3.1262 \(\int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=287 \[ \frac {b \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac {b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {\cot (c+d x)}{a^2 d}-\frac {2 b \left (3 a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac {3 \left (a^2-b^2\right ) \cot (c+d x)}{a^4 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^4 b d (a+b \sin (c+d x))}-\frac {2 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 b^2 d}+\frac {4 \left (a^6-3 a^2 b^4+2 b^6\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 b^2 d \sqrt {a^2-b^2}}-\frac {x}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.38, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2897, 3770, 3767, 8, 3768, 2664, 12, 2660, 618, 204} \[ -\frac {2 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 b^2 d}+\frac {4 \left (-3 a^2 b^4+a^6+2 b^6\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 b^2 d \sqrt {a^2-b^2}}+\frac {3 \left (a^2-b^2\right ) \cot (c+d x)}{a^4 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^4 b d (a+b \sin (c+d x))}-\frac {2 b \left (3 a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac {b \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac {b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {\cot (c+d x)}{a^2 d}-\frac {x}{b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 2664

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n +
1))/(d*(n + 1)*(a^2 - b^2)), x] + Dist[1/((n + 1)*(a^2 - b^2)), Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1
) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integer
Q[2*n]

Rule 2897

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

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

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 {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\int \left (-\frac {1}{b^2}+\frac {2 \left (3 a^2 b-2 b^3\right ) \csc (c+d x)}{a^5}-\frac {3 \left (a^2-b^2\right ) \csc ^2(c+d x)}{a^4}-\frac {2 b \csc ^3(c+d x)}{a^3}+\frac {\csc ^4(c+d x)}{a^2}-\frac {\left (a^2-b^2\right )^3}{a^4 b^2 (a+b \sin (c+d x))^2}+\frac {2 \left (a^6-3 a^2 b^4+2 b^6\right )}{a^5 b^2 (a+b \sin (c+d x))}\right ) \, dx\\ &=-\frac {x}{b^2}+\frac {\int \csc ^4(c+d x) \, dx}{a^2}-\frac {(2 b) \int \csc ^3(c+d x) \, dx}{a^3}+\frac {\left (2 b \left (3 a^2-2 b^2\right )\right ) \int \csc (c+d x) \, dx}{a^5}-\frac {\left (3 \left (a^2-b^2\right )\right ) \int \csc ^2(c+d x) \, dx}{a^4}-\frac {\left (a^2-b^2\right )^3 \int \frac {1}{(a+b \sin (c+d x))^2} \, dx}{a^4 b^2}+\frac {\left (2 \left (a^6-3 a^2 b^4+2 b^6\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^5 b^2}\\ &=-\frac {x}{b^2}-\frac {2 b \left (3 a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac {b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^4 b d (a+b \sin (c+d x))}-\frac {b \int \csc (c+d x) \, dx}{a^3}-\frac {\left (a^2-b^2\right )^2 \int \frac {a}{a+b \sin (c+d x)} \, dx}{a^4 b^2}-\frac {\operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}+\frac {\left (3 \left (a^2-b^2\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^4 d}+\frac {\left (4 \left (a^6-3 a^2 b^4+2 b^6\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^5 b^2 d}\\ &=-\frac {x}{b^2}+\frac {b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {2 b \left (3 a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}-\frac {\cot (c+d x)}{a^2 d}+\frac {3 \left (a^2-b^2\right ) \cot (c+d x)}{a^4 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}+\frac {b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^4 b d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right )^2 \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^3 b^2}-\frac {\left (8 \left (a^6-3 a^2 b^4+2 b^6\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^5 b^2 d}\\ &=-\frac {x}{b^2}+\frac {4 \left (a^6-3 a^2 b^4+2 