3.1134 \(\int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=292 \[ \frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))}-\frac {b \left (11 a^2-15 b^2\right ) \cot (c+d x)}{3 a^5 d}+\frac {\left (13 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}-\frac {2 b \left (2 a^4-7 a^2 b^2+5 b^4\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^6 d \sqrt {a^2-b^2}}-\frac {\left (3 a^4-36 a^2 b^2+40 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))} \]

[Out]

-1/8*(3*a^4-36*a^2*b^2+40*b^4)*arctanh(cos(d*x+c))/a^6/d-1/3*b*(11*a^2-15*b^2)*cot(d*x+c)/a^5/d+1/8*(13*a^2-20
*b^2)*cot(d*x+c)*csc(d*x+c)/a^4/d-1/3*(3*a^2-5*b^2)*cot(d*x+c)*csc(d*x+c)^2/a^3/b/d+1/4*(4*a^2-5*b^2)*cot(d*x+
c)*csc(d*x+c)^2/a^2/b/d/(a+b*sin(d*x+c))-1/4*cot(d*x+c)*csc(d*x+c)^3/a/d/(a+b*sin(d*x+c))-2*b*(2*a^4-7*a^2*b^2
+5*b^4)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^6/d/(a^2-b^2)^(1/2)

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Rubi [A]  time = 1.05, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2890, 3055, 3001, 3770, 2660, 618, 204} \[ -\frac {2 b \left (-7 a^2 b^2+2 a^4+5 b^4\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^6 d \sqrt {a^2-b^2}}-\frac {b \left (11 a^2-15 b^2\right ) \cot (c+d x)}{3 a^5 d}-\frac {\left (-36 a^2 b^2+3 a^4+40 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}+\frac {\left (13 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*(2*a^4 - 7*a^2*b^2 + 5*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^2 - b^2]*d) -
((3*a^4 - 36*a^2*b^2 + 40*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^6*d) - (b*(11*a^2 - 15*b^2)*Cot[c + d*x])/(3*a^5*d)
+ ((13*a^2 - 20*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^4*d) - ((3*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(3*
a^3*b*d) + ((4*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(4*a^2*b*d*(a + b*Sin[c + d*x])) - (Cot[c + d*x]*Csc[
c + d*x]^3)/(4*a*d*(a + b*Sin[c + d*x]))

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 618

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 2660

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 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 3001

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 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 3770

