3.1327 \(\int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=195 \[ \frac {b \cot ^3(c+d x)}{3 a^2 d}+\frac {2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 b d}+\frac {b \left (b^2-2 a^2\right ) \cot (c+d x)}{a^4 d}+\frac {\left (7 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}-\frac {\left (15 a^4-20 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a d}-\frac {x}{b} \]

[Out]

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

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Rubi [A]  time = 0.29, antiderivative size = 275, normalized size of antiderivative = 1.41, number of steps used = 15, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2897, 3770, 3767, 8, 3768, 2660, 618, 204} \[ \frac {2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 b d}-\frac {b \left (3 a^2-b^2\right ) \cot (c+d x)}{a^4 d}+\frac {\left (3 a^2-b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac {\left (-3 a^2 b^2+3 a^4+b^4\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac {\left (3 a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {b \cot ^3(c+d x)}{3 a^2 d}+\frac {b \cot (c+d x)}{a^2 d}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {x}{b} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + b*Sin[c + d*x]),x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 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 (c+d x) \cot ^5(c+d x)}{a+b \sin (c+d x)} \, dx &=\int \left (-\frac {1}{b}+\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \csc (c+d x)}{a^5}+\frac {\left (3 a^2 b-b^3\right ) \csc ^2(c+d x)}{a^4}+\frac {\left (-3 a^2+b^2\right ) \csc ^3(c+d x)}{a^3}-\frac {b \csc ^4(c+d x)}{a^2}+\frac {\csc ^5(c+d x)}{a}+\frac {\left (a^2-b^2\right )^3}{a^5 b (a+b \sin (c+d x))}\right ) \, dx\\ &=-\frac {x}{b}+\frac {\int \csc ^5(c+d x) \, dx}{a}-\frac {b \int \csc ^4(c+d x) \, dx}{a^2}+\frac {\left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^5 b}-\frac {\left (3 a^2-b^2\right ) \int \csc ^3(c+d x) \, dx}{a^3}+\frac {\left (b \left (3 a^2-b^2\right )\right ) \int \csc ^2(c+d x) \, dx}{a^4}+\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \int \csc (c+d x) \, dx}{a^5}\\ &=-\frac {x}{b}-\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac {\left (3 a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {3 \int \csc ^3(c+d x) \, dx}{4 a}-\frac {\left (3 a^2-b^2\right ) \int \csc (c+d x) \, dx}{2 a^3}+\frac {b \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}+\frac {\left (2 \left (a^2-b^2\right )^3\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 d}-\frac {\left (b \left (3 a^2-b^2\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^4 d}\\ &=-\frac {x}{b}+\frac {\left (3 a^2-b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac {b \cot (c+d x)}{a^2 d}-\frac {b \left (3 a^2-b^2\right ) \cot (c+d x)}{a^4 d}+\frac {b \cot ^3(c+d x)}{3 a^2 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}+\frac {\left (3 a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {3 \int \csc (c+d x) \, dx}{8 a}-\frac {\left (4 \left (a^2-b^2\right )^3\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 d}\\ &=-\frac {x}{b}+\frac {2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 b d}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac {\left (3 a^2-b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac {b \cot (c+d x)}{a^2 d}-\frac {b \left (3 a^2-b^2\right ) \cot (c+d x)}{a^4 d}+\frac {b \cot ^3(c+d x)}{3 a^2 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}+\frac {\left (3 a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\\ \end {align*}

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Mathematica [B]  time = 6.19, size = 448, normalized size = 2.30 \[ \frac {b \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^2 d}-\frac {b \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^2 d}+\frac {2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (a \sin \left (\frac {1}{2} (c+d x)\right )+b \cos \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^5 b d}+\frac {\csc \left (\frac {1}{2} (c+d x)\right ) \left (3 b^3 \cos \left (\frac {1}{2} (c+d x)\right )-7 a^2 b \cos \left (\frac {1}{2} (c+d x)\right )\right )}{6 a^4 d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (7 a^2 b \sin \left (\frac {1}{2} (c+d x)\right )-3 b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{6 a^4 d}+\frac {\left (9 a^2-4 b^2\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 a^3 d}+\frac {\left (4 b^2-9 a^2\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 a^3 d}+\frac {\left (15 a^4-20 a^2 b^2+8 b^4\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^5 d}+\frac {\left (-15 a^4+20 a^2 b^2-8 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^5 d}-\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 a d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 a d}-\frac {c+d x}{b d} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + b*Sin[c + d*x]),x]