b^6\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 b^2 \sqrt {a^2-b^2} d}+\frac {b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {2 b \left (3 a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}-\frac {\cot (c+d x)}{a^2 d}+\frac {3 \left (a^2-b^2\right ) \cot (c+d x)}{a^4 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}+\frac {b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^4 b d (a+b \sin (c+d x))}-\frac {\left (2 \left (a^2-b^2\right )^2\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^3 b^2 d}\\ &=-\frac {x}{b^2}+\frac {4 \left (a^6-3 a^2 b^4+2 b^6\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 b^2 \sqrt {a^2-b^2} d}+\frac {b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {2 b \left (3 a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}-\frac {\cot (c+d x)}{a^2 d}+\frac {3 \left (a^2-b^2\right ) \cot (c+d x)}{a^4 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}+\frac {b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^4 b d (a+b \sin (c+d x))}+\frac {\left (4 \left (a^2-b^2\right )^2\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^3 b^2 d}\\ &=-\frac {x}{b^2}-\frac {2 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 b^2 d}+\frac {4 \left (a^6-3 a^2 b^4+2 b^6\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 b^2 \sqrt {a^2-b^2} d}+\frac {b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {2 b \left (3 a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}-\frac {\cot (c+d x)}{a^2 d}+\frac {3 \left (a^2-b^2\right ) \cot (c+d x)}{a^4 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}+\frac {b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^4 b d (a+b \sin (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 6.30, size = 428, normalized size = 1.49 \[ \frac {b \csc ^2\left (\frac {1}{2} (c+d x)\right )}{4 a^3 d}-\frac {b \sec ^2\left (\frac {1}{2} (c+d x)\right )}{4 a^3 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^2 d}+\frac {\tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^2 d}+\frac {\left (5 a^2 b-4 b^3\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d}+\frac {\left (4 b^3-5 a^2 b\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d}+\frac {2 \left (a^2-b^2\right )^{3/2} \left (a^2+4 b^2\right ) \tan ^{-1}\left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (a \sin \left (\frac {1}{2} (c+d x)\right )+b \cos \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^5 b^2 d}+\frac {\csc \left (\frac {1}{2} (c+d x)\right ) \left (7 a^2 \cos \left (\frac {1}{2} (c+d x)\right )-9 b^2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{6 a^4 d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (9 b^2 \sin \left (\frac {1}{2} (c+d x)\right )-7 a^2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{6 a^4 d}+\frac {a^4 (-\cos (c+d x))+2 a^2 b^2 \cos (c+d x)-b^4 \cos (c+d x)}{a^4 b d (a+b \sin (c+d x))}-\frac {c+d x}{b^2 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((c + d*x)/(b^2*d)) + (2*(a^2 - b^2)^(3/2)*(a^2 + 4*b^2)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin
[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^5*b^2*d) + ((7*a^2*Cos[(c + d*x)/2] - 9*b^2*Cos[(c + d*x)/2])*Csc[(c + d*
x)/2])/(6*a^4*d) + (b*Csc[(c + d*x)/2]^2)/(4*a^3*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a^2*d) + ((-5*
a^2*b + 4*b^3)*Log[Cos[(c + d*x)/2]])/(a^5*d) + ((5*a^2*b - 4*b^3)*Log[Sin[(c + d*x)/2]])/(a^5*d) - (b*Sec[(c
+ d*x)/2]^2)/(4*a^3*d) + (Sec[(c + d*x)/2]*(-7*a^2*Sin[(c + d*x)/2] + 9*b^2*Sin[(c + d*x)/2]))/(6*a^4*d) + (-(
a^4*Cos[c + d*x]) + 2*a^2*b^2*Cos[c + d*x] - b^4*Cos[c + d*x])/(a^4*b*d*(a + b*Sin[c + d*x])) + (Sec[(c + d*x)
/2]^2*Tan[(c + d*x)/2])/(24*a^2*d)

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fricas [B]  time = 1.53, size = 1437, normalized size = 5.01 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^4/(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