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

Rubi steps

\begin {align*} \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^4(c+d x) \left (4 \left (3 a^2-5 b^2\right )-a b \sin (c+d x)-\left (8 a^2-15 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 a^2 b}\\ &=-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^3(c+d x) \left (-3 b \left (13 a^2-20 b^2\right )+5 a b^2 \sin (c+d x)+8 b \left (3 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{12 a^3 b}\\ &=\frac {\left (13 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (8 b^2 \left (11 a^2-15 b^2\right )+a b \left (9 a^2-20 b^2\right ) \sin (c+d x)-3 b^2 \left (13 a^2-20 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{24 a^4 b}\\ &=-\frac {b \left (11 a^2-15 b^2\right ) \cot (c+d x)}{3 a^5 d}+\frac {\left (13 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (3 b \left (3 a^4-36 a^2 b^2+40 b^4\right )-3 a b^2 \left (13 a^2-20 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{24 a^5 b}\\ &=-\frac {b \left (11 a^2-15 b^2\right ) \cot (c+d x)}{3 a^5 d}+\frac {\left (13 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac {\left (b \left (2 a^4-7 a^2 b^2+5 b^4\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^6}+\frac {\left (3 a^4-36 a^2 b^2+40 b^4\right ) \int \csc (c+d x) \, dx}{8 a^6}\\ &=-\frac {\left (3 a^4-36 a^2 b^2+40 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac {b \left (11 a^2-15 b^2\right ) \cot (c+d x)}{3 a^5 d}+\frac {\left (13 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))}-\frac {\left (2 b \left (2 a^4-7 a^2 b^2+5 b^4\right )\right ) \operatorname {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 {\left (3 a^4-36 a^2 b^2+40 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac {b \left (11 a^2-15 b^2\right ) \cot (c+d x)}{3 a^5 d}+\frac {\left (13 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))}+\frac {\left (4 b \left (2 a^4-7 a^2 b^2+5 b^4\right )\right ) \operatorname {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 \left (2 a^4-7 a^2 b^2+5 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 \sqrt {a^2-b^2} d}-\frac {\left (3 a^4-36 a^2 b^2+40 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac {b \left (11 a^2-15 b^2\right ) \cot (c+d x)}{3 a^5 d}+\frac {\left (13 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 6.29, size = 496, normalized size = 1.70 \[ \frac {b \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{12 a^3 d}-\frac {b \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{12 a^3 d}-\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 a^2 d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 a^2 d}-\frac {2 \csc \left (\frac {1}{2} (c+d x)\right ) \left (2 a^2 b \cos \left (\frac {1}{2} (c+d x)\right )-3 b^3 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^5 d}+\frac {2 \sec \left (\frac {1}{2} (c+d x)\right ) \left (2 a^2 b \sin \left (\frac {1}{2} (c+d x)\right )-3 b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^5 d}+\frac {b^4 \cos (c+d x)-a^2 b^2 \cos (c+d x)}{a^5 d (a+b \sin (c+d x))}+\frac {\left (5 a^2-12 b^2\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 a^4 d}+\frac {\left (12 b^2-5 a^2\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 a^4 d}+\frac {\left (3 a^4-36 a^2 b^2+40 b^4\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^6 d}+\frac {\left (-3 a^4+36 a^2 b^2-40 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^6 d}-\frac {2 b \left (2 a^4-7 a^2 b^2+5 b^4\right ) \tan ^{-1}\left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (a \sin \left (\frac {1}{2} (c+d x)\right )+b \cos \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^6 d \sqrt {a^2-b^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*b*(2*a^4 - 7*a^2*b^2 + 5*b^4)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2
- b^2]])/(a^6*Sqrt[a^2 - b^2]*d) - (2*(2*a^2*b*Cos[(c + d*x)/2] - 3*b^3*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(
3*a^5*d) + ((5*a^2 - 12*b^2)*Csc[(c + d*x)/2]^2)/(32*a^4*d) + (b*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(12*a^3*
d) - Csc[(c + d*x)/2]^4/(64*a^2*d) + ((-3*a^4 + 36*a^2*b^2 - 40*b^4)*Log[Cos[(c + d*x)/2]])/(8*a^6*d) + ((3*a^
4 - 36*a^2*b^2 + 40*b^4)*Log[Sin[(c + d*x)/2]])/(8*a^6*d) + ((-5*a^2 + 12*b^2)*Sec[(c + d*x)/2]^2)/(32*a^4*d)
+ Sec[(c + d*x)/2]^4/(64*a^2*d) + (2*Sec[(c + d*x)/2]*(2*a^2*b*Sin[(c + d*x)/2] - 3*b^3*Sin[(c + d*x)/2]))/(3*
a^5*d) + (-(a^2*b^2*Cos[c + d*x]) + b^4*Cos[c + d*x])/(a^5*d*(a + b*Sin[c + d*x])) - (b*Sec[(c + d*x)/2]^2*Tan
[(c + d*x)/2])/(12*a^3*d)

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fricas [B]  time = 1.15, size = 1578, normalized size = 5.40 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^5/(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

