3.1141 \(\int \frac{\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=289 \[ \frac{\left (-19 a^2 b^2+2 a^4+20 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^6 d \sqrt{a^2-b^2}}+\frac{\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac{b \left (9 a^2-20 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}-\frac{\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac{\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac{\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2} \]

[Out]

((2*a^4 - 19*a^2*b^2 + 20*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^2 - b^2]*d) - (b*
(9*a^2 - 20*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^6*d) + ((17*a^2 - 60*b^2)*Cot[c + d*x])/(6*a^5*d) - ((a^2 - 5*b^2
)*Cot[c + d*x]*Csc[c + d*x])/(a^4*b*d) + ((3*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x])/(6*a^2*b*d*(a + b*Sin[c +
d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*(a + b*Sin[c + d*x])^2) + ((3*a^2 - 20*b^2)*Cot[c + d*x]*Csc[
c + d*x])/(6*a^3*b*d*(a + b*Sin[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.09865, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2724, 3055, 3001, 3770, 2660, 618, 204} \[ \frac{\left (-19 a^2 b^2+2 a^4+20 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^6 d \sqrt{a^2-b^2}}+\frac{\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac{b \left (9 a^2-20 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}-\frac{\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac{\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac{\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^3,x]

[Out]

((2*a^4 - 19*a^2*b^2 + 20*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^2 - b^2]*d) - (b*
(9*a^2 - 20*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^6*d) + ((17*a^2 - 60*b^2)*Cot[c + d*x])/(6*a^5*d) - ((a^2 - 5*b^2
)*Cot[c + d*x]*Csc[c + d*x])/(a^4*b*d) + ((3*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x])/(6*a^2*b*d*(a + b*Sin[c +
d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*(a + b*Sin[c + d*x])^2) + ((3*a^2 - 20*b^2)*Cot[c + d*x]*Csc[
c + d*x])/(6*a^3*b*d*(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 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 3770

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

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 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 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])

Rubi steps

\begin{align*} \int \frac{\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\frac{\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac{\int \frac{\csc ^3(c+d x) \left (2 \left (3 a^2-10 b^2\right )-2 a b \sin (c+d x)-3 \left (a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{6 a^2 b}\\ &=\frac{\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac{\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^3(c+d x) \left (12 \left (a^4-6 a^2 b^2+5 b^4\right )-5 a b \left (a^2-b^2\right ) \sin (c+d x)-2 \left (3 a^4-23 a^2 b^2+20 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{6 a^3 b \left (a^2-b^2\right )}\\ &=-\frac{\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac{\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac{\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^2(c+d x) \left (-2 b \left (17 a^4-77 a^2 b^2+60 b^4\right )+20 a b^2 \left (a^2-b^2\right ) \sin (c+d x)+12 b \left (a^4-6 a^2 b^2+5 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{12 a^4 b \left (a^2-b^2\right )}\\ &=\frac{\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac{\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac{\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac{\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac{\int \frac{\csc (c+d x) \left (6 b^2 \left (9 a^4-29 a^2 b^2+20 b^4\right )+12 a b \left (a^4-6 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{12 a^5 b \left (a^2-b^2\right )}\\ &=\frac{\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac{\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac{\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac{\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac{\left (b \left (9 a^4-29 a^2 b^2+20 b^4\right )\right ) \int \csc (c+d x) \, dx}{2 a^6 \left (a^2-b^2\right )}+\frac{\left (2 a^6-21 a^4 b^2+39 a^2 b^4-20 b^6\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{2 a^6 \left (a^2-b^2\right )}\\ &=-\frac{b \left (9 a^2-20 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}+\frac{\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac{\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac{\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac{\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac{\left (2 a^4-19 a^2 b^2+20 b^4\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{b \left (9 a^2-20 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}+\frac{\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac{\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac{\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac{\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}-\frac{\left (2 \left (2 a^4-19 a^2 b^2+20 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{\left (2 a^4-19 a^2 b^2+20 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{b \left (9 a^2-20 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}+\frac{\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac{\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac{\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac{\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 6.20106, size = 459, normalized size = 1.59 \[ \frac{\left (9 a^2 b-20 b^3\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac{\left (20 b^3-9 a^2 b\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac{3 a^2 b \cos (c+d x)-8 b^3 \cos (c+d x)}{2 a^5 d (a+b \sin (c+d x))}+\frac{a^2 b \cos (c+d x)-b^3 \cos (c+d x)}{2 a^4 d (a+b \sin (c+d x))^2}+\frac{\csc \left (\frac{1}{2} (c+d x)\right ) \left (2 a^2 \cos \left (\frac{1}{2} (c+d x)\right )-9 b^2 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{3 a^5 d}+\frac{\sec \left (\frac{1}{2} (c+d x)\right ) \left (9 b^2 \sin \left (\frac{1}{2} (c+d x)\right )-2 a^2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{3 a^5 d}+\frac{\left (-19 a^2 b^2+2 a^4+20 b^4\right ) \tan ^{-1}\left (\frac{\sec \left (\frac{1}{2} (c+d x)\right ) \left (a \sin \left (\frac{1}{2} (c+d x)\right )+b \cos \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{a^2-b^2}}\right )}{a^6 d \sqrt{a^2-b^2}}+\frac{3 b \csc ^2\left (\frac{1}{2} (c+d x)\right )}{8 a^4 d}-\frac{3 b \sec ^2\left (\frac{1}{2} (c+d x)\right )}{8 a^4 d}-\frac{\cot \left (\frac{1}{2} (c+d x)\right ) \csc ^2\left (\frac{1}{2} (c+d x)\right )}{24 a^3 d}+\frac{\tan \left (\frac{1}{2} (c+d x)\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )}{24 a^3 d} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^3,x]