[-1/6*(6*a^5*b*d*x*cos(d*x + c)^4 - 12*a^5*b*d*x*cos(d*x + c)^2 + 6*a^5*b*d*x + 2*(7*a^4*b^2 - 6*a^2*b^4)*cos(
d*x + c)^3 + 3*(a^4*b + 3*a^2*b^3 - 4*b^5 + (a^4*b + 3*a^2*b^3 - 4*b^5)*cos(d*x + c)^4 - 2*(a^4*b + 3*a^2*b^3
- 4*b^5)*cos(d*x + c)^2 + (a^5 + 3*a^3*b^2 - 4*a*b^4 - (a^5 + 3*a^3*b^2 - 4*a*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*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)) - 12*(a^4
*b^2 - a^2*b^4)*cos(d*x + c) + 3*(5*a^2*b^4 - 4*b^6 + (5*a^2*b^4 - 4*b^6)*cos(d*x + c)^4 - 2*(5*a^2*b^4 - 4*b^
6)*cos(d*x + c)^2 + (5*a^3*b^3 - 4*a*b^5 - (5*a^3*b^3 - 4*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x
+ c) + 1/2) - 3*(5*a^2*b^4 - 4*b^6 + (5*a^2*b^4 - 4*b^6)*cos(d*x + c)^4 - 2*(5*a^2*b^4 - 4*b^6)*cos(d*x + c)^
2 + (5*a^3*b^3 - 4*a*b^5 - (5*a^3*b^3 - 4*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) -
2*(3*a^6*d*x*cos(d*x + c)^2 - 3*a^6*d*x + (3*a^5*b - 13*a^3*b^3 + 12*a*b^5)*cos(d*x + c)^3 - 3*(a^5*b - 5*a^3*
b^3 + 4*a*b^5)*cos(d*x + c))*sin(d*x + c))/(a^5*b^3*d*cos(d*x + c)^4 - 2*a^5*b^3*d*cos(d*x + c)^2 + a^5*b^3*d
- (a^6*b^2*d*cos(d*x + c)^2 - a^6*b^2*d)*sin(d*x + c)), -1/6*(6*a^5*b*d*x*cos(d*x + c)^4 - 12*a^5*b*d*x*cos(d*
x + c)^2 + 6*a^5*b*d*x + 2*(7*a^4*b^2 - 6*a^2*b^4)*cos(d*x + c)^3 + 6*(a^4*b + 3*a^2*b^3 - 4*b^5 + (a^4*b + 3*
a^2*b^3 - 4*b^5)*cos(d*x + c)^4 - 2*(a^4*b + 3*a^2*b^3 - 4*b^5)*cos(d*x + c)^2 + (a^5 + 3*a^3*b^2 - 4*a*b^4 -
(a^5 + 3*a^3*b^2 - 4*a*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))) - 12*(a^4*b^2 - a^2*b^4)*cos(d*x + c) + 3*(5*a^2*b^4 - 4*b^6 + (5*a^2*b^4 - 4*b^6)*co
s(d*x + c)^4 - 2*(5*a^2*b^4 - 4*b^6)*cos(d*x + c)^2 + (5*a^3*b^3 - 4*a*b^5 - (5*a^3*b^3 - 4*a*b^5)*cos(d*x + c
)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 3*(5*a^2*b^4 - 4*b^6 + (5*a^2*b^4 - 4*b^6)*cos(d*x + c)^4 - 2
*(5*a^2*b^4 - 4*b^6)*cos(d*x + c)^2 + (5*a^3*b^3 - 4*a*b^5 - (5*a^3*b^3 - 4*a*b^5)*cos(d*x + c)^2)*sin(d*x + c
))*log(-1/2*cos(d*x + c) + 1/2) - 2*(3*a^6*d*x*cos(d*x + c)^2 - 3*a^6*d*x + (3*a^5*b - 13*a^3*b^3 + 12*a*b^5)*
cos(d*x + c)^3 - 3*(a^5*b - 5*a^3*b^3 + 4*a*b^5)*cos(d*x + c))*sin(d*x + c))/(a^5*b^3*d*cos(d*x + c)^4 - 2*a^5
*b^3*d*cos(d*x + c)^2 + a^5*b^3*d - (a^6*b^2*d*cos(d*x + c)^2 - a^6*b^2*d)*sin(d*x + c))]

