3.1180 \(\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=294 \[ \frac{\left (2 a^2-3 b^2\right ) \cos (c+d x)}{a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\left (4 a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{a b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (4 a^2-3 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{a^2 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{3 b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{a^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x)}{a d \sqrt{a+b \sin (c+d x)}} \]

[Out]

((2*a^2 - 3*b^2)*Cos[c + d*x])/(a^2*b*d*Sqrt[a + b*Sin[c + d*x]]) - Cot[c + d*x]/(a*d*Sqrt[a + b*Sin[c + d*x]]
) + ((4*a^2 - 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(a^2*b^2*d*Sqrt[(a
+ b*Sin[c + d*x])/(a + b)]) - ((4*a^2 - b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c +
d*x])/(a + b)])/(a*b^2*d*Sqrt[a + b*Sin[c + d*x]]) - (3*b*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sq
rt[(a + b*Sin[c + d*x])/(a + b)])/(a^2*d*Sqrt[a + b*Sin[c + d*x]])

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Rubi [A]  time = 0.705431, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29, Rules used = {2890, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ \frac{\left (2 a^2-3 b^2\right ) \cos (c+d x)}{a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\left (4 a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{a b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (4 a^2-3 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{a^2 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{3 b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{a^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x)}{a d \sqrt{a+b \sin (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a^2 - 3*b^2)*Cos[c + d*x])/(a^2*b*d*Sqrt[a + b*Sin[c + d*x]]) - Cot[c + d*x]/(a*d*Sqrt[a + b*Sin[c + d*x]]
) + ((4*a^2 - 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(a^2*b^2*d*Sqrt[(a
+ b*Sin[c + d*x])/(a + b)]) - ((4*a^2 - b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c +
d*x])/(a + b)])/(a*b^2*d*Sqrt[a + b*Sin[c + d*x]]) - (3*b*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sq
rt[(a + b*Sin[c + d*x])/(a + b)])/(a^2*d*Sqrt[a + b*Sin[c + d*x]])

Rule 2890

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[(Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1))/(a*d*f*(n + 1)), x] +
(Dist[1/(a^2*b*d*(n + 1)*(m + 1)), Int[(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1)*Simp[a^2*(n + 1)
*(n + 2) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 1)*Sin[e + f*x] - (a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m
+ n + 4))*Sin[e + f*x]^2, x], x], x] - Simp[((a^2*(n + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(d*Sin[e + f*x])^(n
+ 2)*(a + b*Sin[e + f*x])^(m + 1))/(a^2*b*d^2*f*(n + 1)*(m + 1)), x]) /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2
- b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] && LtQ[n, -1]

Rule 3059

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) +
(f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Dist[C/(b*d), Int[Sqrt[a + b*Sin[e + f*x]]
, x], x] - Dist[1/(b*d), Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e +
f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2
- b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 3002

Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[
(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[B/d, Int[(a + b*Sin[e + f*x])^m, x], x] - Dist[(B*c - A*d)/d, Int[(a +
b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2807

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d*
Sin[e + f*x])/(c + d)]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 2805

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[c + d]), x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]

Rubi steps

\begin{align*} \int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx &=\frac{\left (2 a^2-3 b^2\right ) \cos (c+d x)}{a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x)}{a d \sqrt{a+b \sin (c+d x)}}+\frac{2 \int \frac{\csc (c+d x) \left (-\frac{3 b^2}{4}-\frac{1}{2} a b \sin (c+d x)+\frac{1}{4} \left (4 a^2-3 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{a^2 b}\\ &=\frac{\left (2 a^2-3 b^2\right ) \cos (c+d x)}{a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x)}{a d \sqrt{a+b \sin (c+d x)}}+\frac{1}{2} \left (-\frac{3}{a^2}+\frac{4}{b^2}\right ) \int \sqrt{a+b \sin (c+d x)} \, dx-\frac{2 \int \frac{\csc (c+d x) \left (\frac{3 b^3}{4}+\frac{1}{4} a \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{a^2 b^2}\\ &=\frac{\left (2 a^2-3 b^2\right ) \cos (c+d x)}{a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x)}{a d \sqrt{a+b \sin (c+d x)}}-\frac{(3 b) \int \frac{\csc (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{2 a^2}-\frac{\left (4 a^2-b^2\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{2 a b^2}+\frac{\left (\left (-\frac{3}{a^2}+\frac{4}{b^2}\right ) \sqrt{a+b \sin (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}} \, dx}{2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}\\ &=\frac{\left (2 a^2-3 b^2\right ) \cos (c+d x)}{a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x)}{a d \sqrt{a+b \sin (c+d x)}}-\frac{\left (\frac{3}{a^2}-\frac{4}{b^2}\right ) E\left (\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )|\frac{2 b}{a+b}\right ) \sqrt{a+b \sin (c+d x)}}{d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left (3 b \sqrt{\frac{a+b \sin (c+d x)}{a+b}}\right ) \int \frac{\csc (c+d x)}{\sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}}} \, dx}{2 a^2 \sqrt{a+b \sin (c+d x)}}-\frac{\left (\left (4 a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}}} \, dx}{2 a b^2 \sqrt{a+b \sin (c+d x)}}\\ &=\frac{\left (2 a^2-3 b^2\right ) \cos (c+d x)}{a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x)}{a d \sqrt{a+b \sin (c+d x)}}-\frac{\left (\frac{3}{a^2}-\frac{4}{b^2}\right ) E\left (\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )|\frac{2 b}{a+b}\right ) \sqrt{a+b \sin (c+d x)}}{d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left (4 a^2-b^2\right ) F\left (\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )|\frac{2 b}{a+b}\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}{a b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{3 b \Pi \left (2;\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )|\frac{2 b}{a+b}\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}{a^2 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 3.42109, size = 433, normalized size = 1.47 \[ \frac{\frac{4 a \left (a^2-b^2\right ) \cos (c+d x)}{b \sqrt{a+b \sin (c+d x)}}-\frac{a \left (4 a^2-9 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{b \sqrt{a+b \sin (c+d x)}}+\frac{i \left (3 b^2-4 a^2\right ) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} \left (b \left (b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )-2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )-2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )}{b^3 \sqrt{-\frac{1}{a+b}}}+\frac{4 a^2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}-2 a \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{2 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I*(-4*a^2 + 3*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/
(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*Ell
ipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sq
rt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)])/(b^3*Sqrt[-(a + b)^(-1)]) + (4*a
*(a^2 - b^2)*Cos[c + d*x])/(b*Sqrt[a + b*Sin[c + d*x]]) - 2*a*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]] + (4*a^2*E
llipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] -
(a*(4*a^2 - 9*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(b*
Sqrt[a + b*Sin[c + d*x]]))/(2*a^3*d)

