3*b^2)*f), 1/2*(2*
(a^2*b*tan(f*x + e)^2 + a^3)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) + (a^3 - 2*
a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3)*tan(f*x + e)^2)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^
2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) - 2*(a^2*b - a*b^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^4*b - 2*a^3*b^2 + a^
2*b^3)*f*tan(f*x + e)^2 + (a^5 - 2*a^4*b + a^3*b^2)*f), 1/2*(2*(a^3 - 2*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3
)*tan(f*x + e)^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) - (a^2*b*tan(f*x + e)^2 + a^3)*sqrt(a
- b)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) - 2*(a
^2*b - a*b^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^4*b - 2*a^3*b^2 + a^2*b^3)*f*tan(f*x + e)^2 + (a^5 - 2*a^4*b + a
^3*b^2)*f), ((a^3 - 2*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3)*tan(f*x + e)^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x +
e)^2 + a)*sqrt(-a)/a) + (a^2*b*tan(f*x + e)^2 + a^3)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a
+ b)/(a - b)) - (a^2*b - a*b^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^4*b - 2*a^3*b^2 + a^2*b^3)*f*tan(f*x + e)^2 +
(a^5 - 2*a^4*b + a^3*b^2)*f)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot {\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)/(a+b*tan(f*x+e)**2)**(3/2),x)

[Out]

Integral(cot(e + f*x)/(a + b*tan(e + f*x)**2)**(3/2), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(t_

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Mupad [B]
time = 12.64, size = 1922, normalized size = 18.13 \begin {gather*} \frac {b}{f\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (a\,b-a^2\right )}-\frac {\mathrm {atanh}\left (\frac {2\,a^2\,b^8\,f^2\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3\,f^2+24\,a^5\,b^4\,f^2-38\,a^4\,b^5\,f^2+30\,a^3\,b^6\,f^2-12\,a^2\,b^7\,f^2+2\,a\,b^8\,f^2\right )}-\frac {12\,a^3\,b^7\,f^2\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3\,f^2+24\,a^5\,b^4\,f^2-38\,a^4\,b^5\,f^2+30\,a^3\,b^6\,f^2-12\,a^2\,b^7\,f^2+2\,a\,b^8\,f^2\right )}+\frac {30\,a^4\,b^6\,f^2\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3\,f^2+24\,a^5\,b^4\,f^2-38\,a^4\,b^5\,f^2+30\,a^3\,b^6\,f^2-12\,a^2\,b^7\,f^2+2\,a\,b^8\,f^2\right )}-\frac {38\,a^5\,b^5\,f^2\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3\,f^2+24\,a^5\,b^4\,f^2-38\,a^4\,b^5\,f^2+30\,a^3\,b^6\,f^2-12\,a^2\,b^7\,f^2+2\,a\,b^8\,f^2\right )}+\frac {24\,a^6\,b^4\,f^2\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3\,f^2+24\,a^5\,b^4\,f^2-38\,a^4\,b^5\,f^2+30\,a^3\,b^6\,f^2-12\,a^2\,b^7\,f^2+2\,a\,b^8\,f^2\right )}-\frac {6\,a^7\,b^3\,f^2\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3\,f^2+24\,a^5\,b^4\,f^2-38\,a^4\,b^5\,f^2+30\,a^3\,b^6\,f^2-12\,a^2\,b^7\,f^2+2\,a\,b^8\,f^2\right )}\right )}{f\,\sqrt {a^3}}+\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (-4\,a^8\,b^2\,f^3+16\,a^7\,b^3\,f^3-26\,a^6\,b^4\,f^3+22\,a^5\,b^5\,f^3-10\,a^4\,b^6\,f^3+2\,a^3\,b^7\,f^3\right )}{2}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (12\,a^5\,b^7\,f^4-2\,a^4\,b^8\,f^4-28\,a^6\,b^6\,f^4+32\,a^7\,b^5\,f^4-18\,a^8\,b^4\,f^4+4\,a^9\,b^3\,f^4+\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\sqrt {{\left (a-b\right )}^3}\,\left (16\,a^{11}\,b^2\,f^5-88\,a^{10}\,b^3\,f^5+200\,a^9\,b^4\,f^5-240\,a^8\,b^5\,f^5+160\,a^7\,b^6\,f^5-56\,a^6\,b^7\,f^5+8\,a^5\,b^8\,f^5\right )}{4\,f\,{\left (a-b\right )}^3}\right )}{2\,f\,{\left (a-b\right )}^3}\right )\,\sqrt {{\left (a-b\right )}^3}\,1{}\mathrm {i}}{f\,{\left (a-b\right )}^3}+\frac {\left (\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (-4\,a^8\,b^2\,f^3+16\,a^7\,b^3\,f^3-26\,a^6\,b^4\,f^3+22\,a^5\,b^5\,f^3-10\,a^4\,b^6\,f^3+2\,a^3\,b^7\,f^3\right )}{2}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (2\,a^4\,b^8\,f^4-12\,a^5\,b^7\,f^4+28\,a^6\,b^6\,f^4-32\,a^7\,b^5\,f^4+18\,a^8\,b^4\,f^4-4\,a^9\,b^3\,f^4+\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\sqrt {{\left (a-b\right )}^3}\,\left (16\,a^{11}\,b^2\,f^5-88\,a^{10}\,b^3\,f^5+200\,a^9\,b^4\,f^5-240\,a^8\,b^5\,f^5+160\,a^7\,b^6\,f^5-56\,a^6\,b^7\,f^5+8\,a^5\,b^8\,f^5\right )}{4\,f\,{\left (a-b\right )}^3}\right )}{2\,f\,{\left (a-b\right )}^3}\right )\,\sqrt {{\left (a-b\right )}^3}\,1{}\mathrm {i}}{f\,{\left (a-b\right )}^3}}{2\,a^3\,b^6\,f^2-6\,a^4\,b^5\,f^2+6\,a^5\,b^4\,f^2-2\,a^6\,b^3\,f^2-\frac {\left (\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (-4\,a^8\,b^2\,f^3+16\,a^7\,b^3\,f^3-26\,a^6\,b^4\,f^3+22\,a^5\,b^5\,f^3-10\,a^4\,b^6\,f^3+2\,a^3\,b^7\,f^3\right )}{2}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (12\,a^5\,b^7\,f^4-2\,a^4\,b^8\,f^4-28\,a^6\,b^6\,f^4+32\,a^7\,b^5\,f^4-18\,a^8\,b^4\,f^4+4\,a^9\,b^3\,f^4+\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\sqrt {{\left (a-b\right )}^3}\,\left (16\,a^{11}\,b^2\,f^5-88\,a^{10}\,b^3\,f^5+200\,a^9\,b^4\,f^5-240\,a^8\,b^5\,f^5+160\,a^7\,b^6\,f^5-56\,a^6\,b^7\,f^5+8\,a^5\,b^8\,f^5\right )}{4\,f\,{\left (a-b\right )}^3}\right )}{2\,f\,{\left (a-b\right )}^3}\right )\,\sqrt {{\left (a-b\right )}^3}}{f\,{\left (a-b\right )}^3}+\frac {\left (\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (-4\,a^8\,b^2\,f^3+16\,a^7\,b^3\,f^3-26\,a^6\,b^4\,f^3+22\,a^5\,b^5\,f^3-10\,a^4\,b^6\,f^3+2\,a^3\,b^7\,f^3\right )}{2}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (2\,a^4\,b^8\,f^4-12\,a^5\,b^7\,f^4+28\,a^6\,b^6\,f^4-32\,a^7\,b^5\,f^4+18\,a^8\,b^4\,f^4-4\,a^9\,b^3\,f^4+\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\sqrt {{\left (a-b\right )}^3}\,\left (16\,a^{11}\,b^2\,f^5-88\,a^{10}\,b^3\,f^5+200\,a^9\,b^4\,f^5-240\,a^8\,b^5\,f^5+160\,a^7\,b^6\,f^5-56\,a^6\,b^7\,f^5+8\,a^5\,b^8\,f^5\right )}{4\,f\,{\left (a-b\right )}^3}\right )}{2\,f\,{\left (a-b\right )}^3}\right )\,\sqrt {{\left (a-b\right )}^3}}{f\,{\left (a-b\right )}^3}}\right )\,\sqrt {{\left (a-b\right )}^3}\,1{}\mathrm {i}}{f\,{\left (a-b\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(e + f*x)/(a + b*tan(e + f*x)^2)^(3/2),x)

[Out]

b/(f*(a + b*tan(e + f*x)^2)^(1/2)*(a*b - a^2)) - atanh((2*a^2*b^8*f^2*(a + b*tan(e + f*x)^2)^(1/2))/((a^3)^(1/
2)*(2*a*b^8*f^2 - 12*a^2*b^7*f^2 + 30*a^3*b^6*f^2 - 38*a^4*b^5*f^2 + 24*a^5*b^4*f^2 - 6*a^6*b^3*f^2)) - (12*a^
3*b^7*f^2*(a + b*tan(e + f*x)^2)^(1/2))/((a^3)^(1/2)*(2*a*b^8*f^2 - 12*a^2*b^7*f^2 + 30*a^3*b^6*f^2 - 38*a^4*b
^5*f^2 + 24*a^5*b^4*f^2 - 6*a^6*b^3*f^2)) + (30*a^4*b^6*f^2*(a + b*tan(e + f*x)^2)^(1/2))/((a^3)^(1/2)*(2*a*b^
8*f^2 - 12*a^2*b^7*f^2 + 30*a^3*b^6*f^2 - 38*a^4*b^5*f^2 + 24*a^5*b^4*f^2 - 6*a^6*b^3*f^2)) - (38*a^5*b^5*f^2*
(a + b*tan(e + f*x)^2)^(1/2))/((a^3)^(1/2)*(2*a*b^8*f^2 - 12*a^2*b^7*f^2 + 30*a^3*b^6*f^2 - 38*a^4*b^5*f^2 + 2
4*a^5*b^4*f^2 - 6*a^6*b^3*f^2)) + (24*a^6*b^4*f^2*(a + b*tan(e + f*x)^2)^(1/2))/((a^3)^(1/2)*(2*a*b^8*f^2 - 12
*a^2*b^7*f^2 + 30*a^3*b^6*f^2 - 38*a^4*b^5*f^2 + 24*a^5*b^4*f^2 - 6*a^6*b^3*f^2)) - (6*a^7*b^3*f^2*(a + b*tan(