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giac [A]  time = 0.28, size = 399, normalized size = 1.39 \[ -\frac {\frac {24 \, {\left (d x + c\right )}}{b^{2}} - \frac {24 \, {\left (5 \, a^{2} b - 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{5}} - \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 27 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}} - \frac {48 \, {\left (a^{6} + 2 \, a^{4} b^{2} - 7 \, a^{2} b^{4} + 4 \, b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{5} b^{2}} + \frac {48 \, {\left (a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )} a^{5} b} + \frac {220 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 176 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 27 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^4/(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/24*(24*(d*x + c)/b^2 - 24*(5*a^2*b - 4*b^3)*log(abs(tan(1/2*d*x + 1/2*c)))/a^5 - (a^4*tan(1/2*d*x + 1/2*c)^
3 - 6*a^3*b*tan(1/2*d*x + 1/2*c)^2 - 27*a^4*tan(1/2*d*x + 1/2*c) + 36*a^2*b^2*tan(1/2*d*x + 1/2*c))/a^6 - 48*(
a^6 + 2*a^4*b^2 - 7*a^2*b^4 + 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^5*b^2) + 48*(a^4*b*tan(1/2*d*x + 1/2*c) - 2*a^2*b^3*tan(1/2*d*x + 1
/2*c) + b^5*tan(1/2*d*x + 1/2*c) + a^5 - 2*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^5*b) + (220*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 176*b^3*tan(1/2*d*x + 1/2*c)^3 - 27*a^3*tan(1/2*d*x + 1/
2*c)^2 + 36*a*b^2*tan(1/2*d*x + 1/2*c)^2 - 6*a^2*b*tan(1/2*d*x + 1/2*c) + a^3)/(a^5*tan(1/2*d*x + 1/2*c)^3))/d

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maple [B]  time = 0.77, size = 678, normalized size = 2.36 \[ \frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d \,a^{2}}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{4 d \,a^{3}}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{2}}+\frac {3 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{4}}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,b^{2}}-\frac {1}{24 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {9}{8 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {3 b^{2}}{2 d \,a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{4 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {5 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}-\frac {4 b^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{5}}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \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 ) a}+\frac {4 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{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} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{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 {2}{d b \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}{d \,a^{2} \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^{3}}{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 a \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 \,b^{2} \sqrt {a^{2}-b^{2}}}+\frac {4 \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 \sqrt {a^{2}-b^{2}}}-\frac {14 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 )}{d \,a^{3} \sqrt {a^{2}-b^{2}}}+\frac {8 b^{4} \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^{5} \sqrt {a^{2}-b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^4/(a+b*sin(d*x+c))^2,x)

[Out]

1/24/d/a^2*tan(1/2*d*x+1/2*c)^3-1/4/d/a^3*tan(1/2*d*x+1/2*c)^2*b-9/8/d/a^2*tan(1/2*d*x+1/2*c)+3/2/d/a^4*b^2*ta
n(1/2*d*x+1/2*c)-2/d/b^2*arctan(tan(1/2*d*x+1/2*c))-1/24/a^2/d/tan(1/2*d*x+1/2*c)^3+9/8/d/a^2/tan(1/2*d*x+1/2*
c)-3/2/d/a^4/tan(1/2*d*x+1/2*c)*b^2+1/4/d/a^3*b/tan(1/2*d*x+1/2*c)^2+5/d/a^3*b*ln(tan(1/2*d*x+1/2*c))-4/d/a^5*
b^3*ln(tan(1/2*d*x+1/2*c))-2/d/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)/a*tan(1/2*d*x+1/2*c)+4/d/a^3*
b^2/(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^5*b^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/b/(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/
(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)-2/d/a^4*b^3/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a
)+2/d/b^2*a/(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))+4/d/a/(a^2-b^2)^(1/2)*arc
tan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))-14/d/a^3*b^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))+8/d/a^5*b^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))

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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)^6*csc(d*x+c)^4/(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 = 12.78, size = 4740, normalized size = 16.52 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6/(sin(c + d*x)^4*(a + b*sin(c + d*x))^2),x)

[Out]

