3.1189 \(\int \frac{\cot ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx\)
Optimal. Leaf size=458 \[ \frac{b \left (32 a^2-105 b^2\right ) \cos (c+d x)}{8 a^5 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (16 a^2-35 b^2\right ) \cot (c+d x)}{8 a^4 d \sqrt{a+b \sin (c+d x)}}-\frac{\left (16 a^2-35 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{8 a^4 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (32 a^2-105 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{8 a^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{15 b \left (4 a^2-7 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{8 a^5 d \sqrt{a+b \sin (c+d x)}}-\frac{\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt{a+b \sin (c+d x)}}+\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}} \]
[Out]
((2*a^2 - 3*b^2)*Cot[c + d*x]*Csc[c + d*x])/(3*a^2*b*d*(a + b*Sin[c + d*x])^(3/2)) - (Cot[c + d*x]*Csc[c + d*x
]^2)/(3*a*d*(a + b*Sin[c + d*x])^(3/2)) + (b*(32*a^2 - 105*b^2)*Cos[c + d*x])/(8*a^5*d*Sqrt[a + b*Sin[c + d*x]
]) + ((16*a^2 - 35*b^2)*Cot[c + d*x])/(8*a^4*d*Sqrt[a + b*Sin[c + d*x]]) - ((8*a^2 - 21*b^2)*Cot[c + d*x]*Csc[
c + d*x])/(12*a^3*b*d*Sqrt[a + b*Sin[c + d*x]]) + ((32*a^2 - 105*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a +
b)]*Sqrt[a + b*Sin[c + d*x]])/(8*a^5*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - ((16*a^2 - 35*b^2)*EllipticF[(c
- Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*a^4*d*Sqrt[a + b*Sin[c + d*x]]) + (15*b
*(4*a^2 - 7*b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*a^5*d
*Sqrt[a + b*Sin[c + d*x]])
________________________________________________________________________________________
Rubi [A] time = 1.56392, antiderivative size = 458, normalized size of antiderivative
= 1., number of steps used = 12, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\)
= 0.435, Rules used = {2724, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}
\[ \frac{b \left (32 a^2-105 b^2\right ) \cos (c+d x)}{8 a^5 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (16 a^2-35 b^2\right ) \cot (c+d x)}{8 a^4 d \sqrt{a+b \sin (c+d x)}}-\frac{\left (16 a^2-35 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{8 a^4 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (32 a^2-105 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{8 a^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{15 b \left (4 a^2-7 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{8 a^5 d \sqrt{a+b \sin (c+d x)}}-\frac{\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt{a+b \sin (c+d x)}}+\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^(5/2),x]
[Out]
((2*a^2 - 3*b^2)*Cot[c + d*x]*Csc[c + d*x])/(3*a^2*b*d*(a + b*Sin[c + d*x])^(3/2)) - (Cot[c + d*x]*Csc[c + d*x
]^2)/(3*a*d*(a + b*Sin[c + d*x])^(3/2)) + (b*(32*a^2 - 105*b^2)*Cos[c + d*x])/(8*a^5*d*Sqrt[a + b*Sin[c + d*x]
]) + ((16*a^2 - 35*b^2)*Cot[c + d*x])/(8*a^4*d*Sqrt[a + b*Sin[c + d*x]]) - ((8*a^2 - 21*b^2)*Cot[c + d*x]*Csc[
c + d*x])/(12*a^3*b*d*Sqrt[a + b*Sin[c + d*x]]) + ((32*a^2 - 105*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a +
b)]*Sqrt[a + b*Sin[c + d*x]])/(8*a^5*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - ((16*a^2 - 35*b^2)*EllipticF[(c
- Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*a^4*d*Sqrt[a + b*Sin[c + d*x]]) + (15*b
*(4*a^2 - 7*b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*a^5*d
*Sqrt[a + b*Sin[c + d*x]])
Rule 2724
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^4, x_Symbol] :> -Simp[(Cos[e + f*x]*(a
+ b*Sin[e + f*x])^(m + 1))/(3*a*f*Sin[e + f*x]^3), x] + (-Dist[1/(3*a^2*b*(m + 1)), Int[((a + b*Sin[e + f*x])
^(m + 1)*Simp[6*a^2 - b^2*(m - 1)*(m - 2) + a*b*(m + 1)*Sin[e + f*x] - (3*a^2 - b^2*m*(m - 2))*Sin[e + f*x]^2,
x])/Sin[e + f*x]^3, x], x] - Simp[((3*a^2 + b^2*(m - 2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(3*a^2*b*
f*(m + 1)*Sin[e + f*x]^2), x]) /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Rule 3055
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, 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] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))
