3.257 \(\int \frac {\cos ^3(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^2} \, dx\)

Optimal. Leaf size=243 \[ \frac {2 b^6 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac {2 b x}{a^3}-\frac {b^5 \sin (c+d x)}{a^2 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)}-\frac {\sin (c+d x)}{a^2 d}+\frac {2 b^4 \left (5 a^2-3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}-\frac {\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}-\frac {\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)} \]

[Out]

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

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Rubi [A]  time = 0.61, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4397, 2897, 2648, 2637, 2664, 12, 2659, 208} \[ -\frac {b^5 \sin (c+d x)}{a^2 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)}+\frac {2 b^6 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac {2 b^4 \left (5 a^2-3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac {2 b x}{a^3}-\frac {\sin (c+d x)}{a^2 d}-\frac {\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}-\frac {\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]

[Out]

(2*b*x)/a^3 + (2*b^6*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d)
+ (2*b^4*(5*a^2 - 3*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)
*d) - Sin[c + d*x]/(a^2*d) - Sin[c + d*x]/(2*(a + b)^2*d*(1 - Cos[c + d*x])) - Sin[c + d*x]/(2*(a - b)^2*d*(1
+ Cos[c + d*x])) - (b^5*Sin[c + d*x])/(a^2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*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 208

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

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2648

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

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*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 4397

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

Rubi steps

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

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Mathematica [A]  time = 2.95, size = 164, normalized size = 0.67 \[ -\frac {-\frac {4 b (c+d x)}{a^3}+\frac {2 b^5 \sin (c+d x)}{a^2 (a-b)^2 (a+b)^2 (a \cos (c+d x)+b)}+\frac {2 \sin (c+d x)}{a^2}+\frac {4 b^4 \left (5 a^2-2 b^2\right ) \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{5/2}}+\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{(a-b)^2}+\frac {\cot \left (\frac {1}{2} (c+d x)\right )}{(a+b)^2}}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]

[Out]

-1/2*((-4*b*(c + d*x))/a^3 + (4*b^4*(5*a^2 - 2*b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3
*(a^2 - b^2)^(5/2)) + Cot[(c + d*x)/2]/(a + b)^2 + (2*Sin[c + d*x])/a^2 + (2*b^5*Sin[c + d*x])/(a^2*(a - b)^2*
(a + b)^2*(b + a*Cos[c + d*x])) + Tan[(c + d*x)/2]/(a - b)^2)/d

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fricas [A]  time = 0.59, size = 857, normalized size = 3.53 \[ \left [-\frac {4 \, a^{7} b - 6 \, a^{5} b^{3} + 6 \, a^{3} b^{5} - 4 \, a b^{7} - 2 \, {\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left (5 \, a^{2} b^{5} - 2 \, b^{7} + {\left (5 \, a^{3} b^{4} - 2 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) \sin \left (d x + c\right ) - 2 \, {\left (3 \, a^{7} b - 5 \, a^{5} b^{3} + 4 \, a^{3} b^{5} - 2 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, a^{8} - 5 \, a^{6} b^{2} + 4 \, a^{4} b^{4} - a^{2} b^{6}\right )} \cos \left (d x + c\right ) - 4 \, {\left ({\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} d x \cos \left (d x + c\right ) + {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} d x\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{10} - 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} - a^{4} b^{6}\right )} d \cos \left (d x + c\right ) + {\left (a^{9} b - 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} - a^{3} b^{7}\right )} d\right )} \sin \left (d x + c\right )}, -\frac {2 \, a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - 2 \, a b^{7} - {\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} \cos \left (d x + c\right )^{3} - {\left (5 \, a^{2} b^{5} - 2 \, b^{7} + {\left (5 \, a^{3} b^{4} - 2 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - {\left (3 \, a^{7} b - 5 \, a^{5} b^{3} + 4 \, a^{3} b^{5} - 2 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, a^{8} - 5 \, a^{6} b^{2} + 4 \, a^{4} b^{4} - a^{2} b^{6}\right )} \cos \left (d x + c\right ) - 2 \, {\left ({\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} d x \cos \left (d x + c\right ) + {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} d x\right )} \sin \left (d x + c\right )}{{\left ({\left (a^{10} - 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} - a^{4} b^{6}\right )} d \cos \left (d x + c\right ) + {\left (a^{9} b - 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} - a^{3} b^{7}\right )} d\right )} \sin \left (d x + c\right )}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3/(a*sin(d*x+c)+b*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