tan(c/2 + (d*x)/2)^3/(24*a^2*d) - (tan(c/2 + (d*x)/2)*((24*a^2 + 32*b^2)/(64*a^4) + 3/(4*a^2) - (2*b^2)/a^4))/
d - (tan(c/2 + (d*x)/2)^2*(8*a*b^2 - (26*a^3)/3) + a^3/3 - (4*a^2*b*tan(c/2 + (d*x)/2))/3 + (tan(c/2 + (d*x)/2
)^4*(7*a^4 + 16*b^4 - 20*a^2*b^2))/a + (4*tan(c/2 + (d*x)/2)^3*(4*a^4 + 10*b^4 - 13*a^2*b^2))/b)/(d*(8*a^5*tan
(c/2 + (d*x)/2)^3 + 8*a^5*tan(c/2 + (d*x)/2)^5 + 16*a^4*b*tan(c/2 + (d*x)/2)^4)) + (2*atan((((((((((32*(4*a^14
*b^7 - 3*a^16*b^5))/(a^12*b^2) + (32*tan(c/2 + (d*x)/2)*(16*a^13*b^10 - 17*a^15*b^8 + 2*a^17*b^6))/(a^12*b^4))
*1i)/b^2 - (32*(32*a^9*b^10 - 64*a^11*b^8 + 30*a^13*b^6 + a^15*b^4))/(a^12*b^2) + (32*tan(c/2 + (d*x)/2)*(148*
a^10*b^11 - 64*a^8*b^13 - 97*a^12*b^9 + 10*a^14*b^7 + 4*a^16*b^5))/(a^12*b^4))*1i)/b^2 - (32*(3*a^16*b - 64*a^
4*b^13 + 208*a^6*b^11 - 220*a^8*b^9 + 71*a^10*b^7 + 5*a^12*b^5 - 4*a^14*b^3))/(a^12*b^2) + (32*tan(c/2 + (d*x)
/2)*(16*a^5*b^14 - 40*a^7*b^12 + 5*a^9*b^10 + 44*a^11*b^8 - 6*a^13*b^6 - 20*a^15*b^4 + 2*a^17*b^2))/(a^12*b^4)
)*1i)/b^2 - (32*(6*a^15 + 16*a^5*b^10 - 56*a^7*b^8 + 97*a^9*b^6 - 84*a^11*b^4 + 20*a^13*b^2))/(a^12*b^2) + (32
*tan(c/2 + (d*x)/2)*(4*a^16*b - 64*b^17 + 304*a^2*b^15 - 540*a^4*b^13 + 405*a^6*b^11 - 124*a^8*b^9 + 78*a^10*b
^7 - 72*a^12*b^5 + 10*a^14*b^3))/(a^12*b^4))/b^2 - ((((((((32*(4*a^14*b^7 - 3*a^16*b^5))/(a^12*b^2) + (32*tan(
c/2 + (d*x)/2)*(16*a^13*b^10 - 17*a^15*b^8 + 2*a^17*b^6))/(a^12*b^4))*1i)/b^2 + (32*(32*a^9*b^10 - 64*a^11*b^8
+ 30*a^13*b^6 + a^15*b^4))/(a^12*b^2) - (32*tan(c/2 + (d*x)/2)*(148*a^10*b^11 - 64*a^8*b^13 - 97*a^12*b^9 + 1
0*a^14*b^7 + 4*a^16*b^5))/(a^12*b^4))*1i)/b^2 - (32*(3*a^16*b - 64*a^4*b^13 + 208*a^6*b^11 - 220*a^8*b^9 + 71*
a^10*b^7 + 5*a^12*b^5 - 4*a^14*b^3))/(a^12*b^2) + (32*tan(c/2 + (d*x)/2)*(16*a^5*b^14 - 40*a^7*b^12 + 5*a^9*b^
10 + 44*a^11*b^8 - 6*a^13*b^6 - 20*a^15*b^4 + 2*a^17*b^2))/(a^12*b^4))*1i)/b^2 + (32*(6*a^15 + 16*a^5*b^10 - 5
6*a^7*b^8 + 97*a^9*b^6 - 84*a^11*b^4 + 20*a^13*b^2))/(a^12*b^2) - (32*tan(c/2 + (d*x)/2)*(4*a^16*b - 64*b^17 +
304*a^2*b^15 - 540*a^4*b^13 + 405*a^6*b^11 - 124*a^8*b^9 + 78*a^10*b^7 - 72*a^12*b^5 + 10*a^14*b^3))/(a^12*b^
4))/b^2)/((((((((((32*(4*a^14*b^7 - 3*a^16*b^5))/(a^12*b^2) + (32*tan(c/2 + (d*x)/2)*(16*a^13*b^10 - 17*a^15*b