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{\cot ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx &=\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}+\frac{2 \int \frac{\csc ^3(c+d x) \left (\frac{3}{4} \left (8 a^2-21 b^2\right )-\frac{3}{2} a b \sin (c+d x)-\frac{3}{4} \left (4 a^2-15 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^{3/2}} \, dx}{9 a^2 b}\\ &=\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}-\frac{\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt{a+b \sin (c+d x)}}+\frac{\int \frac{\csc ^2(c+d x) \left (-\frac{9}{8} b \left (16 a^2-35 b^2\right )+\frac{27}{4} a b^2 \sin (c+d x)+\frac{9}{8} b \left (8 a^2-21 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^{3/2}} \, dx}{9 a^3 b}\\ &=\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}+\frac{\left (16 a^2-35 b^2\right ) \cot (c+d x)}{8 a^4 d \sqrt{a+b \sin (c+d x)}}-\frac{\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt{a+b \sin (c+d x)}}+\frac{\int \frac{\csc (c+d x) \left (\frac{135}{16} b^2 \left (4 a^2-7 b^2\right )+\frac{9}{8} a b \left (8 a^2-21 b^2\right ) \sin (c+d x)-\frac{9}{16} b^2 \left (16 a^2-35 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^{3/2}} \, dx}{9 a^4 b}\\ &=\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}+\frac{b \left (32 a^2-105 b^2\right ) \cos (c+d x)}{8 a^5 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (16 a^2-35 b^2\right ) \cot (c+d x)}{8 a^4 d \sqrt{a+b \sin (c+d x)}}-\frac{\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt{a+b \sin (c+d x)}}+\frac{2 \int \frac{\csc (c+d x) \left (\frac{135}{32} b^2 \left (4 a^4-11 a^2 b^2+7 b^4\right )+\frac{9}{16} a b \left (8 a^4-43 a^2 b^2+35 b^4\right ) \sin (c+d x)+\frac{9}{32} b^2 \left (32 a^4-137 a^2 b^2+105 b^4\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{9 a^5 b \left (a^2-b^2\right )}\\ &=\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}+\frac{b \left (32 a^2-105 b^2\right ) \cos (c+d x)}{8 a^5 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (16 a^2-35 b^2\right ) \cot (c+d x)}{8 a^4 d \sqrt{a+b \sin (c+d x)}}-\frac{\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt{a+b \sin (c+d x)}}-\frac{2 \int \frac{\csc (c+d x) \left (-\frac{135}{32} b^3 \left (4 a^4-11 a^2 b^2+7 b^4\right )+\frac{9}{32} a b^2 \left (16 a^4-51 a^2 b^2+35 b^4\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{9 a^5 b^2 \left (a^2-b^2\right )}+\frac{\left (32 a^4-137 a^2 b^2+105 b^4\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{16 a^5 \left (a^2-b^2\right )}\\ &=\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}+\frac{b \left (32 a^2-105 b^2\right ) \cos (c+d x)}{8 a^5 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (16 a^2-35 b^2\right ) \cot (c+d x)}{8 a^4 d \sqrt{a+b \sin (c+d x)}}-\frac{\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt{a+b \sin (c+d x)}}+\frac{\left (15 b \left (4 a^2-7 b^2\right )\right ) \int \frac{\csc (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{16 a^5}-\frac{\left (16 a^4-51 a^2 b^2+35 b^4\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{16 a^4 \left (a^2-b^2\right )}+\frac{\left (\left (32 a^4-137 a^2 b^2+105 b^4\right ) \sqrt{a+b \sin (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}} \, dx}{16 a^5 \left (a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}\\ &=\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}+\frac{b \left (32 a^2-105 b^2\right ) \cos (c+d x)}{8 a^5 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (16 a^2-35 b^2\right ) \cot (c+d x)}{8 a^4 d \sqrt{a+b \sin (c+d x)}}-\frac{\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt{a+b \sin (c+d x)}}+\frac{\left (32 a^2-105 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)}}{8 a^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (15 b \left (4 a^2-7 b^2\right ) \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}{16 a^5 \sqrt{a+b \sin (c+d x)}}-\frac{\left (\left (16 a^4-51 a^2 b^2+35 b^4\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}{16 a^4 \left (a^2-b^2\right ) \sqrt{a+b \sin (c+d x)}}\\ &=\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}+\frac{b \left (32 a^2-105 b^2\right ) \cos (c+d x)}{8 a^5 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (16 a^2-35 b^2\right ) \cot (c+d x)}{8 a^4 d \sqrt{a+b \sin (c+d x)}}-\frac{\left (8 a^2-21 b^2\right ) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt{a+b \sin (c+d x)}}+\frac{\left (32 a^2-105 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)}}{8 a^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left (16 a^2-35 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}}}{8 a^4 d \sqrt{a+b \sin (c+d x)}}+\frac{15 b \left (4 a^2-7 b^2\right ) \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}}}{8 a^5 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 6.72733, size = 680, normalized size = 1.48 \[ \frac{\sqrt{a+b \sin (c+d x)} \left (\frac{8 \left (a^2 b \cos (c+d x)-3 b^3 \cos (c+d x)\right )}{3 a^5 (a+b \sin (c+d x))}+\frac{2 \left (a^2 b \cos (c+d x)-b^3 \cos (c+d x)\right )}{3 a^4 (a+b \sin (c+d x))^2}+\frac{\csc (c+d x) \left (32 a^2 \cos (c+d x)-123 b^2 \cos (c+d x)\right )}{24 a^5}+\frac{17 b \cot (c+d x) \csc (c+d x)}{12 a^4}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a^3}\right )}{d}+\frac{-\frac{2 \left (32 a^3-140 a b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (-c-d x+\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}-\frac{2 \left (152 a^2 b-315 b^3\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (-c-d x+\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left (105 b^3-32 a^2 b\right ) \cos (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \sin (c+d x)}{a+b}} \sqrt{-\frac{b \sin (c+d x)+b}{a-b}} \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )-b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )\right )}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(c+d x)} \left (-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right ) \sqrt{-\frac{a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}}{32 a^5 d} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^(5/2),x]
[Out]
(Sqrt[a + b*Sin[c + d*x]]*(((32*a^2*Cos[c + d*x] - 123*b^2*Cos[c + d*x])*Csc[c + d*x])/(24*a^5) + (17*b*Cot[c
+ d*x]*Csc[c + d*x])/(12*a^4) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a^3) + (2*(a^2*b*Cos[c + d*x] - b^3*Cos[c + d
*x]))/(3*a^4*(a + b*Sin[c + d*x])^2) + (8*(a^2*b*Cos[c + d*x] - 3*b^3*Cos[c + d*x]))/(3*a^5*(a + b*Sin[c + d*x
]))))/d + ((-2*(32*a^3 - 140*a*b^2)*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a
+ b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(152*a^2*b - 315*b^3)*EllipticPi[2, (-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*
Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - ((2*I)*(-32*a^2*b + 105*b^3)*Cos[c + d*x]*Cos[2
*(c + d*x)]*(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*EllipticPi[(a
+ b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sqrt[(b - b*Sin[c + d*x])/
(a + b)]*Sqrt[-((b + b*Sin[c + d*x])/(a - b))])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Sin[c + d*x]^2]*(-2*a^2 + b^2
+ 4*a*(a + b*Sin[c + d*x]) - 2*(a + b*Sin[c + d*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[c + d*x]) + (a + b*S
in[c + d*x])^2)/b^2)]))/(32*a^5*d)
________________________________________________________________________________________
Maple [B] time = 2.266, size = 2870, normalized size = 6.3 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(d*x+c)^4/(a+b*sin(d*x+c))^(5/2),x)
[Out]
-1/24*(-315*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*Ell
ipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a^2*b^4*sin(d*x+c)^3+315*((a+b*sin(d*x+c))
/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticPi(((a+b*sin(d*x+c))/(a-
b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a*b^5*sin(d*x+c)^3+315*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*
b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))
^(1/2))*b^6*sin(d*x+c)^4+96*a^3*b^3*sin(d*x+c)^6+8*a^5*b-159*a^3*b^3*sin(d*x+c)^4+315*a*b^5*sin(d*x+c)^4-126*a
