3.559 \(\int \frac{\cos ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx\)

Optimal. Leaf size=235 \[ \frac{3 b \left (a^2-b^2\right ) \left (a^2+5 b^2\right )}{8 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left (b \left (a^2-5 b^2\right )-3 a \left (a^2+3 b^2\right ) \tan (c+d x)\right )}{8 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) (a \tan (c+d x)+b)}{4 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac{6 a b^5 \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{3 x \left (5 a^4 b^2+15 a^2 b^4+a^6-5 b^6\right )}{8 \left (a^2+b^2\right )^4} \]

[Out]

(3*(a^6 + 5*a^4*b^2 + 15*a^2*b^4 - 5*b^6)*x)/(8*(a^2 + b^2)^4) + (6*a*b^5*Log[a*Cos[c + d*x] + b*Sin[c + d*x]]
)/((a^2 + b^2)^4*d) + (3*b*(a^2 - b^2)*(a^2 + 5*b^2))/(8*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x])) + (Cos[c + d*x]
^4*(b + a*Tan[c + d*x]))/(4*(a^2 + b^2)*d*(a + b*Tan[c + d*x])) - (Cos[c + d*x]^2*(b*(a^2 - 5*b^2) - 3*a*(a^2
+ 3*b^2)*Tan[c + d*x]))/(8*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))

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Rubi [A]  time = 0.265665, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3506, 741, 823, 801, 635, 203, 260} \[ \frac{3 b \left (a^2-b^2\right ) \left (a^2+5 b^2\right )}{8 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left (b \left (a^2-5 b^2\right )-3 a \left (a^2+3 b^2\right ) \tan (c+d x)\right )}{8 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) (a \tan (c+d x)+b)}{4 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac{6 a b^5 \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{3 x \left (5 a^4 b^2+15 a^2 b^4+a^6-5 b^6\right )}{8 \left (a^2+b^2\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(a^6 + 5*a^4*b^2 + 15*a^2*b^4 - 5*b^6)*x)/(8*(a^2 + b^2)^4) + (6*a*b^5*Log[a*Cos[c + d*x] + b*Sin[c + d*x]]
)/((a^2 + b^2)^4*d) + (3*b*(a^2 - b^2)*(a^2 + 5*b^2))/(8*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x])) + (Cos[c + d*x]
^4*(b + a*Tan[c + d*x]))/(4*(a^2 + b^2)*d*(a + b*Tan[c + d*x])) - (Cos[c + d*x]^2*(b*(a^2 - 5*b^2) - 3*a*(a^2
+ 3*b^2)*Tan[c + d*x]))/(8*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))

Rule 3506

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(b*f), Subst
[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b
^2, 0] && IntegerQ[m/2]