^8 + 2*a^17*b^6))/(a^12*b^4))*1i)/b^2 - (32*(32*a^9*b^10 - 64*a^11*b^8 + 30*a^13*b^6 + a^15*b^4))/(a^12*b^2) +
(32*tan(c/2 + (d*x)/2)*(148*a^10*b^11 - 64*a^8*b^13 - 97*a^12*b^9 + 10*a^14*b^7 + 4*a^16*b^5))/(a^12*b^4))*1i
)/b^2 - (32*(3*a^16*b - 64*a^4*b^13 + 208*a^6*b^11 - 220*a^8*b^9 + 71*a^10*b^7 + 5*a^12*b^5 - 4*a^14*b^3))/(a^
12*b^2) + (32*tan(c/2 + (d*x)/2)*(16*a^5*b^14 - 40*a^7*b^12 + 5*a^9*b^10 + 44*a^11*b^8 - 6*a^13*b^6 - 20*a^15*
b^4 + 2*a^17*b^2))/(a^12*b^4))*1i)/b^2 - (32*(6*a^15 + 16*a^5*b^10 - 56*a^7*b^8 + 97*a^9*b^6 - 84*a^11*b^4 + 2
0*a^13*b^2))/(a^12*b^2) + (32*tan(c/2 + (d*x)/2)*(4*a^16*b - 64*b^17 + 304*a^2*b^15 - 540*a^4*b^13 + 405*a^6*b
^11 - 124*a^8*b^9 + 78*a^10*b^7 - 72*a^12*b^5 + 10*a^14*b^3))/(a^12*b^4))*1i)/b^2 + (((((((((32*(4*a^14*b^7 -
3*a^16*b^5))/(a^12*b^2) + (32*tan(c/2 + (d*x)/2)*(16*a^13*b^10 - 17*a^15*b^8 + 2*a^17*b^6))/(a^12*b^4))*1i)/b^
2 + (32*(32*a^9*b^10 - 64*a^11*b^8 + 30*a^13*b^6 + a^15*b^4))/(a^12*b^2) - (32*tan(c/2 + (d*x)/2)*(148*a^10*b^
11 - 64*a^8*b^13 - 97*a^12*b^9 + 10*a^14*b^7 + 4*a^16*b^5))/(a^12*b^4))*1i)/b^2 - (32*(3*a^16*b - 64*a^4*b^13
+ 208*a^6*b^11 - 220*a^8*b^9 + 71*a^10*b^7 + 5*a^12*b^5 - 4*a^14*b^3))/(a^12*b^2) + (32*tan(c/2 + (d*x)/2)*(16
*a^5*b^14 - 40*a^7*b^12 + 5*a^9*b^10 + 44*a^11*b^8 - 6*a^13*b^6 - 20*a^15*b^4 + 2*a^17*b^2))/(a^12*b^4))*1i)/b
^2 + (32*(6*a^15 + 16*a^5*b^10 - 56*a^7*b^8 + 97*a^9*b^6 - 84*a^11*b^4 + 20*a^13*b^2))/(a^12*b^2) - (32*tan(c/
2 + (d*x)/2)*(4*a^16*b - 64*b^17 + 304*a^2*b^15 - 540*a^4*b^13 + 405*a^6*b^11 - 124*a^8*b^9 + 78*a^10*b^7 - 72
*a^12*b^5 + 10*a^14*b^3))/(a^12*b^4))*1i)/b^2 - (64*(30*a^12*b - 64*b^13 + 304*a^2*b^11 - 604*a^4*b^9 + 613*a^
6*b^7 - 280*a^8*b^5 + a^10*b^3))/(a^12*b^2) - (64*tan(c/2 + (d*x)/2)*(8*a^15 - 16*a^7*b^8 + 60*a^9*b^6 - 64*a^
11*b^4 + 12*a^13*b^2))/(a^12*b^4))))/(b^2*d) - (b*tan(c/2 + (d*x)/2)^2)/(4*a^3*d) + (b*log(tan(c/2 + (d*x)/2))
*(5*a^2 - 4*b^2))/(a^5*d) - (atan((((a^2 + 4*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((32*tan(c/2 + (d*x)/2)*(4*a^16
*b - 64*b^17 + 304*a^2*b^15 - 540*a^4*b^13 + 405*a^6*b^11 - 124*a^8*b^9 + 78*a^10*b^7 - 72*a^12*b^5 + 10*a^14*
b^3))/(a^12*b^4) - (32*(6*a^15 + 16*a^5*b^10 - 56*a^7*b^8 + 97*a^9*b^6 - 84*a^11*b^4 + 20*a^13*b^2))/(a^12*b^2