[-1/48*(16*(11*a^3*b^2 - 15*a*b^4)*cos(d*x + c)^5 + 2*(15*a^5 - 196*a^3*b^2 + 240*a*b^4)*cos(d*x + c)^3 + 24*(
(2*a^3*b - 5*a*b^3)*cos(d*x + c)^4 + 2*a^3*b - 5*a*b^3 - 2*(2*a^3*b - 5*a*b^3)*cos(d*x + c)^2 + ((2*a^2*b^2 -
5*b^4)*cos(d*x + c)^4 + 2*a^2*b^2 - 5*b^4 - 2*(2*a^2*b^2 - 5*b^4)*cos(d*x + c)^2)*sin(d*x + c))*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*co
s(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)) - 6*(3*a^5 - 36*a^3*b^2 +
40*a*b^4)*cos(d*x + c) + 3*(3*a^5 - 36*a^3*b^2 + 40*a*b^4 + (3*a^5 - 36*a^3*b^2 + 40*a*b^4)*cos(d*x + c)^4 -
2*(3*a^5 - 36*a^3*b^2 + 40*a*b^4)*cos(d*x + c)^2 + (3*a^4*b - 36*a^2*b^3 + 40*b^5 + (3*a^4*b - 36*a^2*b^3 + 40
*b^5)*cos(d*x + c)^4 - 2*(3*a^4*b - 36*a^2*b^3 + 40*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) +
1/2) - 3*(3*a^5 - 36*a^3*b^2 + 40*a*b^4 + (3*a^5 - 36*a^3*b^2 + 40*a*b^4)*cos(d*x + c)^4 - 2*(3*a^5 - 36*a^3*b
^2 + 40*a*b^4)*cos(d*x + c)^2 + (3*a^4*b - 36*a^2*b^3 + 40*b^5 + (3*a^4*b - 36*a^2*b^3 + 40*b^5)*cos(d*x + c)^
4 - 2*(3*a^4*b - 36*a^2*b^3 + 40*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) - 2*((49*a^4*
b - 60*a^2*b^3)*cos(d*x + c)^3 - 3*(13*a^4*b - 20*a^2*b^3)*cos(d*x + c))*sin(d*x + c))/(a^7*d*cos(d*x + c)^4 -
2*a^7*d*cos(d*x + c)^2 + a^7*d + (a^6*b*d*cos(d*x + c)^4 - 2*a^6*b*d*cos(d*x + c)^2 + a^6*b*d)*sin(d*x + c)),
-1/48*(16*(11*a^3*b^2 - 15*a*b^4)*cos(d*x + c)^5 + 2*(15*a^5 - 196*a^3*b^2 + 240*a*b^4)*cos(d*x + c)^3 - 48*(
(2*a^3*b - 5*a*b^3)*cos(d*x + c)^4 + 2*a^3*b - 5*a*b^3 - 2*(2*a^3*b - 5*a*b^3)*cos(d*x + c)^2 + ((2*a^2*b^2 -
5*b^4)*cos(d*x + c)^4 + 2*a^2*b^2 - 5*b^4 - 2*(2*a^2*b^2 - 5*b^4)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(a^2 - b^2
)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 6*(3*a^5 - 36*a^3*b^2 + 40*a*b^4)*cos(d*x + c
) + 3*(3*a^5 - 36*a^3*b^2 + 40*a*b^4 + (3*a^5 - 36*a^3*b^2 + 40*a*b^4)*cos(d*x + c)^4 - 2*(3*a^5 - 36*a^3*b^2
+ 40*a*b^4)*cos(d*x + c)^2 + (3*a^4*b - 36*a^2*b^3 + 40*b^5 + (3*a^4*b - 36*a^2*b^3 + 40*b^5)*cos(d*x + c)^4 -
2*(3*a^4*b - 36*a^2*b^3 + 40*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 3*(3*a^5 - 36*a
^3*b^2 + 40*a*b^4 + (3*a^5 - 36*a^3*b^2 + 40*a*b^4)*cos(d*x + c)^4 - 2*(3*a^5 - 36*a^3*b^2 + 40*a*b^4)*cos(d*x
+ c)^2 + (3*a^4*b - 36*a^2*b^3 + 40*b^5 + (3*a^4*b - 36*a^2*b^3 + 40*b^5)*cos(d*x + c)^4 - 2*(3*a^4*b - 36*a^
2*b^3 + 40*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) - 2*((49*a^4*b - 60*a^2*b^3)*cos(d*
x + c)^3 - 3*(13*a^4*b - 20*a^2*b^3)*cos(d*x + c))*sin(d*x + c))/(a^7*d*cos(d*x + c)^4 - 2*a^7*d*cos(d*x + c)^
2 + a^7*d + (a^6*b*d*cos(d*x + c)^4 - 2*a^6*b*d*cos(d*x + c)^2 + a^6*b*d)*sin(d*x + c))]