[-1/2*(4*a^7*b - 6*a^5*b^3 + 6*a^3*b^5 - 4*a*b^7 - 2*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*cos(d*x + c)^3 +
(5*a^2*b^5 - 2*b^7 + (5*a^3*b^4 - 2*a*b^6)*cos(d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^
2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2
*a*b*cos(d*x + c) + b^2))*sin(d*x + c) - 2*(3*a^7*b - 5*a^5*b^3 + 4*a^3*b^5 - 2*a*b^7)*cos(d*x + c)^2 + 2*(2*a
^8 - 5*a^6*b^2 + 4*a^4*b^4 - a^2*b^6)*cos(d*x + c) - 4*((a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*x*cos(d*x +
c) + (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*d*x)*sin(d*x + c))/(((a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*d*c
os(d*x + c) + (a^9*b - 3*a^7*b^3 + 3*a^5*b^5 - a^3*b^7)*d)*sin(d*x + c)), -(2*a^7*b - 3*a^5*b^3 + 3*a^3*b^5 -
2*a*b^7 - (a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*cos(d*x + c)^3 - (5*a^2*b^5 - 2*b^7 + (5*a^3*b^4 - 2*a*b^6)*
cos(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c)))*sin(d
*x + c) - (3*a^7*b - 5*a^5*b^3 + 4*a^3*b^5 - 2*a*b^7)*cos(d*x + c)^2 + (2*a^8 - 5*a^6*b^2 + 4*a^4*b^4 - a^2*b^
6)*cos(d*x + c) - 2*((a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*x*cos(d*x + c) + (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b
^6 - b^8)*d*x)*sin(d*x + c))/(((a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*d*cos(d*x + c) + (a^9*b - 3*a^7*b^3 +
3*a^5*b^5 - a^3*b^7)*d)*sin(d*x + c))]

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giac [B]  time = 1.13, size = 1362, normalized size = 5.60 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3/(a*sin(d*x+c)+b*tan(d*x+c))^2,x, algorithm="giac")

[Out]

-1/2*(2*((2*a^4*b - 2*a^3*b^2 - 4*a^2*b^3 - a*b^4 + 2*b^5)*sqrt(-a^2 + b^2)*abs(a^7 - 2*a^5*b^2 + a^3*b^4)*abs
(a - b) - (2*a^11*b - 2*a^10*b^2 - 8*a^9*b^3 + 13*a^8*b^4 + 12*a^7*b^5 - 24*a^6*b^6 - 8*a^5*b^7 + 17*a^4*b^8 +
2*a^3*b^9 - 4*a^2*b^10)*sqrt(-a^2 + b^2)*abs(a - b))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(tan(1/2*d*x +
1/2*c)/sqrt(-(a^6*b - 2*a^4*b^3 + a^2*b^5 + sqrt((a^7 + a^6*b - 2*a^5*b^2 - 2*a^4*b^3 + a^3*b^4 + a^2*b^5)*(a
^7 - a^6*b - 2*a^5*b^2 + 2*a^4*b^3 + a^3*b^4 - a^2*b^5) + (a^6*b - 2*a^4*b^3 + a^2*b^5)^2))/(a^7 - a^6*b - 2*a
^5*b^2 + 2*a^4*b^3 + a^3*b^4 - a^2*b^5))))/((a^7 - 2*a^5*b^2 + a^3*b^4)^2*(a^2 - 2*a*b + b^2) + (a^8*b - 2*a^7
*b^2 - a^6*b^3 + 4*a^5*b^4 - a^4*b^5 - 2*a^3*b^6 + a^2*b^7)*abs(a^7 - 2*a^5*b^2 + a^3*b^4)) + 2*(2*a^11*b - 2*
a^10*b^2 - 8*a^9*b^3 + 13*a^8*b^4 + 12*a^7*b^5 - 24*a^6*b^6 - 8*a^5*b^7 + 17*a^4*b^8 + 2*a^3*b^9 - 4*a^2*b^10
+ 2*a^4*b*abs(a^7 - 2*a^5*b^2 + a^3*b^4) - 2*a^3*b^2*abs(a^7 - 2*a^5*b^2 + a^3*b^4) - 4*a^2*b^3*abs(a^7 - 2*a^
5*b^2 + a^3*b^4) - a*b^4*abs(a^7 - 2*a^5*b^2 + a^3*b^4) + 2*b^5*abs(a^7 - 2*a^5*b^2 + a^3*b^4))*(pi*floor(1/2*
(d*x + c)/pi + 1/2) + arctan(tan(1/2*d*x + 1/2*c)/sqrt(-(a^6*b - 2*a^4*b^3 + a^2*b^5 - sqrt((a^7 + a^6*b - 2*a
^5*b^2 - 2*a^4*b^3 + a^3*b^4 + a^2*b^5)*(a^7 - a^6*b - 2*a^5*b^2 + 2*a^4*b^3 + a^3*b^4 - a^2*b^5) + (a^6*b - 2
*a^4*b^3 + a^2*b^5)^2))/(a^7 - a^6*b - 2*a^5*b^2 + 2*a^4*b^3 + a^3*b^4 - a^2*b^5))))/(a^6*b*abs(a^7 - 2*a^5*b^
2 + a^3*b^4) - 2*a^4*b^3*abs(a^7 - 2*a^5*b^2 + a^3*b^4) + a^2*b^5*abs(a^7 - 2*a^5*b^2 + a^3*b^4) - (a^7 - 2*a^
5*b^2 + a^3*b^4)^2) + tan(1/2*d*x + 1/2*c)/(a^2 - 2*a*b + b^2) + (5*a^5*tan(1/2*d*x + 1/2*c)^4 - 7*a^4*b*tan(1
/2*d*x + 1/2*c)^4 - 5*a^3*b^2*tan(1/2*d*x + 1/2*c)^4 + 7*a^2*b^3*tan(1/2*d*x + 1/2*c)^4 + 4*a*b^4*tan(1/2*d*x
+ 1/2*c)^4 - 8*b^5*tan(1/2*d*x + 1/2*c)^4 - 4*a^5*tan(1/2*d*x + 1/2*c)^2 - 6*a^4*b*tan(1/2*d*x + 1/2*c)^2 + 12
*a^3*b^2*tan(1/2*d*x + 1/2*c)^2 + 6*a^2*b^3*tan(1/2*d*x + 1/2*c)^2 - 4*a*b^4*tan(1/2*d*x + 1/2*c)^2 - 8*b^5*ta
n(1/2*d*x + 1/2*c)^2 - a^5 + a^4*b + a^3*b^2 - a^2*b^3)/((a^6 - 2*a^4*b^2 + a^2*b^4)*(a*tan(1/2*d*x + 1/2*c)^5
- b*tan(1/2*d*x + 1/2*c)^5 - 2*b*tan(1/2*d*x + 1/2*c)^3 - a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))))/
d