) + ((a^2 + 4*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((32*tan(c/2 + (d*x)/2)*(16*a^5*b^14 - 40*a^7*b^12 + 5*a^9*b^1
0 + 44*a^11*b^8 - 6*a^13*b^6 - 20*a^15*b^4 + 2*a^17*b^2))/(a^12*b^4) - (32*(3*a^16*b - 64*a^4*b^13 + 208*a^6*b
^11 - 220*a^8*b^9 + 71*a^10*b^7 + 5*a^12*b^5 - 4*a^14*b^3))/(a^12*b^2) + ((a^2 + 4*b^2)*(-(a + b)^3*(a - b)^3)
^(1/2)*((32*tan(c/2 + (d*x)/2)*(148*a^10*b^11 - 64*a^8*b^13 - 97*a^12*b^9 + 10*a^14*b^7 + 4*a^16*b^5))/(a^12*b
^4) - (32*(32*a^9*b^10 - 64*a^11*b^8 + 30*a^13*b^6 + a^15*b^4))/(a^12*b^2) + (((32*(4*a^14*b^7 - 3*a^16*b^5))/
(a^12*b^2) + (32*tan(c/2 + (d*x)/2)*(16*a^13*b^10 - 17*a^15*b^8 + 2*a^17*b^6))/(a^12*b^4))*(a^2 + 4*b^2)*(-(a
+ b)^3*(a - b)^3)^(1/2))/(a^5*b^2)))/(a^5*b^2)))/(a^5*b^2))*1i)/(a^5*b^2) - ((a^2 + 4*b^2)*(-(a + b)^3*(a - b)
^3)^(1/2)*((32*(6*a^15 + 16*a^5*b^10 - 56*a^7*b^8 + 97*a^9*b^6 - 84*a^11*b^4 + 20*a^13*b^2))/(a^12*b^2) - (32*
tan(c/2 + (d*x)/2)*(4*a^16*b - 64*b^17 + 304*a^2*b^15 - 540*a^4*b^13 + 405*a^6*b^11 - 124*a^8*b^9 + 78*a^10*b^
7 - 72*a^12*b^5 + 10*a^14*b^3))/(a^12*b^4) + ((a^2 + 4*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((32*tan(c/2 + (d*x)/
2)*(16*a^5*b^14 - 40*a^7*b^12 + 5*a^9*b^10 + 44*a^11*b^8 - 6*a^13*b^6 - 20*a^15*b^4 + 2*a^17*b^2))/(a^12*b^4)
- (32*(3*a^16*b - 64*a^4*b^13 + 208*a^6*b^11 - 220*a^8*b^9 + 71*a^10*b^7 + 5*a^12*b^5 - 4*a^14*b^3))/(a^12*b^2
) + ((a^2 + 4*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((32*(32*a^9*b^10 - 64*a^11*b^8 + 30*a^13*b^6 + a^15*b^4))/(a^
12*b^2) - (32*tan(c/2 + (d*x)/2)*(148*a^10*b^11 - 64*a^8*b^13 - 97*a^12*b^9 + 10*a^14*b^7 + 4*a^16*b^5))/(a^12
*b^4) + (((32*(4*a^14*b^7 - 3*a^16*b^5))/(a^12*b^2) + (32*tan(c/2 + (d*x)/2)*(16*a^13*b^10 - 17*a^15*b^8 + 2*a
^17*b^6))/(a^12*b^4))*(a^2 + 4*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))/(a^5*b^2)))/(a^5*b^2)))/(a^5*b^2))*1i)/(a^5*
b^2))/((64*(30*a^12*b - 64*b^13 + 304*a^2*b^11 - 604*a^4*b^9 + 613*a^6*b^7 - 280*a^8*b^5 + a^10*b^3))/(a^12*b^
2) + (64*tan(c/2 + (d*x)/2)*(8*a^15 - 16*a^7*b^8 + 60*a^9*b^6 - 64*a^11*b^4 + 12*a^13*b^2))/(a^12*b^4) - ((a^2
+ 4*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((32*tan(c/2 + (d*x)/2)*(4*a^16*b - 64*b^17 + 304*a^2*b^15 - 540*a^4*b^