^4*b^2*sin(d*x+c)^3+420*a^2*b^4*sin(d*x+c)^3+63*a^3*b^3*sin(d*x+c)^2-18*a^4*b^2*sin(d*x+c)-315*a*b^5*sin(d*x+c
)^6+144*a^4*b^2*sin(d*x+c)^5-420*a^2*b^4*sin(d*x+c)^5+32*a^5*b*sin(d*x+c)^4-40*a^5*b*sin(d*x+c)^2+306*((a+b*si
n(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x
+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^2*sin(d*x+c)^3+105*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1
)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))
*a^3*b^3*sin(d*x+c)^3-315*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a
-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*b^4*sin(d*x+c)^3-180*((a+b*sin(d*
x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticPi(((a+b*sin(d*x+c)
)/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a^3*b^3*sin(d*x+c)^3+96*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+
c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1
/2))*a^6*sin(d*x+c)^3-48*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-
b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^6*sin(d*x+c)^3-411*((a+b*sin(d*x+c))
/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b
))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^3*sin(d*x+c)^4+315*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b)
)^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^5*si
n(d*x+c)^4-48*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*E
llipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b*sin(d*x+c)^4-48*((a+b*sin(d*x+c))/(a-b))^(1
/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),(
(a-b)/(a+b))^(1/2))*a^4*b^2*sin(d*x+c)^4+306*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-
(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^3*sin(d*x+c)
^4+105*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*Elliptic
F(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*b^4*sin(d*x+c)^4-315*((a+b*sin(d*x+c))/(a-b))^(1/2)*
(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b
)/(a+b))^(1/2))*a*b^5*sin(d*x+c)^4+180*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin
(d*x+c))*b/(a-b))^(1/2)*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a^3*b^3*sin(d*x
+c)^4-180*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*Ellip
ticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a^2*b^4*sin(d*x+c)^4-315*((a+b*sin(d*x+c))/(
a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticPi(((a+b*sin(d*x+c))/(a-b)
)^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a*b^5*sin(d*x+c)^4-411*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/
(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4
*b^2*sin(d*x+c)^3+315*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))
^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*b^4*sin(d*x+c)^3-48*((a+b*sin(d*x+c))
/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b
))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b*sin(d*x+c)^3+180*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^
(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a
^4*b^2*sin(d*x+c)^3+96*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b)
)^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b*sin(d*x+c)^4)/sin(d*x+c)^3/a^6/(a+
b*sin(d*x+c))^(3/2)/b/cos(d*x+c)/d
________________________________________________________________________________________
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(cot(d*x+c)^4/(a+b*sin(d*x+c))^(5/2),x, algorithm="maxima")
[Out]
Timed out
________________________________________________________________________________________
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(cot(d*x+c)^4/(a+b*sin(d*x+c))^(5/2),x, algorithm="fricas")
[Out]
Timed out
________________________________________________________________________________________
Sympy [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(cot(d*x+c)**4/(a+b*sin(d*x+c))**(5/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (d x + c\right )^{4}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^4/(a+b*sin(d*x+c))^(5/2),x, algorithm="giac")
[Out]
integrate(cot(d*x + c)^4/(b*sin(d*x + c) + a)^(5/2), x)