[Out]

((2*a^4 - 19*a^2*b^2 + 20*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*a^2*Cos[(c + d*x)/2] - 9*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(3*a^5*d
) + (3*b*Csc[(c + d*x)/2]^2)/(8*a^4*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a^3*d) + ((-9*a^2*b + 20*b^
3)*Log[Cos[(c + d*x)/2]])/(2*a^6*d) + ((9*a^2*b - 20*b^3)*Log[Sin[(c + d*x)/2]])/(2*a^6*d) - (3*b*Sec[(c + d*x
)/2]^2)/(8*a^4*d) + (Sec[(c + d*x)/2]*(-2*a^2*Sin[(c + d*x)/2] + 9*b^2*Sin[(c + d*x)/2]))/(3*a^5*d) + (a^2*b*C
os[c + d*x] - b^3*Cos[c + d*x])/(2*a^4*d*(a + b*Sin[c + d*x])^2) + (3*a^2*b*Cos[c + d*x] - 8*b^3*Cos[c + d*x])
/(2*a^5*d*(a + b*Sin[c + d*x])) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a^3*d)

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Maple [B]  time = 0.221, size = 780, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^4/(a+b*sin(d*x+c))^3,x)

[Out]

1/24/d/a^3*tan(1/2*d*x+1/2*c)^3-3/8/d/a^4*tan(1/2*d*x+1/2*c)^2*b-5/8/d/a^3*tan(1/2*d*x+1/2*c)+3/d/a^5*b^2*tan(
1/2*d*x+1/2*c)+5/d/a^3/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^3*b^2-10/d/a^5/(
tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^3*b^4+4/d/a^2/(tan(1/2*d*x+1/2*c)^2*a+2*
tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^2*b-1/d/a^4/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*t
an(1/2*d*x+1/2*c)^2*b^3-18/d/a^6/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^2*b^5+
11/d/a^3/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)*b^2-26/d/a^5/(tan(1/2*d*x+1/2*
c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)*b^4+4/d/a^2/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)
*b+a)^2*b-9/d/a^4/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*b^3+2/d/a^2/(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))-19/d/a^4/(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))*b^2+20/d/a^6/(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))*b^4-1/2
4/d/a^3/tan(1/2*d*x+1/2*c)^3+5/8/d/a^3/tan(1/2*d*x+1/2*c)-3/d/a^5/tan(1/2*d*x+1/2*c)*b^2+3/8/d/a^4*b/tan(1/2*d
*x+1/2*c)^2+9/2/d/a^4*b*ln(tan(1/2*d*x+1/2*c))-10/d/a^6*b^3*ln(tan(1/2*d*x+1/2*c))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^4/(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.47259, size = 4578, normalized size = 15.84 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^4/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

