\(\int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx\) [1294]

Optimal result

Integrand size = 29, antiderivative size = 238 \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 b^4 \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 d}-\frac {b \left (a^4+4 a^2 b^2-8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^6 d}+\frac {\left (2 a^4+5 a^2 b^2-15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d} \]

[Out]

-1/8*b*(a^4+4*a^2*b^2-8*b^4)*arctanh(cos(d*x+c))/a^6/d+1/15*(2*a^4+5*a^2*b^2-15*b^4)*cot(d*x+c)/a^5/d-1/8*b*(a
^2-4*b^2)*cot(d*x+c)*csc(d*x+c)/a^4/d+1/15*(a^2-5*b^2)*cot(d*x+c)*csc(d*x+c)^2/a^3/d+1/4*b*cot(d*x+c)*csc(d*x+
c)^3/a^2/d-1/5*cot(d*x+c)*csc(d*x+c)^4/a/d-2*b^4*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))*(a^2-b^2)^(1
/2)/a^6/d

Rubi [A] (verified)

Time = 1.53 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2968, 3135, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {2 b^4 \sqrt {a^2-b^2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^6 d}-\frac {b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}-\frac {b \left (a^4+4 a^2 b^2-8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^6 d}+\frac {\left (2 a^4+5 a^2 b^2-15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d} \]

[In]

Int[(Cot[c + d*x]^2*Csc[c + d*x]^4)/(a + b*Sin[c + d*x]),x]

[Out]

(-2*b^4*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*d) - (b*(a^4 + 4*a^2*b^2 - 8*b^
4)*ArcTanh[Cos[c + d*x]])/(8*a^6*d) + ((2*a^4 + 5*a^2*b^2 - 15*b^4)*Cot[c + d*x])/(15*a^5*d) - (b*(a^2 - 4*b^2
)*Cot[c + d*x]*Csc[c + d*x])/(8*a^4*d) + ((a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(15*a^3*d) + (b*Cot[c + d
*x]*Csc[c + d*x]^3)/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d)

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 2968

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

Rule 3080

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

Rule 3134

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] + D
ist[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] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3135

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 + 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] + Dist[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[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C
)*(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d*(A*b^2 + a^2*C)*(m + n +
3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2,
0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && L
tQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\csc ^6(c+d x) \left (1-\sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx \\ & = -\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac {\int \frac {\csc ^5(c+d x) \left (-5 b-a \sin (c+d x)+4 b \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{5 a} \\ & = \frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac {\int \frac {\csc ^4(c+d x) \left (-4 \left (a^2-5 b^2\right )+a b \sin (c+d x)-15 b^2 \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{20 a^2} \\ & = \frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac {\int \frac {\csc ^3(c+d x) \left (15 b \left (a^2-4 b^2\right )-a \left (8 a^2+5 b^2\right ) \sin (c+d x)-8 b \left (a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{60 a^3} \\ & = -\frac {b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac {\int \frac {\csc ^2(c+d x) \left (-8 \left (2 a^4+5 a^2 b^2-15 b^4\right )-a b \left (a^2-20 b^2\right ) \sin (c+d x)+15 b^2 \left (a^2-4 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{120 a^4} \\ & = \frac {\left (2 a^4+5 a^2 b^2-15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac {\int \frac {\csc (c+d x) \left (15 b \left (a^4+4 a^2 b^2-8 b^4\right )+15 a b^2 \left (a^2-4 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{120 a^5} \\ & = \frac {\left (2 a^4+5 a^2 b^2-15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\left (b^4 \left (a^2-b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^6}+\frac {\left (b \left (a^4+4 a^2 b^2-8 b^4\right )\right ) \int \csc (c+d x) \, dx}{8 a^6} \\ & = -\frac {b \left (a^4+4 a^2 b^2-8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^6 d}+\frac {\left (2 a^4+5 a^2 b^2-15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\left (2 b^4 \left (a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^6 d} \\ & = -\frac {b \left (a^4+4 a^2 b^2-8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^6 d}+\frac {\left (2 a^4+5 a^2 b^2-15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac {\left (4 b^4 \left (a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^6 d} \\ & = -\frac {2 b^4 \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 d}-\frac {b \left (a^4+4 a^2 b^2-8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^6 d}+\frac {\left (2 a^4+5 a^2 b^2-15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {b \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(506\) vs. \(2(238)=476\).