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maple [A]  time = 0.24, size = 291, normalized size = 1.20 \[ -\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \left (a^{2}-2 a b +b^{2}\right )}+\frac {2 b^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (a +b \right )^{2} \left (a -b \right )^{2} a^{2} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )}+\frac {10 b^{4} \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{d \left (a +b \right )^{2} \left (a -b \right )^{2} a \sqrt {\left (a +b \right ) \left (a -b \right )}}-\frac {4 b^{6} \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{d \left (a +b \right )^{2} \left (a -b \right )^{2} a^{3} \sqrt {\left (a +b \right ) \left (a -b \right )}}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {4 b \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}-\frac {1}{2 d \left (a +b \right )^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3/(a*sin(d*x+c)+b*tan(d*x+c))^2,x)

[Out]

-1/2/d/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)+2/d*b^5/(a+b)^2/(a-b)^2/a^2*tan(1/2*d*x+1/2*c)/(tan(1/2*d*x+1/2*c)^2
*a-b*tan(1/2*d*x+1/2*c)^2-a-b)+10/d*b^4/(a+b)^2/(a-b)^2/a/((a+b)*(a-b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)
/((a+b)*(a-b))^(1/2))-4/d*b^6/(a+b)^2/(a-b)^2/a^3/((a+b)*(a-b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a+b)*
(a-b))^(1/2))-2/d/a^2*tan(1/2*d*x+1/2*c)/(tan(1/2*d*x+1/2*c)^2+1)+4/d/a^3*b*arctan(tan(1/2*d*x+1/2*c))-1/2/d/(
a+b)^2/tan(1/2*d*x+1/2*c)

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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)^3/(a*sin(d*x+c)+b*tan(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*a^2-4*b^2>0)', see `assume?`
for more details)Is 4*a^2-4*b^2 positive or negative?

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mupad [B]  time = 5.44, size = 7329, normalized size = 30.16 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^3/(a*sin(c + d*x) + b*tan(c + d*x))^2,x)

[Out]