[1/12*(2*(17*a^5*b^2 - 77*a^3*b^4 + 60*a*b^6)*cos(d*x + c)^5 - 4*(4*a^7 + 3*a^5*b^2 - 67*a^3*b^4 + 60*a*b^6)*c
os(d*x + c)^3 - 3*(4*a^5*b - 38*a^3*b^3 + 40*a*b^5 + 2*(2*a^5*b - 19*a^3*b^3 + 20*a*b^5)*cos(d*x + c)^4 - 4*(2
*a^5*b - 19*a^3*b^3 + 20*a*b^5)*cos(d*x + c)^2 + (2*a^6 - 17*a^4*b^2 + a^2*b^4 + 20*b^6 + (2*a^4*b^2 - 19*a^2*
b^4 + 20*b^6)*cos(d*x + c)^4 - (2*a^6 - 15*a^4*b^2 - 18*a^2*b^4 + 40*b^6)*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*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)) + 6*(2*a^7 - 3*a^5
*b^2 - 19*a^3*b^4 + 20*a*b^6)*cos(d*x + c) - 3*(18*a^5*b^2 - 58*a^3*b^4 + 40*a*b^6 + 2*(9*a^5*b^2 - 29*a^3*b^4
+ 20*a*b^6)*cos(d*x + c)^4 - 4*(9*a^5*b^2 - 29*a^3*b^4 + 20*a*b^6)*cos(d*x + c)^2 + (9*a^6*b - 20*a^4*b^3 - 9
*a^2*b^5 + 20*b^7 + (9*a^4*b^3 - 29*a^2*b^5 + 20*b^7)*cos(d*x + c)^4 - (9*a^6*b - 11*a^4*b^3 - 38*a^2*b^5 + 40
*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 3*(18*a^5*b^2 - 58*a^3*b^4 + 40*a*b^6 + 2*(9
*a^5*b^2 - 29*a^3*b^4 + 20*a*b^6)*cos(d*x + c)^4 - 4*(9*a^5*b^2 - 29*a^3*b^4 + 20*a*b^6)*cos(d*x + c)^2 + (9*a
^6*b - 20*a^4*b^3 - 9*a^2*b^5 + 20*b^7 + (9*a^4*b^3 - 29*a^2*b^5 + 20*b^7)*cos(d*x + c)^4 - (9*a^6*b - 11*a^4*
b^3 - 38*a^2*b^5 + 40*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) - 2*(2*(14*a^6*b - 59*a^
4*b^3 + 45*a^2*b^5)*cos(d*x + c)^3 - 3*(11*a^6*b - 41*a^4*b^3 + 30*a^2*b^5)*cos(d*x + c))*sin(d*x + c))/(2*(a^
9*b - a^7*b^3)*d*cos(d*x + c)^4 - 4*(a^9*b - a^7*b^3)*d*cos(d*x + c)^2 + 2*(a^9*b - a^7*b^3)*d + ((a^8*b^2 - a
^6*b^4)*d*cos(d*x + c)^4 - (a^10 + a^8*b^2 - 2*a^6*b^4)*d*cos(d*x + c)^2 + (a^10 - a^6*b^4)*d)*sin(d*x + c)),
1/12*(2*(17*a^5*b^2 - 77*a^3*b^4 + 60*a*b^6)*cos(d*x + c)^5 - 4*(4*a^7 + 3*a^5*b^2 - 67*a^3*b^4 + 60*a*b^6)*co
s(d*x + c)^3 - 6*(4*a^5*b - 38*a^3*b^3 + 40*a*b^5 + 2*(2*a^5*b - 19*a^3*b^3 + 20*a*b^5)*cos(d*x + c)^4 - 4*(2*
a^5*b - 19*a^3*b^3 + 20*a*b^5)*cos(d*x + c)^2 + (2*a^6 - 17*a^4*b^2 + a^2*b^4 + 20*b^6 + (2*a^4*b^2 - 19*a^2*b
^4 + 20*b^6)*cos(d*x + c)^4 - (2*a^6 - 15*a^4*b^2 - 18*a^2*b^4 + 40*b^6)*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*(2*a^7 - 3*a^5*b^2 - 19*a^3*b^4 + 20
*a*b^6)*cos(d*x + c) - 3*(18*a^5*b^2 - 58*a^3*b^4 + 40*a*b^6 + 2*(9*a^5*b^2 - 29*a^3*b^4 + 20*a*b^6)*cos(d*x +
c)^4 - 4*(9*a^5*b^2 - 29*a^3*b^4 + 20*a*b^6)*cos(d*x + c)^2 + (9*a^6*b - 20*a^4*b^3 - 9*a^2*b^5 + 20*b^7 + (9
*a^4*b^3 - 29*a^2*b^5 + 20*b^7)*cos(d*x + c)^4 - (9*a^6*b - 11*a^4*b^3 - 38*a^2*b^5 + 40*b^7)*cos(d*x + c)^2)*
sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 3*(18*a^5*b^2 - 58*a^3*b^4 + 40*a*b^6 + 2*(9*a^5*b^2 - 29*a^3*b^4
+ 20*a*b^6)*cos(d*x + c)^4 - 4*(9*a^5*b^2 - 29*a^3*b^4 + 20*a*b^6)*cos(d*x + c)^2 + (9*a^6*b - 20*a^4*b^3 - 9*
a^2*b^5 + 20*b^7 + (9*a^4*b^3 - 29*a^2*b^5 + 20*b^7)*cos(d*x + c)^4 - (9*a^6*b - 11*a^4*b^3 - 38*a^2*b^5 + 40*
b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) - 2*(2*(14*a^6*b - 59*a^4*b^3 + 45*a^2*b^5)*co
s(d*x + c)^3 - 3*(11*a^6*b - 41*a^4*b^3 + 30*a^2*b^5)*cos(d*x + c))*sin(d*x + c))/(2*(a^9*b - a^7*b^3)*d*cos(d
*x + c)^4 - 4*(a^9*b - a^7*b^3)*d*cos(d*x + c)^2 + 2*(a^9*b - a^7*b^3)*d + ((a^8*b^2 - a^6*b^4)*d*cos(d*x + c)
^4 - (a^10 + a^8*b^2 - 2*a^6*b^4)*d*cos(d*x + c)^2 + (a^10 - a^6*b^4)*d)*sin(d*x + c))]

