3.234 \(\int (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx\)

Optimal. Leaf size=346 \[ \frac {a \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt {2} d}-\frac {a \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right ) \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt {2} d}-\frac {a \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {a \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {2 a \sin (c+d x) \tan ^2(c+d x) (e \cot (c+d x))^{3/2}}{d}-\frac {2 \tan (c+d x) (a \sec (c+d x)+a) (e \cot (c+d x))^{3/2}}{d}-\frac {2 a \sin (c+d x) \tan (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) (e \cot (c+d x))^{3/2}}{d \sqrt {\sin (2 c+2 d x)}} \]

[Out]

-2*(e*cot(d*x+c))^(3/2)*(a+a*sec(d*x+c))*tan(d*x+c)/d+2*a*(e*cot(d*x+c))^(3/2)*(sin(c+1/4*Pi+d*x)^2)^(1/2)/sin
(c+1/4*Pi+d*x)*EllipticE(cos(c+1/4*Pi+d*x),2^(1/2))*sin(d*x+c)*tan(d*x+c)/d/sin(2*d*x+2*c)^(1/2)-1/2*a*arctan(
-1+2^(1/2)*tan(d*x+c)^(1/2))*(e*cot(d*x+c))^(3/2)*tan(d*x+c)^(3/2)/d*2^(1/2)-1/2*a*arctan(1+2^(1/2)*tan(d*x+c)
^(1/2))*(e*cot(d*x+c))^(3/2)*tan(d*x+c)^(3/2)/d*2^(1/2)-1/4*a*(e*cot(d*x+c))^(3/2)*ln(1-2^(1/2)*tan(d*x+c)^(1/
2)+tan(d*x+c))*tan(d*x+c)^(3/2)/d*2^(1/2)+1/4*a*(e*cot(d*x+c))^(3/2)*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))
*tan(d*x+c)^(3/2)/d*2^(1/2)+2*a*(e*cot(d*x+c))^(3/2)*sin(d*x+c)*tan(d*x+c)^2/d

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Rubi [A]  time = 0.28, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 15, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {3900, 3882, 3884, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639} \[ \frac {a \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt {2} d}-\frac {a \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right ) \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt {2} d}-\frac {a \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {a \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {2 a \sin (c+d x) \tan ^2(c+d x) (e \cot (c+d x))^{3/2}}{d}-\frac {2 \tan (c+d x) (a \sec (c+d x)+a) (e \cot (c+d x))^{3/2}}{d}-\frac {2 a \sin (c+d x) \tan (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) (e \cot (c+d x))^{3/2}}{d \sqrt {\sin (2 c+2 d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(e*Cot[c + d*x])^(3/2)*(a + a*Sec[c + d*x])*Tan[c + d*x])/d - (2*a*(e*Cot[c + d*x])^(3/2)*EllipticE[c - Pi
/4 + d*x, 2]*Sin[c + d*x]*Tan[c + d*x])/(d*Sqrt[Sin[2*c + 2*d*x]]) + (a*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]
*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))/(Sqrt[2]*d) - (a*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c +
d*x])^(3/2)*Tan[c + d*x]^(3/2))/(Sqrt[2]*d) - (a*(e*Cot[c + d*x])^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] +
Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*d) + (a*(e*Cot[c + d*x])^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]
+ Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*d) + (2*a*(e*Cot[c + d*x])^(3/2)*Sin[c + d*x]*Tan[c + d*x]^2)/
d

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 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

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 2572

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(Sqrt[a*Sin[e +
f*x]]*Sqrt[b*Cos[e + f*x]])/Sqrt[Sin[2*e + 2*f*x]], Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f},
x]

Rule 2613

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a^2*(a*Sec[
e + f*x])^(m - 2)*(b*Tan[e + f*x])^(n + 1))/(b*f*(m + n - 1)), x] + Dist[(a^2*(m - 2))/(m + n - 1), Int[(a*Sec
[e + f*x])^(m - 2)*(b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (GtQ[m, 1] || (EqQ[m, 1] && EqQ[
n, 1/2])) && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]

Rule 2615

Int[Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]]/sec[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[(Sqrt[Cos[e + f*x]]*Sqrt[b*
Tan[e + f*x]])/Sqrt[Sin[e + f*x]], Int[Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]], x], x] /; FreeQ[{b, e, f}, x]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rule 3476