[Out]

-((c + d*x)/(b*d)) + (2*(a^2 - b^2)^(5/2)*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*d) + ((-7*a^2*b*Cos[(c + d*x)/2] + 3*b^3*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^4*d
) + ((9*a^2 - 4*b^2)*Csc[(c + d*x)/2]^2)/(32*a^3*d) + (b*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a^2*d) - Csc
[(c + d*x)/2]^4/(64*a*d) + ((-15*a^4 + 20*a^2*b^2 - 8*b^4)*Log[Cos[(c + d*x)/2]])/(8*a^5*d) + ((15*a^4 - 20*a^
2*b^2 + 8*b^4)*Log[Sin[(c + d*x)/2]])/(8*a^5*d) + ((-9*a^2 + 4*b^2)*Sec[(c + d*x)/2]^2)/(32*a^3*d) + Sec[(c +
d*x)/2]^4/(64*a*d) + (Sec[(c + d*x)/2]*(7*a^2*b*Sin[(c + d*x)/2] - 3*b^3*Sin[(c + d*x)/2]))/(6*a^4*d) - (b*Sec
[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a^2*d)

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fricas [B]  time = 1.51, size = 1034, normalized size = 5.30 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^5/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

[-1/48*(48*a^5*d*x*cos(d*x + c)^4 - 96*a^5*d*x*cos(d*x + c)^2 + 48*a^5*d*x + 6*(9*a^4*b - 4*a^2*b^3)*cos(d*x +
c)^3 - 24*((a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*cos(d*x
+ c)^2)*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*(7*a^4*b - 4*a^2*b^3)*cos(d*x + c) + 3*(15*a^4*b - 20*a^2*b^3 + 8*b^5 + (15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(
d*x + c)^4 - 2*(15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) - 3*(15*a^4*b - 20*
a^2*b^3 + 8*b^5 + (15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^4 - 2*(15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x +
c)^2)*log(-1/2*cos(d*x + c) + 1/2) - 16*((7*a^3*b^2 - 3*a*b^4)*cos(d*x + c)^3 - 3*(2*a^3*b^2 - a*b^4)*cos(d*x
+ c))*sin(d*x + c))/(a^5*b*d*cos(d*x + c)^4 - 2*a^5*b*d*cos(d*x + c)^2 + a^5*b*d), -1/48*(48*a^5*d*x*cos(d*x
+ c)^4 - 96*a^5*d*x*cos(d*x + c)^2 + 48*a^5*d*x + 6*(9*a^4*b - 4*a^2*b^3)*cos(d*x + c)^3 + 48*((a^4 - 2*a^2*b^
2 + b^4)*cos(d*x + c)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(a^2 - b^2)*ar
ctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 6*(7*a^4*b - 4*a^2*b^3)*cos(d*x + c) + 3*(15*a^4*
b - 20*a^2*b^3 + 8*b^5 + (15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^4 - 2*(15*a^4*b - 20*a^2*b^3 + 8*b^5)*co
s(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) - 3*(15*a^4*b - 20*a^2*b^3 + 8*b^5 + (15*a^4*b - 20*a^2*b^3 + 8*b^5)
*cos(d*x + c)^4 - 2*(15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2) - 16*((7*a^3*
b^2 - 3*a*b^4)*cos(d*x + c)^3 - 3*(2*a^3*b^2 - a*b^4)*cos(d*x + c))*sin(d*x + c))/(a^5*b*d*cos(d*x + c)^4 - 2*
a^5*b*d*cos(d*x + c)^2 + a^5*b*d)]