Time = 2.11 (sec) , antiderivative size = 506, normalized size of antiderivative = 2.13 \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-1920 b^4 \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+32 \left (2 a^5+5 a^3 b^2-15 a b^4\right ) \cot \left (\frac {1}{2} (c+d x)\right )-30 a^4 b \csc ^2\left (\frac {1}{2} (c+d x)\right )+120 a^2 b^3 \csc ^2\left (\frac {1}{2} (c+d x)\right )+15 a^4 b \csc ^4\left (\frac {1}{2} (c+d x)\right )-120 a^4 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-480 a^2 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+960 b^5 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+120 a^4 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+480 a^2 b^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-960 b^5 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+30 a^4 b \sec ^2\left (\frac {1}{2} (c+d x)\right )-120 a^2 b^3 \sec ^2\left (\frac {1}{2} (c+d x)\right )-15 a^4 b \sec ^4\left (\frac {1}{2} (c+d x)\right )-16 a^5 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+320 a^3 b^2 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+a^5 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-20 a^3 b^2 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-3 a^5 \csc ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-64 a^5 \tan \left (\frac {1}{2} (c+d x)\right )-160 a^3 b^2 \tan \left (\frac {1}{2} (c+d x)\right )+480 a b^4 \tan \left (\frac {1}{2} (c+d x)\right )+6 a^5 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{960 a^6 d} \]

[In]

Integrate[(Cot[c + d*x]^2*Csc[c + d*x]^4)/(a + b*Sin[c + d*x]),x]

[Out]

(-1920*b^4*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 32*(2*a^5 + 5*a^3*b^2 - 15*a*b^4
)*Cot[(c + d*x)/2] - 30*a^4*b*Csc[(c + d*x)/2]^2 + 120*a^2*b^3*Csc[(c + d*x)/2]^2 + 15*a^4*b*Csc[(c + d*x)/2]^
4 - 120*a^4*b*Log[Cos[(c + d*x)/2]] - 480*a^2*b^3*Log[Cos[(c + d*x)/2]] + 960*b^5*Log[Cos[(c + d*x)/2]] + 120*
a^4*b*Log[Sin[(c + d*x)/2]] + 480*a^2*b^3*Log[Sin[(c + d*x)/2]] - 960*b^5*Log[Sin[(c + d*x)/2]] + 30*a^4*b*Sec
[(c + d*x)/2]^2 - 120*a^2*b^3*Sec[(c + d*x)/2]^2 - 15*a^4*b*Sec[(c + d*x)/2]^4 - 16*a^5*Csc[c + d*x]^3*Sin[(c
+ d*x)/2]^4 + 320*a^3*b^2*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + a^5*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 20*a^3*b^2
*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 3*a^5*Csc[(c + d*x)/2]^6*Sin[c + d*x] - 64*a^5*Tan[(c + d*x)/2] - 160*a^3*b
^2*Tan[(c + d*x)/2] + 480*a*b^4*Tan[(c + d*x)/2] + 6*a^5*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(960*a^6*d)