((a^2 - 2*a*b + b^2)/(a + b) + (2*tan(c/2 + (d*x)/2)^2*(2*a*b^4 + 3*a^4*b + 2*a^5 + 4*b^5 - 3*a^2*b^3 - 6*a^3*
b^2))/(a^2*(a + b)^2) - (tan(c/2 + (d*x)/2)^4*(4*a*b^4 - 7*a^4*b + 5*a^5 - 8*b^5 + 7*a^2*b^3 - 5*a^3*b^2))/(a^
2*(a + b)^2))/(d*(tan(c/2 + (d*x)/2)^5*(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3) - tan(c/2 + (d*x)/2)^3*(4*a^2*b - 8
*a*b^2 + 4*b^3) + tan(c/2 + (d*x)/2)*(2*a*b^2 + 2*a^2*b - 2*a^3 - 2*b^3))) - tan(c/2 + (d*x)/2)/(2*d*(a - b)^2
) + (4*b*atan((1920*a^7*b^22*tan(c/2 + (d*x)/2))/(1920*a^7*b^22 - 1920*a^8*b^21 - 16640*a^9*b^20 + 16640*a^10*
b^19 + 62080*a^11*b^18 - 62080*a^12*b^17 - 131072*a^13*b^16 + 131072*a^14*b^15 + 172672*a^15*b^14 - 172672*a^1
6*b^13 - 147200*a^17*b^12 + 147200*a^18*b^11 + 81280*a^19*b^10 - 81280*a^20*b^9 - 28160*a^21*b^8 + 28160*a^22*
b^7 + 5632*a^23*b^6 - 5632*a^24*b^5 - 512*a^25*b^4 + 512*a^26*b^3) - (1920*a^8*b^21*tan(c/2 + (d*x)/2))/(1920*
a^7*b^22 - 1920*a^8*b^21 - 16640*a^9*b^20 + 16640*a^10*b^19 + 62080*a^11*b^18 - 62080*a^12*b^17 - 131072*a^13*
b^16 + 131072*a^14*b^15 + 172672*a^15*b^14 - 172672*a^16*b^13 - 147200*a^17*b^12 + 147200*a^18*b^11 + 81280*a^
19*b^10 - 81280*a^20*b^9 - 28160*a^21*b^8 + 28160*a^22*b^7 + 5632*a^23*b^6 - 5632*a^24*b^5 - 512*a^25*b^4 + 51
2*a^26*b^3) - (16640*a^9*b^20*tan(c/2 + (d*x)/2))/(1920*a^7*b^22 - 1920*a^8*b^21 - 16640*a^9*b^20 + 16640*a^10
*b^19 + 62080*a^11*b^18 - 62080*a^12*b^17 - 131072*a^13*b^16 + 131072*a^14*b^15 + 172672*a^15*b^14 - 172672*a^
16*b^13 - 147200*a^17*b^12 + 147200*a^18*b^11 + 81280*a^19*b^10 - 81280*a^20*b^9 - 28160*a^21*b^8 + 28160*a^22
*b^7 + 5632*a^23*b^6 - 5632*a^24*b^5 - 512*a^25*b^4 + 512*a^26*b^3) + (16640*a^10*b^19*tan(c/2 + (d*x)/2))/(19
20*a^7*b^22 - 1920*a^8*b^21 - 16640*a^9*b^20 + 16640*a^10*b^19 + 62080*a^11*b^18 - 62080*a^12*b^17 - 131072*a^
13*b^16 + 131072*a^14*b^15 + 172672*a^15*b^14 - 172672*a^16*b^13 - 147200*a^17*b^12 + 147200*a^18*b^11 + 81280
*a^19*b^10 - 81280*a^20*b^9 - 28160*a^21*b^8 + 28160*a^22*b^7 + 5632*a^23*b^6 - 5632*a^24*b^5 - 512*a^25*b^4 +
512*a^26*b^3) + (62080*a^11*b^18*tan(c/2 + (d*x)/2))/(1920*a^7*b^22 - 1920*a^8*b^21 - 16640*a^9*b^20 + 16640*
a^10*b^19 + 62080*a^11*b^18 - 62080*a^12*b^17 - 131072*a^13*b^16 + 131072*a^14*b^15 + 172672*a^15*b^14 - 17267
2*a^16*b^13 - 147200*a^17*b^12 + 147200*a^18*b^11 + 81280*a^19*b^10 - 81280*a^20*b^9 - 28160*a^21*b^8 + 28160*
a^22*b^7 + 5632*a^23*b^6 - 5632*a^24*b^5 - 512*a^25*b^4 + 512*a^26*b^3) - (62080*a^12*b^17*tan(c/2 + (d*x)/2))
/(1920*a^7*b^22 - 1920*a^8*b^21 - 16640*a^9*b^20 + 16640*a^10*b^19 + 62080*a^11*b^18 - 62080*a^12*b^17 - 13107
2*a^13*b^16 + 131072*a^14*b^15 + 172672*a^15*b^14 - 172672*a^16*b^13 - 147200*a^17*b^12 + 147200*a^18*b^11 + 8
1280*a^19*b^10 - 81280*a^20*b^9 - 28160*a^21*b^8 + 28160*a^22*b^7 + 5632*a^23*b^6 - 5632*a^24*b^5 - 512*a^25*b
^4 + 512*a^26*b^3) - (131072*a^13*b^16*tan(c/2 + (d*x)/2))/(1920*a^7*b^22 - 1920*a^8*b^21 - 16640*a^9*b^20 + 1
6640*a^10*b^19 + 62080*a^11*b^18 - 62080*a^12*b^17 - 131072*a^13*b^16 + 131072*a^14*b^15 + 172672*a^15*b^14 -