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giac [A]  time = 0.28, size = 461, normalized size = 1.58 \[ \frac {\frac {24 \, {\left (3 \, a^{4} - 36 \, a^{2} b^{2} + 40 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{6}} - \frac {384 \, {\left (2 \, a^{4} b - 7 \, a^{2} b^{3} + 5 \, b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \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 {384 \, {\left (a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3} b^{2} - a b^{4}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )} a^{6}} + \frac {3 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 16 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 240 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 384 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{8}} - \frac {150 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1800 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2000 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 240 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 384 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{4}}{a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^5/(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/192*(24*(3*a^4 - 36*a^2*b^2 + 40*b^4)*log(abs(tan(1/2*d*x + 1/2*c)))/a^6 - 384*(2*a^4*b - 7*a^2*b^3 + 5*b^5)
*(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) - 384*(a^2*b^3*tan(1/2*d*x + 1/2*c) - b^5*tan(1/2*d*x + 1/2*c) + a^3*b^2 - a*b^4)/((a*tan(1/2*d*x +
1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)*a^6) + (3*a^6*tan(1/2*d*x + 1/2*c)^4 - 16*a^5*b*tan(1/2*d*x + 1/2*c)^
3 - 24*a^6*tan(1/2*d*x + 1/2*c)^2 + 72*a^4*b^2*tan(1/2*d*x + 1/2*c)^2 + 240*a^5*b*tan(1/2*d*x + 1/2*c) - 384*a
^3*b^3*tan(1/2*d*x + 1/2*c))/a^8 - (150*a^4*tan(1/2*d*x + 1/2*c)^4 - 1800*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 200
0*b^4*tan(1/2*d*x + 1/2*c)^4 + 240*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 384*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 24*a^4*ta
n(1/2*d*x + 1/2*c)^2 + 72*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 - 16*a^3*b*tan(1/2*d*x + 1/2*c) + 3*a^4)/(a^6*tan(1/2
*d*x + 1/2*c)^4))/d

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maple [B]  time = 0.84, size = 634, normalized size = 2.17 \[ \frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a^{2} d}-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{12 d \,a^{3}}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{2} d}+\frac {3 b^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{4 d \,a^{3}}-\frac {2 b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{5}}-\frac {1}{64 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {1}{8 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {3 b^{2}}{8 d \,a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{2}}-\frac {9 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{2 d \,a^{4}}+\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}}{d \,a^{6}}+\frac {b}{12 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5 b}{4 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 b^{3}}{d \,a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{4} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right )}+\frac {2 b^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{6} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right )}-\frac {2 b^{2}}{d \,a^{3} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right )}+\frac {2 b^{4}}{d \,a^{5} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right )}-\frac {4 b \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d \,a^{2} \sqrt {a^{2}-b^{2}}}+\frac {14 b^{3} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d \,a^{4} \sqrt {a^{2}-b^{2}}}-\frac {10 b^{5} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d \,a^{6} \sqrt {a^{2}-b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^5/(a+b*sin(d*x+c))^2,x)

[Out]

1/64/d/a^2*tan(1/2*d*x+1/2*c)^4-1/12/d/a^3*tan(1/2*d*x+1/2*c)^3*b-1/8/d/a^2*tan(1/2*d*x+1/2*c)^2+3/8/d/a^4*b^2
*tan(1/2*d*x+1/2*c)^2+5/4/d/a^3*tan(1/2*d*x+1/2*c)*b-2/d/a^5*b^3*tan(1/2*d*x+1/2*c)-1/64/a^2/d/tan(1/2*d*x+1/2
*c)^4+1/8/a^2/d/tan(1/2*d*x+1/2*c)^2-3/8/d/a^4/tan(1/2*d*x+1/2*c)^2*b^2+3/8/d/a^2*ln(tan(1/2*d*x+1/2*c))-9/2/d
/a^4*ln(tan(1/2*d*x+1/2*c))*b^2+5/d/a^6*ln(tan(1/2*d*x+1/2*c))*b^4+1/12/d/a^3*b/tan(1/2*d*x+1/2*c)^3-5/4/d*b/a
^3/tan(1/2*d*x+1/2*c)+2/d*b^3/a^5/tan(1/2*d*x+1/2*c)-2/d*b^3/a^4/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*
b+a)*tan(1/2*d*x+1/2*c)+2/d/a^6*b^5/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)*tan(1/2*d*x+1/2*c)-2/d*b
^2/a^3/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)+2/d/a^5*b^4/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2
*c)*b+a)-4/d/a^2*b/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))+14/d/a^4*b^3/(a^2-
b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))-10/d/a^6*b^5/(a^2-b^2)^(1/2)*arctan(1/2*(2
*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^5/(a+b*sin(d*x+c))^2,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 details)Is 4*b^2-4*a^2 positive or negative?