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Maple [A]  time = 1.711, size = 620, normalized size = 2.1 \begin{align*} -{\frac{1}{{a}^{3}{b}^{3}\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) d} \left ( \left ( -2\,{a}^{3}{b}^{2}+3\,a{b}^{4} \right ) \sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}-\sqrt{-{\frac{b\sin \left ( dx+c \right ) }{a-b}}-{\frac{b}{a-b}}}\sqrt{-{\frac{b\sin \left ( dx+c \right ) }{a+b}}+{\frac{b}{a+b}}}\sqrt{{\frac{b\sin \left ( dx+c \right ) }{a-b}}+{\frac{a}{a-b}}} \left ( 4\,{\it EllipticF} \left ( \sqrt{{\frac{b\sin \left ( dx+c \right ) }{a-b}}+{\frac{a}{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ){a}^{4}b-6\,{\it EllipticF} \left ( \sqrt{{\frac{b\sin \left ( dx+c \right ) }{a-b}}+{\frac{a}{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ){a}^{3}{b}^{2}-{\it EllipticF} \left ( \sqrt{{\frac{b\sin \left ( dx+c \right ) }{a-b}}+{\frac{a}{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ){a}^{2}{b}^{3}+3\,{\it EllipticF} \left ( \sqrt{{\frac{b\sin \left ( dx+c \right ) }{a-b}}+{\frac{a}{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ) a{b}^{4}+3\,{\it EllipticPi} \left ( \sqrt{{\frac{b\sin \left ( dx+c \right ) }{a-b}}+{\frac{a}{a-b}}},{\frac{a-b}{a}},\sqrt{{\frac{a-b}{a+b}}} \right ) a{b}^{4}-3\,{\it EllipticPi} \left ( \sqrt{{\frac{b\sin \left ( dx+c \right ) }{a-b}}+{\frac{a}{a-b}}},{\frac{a-b}{a}},\sqrt{{\frac{a-b}{a+b}}} \right ){b}^{5}-4\,{\it EllipticE} \left ( \sqrt{{\frac{b\sin \left ( dx+c \right ) }{a-b}}+{\frac{a}{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ){a}^{5}+7\,{\it EllipticE} \left ( \sqrt{{\frac{b\sin \left ( dx+c \right ) }{a-b}}+{\frac{a}{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ){a}^{3}{b}^{2}-3\,{\it EllipticE} \left ( \sqrt{{\frac{b\sin \left ( dx+c \right ) }{a-b}}+{\frac{a}{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ) a{b}^{4} \right ) \sin \left ( dx+c \right ) +{a}^{2}{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ){\frac{1}{\sqrt{a+b\sin \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((-2*a^3*b^2+3*a*b^4)*sin(d*x+c)*cos(d*x+c)^2-(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b
))^(1/2)*(b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2)*(4*EllipticF((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^
(1/2))*a^4*b-6*EllipticF((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^2-EllipticF((b/(a-b)*
sin(d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*a^2*b^3+3*EllipticF((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),((a-
b)/(a+b))^(1/2))*a*b^4+3*EllipticPi((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a*b^4-3*
EllipticPi((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*b^5-4*EllipticE((b/(a-b)*sin(d*x+
c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*a^5+7*EllipticE((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1
/2))*a^3*b^2-3*EllipticE((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*a*b^4)*sin(d*x+c)+a^2*b^3*c
os(d*x+c)^2)/b^3/sin(d*x+c)/a^3/cos(d*x+c)/(a+b*sin(d*x+c))^(1/2)/d

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Maxima [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(cos(d*x+c)^2*cot(d*x+c)^2/(a+b*sin(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [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(cos(d*x+c)^2*cot(d*x+c)^2/(a+b*sin(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos ^{2}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}}{\left (a + b \sin{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**2*cot(d*x+c)**2/(a+b*sin(d*x+c))**(3/2),x)

[Out]

Integral(cos(c + d*x)**2*cot(c + d*x)**2/(a + b*sin(c + d*x))**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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