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.41

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{5}-\frac {b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{2}+\frac {a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {4 a^{2} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-4 a \,b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{2}+16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4}}{32 a^{5}}-\frac {2 b^{4} \sqrt {a^{2}-b^{2}}\, \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{6}}-\frac {1}{160 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {a^{2}+4 b^{2}}{96 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-2 a^{4}-4 a^{2} b^{2}+16 b^{4}}{32 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{64 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {b^{3}}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (a^{4}+4 a^{2} b^{2}-8 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{6}}}{d}\)	\(336\)
default	\(\frac {\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{5}-\frac {b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{2}+\frac {a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {4 a^{2} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-4 a \,b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{2}+16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4}}{32 a^{5}}-\frac {2 b^{4} \sqrt {a^{2}-b^{2}}\, \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{6}}-\frac {1}{160 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {a^{2}+4 b^{2}}{96 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-2 a^{4}-4 a^{2} b^{2}+16 b^{4}}{32 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{64 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {b^{3}}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (a^{4}+4 a^{2} b^{2}-8 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{6}}}{d}\)	\(336\)
risch	\(\frac {160 i a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-80 i a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+15 a^{3} b \,{\mathrm e}^{9 i \left (d x +c \right )}-60 a \,b^{3} {\mathrm e}^{9 i \left (d x +c \right )}-120 i b^{4} {\mathrm e}^{8 i \left (d x +c \right )}-80 i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+480 i b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+90 a^{3} b \,{\mathrm e}^{7 i \left (d x +c \right )}+120 b^{3} a \,{\mathrm e}^{7 i \left (d x +c \right )}-240 i a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-120 i b^{4}+40 i a^{2} b^{2}-80 i a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+480 i b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-240 i a^{4} {\mathrm e}^{6 i \left (d x +c \right )}-90 b \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-120 b^{3} a \,{\mathrm e}^{3 i \left (d x +c \right )}-720 i b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+16 i a^{4}+120 i a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-15 b \,a^{3} {\mathrm e}^{i \left (d x +c \right )}+60 \,{\mathrm e}^{i \left (d x +c \right )} b^{3} a}{60 d \,a^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 a^{2} d}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 a^{4} d}+\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{6} d}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 a^{2} d}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 a^{4} d}-\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{6} d}-\frac {\sqrt {-a^{2}+b^{2}}\, b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{6}}+\frac {\sqrt {-a^{2}+b^{2}}\, b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{6}}\)	\(582\)

[In]

int(cos(d*x+c)^2*csc(d*x+c)^6/(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/32/a^5*(1/5*tan(1/2*d*x+1/2*c)^5*a^4-1/2*b*tan(1/2*d*x+1/2*c)^4*a^3+1/3*a^4*tan(1/2*d*x+1/2*c)^3+4/3*a^
2*b^2*tan(1/2*d*x+1/2*c)^3-4*a*b^3*tan(1/2*d*x+1/2*c)^2-2*tan(1/2*d*x+1/2*c)*a^4-4*tan(1/2*d*x+1/2*c)*a^2*b^2+
16*tan(1/2*d*x+1/2*c)*b^4)-2*b^4*(a^2-b^2)^(1/2)/a^6*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))-
1/160/a/tan(1/2*d*x+1/2*c)^5-1/96/a^3*(a^2+4*b^2)/tan(1/2*d*x+1/2*c)^3-1/32*(-2*a^4-4*a^2*b^2+16*b^4)/a^5/tan(
1/2*d*x+1/2*c)+1/64/a^2*b/tan(1/2*d*x+1/2*c)^4+1/8/a^4*b^3/tan(1/2*d*x+1/2*c)^2+1/8/a^6*b*(a^4+4*a^2*b^2-8*b^4
)*ln(tan(1/2*d*x+1/2*c)))

Fricas [A] (verification not implemented)

none

Time = 0.54 (sec) , antiderivative size = 959, normalized size of antiderivative = 4.03 \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^2*csc(d*x+c)^6/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