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giac [B]  time = 0.24, size = 396, normalized size = 2.03 \[ -\frac {\frac {192 \, {\left (d x + c\right )}}{b} - \frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 216 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{4}} - \frac {24 \, {\left (15 \, a^{4} - 20 \, a^{2} b^{2} + 8 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{5}} - \frac {384 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{5} b} + \frac {750 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1000 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 400 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 216 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 96 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{4}}{a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^5/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/192*(192*(d*x + c)/b - (3*a^3*tan(1/2*d*x + 1/2*c)^4 - 8*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 48*a^3*tan(1/2*d*x
+ 1/2*c)^2 + 24*a*b^2*tan(1/2*d*x + 1/2*c)^2 + 216*a^2*b*tan(1/2*d*x + 1/2*c) - 96*b^3*tan(1/2*d*x + 1/2*c))/a
^4 - 24*(15*a^4 - 20*a^2*b^2 + 8*b^4)*log(abs(tan(1/2*d*x + 1/2*c)))/a^5 - 384*(a^6 - 3*a^4*b^2 + 3*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^5*b) + (750*a^4*tan(1/2*d*x + 1/2*c)^4 - 1000*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 400*b^4*tan(1/2*d*x
+ 1/2*c)^4 + 216*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 96*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 48*a^4*tan(1/2*d*x + 1/2*c)^
2 + 24*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 - 8*a^3*b*tan(1/2*d*x + 1/2*c) + 3*a^4)/(a^5*tan(1/2*d*x + 1/2*c)^4))/d

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maple [B]  time = 0.49, size = 523, normalized size = 2.68 \[ \frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d a}-\frac {b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \,a^{2}}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {b^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{3}}+\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{8 d \,a^{2}}-\frac {b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{4}}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d b}-\frac {1}{64 d a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {1}{4 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {b^{2}}{8 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {15 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d}-\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{2 d \,a^{3}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}}{d \,a^{5}}+\frac {b}{24 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {9 b}{8 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b^{3}}{2 d \,a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 a \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d b \sqrt {a^{2}-b^{2}}}-\frac {6 b \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d a \sqrt {a^{2}-b^{2}}}+\frac {6 b^{3} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d \,a^{3} \sqrt {a^{2}-b^{2}}}-\frac {2 b^{5} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d \,a^{5} \sqrt {a^{2}-b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^5/(a+b*sin(d*x+c)),x)

[Out]

1/64/d/a*tan(1/2*d*x+1/2*c)^4-1/24/d/a^2*b*tan(1/2*d*x+1/2*c)^3-1/4/a/d*tan(1/2*d*x+1/2*c)^2+1/8/d/a^3*b^2*tan
(1/2*d*x+1/2*c)^2+9/8/d/a^2*tan(1/2*d*x+1/2*c)*b-1/2/d/a^4*b^3*tan(1/2*d*x+1/2*c)-2/d/b*arctan(tan(1/2*d*x+1/2
*c))-1/64/d/a/tan(1/2*d*x+1/2*c)^4+1/4/a/d/tan(1/2*d*x+1/2*c)^2-1/8/d*b^2/a^3/tan(1/2*d*x+1/2*c)^2+15/8/a/d*ln
(tan(1/2*d*x+1/2*c))-5/2/d/a^3*ln(tan(1/2*d*x+1/2*c))*b^2+1/d/a^5*ln(tan(1/2*d*x+1/2*c))*b^4+1/24/d/a^2*b/tan(
1/2*d*x+1/2*c)^3-9/8/d/a^2*b/tan(1/2*d*x+1/2*c)+1/2/d*b^3/a^4/tan(1/2*d*x+1/2*c)+2/d*a/b/(a^2-b^2)^(1/2)*arcta
n(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))-6/d*b/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))+6/d/a^3*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))-2/
d/a^5*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)^6*csc(d*x+c)^5/(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 details)Is 4*b^2-4*a^2 positive or negative?