172672*a^16*b^13 - 147200*a^17*b^12 + 147200*a^18*b^11 + 81280*a^19*b^10 - 81280*a^20*b^9 - 28160*a^21*b^8 + 2
8160*a^22*b^7 + 5632*a^23*b^6 - 5632*a^24*b^5 - 512*a^25*b^4 + 512*a^26*b^3) + (131072*a^14*b^15*tan(c/2 + (d*
x)/2))/(1920*a^7*b^22 - 1920*a^8*b^21 - 16640*a^9*b^20 + 16640*a^10*b^19 + 62080*a^11*b^18 - 62080*a^12*b^17 -
131072*a^13*b^16 + 131072*a^14*b^15 + 172672*a^15*b^14 - 172672*a^16*b^13 - 147200*a^17*b^12 + 147200*a^18*b^
11 + 81280*a^19*b^10 - 81280*a^20*b^9 - 28160*a^21*b^8 + 28160*a^22*b^7 + 5632*a^23*b^6 - 5632*a^24*b^5 - 512*
a^25*b^4 + 512*a^26*b^3) + (172672*a^15*b^14*tan(c/2 + (d*x)/2))/(1920*a^7*b^22 - 1920*a^8*b^21 - 16640*a^9*b^
20 + 16640*a^10*b^19 + 62080*a^11*b^18 - 62080*a^12*b^17 - 131072*a^13*b^16 + 131072*a^14*b^15 + 172672*a^15*b
^14 - 172672*a^16*b^13 - 147200*a^17*b^12 + 147200*a^18*b^11 + 81280*a^19*b^10 - 81280*a^20*b^9 - 28160*a^21*b
^8 + 28160*a^22*b^7 + 5632*a^23*b^6 - 5632*a^24*b^5 - 512*a^25*b^4 + 512*a^26*b^3) - (172672*a^16*b^13*tan(c/2
+ (d*x)/2))/(1920*a^7*b^22 - 1920*a^8*b^21 - 16640*a^9*b^20 + 16640*a^10*b^19 + 62080*a^11*b^18 - 62080*a^12*
b^17 - 131072*a^13*b^16 + 131072*a^14*b^15 + 172672*a^15*b^14 - 172672*a^16*b^13 - 147200*a^17*b^12 + 147200*a
^18*b^11 + 81280*a^19*b^10 - 81280*a^20*b^9 - 28160*a^21*b^8 + 28160*a^22*b^7 + 5632*a^23*b^6 - 5632*a^24*b^5
- 512*a^25*b^4 + 512*a^26*b^3) - (147200*a^17*b^12*tan(c/2 + (d*x)/2))/(1920*a^7*b^22 - 1920*a^8*b^21 - 16640*
a^9*b^20 + 16640*a^10*b^19 + 62080*a^11*b^18 - 62080*a^12*b^17 - 131072*a^13*b^16 + 131072*a^14*b^15 + 172672*
a^15*b^14 - 172672*a^16*b^13 - 147200*a^17*b^12 + 147200*a^18*b^11 + 81280*a^19*b^10 - 81280*a^20*b^9 - 28160*
a^21*b^8 + 28160*a^22*b^7 + 5632*a^23*b^6 - 5632*a^24*b^5 - 512*a^25*b^4 + 512*a^26*b^3) + (147200*a^18*b^11*t
an(c/2 + (d*x)/2))/(1920*a^7*b^22 - 1920*a^8*b^21 - 16640*a^9*b^20 + 16640*a^10*b^19 + 62080*a^11*b^18 - 62080
*a^12*b^17 - 131072*a^13*b^16 + 131072*a^14*b^15 + 172672*a^15*b^14 - 172672*a^16*b^13 - 147200*a^17*b^12 + 14
7200*a^18*b^11 + 81280*a^19*b^10 - 81280*a^20*b^9 - 28160*a^21*b^8 + 28160*a^22*b^7 + 5632*a^23*b^6 - 5632*a^2
4*b^5 - 512*a^25*b^4 + 512*a^26*b^3) + (81280*a^19*b^10*tan(c/2 + (d*x)/2))/(1920*a^7*b^22 - 1920*a^8*b^21 - 1
6640*a^9*b^20 + 16640*a^10*b^19 + 62080*a^11*b^18 - 62080*a^12*b^17 - 131072*a^13*b^16 + 131072*a^14*b^15 + 17
2672*a^15*b^14 - 172672*a^16*b^13 - 147200*a^17*b^12 + 147200*a^18*b^11 + 81280*a^19*b^10 - 81280*a^20*b^9 - 2
8160*a^21*b^8 + 28160*a^22*b^7 + 5632*a^23*b^6 - 5632*a^24*b^5 - 512*a^25*b^4 + 512*a^26*b^3) - (81280*a^20*b^
9*tan(c/2 + (d*x)/2))/(1920*a^7*b^22 - 1920*a^8*b^21 - 16640*a^9*b^20 + 16640*a^10*b^19 + 62080*a^11*b^18 - 62
080*a^12*b^17 - 131072*a^13*b^16 + 131072*a^14*b^15 + 172672*a^15*b^14 - 172672*a^16*b^13 - 147200*a^17*b^12 +
147200*a^18*b^11 + 81280*a^19*b^10 - 81280*a^20*b^9 - 28160*a^21*b^8 + 28160*a^22*b^7 + 5632*a^23*b^6 - 5632*
a^24*b^5 - 512*a^25*b^4 + 512*a^26*b^3) - (28160*a^21*b^8*tan(c/2 + (d*x)/2))/(1920*a^7*b^22 - 1920*a^8*b^21 -