13 + 405*a^6*b^11 - 124*a^8*b^9 + 78*a^10*b^7 - 72*a^12*b^5 + 10*a^14*b^3))/(a^12*b^4) - (32*(6*a^15 + 16*a^5*
b^10 - 56*a^7*b^8 + 97*a^9*b^6 - 84*a^11*b^4 + 20*a^13*b^2))/(a^12*b^2) + ((a^2 + 4*b^2)*(-(a + b)^3*(a - b)^3
)^(1/2)*((32*tan(c/2 + (d*x)/2)*(16*a^5*b^14 - 40*a^7*b^12 + 5*a^9*b^10 + 44*a^11*b^8 - 6*a^13*b^6 - 20*a^15*b
^4 + 2*a^17*b^2))/(a^12*b^4) - (32*(3*a^16*b - 64*a^4*b^13 + 208*a^6*b^11 - 220*a^8*b^9 + 71*a^10*b^7 + 5*a^12
*b^5 - 4*a^14*b^3))/(a^12*b^2) + ((a^2 + 4*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((32*tan(c/2 + (d*x)/2)*(148*a^10
*b^11 - 64*a^8*b^13 - 97*a^12*b^9 + 10*a^14*b^7 + 4*a^16*b^5))/(a^12*b^4) - (32*(32*a^9*b^10 - 64*a^11*b^8 + 3
0*a^13*b^6 + a^15*b^4))/(a^12*b^2) + (((32*(4*a^14*b^7 - 3*a^16*b^5))/(a^12*b^2) + (32*tan(c/2 + (d*x)/2)*(16*
a^13*b^10 - 17*a^15*b^8 + 2*a^17*b^6))/(a^12*b^4))*(a^2 + 4*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))/(a^5*b^2)))/(a^
5*b^2)))/(a^5*b^2)))/(a^5*b^2) - ((a^2 + 4*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((32*(6*a^15 + 16*a^5*b^10 - 56*a
^7*b^8 + 97*a^9*b^6 - 84*a^11*b^4 + 20*a^13*b^2))/(a^12*b^2) - (32*tan(c/2 + (d*x)/2)*(4*a^16*b - 64*b^17 + 30
4*a^2*b^15 - 540*a^4*b^13 + 405*a^6*b^11 - 124*a^8*b^9 + 78*a^10*b^7 - 72*a^12*b^5 + 10*a^14*b^3))/(a^12*b^4)
+ ((a^2 + 4*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((32*tan(c/2 + (d*x)/2)*(16*a^5*b^14 - 40*a^7*b^12 + 5*a^9*b^10
+ 44*a^11*b^8 - 6*a^13*b^6 - 20*a^15*b^4 + 2*a^17*b^2))/(a^12*b^4) - (32*(3*a^16*b - 64*a^4*b^13 + 208*a^6*b^1
1 - 220*a^8*b^9 + 71*a^10*b^7 + 5*a^12*b^5 - 4*a^14*b^3))/(a^12*b^2) + ((a^2 + 4*b^2)*(-(a + b)^3*(a - b)^3)^(
1/2)*((32*(32*a^9*b^10 - 64*a^11*b^8 + 30*a^13*b^6 + a^15*b^4))/(a^12*b^2) - (32*tan(c/2 + (d*x)/2)*(148*a^10*
b^11 - 64*a^8*b^13 - 97*a^12*b^9 + 10*a^14*b^7 + 4*a^16*b^5))/(a^12*b^4) + (((32*(4*a^14*b^7 - 3*a^16*b^5))/(a
^12*b^2) + (32*tan(c/2 + (d*x)/2)*(16*a^13*b^10 - 17*a^15*b^8 + 2*a^17*b^6))/(a^12*b^4))*(a^2 + 4*b^2)*(-(a +
b)^3*(a - b)^3)^(1/2))/(a^5*b^2)))/(a^5*b^2)))/(a^5*b^2)))/(a^5*b^2)))*(a^2 + 4*b^2)*(-(a + b)^3*(a - b)^3)^(1
/2)*2i)/(a^5*b^2*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)**6*csc(d*x+c)**4/(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

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