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mupad [B]  time = 13.40, size = 4334, normalized size = 22.23 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6/(sin(c + d*x)^5*(a + b*sin(c + d*x))),x)

[Out]

tan(c/2 + (d*x)/2)^4/(64*a*d) - (2*atan(-(((((((4*(64*a^9*b^9 - 208*a^11*b^7 + 240*a^13*b^5 - 93*a^15*b^3))/a^
12 + (((4*(32*a^14*b^5 - 24*a^16*b^3))/a^12 + (4*tan(c/2 + (d*x)/2)*(128*a^13*b^6 - 136*a^15*b^4 + 16*a^17*b^2
))/a^12)*1i)/b + (4*tan(c/2 + (d*x)/2)*(128*a^8*b^10 - 456*a^10*b^8 + 604*a^12*b^6 - 335*a^14*b^4 + 62*a^16*b^
2))/a^12)*1i)/b - (4*(24*a^16*b - 32*a^4*b^13 + 184*a^6*b^11 - 440*a^8*b^9 + 543*a^10*b^7 - 345*a^12*b^5 + 58*
a^14*b^3))/a^12 + (4*tan(c/2 + (d*x)/2)*(16*a^17 + 8*a^5*b^12 - 40*a^7*b^10 + 100*a^9*b^8 - 148*a^11*b^6 + 252
*a^13*b^4 - 180*a^15*b^2))/a^12)*1i)/b - (4*(53*a^15*b + 8*a^5*b^11 - 48*a^7*b^9 + 56*a^9*b^7 + 48*a^11*b^5 -
120*a^13*b^3))/a^12 + (4*tan(c/2 + (d*x)/2)*(62*a^16 + 8*b^16 - 68*a^2*b^14 + 255*a^4*b^12 - 550*a^6*b^10 + 87
3*a^8*b^8 - 1096*a^10*b^6 + 929*a^12*b^4 - 410*a^14*b^2))/a^12)/b + ((((4*(24*a^16*b - 32*a^4*b^13 + 184*a^6*b
^11 - 440*a^8*b^9 + 543*a^10*b^7 - 345*a^12*b^5 + 58*a^14*b^3))/a^12 + (((4*(64*a^9*b^9 - 208*a^11*b^7 + 240*a
^13*b^5 - 93*a^15*b^3))/a^12 - (((4*(32*a^14*b^5 - 24*a^16*b^3))/a^12 + (4*tan(c/2 + (d*x)/2)*(128*a^13*b^6 -
136*a^15*b^4 + 16*a^17*b^2))/a^12)*1i)/b + (4*tan(c/2 + (d*x)/2)*(128*a^8*b^10 - 456*a^10*b^8 + 604*a^12*b^6 -
335*a^14*b^4 + 62*a^16*b^2))/a^12)*1i)/b - (4*tan(c/2 + (d*x)/2)*(16*a^17 + 8*a^5*b^12 - 40*a^7*b^10 + 100*a^
9*b^8 - 148*a^11*b^6 + 252*a^13*b^4 - 180*a^15*b^2))/a^12)*1i)/b - (4*(53*a^15*b + 8*a^5*b^11 - 48*a^7*b^9 + 5
6*a^9*b^7 + 48*a^11*b^5 - 120*a^13*b^3))/a^12 + (4*tan(c/2 + (d*x)/2)*(62*a^16 + 8*b^16 - 68*a^2*b^14 + 255*a^
4*b^12 - 550*a^6*b^10 + 873*a^8*b^8 - 1096*a^10*b^6 + 929*a^12*b^4 - 410*a^14*b^2))/a^12)/b)/((((((((4*(64*a^9
*b^9 - 208*a^11*b^7 + 240*a^13*b^5 - 93*a^15*b^3))/a^12 + (((4*(32*a^14*b^5 - 24*a^16*b^3))/a^12 + (4*tan(c/2
+ (d*x)/2)*(128*a^13*b^6 - 136*a^15*b^4 + 16*a^17*b^2))/a^12)*1i)/b + (4*tan(c/2 + (d*x)/2)*(128*a^8*b^10 - 45