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rule 3882

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[((e*Cot[c
+ d*x])^(m + 1)*(a + b*Csc[c + d*x]))/(d*e*(m + 1)), x] - Dist[1/(e^2*(m + 1)), Int[(e*Cot[c + d*x])^(m + 2)*(
a*(m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1]

Rule 3884

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(e*
Cot[c + d*x])^m, x], x] + Dist[b, Int[(e*Cot[c + d*x])^m*Csc[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, m}, x]

Rule 3900

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*((a_) + (b_.)*sec[(c_.) + (d_.)*(x_)])^(n_.), x_Symbol] :> Dist[(e*Co
t[c + d*x])^m*Tan[c + d*x]^m, Int[(a + b*Sec[c + d*x])^n/Tan[c + d*x]^m, x], x] /; FreeQ[{a, b, c, d, e, m, n}
, x] &&  !IntegerQ[m]

Rubi steps

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

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Mathematica [C]  time = 1.20, size = 191, normalized size = 0.55 \[ \frac {a e (\cos (c+d x)+1) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \sqrt {e \cot (c+d x)} \left (8 \cot ^2(c+d x) \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\cot ^2(c+d x)\right )+3 \sqrt {\csc ^2(c+d x)} \left (-4 \cos ^2(c+d x)-4 \cos (c+d x)+\sqrt {\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+\sqrt {\sin (2 (c+d x))} \log \left (\sin (c+d x)+\sqrt {\sin (2 (c+d x))}+\cos (c+d x)\right )\right )\right )}{12 d \sqrt {\csc ^2(c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*e*(1 + Cos[c + d*x])*Sqrt[e*Cot[c + d*x]]*Sec[(c + d*x)/2]^2*Sec[c + d*x]*(8*Cot[c + d*x]^2*Hypergeometric2
F1[3/4, 3/2, 7/4, -Cot[c + d*x]^2] + 3*Sqrt[Csc[c + d*x]^2]*(-4*Cos[c + d*x] - 4*Cos[c + d*x]^2 + ArcSin[Cos[c
+ d*x] - Sin[c + d*x]]*Sqrt[Sin[2*(c + d*x)]] + Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqr
t[Sin[2*(c + d*x)]])))/(12*d*Sqrt[Csc[c + d*x]^2])

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 2.35, size = 1390, normalized size = 4.02 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a/d*(I*EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*cos(d*x+c)*((-1+cos
(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c)
)^(1/2)-I*EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*cos(d*x+c)*((-1+cos(d
*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^
(1/2)+I*EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*
x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)-I*Ellip
ticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*(
(-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)-EllipticPi(((1-cos(d
*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*cos(d*x+c)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+
cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)-EllipticPi(((1-cos(d*x+c
)+sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*cos(d*x+c)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(
d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)-4*EllipticE(((1-cos(d*x+c)+s
in(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))*cos(d*x+c)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*
x+c))/sin(d*x+c))^(1/2)*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)+2*EllipticF(((1-cos(d*x+c)+sin(d*x+c))/si
n(d*x+c))^(1/2),1/2*2^(1/2))*cos(d*x+c)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x
+c))^(1/2)*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)-EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2
),1/2-1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((1-
cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)-EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2
*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((1-cos(d*x+c)+sin(
d*x+c))/sin(d*x+c))^(1/2)-4*EllipticE(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))*((-1+cos(d*x+c
))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2
)+2*EllipticF(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-
1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)+4*cos(d*x+c)*2^(1/2))*
sin(d*x+c)*(e*cos(d*x+c)/sin(d*x+c))^(3/2)/cos(d*x+c)^2*2^(1/2)

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maxima [A]  time = 0.50, size = 180, normalized size = 0.52 \[ \frac {{\left (2 \, \sqrt {2} \sqrt {e} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right ) + 2 \, \sqrt {2} \sqrt {e} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right ) + \sqrt {2} \sqrt {e} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right ) - \sqrt {2} \sqrt {e} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right ) - 8 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )} a e}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}\,\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [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((e*cot(d*x+c))**(3/2)*(a+a*sec(d*x+c)),x)

[Out]

Timed out

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