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mupad [B]  time = 9.83, size = 1158, normalized size = 3.97 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^4/(sin(c + d*x)^5*(a + b*sin(c + d*x))^2),x)

[Out]

tan(c/2 + (d*x)/2)^4/(64*a^2*d) + (tan(c/2 + (d*x)/2)^4*(2*a^4 + 96*b^4 - 78*a^2*b^2) - a^4/4 + tan(c/2 + (d*x
)/2)^2*((7*a^4)/4 - (10*a^2*b^2)/3) + tan(c/2 + (d*x)/2)^3*(20*a*b^3 - (44*a^3*b)/3) - (4*tan(c/2 + (d*x)/2)^5
*(5*a^4*b - 8*b^5))/a + (5*a^3*b*tan(c/2 + (d*x)/2))/6)/(d*(16*a^6*tan(c/2 + (d*x)/2)^4 + 16*a^6*tan(c/2 + (d*
x)/2)^6 + 32*a^5*b*tan(c/2 + (d*x)/2)^5)) - (tan(c/2 + (d*x)/2)^2*((32*a^2 + 64*b^2)/(512*a^4) + 1/(16*a^2) -
b^2/(2*a^4)))/d + (tan(c/2 + (d*x)/2)*((b*(32*a^2 + 64*b^2))/(64*a^5) - b/(4*a^3) + (4*b*((32*a^2 + 64*b^2)/(2
56*a^4) + 1/(8*a^2) - b^2/a^4))/a))/d - (b*tan(c/2 + (d*x)/2)^3)/(12*a^3*d) + (log(tan(c/2 + (d*x)/2))*(3*a^4
+ 40*b^4 - 36*a^2*b^2))/(8*a^6*d) - (b*atan(((b*(b^2 - a^2)^(1/2)*(2*a^2 - 5*b^2)*((tan(c/2 + (d*x)/2)*(3*a^10
- 160*a^4*b^6 + 224*a^6*b^4 - 74*a^8*b^2))/(4*a^9) - (19*a^10*b + 80*a^6*b^5 - 92*a^8*b^3)/(4*a^10) + (b*(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)*(2*a^2 - 5*b^2))/a^6)*1i)/a^6 -
(b*(b^2 - a^2)^(1/2)*(2*a^2 - 5*b^2)*((19*a^10*b + 80*a^6*b^5 - 92*a^8*b^3)/(4*a^10) - (tan(c/2 + (d*x)/2)*(3
*a^10 - 160*a^4*b^6 + 224*a^6*b^4 - 74*a^8*b^2))/(4*a^9) + (b*(2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^12 - 32*a^1
0*b^2))/(4*a^9))*(b^2 - a^2)^(1/2)*(2*a^2 - 5*b^2))/a^6)*1i)/a^6)/((6*a^8*b + 200*b^9 - 460*a^2*b^7 + 347*a^4*
b^5 - 93*a^6*b^3)/(2*a^10) + (tan(c/2 + (d*x)/2)*(200*b^8 - 410*a^2*b^6 + 262*a^4*b^4 - 52*a^6*b^2))/(2*a^9) +
(b*(b^2 - a^2)^(1/2)*(2*a^2 - 5*b^2)*((tan(c/2 + (d*x)/2)*(3*a^10 - 160*a^4*b^6 + 224*a^6*b^4 - 74*a^8*b^2))/
(4*a^9) - (19*a^10*b + 80*a^6*b^5 - 92*a^8*b^3)/(4*a^10) + (b*(2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^12 - 32*a^1
0*b^2))/(4*a^9))*(b^2 - a^2)^(1/2)*(2*a^2 - 5*b^2))/a^6))/a^6 + (b*(b^2 - a^2)^(1/2)*(2*a^2 - 5*b^2)*((19*a^10
*b + 80*a^6*b^5 - 92*a^8*b^3)/(4*a^10) - (tan(c/2 + (d*x)/2)*(3*a^10 - 160*a^4*b^6 + 224*a^6*b^4 - 74*a^8*b^2)
)/(4*a^9) + (b*(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)*(2*a^2 - 5*b
^2))/a^6))/a^6))*(b^2 - a^2)^(1/2)*(2*a^2 - 5*b^2)*2i)/(a^6*d)

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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(cos(d*x+c)**4*csc(d*x+c)**5/(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

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