16640*a^9*b^20 + 16640*a^10*b^19 + 62080*a^11*b^18 - 62080*a^12*b^17 - 131072*a^13*b^16 + 131072*a^14*b^15 +
172672*a^15*b^14 - 172672*a^16*b^13 - 147200*a^17*b^12 + 147200*a^18*b^11 + 81280*a^19*b^10 - 81280*a^20*b^9 -
28160*a^21*b^8 + 28160*a^22*b^7 + 5632*a^23*b^6 - 5632*a^24*b^5 - 512*a^25*b^4 + 512*a^26*b^3) + (28160*a^22*
b^7*tan(c/2 + (d*x)/2))/(1920*a^7*b^22 - 1920*a^8*b^21 - 16640*a^9*b^20 + 16640*a^10*b^19 + 62080*a^11*b^18 -
62080*a^12*b^17 - 131072*a^13*b^16 + 131072*a^14*b^15 + 172672*a^15*b^14 - 172672*a^16*b^13 - 147200*a^17*b^12
+ 147200*a^18*b^11 + 81280*a^19*b^10 - 81280*a^20*b^9 - 28160*a^21*b^8 + 28160*a^22*b^7 + 5632*a^23*b^6 - 563
2*a^24*b^5 - 512*a^25*b^4 + 512*a^26*b^3) + (5632*a^23*b^6*tan(c/2 + (d*x)/2))/(1920*a^7*b^22 - 1920*a^8*b^21
- 16640*a^9*b^20 + 16640*a^10*b^19 + 62080*a^11*b^18 - 62080*a^12*b^17 - 131072*a^13*b^16 + 131072*a^14*b^15 +
172672*a^15*b^14 - 172672*a^16*b^13 - 147200*a^17*b^12 + 147200*a^18*b^11 + 81280*a^19*b^10 - 81280*a^20*b^9
- 28160*a^21*b^8 + 28160*a^22*b^7 + 5632*a^23*b^6 - 5632*a^24*b^5 - 512*a^25*b^4 + 512*a^26*b^3) - (5632*a^24*
b^5*tan(c/2 + (d*x)/2))/(1920*a^7*b^22 - 1920*a^8*b^21 - 16640*a^9*b^20 + 16640*a^10*b^19 + 62080*a^11*b^18 -
62080*a^12*b^17 - 131072*a^13*b^16 + 131072*a^14*b^15 + 172672*a^15*b^14 - 172672*a^16*b^13 - 147200*a^17*b^12
+ 147200*a^18*b^11 + 81280*a^19*b^10 - 81280*a^20*b^9 - 28160*a^21*b^8 + 28160*a^22*b^7 + 5632*a^23*b^6 - 563
2*a^24*b^5 - 512*a^25*b^4 + 512*a^26*b^3) - (512*a^25*b^4*tan(c/2 + (d*x)/2))/(1920*a^7*b^22 - 1920*a^8*b^21 -
16640*a^9*b^20 + 16640*a^10*b^19 + 62080*a^11*b^18 - 62080*a^12*b^17 - 131072*a^13*b^16 + 131072*a^14*b^15 +
172672*a^15*b^14 - 172672*a^16*b^13 - 147200*a^17*b^12 + 147200*a^18*b^11 + 81280*a^19*b^10 - 81280*a^20*b^9 -
28160*a^21*b^8 + 28160*a^22*b^7 + 5632*a^23*b^6 - 5632*a^24*b^5 - 512*a^25*b^4 + 512*a^26*b^3) + (512*a^26*b^
3*tan(c/2 + (d*x)/2))/(1920*a^7*b^22 - 1920*a^8*b^21 - 16640*a^9*b^20 + 16640*a^10*b^19 + 62080*a^11*b^18 - 62
080*a^12*b^17 - 131072*a^13*b^16 + 131072*a^14*b^15 + 172672*a^15*b^14 - 172672*a^16*b^13 - 147200*a^17*b^12 +
147200*a^18*b^11 + 81280*a^19*b^10 - 81280*a^20*b^9 - 28160*a^21*b^8 + 28160*a^22*b^7 + 5632*a^23*b^6 - 5632*
a^24*b^5 - 512*a^25*b^4 + 512*a^26*b^3)))/(a^3*d) + (b^4*atan(((b^4*(tan(c/2 + (d*x)/2)*(256*a^6*b^25 - 512*a^
7*b^24 - 2304*a^8*b^23 + 5120*a^9*b^22 + 8480*a^10*b^21 - 22560*a^11*b^20 - 15040*a^12*b^19 + 57280*a^13*b^18
+ 7520*a^14*b^17 - 92000*a^15*b^16 + 22016*a^16*b^15 + 96256*a^17*b^14 - 53920*a^18*b^13 - 64352*a^19*b^12 + 5
9840*a^20*b^11 + 24640*a^21*b^10 - 39520*a^22*b^9 - 2720*a^23*b^8 + 16000*a^24*b^7 - 1920*a^25*b^6 - 3712*a^26
*b^5 + 896*a^27*b^4 + 384*a^28*b^3 - 128*a^29*b^2) - (b^4*(5*a^2 - 2*b^2)*((a + b)^5*(a - b)^5)^(1/2)*(96*a^11
*b^22 - 64*a^10*b^23 - 64*a^32*b + 640*a^12*b^21 - 1120*a^13*b^20 - 2624*a^14*b^19 + 5568*a^15*b^18 + 5568*a^1
6*b^17 - 15744*a^17*b^16 - 5760*a^18*b^15 + 28224*a^19*b^14 - 33600*a^21*b^12 + 8064*a^22*b^11 + 26880*a^23*b^