e + f*x)^2)^(1/2))/((a^3)^(1/2)*(2*a*b^8*f^2 - 12*a^2*b^7*f^2 + 30*a^3*b^6*f^2 - 38*a^4*b^5*f^2 + 24*a^5*b^4*f
^2 - 6*a^6*b^3*f^2)))/(f*(a^3)^(1/2)) + (atan((((((a + b*tan(e + f*x)^2)^(1/2)*(2*a^3*b^7*f^3 - 10*a^4*b^6*f^3
+ 22*a^5*b^5*f^3 - 26*a^6*b^4*f^3 + 16*a^7*b^3*f^3 - 4*a^8*b^2*f^3))/2 + (((a - b)^3)^(1/2)*(12*a^5*b^7*f^4 -
2*a^4*b^8*f^4 - 28*a^6*b^6*f^4 + 32*a^7*b^5*f^4 - 18*a^8*b^4*f^4 + 4*a^9*b^3*f^4 + ((a + b*tan(e + f*x)^2)^(1
/2)*((a - b)^3)^(1/2)*(8*a^5*b^8*f^5 - 56*a^6*b^7*f^5 + 160*a^7*b^6*f^5 - 240*a^8*b^5*f^5 + 200*a^9*b^4*f^5 -
88*a^10*b^3*f^5 + 16*a^11*b^2*f^5))/(4*f*(a - b)^3)))/(2*f*(a - b)^3))*((a - b)^3)^(1/2)*1i)/(f*(a - b)^3) + (
(((a + b*tan(e + f*x)^2)^(1/2)*(2*a^3*b^7*f^3 - 10*a^4*b^6*f^3 + 22*a^5*b^5*f^3 - 26*a^6*b^4*f^3 + 16*a^7*b^3*
f^3 - 4*a^8*b^2*f^3))/2 + (((a - b)^3)^(1/2)*(2*a^4*b^8*f^4 - 12*a^5*b^7*f^4 + 28*a^6*b^6*f^4 - 32*a^7*b^5*f^4
+ 18*a^8*b^4*f^4 - 4*a^9*b^3*f^4 + ((a + b*tan(e + f*x)^2)^(1/2)*((a - b)^3)^(1/2)*(8*a^5*b^8*f^5 - 56*a^6*b^
7*f^5 + 160*a^7*b^6*f^5 - 240*a^8*b^5*f^5 + 200*a^9*b^4*f^5 - 88*a^10*b^3*f^5 + 16*a^11*b^2*f^5))/(4*f*(a - b)
^3)))/(2*f*(a - b)^3))*((a - b)^3)^(1/2)*1i)/(f*(a - b)^3))/(2*a^3*b^6*f^2 - 6*a^4*b^5*f^2 + 6*a^5*b^4*f^2 - 2
*a^6*b^3*f^2 - ((((a + b*tan(e + f*x)^2)^(1/2)*(2*a^3*b^7*f^3 - 10*a^4*b^6*f^3 + 22*a^5*b^5*f^3 - 26*a^6*b^4*f
^3 + 16*a^7*b^3*f^3 - 4*a^8*b^2*f^3))/2 + (((a - b)^3)^(1/2)*(12*a^5*b^7*f^4 - 2*a^4*b^8*f^4 - 28*a^6*b^6*f^4
+ 32*a^7*b^5*f^4 - 18*a^8*b^4*f^4 + 4*a^9*b^3*f^4 + ((a + b*tan(e + f*x)^2)^(1/2)*((a - b)^3)^(1/2)*(8*a^5*b^8
*f^5 - 56*a^6*b^7*f^5 + 160*a^7*b^6*f^5 - 240*a^8*b^5*f^5 + 200*a^9*b^4*f^5 - 88*a^10*b^3*f^5 + 16*a^11*b^2*f^
5))/(4*f*(a - b)^3)))/(2*f*(a - b)^3))*((a - b)^3)^(1/2))/(f*(a - b)^3) + ((((a + b*tan(e + f*x)^2)^(1/2)*(2*a
^3*b^7*f^3 - 10*a^4*b^6*f^3 + 22*a^5*b^5*f^3 - 26*a^6*b^4*f^3 + 16*a^7*b^3*f^3 - 4*a^8*b^2*f^3))/2 + (((a - b)
^3)^(1/2)*(2*a^4*b^8*f^4 - 12*a^5*b^7*f^4 + 28*a^6*b^6*f^4 - 32*a^7*b^5*f^4 + 18*a^8*b^4*f^4 - 4*a^9*b^3*f^4 +
((a + b*tan(e + f*x)^2)^(1/2)*((a - b)^3)^(1/2)*(8*a^5*b^8*f^5 - 56*a^6*b^7*f^5 + 160*a^7*b^6*f^5 - 240*a^8*b
^5*f^5 + 200*a^9*b^4*f^5 - 88*a^10*b^3*f^5 + 16*a^11*b^2*f^5))/(4*f*(a - b)^3)))/(2*f*(a - b)^3))*((a - b)^3)^
(1/2))/(f*(a - b)^3)))*((a - b)^3)^(1/2)*1i)/(f*(a - b)^3)

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