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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(cos(d*x+c)**4*csc(d*x+c)**4/(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.29902, size = 609, normalized size = 2.11 \begin{align*} \frac{\frac{12 \,{\left (9 \, a^{2} b - 20 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{6}} + \frac{24 \,{\left (2 \, a^{4} - 19 \, a^{2} b^{2} + 20 \, b^{4}\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{24 \,{\left (5 \, a^{3} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 10 \, a b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 4 \, a^{4} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a^{2} b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 18 \, b^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 11 \, a^{3} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 26 \, a b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 4 \, a^{4} b - 9 \, a^{2} b^{3}\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 )}^{2} a^{6}} + \frac{a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 9 \, a^{5} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 72 \, a^{4} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{9}} - \frac{198 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 440 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 72 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 9 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a^{3}}{a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^4/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/24*(12*(9*a^2*b - 20*b^3)*log(abs(tan(1/2*d*x + 1/2*c)))/a^6 + 24*(2*a^4 - 19*a^2*b^2 + 20*b^4)*(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) + 2
4*(5*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - 10*a*b^4*tan(1/2*d*x + 1/2*c)^3 + 4*a^4*b*tan(1/2*d*x + 1/2*c)^2 - a^2*b
^3*tan(1/2*d*x + 1/2*c)^2 - 18*b^5*tan(1/2*d*x + 1/2*c)^2 + 11*a^3*b^2*tan(1/2*d*x + 1/2*c) - 26*a*b^4*tan(1/2
*d*x + 1/2*c) + 4*a^4*b - 9*a^2*b^3)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^2*a^6) + (a^6*
tan(1/2*d*x + 1/2*c)^3 - 9*a^5*b*tan(1/2*d*x + 1/2*c)^2 - 15*a^6*tan(1/2*d*x + 1/2*c) + 72*a^4*b^2*tan(1/2*d*x
+ 1/2*c))/a^9 - (198*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 440*b^3*tan(1/2*d*x + 1/2*c)^3 - 15*a^3*tan(1/2*d*x + 1/2
*c)^2 + 72*a*b^2*tan(1/2*d*x + 1/2*c)^2 - 9*a^2*b*tan(1/2*d*x + 1/2*c) + a^3)/(a^6*tan(1/2*d*x + 1/2*c)^3))/d