[-1/240*(240*a*b^4*cos(d*x + c) - 16*(2*a^5 + 5*a^3*b^2 - 15*a*b^4)*cos(d*x + c)^5 + 80*(a^5 + a^3*b^2 - 6*a*b
^4)*cos(d*x + c)^3 - 120*(b^4*cos(d*x + c)^4 - 2*b^4*cos(d*x + c)^2 + b^4)*sqrt(-a^2 + b^2)*log(((2*a^2 - b^2)
*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 + 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2
+ b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2))*sin(d*x + c) + 15*(a^4*b + 4*a^2*b^3 - 8*b^5 +
(a^4*b + 4*a^2*b^3 - 8*b^5)*cos(d*x + c)^4 - 2*(a^4*b + 4*a^2*b^3 - 8*b^5)*cos(d*x + c)^2)*log(1/2*cos(d*x + c
) + 1/2)*sin(d*x + c) - 15*(a^4*b + 4*a^2*b^3 - 8*b^5 + (a^4*b + 4*a^2*b^3 - 8*b^5)*cos(d*x + c)^4 - 2*(a^4*b
+ 4*a^2*b^3 - 8*b^5)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 30*((a^4*b - 4*a^2*b^3)*cos(d
*x + c)^3 + (a^4*b + 4*a^2*b^3)*cos(d*x + c))*sin(d*x + c))/((a^6*d*cos(d*x + c)^4 - 2*a^6*d*cos(d*x + c)^2 +
a^6*d)*sin(d*x + c)), -1/240*(240*a*b^4*cos(d*x + c) - 16*(2*a^5 + 5*a^3*b^2 - 15*a*b^4)*cos(d*x + c)^5 + 80*(
a^5 + a^3*b^2 - 6*a*b^4)*cos(d*x + c)^3 - 240*(b^4*cos(d*x + c)^4 - 2*b^4*cos(d*x + c)^2 + b^4)*sqrt(a^2 - b^2
)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c)))*sin(d*x + c) + 15*(a^4*b + 4*a^2*b^3 - 8*b^5 +
(a^4*b + 4*a^2*b^3 - 8*b^5)*cos(d*x + c)^4 - 2*(a^4*b + 4*a^2*b^3 - 8*b^5)*cos(d*x + c)^2)*log(1/2*cos(d*x + c
) + 1/2)*sin(d*x + c) - 15*(a^4*b + 4*a^2*b^3 - 8*b^5 + (a^4*b + 4*a^2*b^3 - 8*b^5)*cos(d*x + c)^4 - 2*(a^4*b
+ 4*a^2*b^3 - 8*b^5)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 30*((a^4*b - 4*a^2*b^3)*cos(d
*x + c)^3 + (a^4*b + 4*a^2*b^3)*cos(d*x + c))*sin(d*x + c))/((a^6*d*cos(d*x + c)^4 - 2*a^6*d*cos(d*x + c)^2 +
a^6*d)*sin(d*x + c))]

Sympy [F]

\[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\cos ^{2}{\left (c + d x \right )} \csc ^{6}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]

[In]

integrate(cos(d*x+c)**2*csc(d*x+c)**6/(a+b*sin(d*x+c)),x)

[Out]

Integral(cos(c + d*x)**2*csc(c + d*x)**6/(a + b*sin(c + d*x)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(cos(d*x+c)^2*csc(d*x+c)^6/(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (221) = 442\).

Time = 0.37 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.87 \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {6 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 10 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 60 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 120 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 480 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{5}} + \frac {120 \, {\left (a^{4} b + 4 \, a^{2} b^{3} - 8 \, b^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{6}} - \frac {1920 \, {\left (a^{2} b^{4} - b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{6}} - \frac {274 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1096 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2192 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 480 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 40 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a^{5}}{a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \]

[In]

integrate(cos(d*x+c)^2*csc(d*x+c)^6/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

1/960*((6*a^4*tan(1/2*d*x + 1/2*c)^5 - 15*a^3*b*tan(1/2*d*x + 1/2*c)^4 + 10*a^4*tan(1/2*d*x + 1/2*c)^3 + 40*a^
2*b^2*tan(1/2*d*x + 1/2*c)^3 - 120*a*b^3*tan(1/2*d*x + 1/2*c)^2 - 60*a^4*tan(1/2*d*x + 1/2*c) - 120*a^2*b^2*ta
n(1/2*d*x + 1/2*c) + 480*b^4*tan(1/2*d*x + 1/2*c))/a^5 + 120*(a^4*b + 4*a^2*b^3 - 8*b^5)*log(abs(tan(1/2*d*x +
1/2*c)))/a^6 - 1920*(a^2*b^4 - b^6)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c)
+ b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*a^6) - (274*a^4*b*tan(1/2*d*x + 1/2*c)^5 + 1096*a^2*b^3*tan(1/2*d*x +
1/2*c)^5 - 2192*b^5*tan(1/2*d*x + 1/2*c)^5 - 60*a^5*tan(1/2*d*x + 1/2*c)^4 - 120*a^3*b^2*tan(1/2*d*x + 1/2*c)
^4 + 480*a*b^4*tan(1/2*d*x + 1/2*c)^4 - 120*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 10*a^5*tan(1/2*d*x + 1/2*c)^2 + 4
0*a^3*b^2*tan(1/2*d*x + 1/2*c)^2 - 15*a^4*b*tan(1/2*d*x + 1/2*c) + 6*a^5)/(a^6*tan(1/2*d*x + 1/2*c)^5))/d