10 - 11136*a^24*b^9 - 14208*a^25*b^8 + 7872*a^26*b^7 + 4704*a^27*b^6 - 3200*a^28*b^5 - 864*a^29*b^4 + 704*a^30
*b^3 + 64*a^31*b^2 + (b^4*tan(c/2 + (d*x)/2)*(5*a^2 - 2*b^2)*((a + b)^5*(a - b)^5)^(1/2)*(128*a^13*b^22 - 64*a
^12*b^23 - 64*a^34*b + 576*a^14*b^21 - 1280*a^15*b^20 - 2240*a^16*b^19 + 5760*a^17*b^18 + 4800*a^18*b^17 - 153
60*a^19*b^16 - 5760*a^20*b^15 + 26880*a^21*b^14 + 2688*a^22*b^13 - 32256*a^23*b^12 + 2688*a^24*b^11 + 26880*a^
25*b^10 - 5760*a^26*b^9 - 15360*a^27*b^8 + 4800*a^28*b^7 + 5760*a^29*b^6 - 2240*a^30*b^5 - 1280*a^31*b^4 + 576
*a^32*b^3 + 128*a^33*b^2))/(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)))/(a^13 - a^3*
b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2))*(5*a^2 - 2*b^2)*((a + b)^5*(a - b)^5)^(1/2)*1i)/(a^1
3 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2) + (b^4*(tan(c/2 + (d*x)/2)*(256*a^6*b^25 - 51
2*a^7*b^24 - 2304*a^8*b^23 + 5120*a^9*b^22 + 8480*a^10*b^21 - 22560*a^11*b^20 - 15040*a^12*b^19 + 57280*a^13*b
^18 + 7520*a^14*b^17 - 92000*a^15*b^16 + 22016*a^16*b^15 + 96256*a^17*b^14 - 53920*a^18*b^13 - 64352*a^19*b^12
+ 59840*a^20*b^11 + 24640*a^21*b^10 - 39520*a^22*b^9 - 2720*a^23*b^8 + 16000*a^24*b^7 - 1920*a^25*b^6 - 3712*
a^26*b^5 + 896*a^27*b^4 + 384*a^28*b^3 - 128*a^29*b^2) - (b^4*(5*a^2 - 2*b^2)*((a + b)^5*(a - b)^5)^(1/2)*(64*
a^32*b + 64*a^10*b^23 - 96*a^11*b^22 - 640*a^12*b^21 + 1120*a^13*b^20 + 2624*a^14*b^19 - 5568*a^15*b^18 - 5568
*a^16*b^17 + 15744*a^17*b^16 + 5760*a^18*b^15 - 28224*a^19*b^14 + 33600*a^21*b^12 - 8064*a^22*b^11 - 26880*a^2
3*b^10 + 11136*a^24*b^9 + 14208*a^25*b^8 - 7872*a^26*b^7 - 4704*a^27*b^6 + 3200*a^28*b^5 + 864*a^29*b^4 - 704*
a^30*b^3 - 64*a^31*b^2 + (b^4*tan(c/2 + (d*x)/2)*(5*a^2 - 2*b^2)*((a + b)^5*(a - b)^5)^(1/2)*(128*a^13*b^22 -
64*a^12*b^23 - 64*a^34*b + 576*a^14*b^21 - 1280*a^15*b^20 - 2240*a^16*b^19 + 5760*a^17*b^18 + 4800*a^18*b^17 -
15360*a^19*b^16 - 5760*a^20*b^15 + 26880*a^21*b^14 + 2688*a^22*b^13 - 32256*a^23*b^12 + 2688*a^24*b^11 + 2688
0*a^25*b^10 - 5760*a^26*b^9 - 15360*a^27*b^8 + 4800*a^28*b^7 + 5760*a^29*b^6 - 2240*a^30*b^5 - 1280*a^31*b^4 +
576*a^32*b^3 + 128*a^33*b^2))/(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)))/(a^13 -
a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2))*(5*a^2 - 2*b^2)*((a + b)^5*(a - b)^5)^(1/2)*1i)/
(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2))/(512*a^4*b^25 - 768*a^5*b^24 - 5120*a^6*
b^23 + 7040*a^7*b^22 + 22912*a^8*b^21 - 27904*a^9*b^20 - 61184*a^10*b^19 + 63872*a^11*b^18 + 108160*a^12*b^17
- 94720*a^13*b^16 - 131072*a^14*b^15 + 96128*a^15*b^14 + 108160*a^16*b^13 - 67840*a^17*b^12 - 58112*a^18*b^11
+ 32384*a^19*b^10 + 18304*a^20*b^9 - 9472*a^21*b^8 - 2560*a^22*b^7 + 1280*a^23*b^6 + (b^4*(tan(c/2 + (d*x)/2)*
(256*a^6*b^25 - 512*a^7*b^24 - 2304*a^8*b^23 + 5120*a^9*b^22 + 8480*a^10*b^21 - 22560*a^11*b^20 - 15040*a^12*b
^19 + 57280*a^13*b^18 + 7520*a^14*b^17 - 92000*a^15*b^16 + 22016*a^16*b^15 + 96256*a^17*b^14 - 53920*a^18*b^13
- 64352*a^19*b^12 + 59840*a^20*b^11 + 24640*a^21*b^10 - 39520*a^22*b^9 - 2720*a^23*b^8 + 16000*a^24*b^7 - 192