6*a^10*b^8 + 604*a^12*b^6 - 335*a^14*b^4 + 62*a^16*b^2))/a^12)*1i)/b - (4*(24*a^16*b - 32*a^4*b^13 + 184*a^6*b
^11 - 440*a^8*b^9 + 543*a^10*b^7 - 345*a^12*b^5 + 58*a^14*b^3))/a^12 + (4*tan(c/2 + (d*x)/2)*(16*a^17 + 8*a^5*
b^12 - 40*a^7*b^10 + 100*a^9*b^8 - 148*a^11*b^6 + 252*a^13*b^4 - 180*a^15*b^2))/a^12)*1i)/b - (4*(53*a^15*b +
8*a^5*b^11 - 48*a^7*b^9 + 56*a^9*b^7 + 48*a^11*b^5 - 120*a^13*b^3))/a^12 + (4*tan(c/2 + (d*x)/2)*(62*a^16 + 8*
b^16 - 68*a^2*b^14 + 255*a^4*b^12 - 550*a^6*b^10 + 873*a^8*b^8 - 1096*a^10*b^6 + 929*a^12*b^4 - 410*a^14*b^2))
/a^12)*1i)/b - (((((4*(24*a^16*b - 32*a^4*b^13 + 184*a^6*b^11 - 440*a^8*b^9 + 543*a^10*b^7 - 345*a^12*b^5 + 58
*a^14*b^3))/a^12 + (((4*(64*a^9*b^9 - 208*a^11*b^7 + 240*a^13*b^5 - 93*a^15*b^3))/a^12 - (((4*(32*a^14*b^5 - 2
4*a^16*b^3))/a^12 + (4*tan(c/2 + (d*x)/2)*(128*a^13*b^6 - 136*a^15*b^4 + 16*a^17*b^2))/a^12)*1i)/b + (4*tan(c/
2 + (d*x)/2)*(128*a^8*b^10 - 456*a^10*b^8 + 604*a^12*b^6 - 335*a^14*b^4 + 62*a^16*b^2))/a^12)*1i)/b - (4*tan(c
/2 + (d*x)/2)*(16*a^17 + 8*a^5*b^12 - 40*a^7*b^10 + 100*a^9*b^8 - 148*a^11*b^6 + 252*a^13*b^4 - 180*a^15*b^2))
/a^12)*1i)/b - (4*(53*a^15*b + 8*a^5*b^11 - 48*a^7*b^9 + 56*a^9*b^7 + 48*a^11*b^5 - 120*a^13*b^3))/a^12 + (4*t
an(c/2 + (d*x)/2)*(62*a^16 + 8*b^16 - 68*a^2*b^14 + 255*a^4*b^12 - 550*a^6*b^10 + 873*a^8*b^8 - 1096*a^10*b^6
+ 929*a^12*b^4 - 410*a^14*b^2))/a^12)*1i)/b - (8*(15*a^14*b + 8*b^15 - 68*a^2*b^13 + 223*a^4*b^11 - 366*a^6*b^
9 + 305*a^8*b^7 - 97*a^10*b^5 - 20*a^12*b^3))/a^12 + (8*tan(c/2 + (d*x)/2)*(8*a^7*b^8 - 4*a^15 - 20*a^9*b^6 +
12*a^11*b^4 + 4*a^13*b^2))/a^12)))/(b*d) - (tan(c/2 + (d*x)/2)^2*(1/(4*a) - b^2/(8*a^3)))/d + (tan(c/2 + (d*x)
/2)*(b/(8*a^2) + (2*b*(1/(2*a) - b^2/(4*a^3)))/a))/d - (tan(c/2 + (d*x)/2)^2*(2*a*b^2 - 4*a^3) + tan(c/2 + (d*
x)/2)^3*(18*a^2*b - 8*b^3) + a^3/4 - (2*a^2*b*tan(c/2 + (d*x)/2))/3)/(16*a^4*d*tan(c/2 + (d*x)/2)^4) + (log(ta
n(c/2 + (d*x)/2))*((15*a^4)/8 + b^4 - (5*a^2*b^2)/2))/(a^5*d) - (b*tan(c/2 + (d*x)/2)^3)/(24*a^2*d) - (atan(((