Mupad [B] (verification not implemented)

Time = 12.17 (sec) , antiderivative size = 1007, normalized size of antiderivative = 4.23 \[ \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

int(cos(c + d*x)^2/(sin(c + d*x)^6*(a + b*sin(c + d*x))),x)

[Out]

tan(c/2 + (d*x)/2)^5/(160*a*d) + (tan(c/2 + (d*x)/2)^2*(b/(32*a^2) - (b*(1/(32*a) + b^2/(8*a^3)))/a))/d - (tan
(c/2 + (d*x)/2)*(1/(16*a) + b^2/(8*a^3) + (2*b*(b/(16*a^2) - (2*b*(1/(32*a) + b^2/(8*a^3)))/a))/a))/d + (tan(c
/2 + (d*x)/2)^3*(1/(96*a) + b^2/(24*a^3)))/d + (log(tan(c/2 + (d*x)/2))*((a^4*b)/8 - b^5 + (a^2*b^3)/2))/(a^6*
d) - (b*tan(c/2 + (d*x)/2)^4)/(64*a^2*d) + (tan(c/2 + (d*x)/2)^4*(2*a^4 - 16*b^4 + 4*a^2*b^2) - a^4/5 - tan(c/
2 + (d*x)/2)^2*(a^4/3 + (4*a^2*b^2)/3) + (a^3*b*tan(c/2 + (d*x)/2))/2 + 4*a*b^3*tan(c/2 + (d*x)/2)^3)/(32*a^5*
d*tan(c/2 + (d*x)/2)^5) - (b^4*atan(((b^4*(b^2 - a^2)^(1/2)*((tan(c/2 + (d*x)/2)*(a^10*b + 32*a^4*b^7 - 32*a^6
*b^5 + 2*a^8*b^3))/(4*a^9) - (12*a^8*b^4 - 16*a^6*b^6 + a^10*b^2)/(4*a^10) + (b^4*(2*a^2*b - (tan(c/2 + (d*x)/
2)*(24*a^12 - 32*a^10*b^2))/(4*a^9))*(b^2 - a^2)^(1/2))/a^6)*1i)/a^6 - (b^4*(b^2 - a^2)^(1/2)*((12*a^8*b^4 - 1
6*a^6*b^6 + a^10*b^2)/(4*a^10) - (tan(c/2 + (d*x)/2)*(a^10*b + 32*a^4*b^7 - 32*a^6*b^5 + 2*a^8*b^3))/(4*a^9) +
(b^4*(2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^12 - 32*a^10*b^2))/(4*a^9))*(b^2 - a^2)^(1/2))/a^6)*1i)/a^6)/((8*b^
11 - 12*a^2*b^9 + 3*a^4*b^7 + a^6*b^5)/(2*a^10) + (tan(c/2 + (d*x)/2)*(8*b^10 - 10*a^2*b^8 + 2*a^4*b^6))/(2*a^
9) + (b^4*(b^2 - a^2)^(1/2)*((tan(c/2 + (d*x)/2)*(a^10*b + 32*a^4*b^7 - 32*a^6*b^5 + 2*a^8*b^3))/(4*a^9) - (12
*a^8*b^4 - 16*a^6*b^6 + a^10*b^2)/(4*a^10) + (b^4*(2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^12 - 32*a^10*b^2))/(4*a
^9))*(b^2 - a^2)^(1/2))/a^6))/a^6 + (b^4*(b^2 - a^2)^(1/2)*((12*a^8*b^4 - 16*a^6*b^6 + a^10*b^2)/(4*a^10) - (t
an(c/2 + (d*x)/2)*(a^10*b + 32*a^4*b^7 - 32*a^6*b^5 + 2*a^8*b^3))/(4*a^9) + (b^4*(2*a^2*b - (tan(c/2 + (d*x)/2
)*(24*a^12 - 32*a^10*b^2))/(4*a^9))*(b^2 - a^2)^(1/2))/a^6))/a^6))*(b^2 - a^2)^(1/2)*2i)/(a^6*d)