0*a^25*b^6 - 3712*a^26*b^5 + 896*a^27*b^4 + 384*a^28*b^3 - 128*a^29*b^2) - (b^4*(5*a^2 - 2*b^2)*((a + b)^5*(a
- b)^5)^(1/2)*(96*a^11*b^22 - 64*a^10*b^23 - 64*a^32*b + 640*a^12*b^21 - 1120*a^13*b^20 - 2624*a^14*b^19 + 556
8*a^15*b^18 + 5568*a^16*b^17 - 15744*a^17*b^16 - 5760*a^18*b^15 + 28224*a^19*b^14 - 33600*a^21*b^12 + 8064*a^2
2*b^11 + 26880*a^23*b^10 - 11136*a^24*b^9 - 14208*a^25*b^8 + 7872*a^26*b^7 + 4704*a^27*b^6 - 3200*a^28*b^5 - 8
64*a^29*b^4 + 704*a^30*b^3 + 64*a^31*b^2 + (b^4*tan(c/2 + (d*x)/2)*(5*a^2 - 2*b^2)*((a + b)^5*(a - b)^5)^(1/2)
*(128*a^13*b^22 - 64*a^12*b^23 - 64*a^34*b + 576*a^14*b^21 - 1280*a^15*b^20 - 2240*a^16*b^19 + 5760*a^17*b^18
+ 4800*a^18*b^17 - 15360*a^19*b^16 - 5760*a^20*b^15 + 26880*a^21*b^14 + 2688*a^22*b^13 - 32256*a^23*b^12 + 268
8*a^24*b^11 + 26880*a^25*b^10 - 5760*a^26*b^9 - 15360*a^27*b^8 + 4800*a^28*b^7 + 5760*a^29*b^6 - 2240*a^30*b^5
- 1280*a^31*b^4 + 576*a^32*b^3 + 128*a^33*b^2))/(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^
11*b^2)))/(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2))*(5*a^2 - 2*b^2)*((a + b)^5*(a
- b)^5)^(1/2))/(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2) - (b^4*(tan(c/2 + (d*x)/2)
*(256*a^6*b^25 - 512*a^7*b^24 - 2304*a^8*b^23 + 5120*a^9*b^22 + 8480*a^10*b^21 - 22560*a^11*b^20 - 15040*a^12*
b^19 + 57280*a^13*b^18 + 7520*a^14*b^17 - 92000*a^15*b^16 + 22016*a^16*b^15 + 96256*a^17*b^14 - 53920*a^18*b^1
3 - 64352*a^19*b^12 + 59840*a^20*b^11 + 24640*a^21*b^10 - 39520*a^22*b^9 - 2720*a^23*b^8 + 16000*a^24*b^7 - 19
20*a^25*b^6 - 3712*a^26*b^5 + 896*a^27*b^4 + 384*a^28*b^3 - 128*a^29*b^2) - (b^4*(5*a^2 - 2*b^2)*((a + b)^5*(a
- b)^5)^(1/2)*(64*a^32*b + 64*a^10*b^23 - 96*a^11*b^22 - 640*a^12*b^21 + 1120*a^13*b^20 + 2624*a^14*b^19 - 55
68*a^15*b^18 - 5568*a^16*b^17 + 15744*a^17*b^16 + 5760*a^18*b^15 - 28224*a^19*b^14 + 33600*a^21*b^12 - 8064*a^
22*b^11 - 26880*a^23*b^10 + 11136*a^24*b^9 + 14208*a^25*b^8 - 7872*a^26*b^7 - 4704*a^27*b^6 + 3200*a^28*b^5 +
864*a^29*b^4 - 704*a^30*b^3 - 64*a^31*b^2 + (b^4*tan(c/2 + (d*x)/2)*(5*a^2 - 2*b^2)*((a + b)^5*(a - b)^5)^(1/2
)*(128*a^13*b^22 - 64*a^12*b^23 - 64*a^34*b + 576*a^14*b^21 - 1280*a^15*b^20 - 2240*a^16*b^19 + 5760*a^17*b^18
+ 4800*a^18*b^17 - 15360*a^19*b^16 - 5760*a^20*b^15 + 26880*a^21*b^14 + 2688*a^22*b^13 - 32256*a^23*b^12 + 26
88*a^24*b^11 + 26880*a^25*b^10 - 5760*a^26*b^9 - 15360*a^27*b^8 + 4800*a^28*b^7 + 5760*a^29*b^6 - 2240*a^30*b^
5 - 1280*a^31*b^4 + 576*a^32*b^3 + 128*a^33*b^2))/(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a
^11*b^2)))/(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2))*(5*a^2 - 2*b^2)*((a + b)^5*(a
- b)^5)^(1/2))/(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)))*(5*a^2 - 2*b^2)*((a + b
)^5*(a - b)^5)^(1/2)*2i)/(d*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2))

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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)**3/(a*sin(d*x+c)+b*tan(d*x+c))**2,x)

[Out]

Timed out

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