(-(a + b)^5*(a - b)^5)^(1/2)*((4*tan(c/2 + (d*x)/2)*(62*a^16 + 8*b^16 - 68*a^2*b^14 + 255*a^4*b^12 - 550*a^6*b
^10 + 873*a^8*b^8 - 1096*a^10*b^6 + 929*a^12*b^4 - 410*a^14*b^2))/a^12 - (4*(53*a^15*b + 8*a^5*b^11 - 48*a^7*b
^9 + 56*a^9*b^7 + 48*a^11*b^5 - 120*a^13*b^3))/a^12 + ((-(a + b)^5*(a - b)^5)^(1/2)*((4*tan(c/2 + (d*x)/2)*(16
*a^17 + 8*a^5*b^12 - 40*a^7*b^10 + 100*a^9*b^8 - 148*a^11*b^6 + 252*a^13*b^4 - 180*a^15*b^2))/a^12 - (4*(24*a^
16*b - 32*a^4*b^13 + 184*a^6*b^11 - 440*a^8*b^9 + 543*a^10*b^7 - 345*a^12*b^5 + 58*a^14*b^3))/a^12 + ((-(a + b
)^5*(a - b)^5)^(1/2)*((4*(64*a^9*b^9 - 208*a^11*b^7 + 240*a^13*b^5 - 93*a^15*b^3))/a^12 + (4*tan(c/2 + (d*x)/2
)*(128*a^8*b^10 - 456*a^10*b^8 + 604*a^12*b^6 - 335*a^14*b^4 + 62*a^16*b^2))/a^12 + (((4*(32*a^14*b^5 - 24*a^1
6*b^3))/a^12 + (4*tan(c/2 + (d*x)/2)*(128*a^13*b^6 - 136*a^15*b^4 + 16*a^17*b^2))/a^12)*(-(a + b)^5*(a - b)^5)
^(1/2))/(a^5*b)))/(a^5*b)))/(a^5*b))*1i)/(a^5*b) + ((-(a + b)^5*(a - b)^5)^(1/2)*((4*tan(c/2 + (d*x)/2)*(62*a^
16 + 8*b^16 - 68*a^2*b^14 + 255*a^4*b^12 - 550*a^6*b^10 + 873*a^8*b^8 - 1096*a^10*b^6 + 929*a^12*b^4 - 410*a^1
4*b^2))/a^12 - (4*(53*a^15*b + 8*a^5*b^11 - 48*a^7*b^9 + 56*a^9*b^7 + 48*a^11*b^5 - 120*a^13*b^3))/a^12 + ((-(
a + b)^5*(a - b)^5)^(1/2)*((4*(24*a^16*b - 32*a^4*b^13 + 184*a^6*b^11 - 440*a^8*b^9 + 543*a^10*b^7 - 345*a^12*
b^5 + 58*a^14*b^3))/a^12 - (4*tan(c/2 + (d*x)/2)*(16*a^17 + 8*a^5*b^12 - 40*a^7*b^10 + 100*a^9*b^8 - 148*a^11*
b^6 + 252*a^13*b^4 - 180*a^15*b^2))/a^12 + ((-(a + b)^5*(a - b)^5)^(1/2)*((4*(64*a^9*b^9 - 208*a^11*b^7 + 240*
a^13*b^5 - 93*a^15*b^3))/a^12 + (4*tan(c/2 + (d*x)/2)*(128*a^8*b^10 - 456*a^10*b^8 + 604*a^12*b^6 - 335*a^14*b
^4 + 62*a^16*b^2))/a^12 - (((4*(32*a^14*b^5 - 24*a^16*b^3))/a^12 + (4*tan(c/2 + (d*x)/2)*(128*a^13*b^6 - 136*a
^15*b^4 + 16*a^17*b^2))/a^12)*(-(a + b)^5*(a - b)^5)^(1/2))/(a^5*b)))/(a^5*b)))/(a^5*b))*1i)/(a^5*b))/((8*(15*
a^14*b + 8*b^15 - 68*a^2*b^13 + 223*a^4*b^11 - 366*a^6*b^9 + 305*a^8*b^7 - 97*a^10*b^5 - 20*a^12*b^3))/a^12 -
(8*tan(c/2 + (d*x)/2)*(8*a^7*b^8 - 4*a^15 - 20*a^9*b^6 + 12*a^11*b^4 + 4*a^13*b^2))/a^12 - ((-(a + b)^5*(a - b
)^5)^(1/2)*((4*tan(c/2 + (d*x)/2)*(62*a^16 + 8*b^16 - 68*a^2*b^14 + 255*a^4*b^12 - 550*a^6*b^10 + 873*a^8*b^8
- 1096*a^10*b^6 + 929*a^12*b^4 - 410*a^14*b^2))/a^12 - (4*(53*a^15*b + 8*a^5*b^11 - 48*a^7*b^9 + 56*a^9*b^7 +
48*a^11*b^5 - 120*a^13*b^3))/a^12 + ((-(a + b)^5*(a - b)^5)^(1/2)*((4*tan(c/2 + (d*x)/2)*(16*a^17 + 8*a^5*b^12
- 40*a^7*b^10 + 100*a^9*b^8 - 148*a^11*b^6 + 252*a^13*b^4 - 180*a^15*b^2))/a^12 - (4*(24*a^16*b - 32*a^4*b^13
+ 184*a^6*b^11 - 440*a^8*b^9 + 543*a^10*b^7 - 345*a^12*b^5 + 58*a^14*b^3))/a^12 + ((-(a + b)^5*(a - b)^5)^(1/
2)*((4*(64*a^9*b^9 - 208*a^11*b^7 + 240*a^13*b^5 - 93*a^15*b^3))/a^12 + (4*tan(c/2 + (d*x)/2)*(128*a^8*b^10 -
456*a^10*b^8 + 604*a^12*b^6 - 335*a^14*b^4 + 62*a^16*b^2))/a^12 + (((4*(32*a^14*b^5 - 24*a^16*b^3))/a^12 + (4*
tan(c/2 + (d*x)/2)*(128*a^13*b^6 - 136*a^15*b^4 + 16*a^17*b^2))/a^12)*(-(a + b)^5*(a - b)^5)^(1/2))/(a^5*b)))/
(a^5*b)))/(a^5*b)))/(a^5*b) + ((-(a + b)^5*(a - b)^5)^(1/2)*((4*tan(c/2 + (d*x)/2)*(62*a^16 + 8*b^16 - 68*a^2*
b^14 + 255*a^4*b^12 - 550*a^6*b^10 + 873*a^8*b^8 - 1096*a^10*b^6 + 929*a^12*b^4 - 410*a^14*b^2))/a^12 - (4*(53
*a^15*b + 8*a^5*b^11 - 48*a^7*b^9 + 56*a^9*b^7 + 48*a^11*b^5 - 120*a^13*b^3))/a^12 + ((-(a + b)^5*(a - b)^5)^(
1/2)*((4*(24*a^16*b - 32*a^4*b^13 + 184*a^6*b^11 - 440*a^8*b^9 + 543*a^10*b^7 - 345*a^12*b^5 + 58*a^14*b^3))/a
^12 - (4*tan(c/2 + (d*x)/2)*(16*a^17 + 8*a^5*b^12 - 40*a^7*b^10 + 100*a^9*b^8 - 148*a^11*b^6 + 252*a^13*b^4 -
180*a^15*b^2))/a^12 + ((-(a + b)^5*(a - b)^5)^(1/2)*((4*(64*a^9*b^9 - 208*a^11*b^7 + 240*a^13*b^5 - 93*a^15*b^
3))/a^12 + (4*tan(c/2 + (d*x)/2)*(128*a^8*b^10 - 456*a^10*b^8 + 604*a^12*b^6 - 335*a^14*b^4 + 62*a^16*b^2))/a^
12 - (((4*(32*a^14*b^5 - 24*a^16*b^3))/a^12 + (4*tan(c/2 + (d*x)/2)*(128*a^13*b^6 - 136*a^15*b^4 + 16*a^17*b^2
))/a^12)*(-(a + b)^5*(a - b)^5)^(1/2))/(a^5*b)))/(a^5*b)))/(a^5*b)))/(a^5*b)))*(-(a + b)^5*(a - b)^5)^(1/2)*2i
)/(a^5*b*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)**5/(a+b*sin(d*x+c)),x)

[Out]

Timed out

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