Rule 741

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(a*e + c*d*x)*(
a + c*x^2)^(p + 1))/(2*a*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*
Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a
, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{\cos ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x)^2 \left (1+\frac{x^2}{b^2}\right )^3} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac{\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac{b \operatorname{Subst}\left (\int \frac{-5-\frac{3 a^2}{b^2}-\frac{4 a x}{b^2}}{(a+x)^2 \left (1+\frac{x^2}{b^2}\right )^2} \, dx,x,b \tan (c+d x)\right )}{4 \left (a^2+b^2\right ) d}\\ &=\frac{\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left (b \left (a^2-5 b^2\right )-3 a \left (a^2+3 b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac{b^5 \operatorname{Subst}\left (\int \frac{\frac{3 \left (a^4+2 a^2 b^2+5 b^4\right )}{b^6}+\frac{6 a \left (a^2+3 b^2\right ) x}{b^6}}{(a+x)^2 \left (1+\frac{x^2}{b^2}\right )} \, dx,x,b \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}\\ &=\frac{\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left (b \left (a^2-5 b^2\right )-3 a \left (a^2+3 b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac{b^5 \operatorname{Subst}\left (\int \left (\frac{3 \left (-a^4-4 a^2 b^2+5 b^4\right )}{b^4 \left (a^2+b^2\right ) (a+x)^2}+\frac{48 a}{\left (a^2+b^2\right )^2 (a+x)}-\frac{3 \left (-a^6-5 a^4 b^2-15 a^2 b^4+5 b^6+16 a b^4 x\right )}{b^4 \left (a^2+b^2\right )^2 \left (b^2+x^2\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}\\ &=\frac{6 a b^5 \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac{3 b \left (a^2-b^2\right ) \left (a^2+5 b^2\right )}{8 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left (b \left (a^2-5 b^2\right )-3 a \left (a^2+3 b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{-a^6-5 a^4 b^2-15 a^2 b^4+5 b^6+16 a b^4 x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d}\\ &=\frac{6 a b^5 \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac{3 b \left (a^2-b^2\right ) \left (a^2+5 b^2\right )}{8 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left (b \left (a^2-5 b^2\right )-3 a \left (a^2+3 b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac{\left (6 a b^5\right ) \operatorname{Subst}\left (\int \frac{x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{\left (a^2+b^2\right )^4 d}+\frac{\left (3 b \left (a^6+5 a^4 b^2+15 a^2 b^4-5 b^6\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d}\\ &=\frac{3 \left (a^6+5 a^4 b^2+15 a^2 b^4-5 b^6\right ) x}{8 \left (a^2+b^2\right )^4}+\frac{6 a b^5 \log (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac{6 a b^5 \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac{3 b \left (a^2-b^2\right ) \left (a^2+5 b^2\right )}{8 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left (b \left (a^2-5 b^2\right )-3 a \left (a^2+3 b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end{align*}

Mathematica [A]  time = 3.16456, size = 416, normalized size = 1.77 \[ \frac{-\frac{\sqrt{-b^2} \left (6 a \left (a^2+b^2\right ) \left (a^2+3 b^2\right ) (a+b \tan (c+d x)) \left (\left (a-\sqrt{-b^2}\right ) \log \left (\sqrt{-b^2}-b \tan (c+d x)\right )+2 \sqrt{-b^2} \log (a+b \tan (c+d x))-\left (a+\sqrt{-b^2}\right ) \log \left (\sqrt{-b^2}+b \tan (c+d x)\right )\right )+3 \left (4 a^2 b^2+a^4-5 b^4\right ) \left (\left (-a^2+2 a \sqrt{-b^2}+b^2\right ) (a+b \tan (c+d x)) \log \left (\sqrt{-b^2}-b \tan (c+d x)\right )+\left (a^2+2 a \sqrt{-b^2}-b^2\right ) (a+b \tan (c+d x)) \log \left (\sqrt{-b^2}+b \tan (c+d x)\right )+2 \sqrt{-b^2} \left (a^2+b^2\right )-4 a \sqrt{-b^2} (a+b \tan (c+d x)) \log (a+b \tan (c+d x))\right )\right )}{\left (a^2+b^2\right )^3}+\frac{2 b \cos ^2(c+d x) \left (3 a \left (a^2+3 b^2\right ) \tan (c+d x)-a^2 b+5 b^3\right )}{a^2+b^2}+4 b \cos ^4(c+d x) (a \tan (c+d x)+b)}{16 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*b*Cos[c + d*x]^4*(b + a*Tan[c + d*x]) + (2*b*Cos[c + d*x]^2*(-(a^2*b) + 5*b^3 + 3*a*(a^2 + 3*b^2)*Tan[c + d
*x]))/(a^2 + b^2) - (Sqrt[-b^2]*(6*a*(a^2 + b^2)*(a^2 + 3*b^2)*((a - Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*
x]] + 2*Sqrt[-b^2]*Log[a + b*Tan[c + d*x]] - (a + Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])*(a + b*Tan[c +
d*x]) + 3*(a^4 + 4*a^2*b^2 - 5*b^4)*(2*Sqrt[-b^2]*(a^2 + b^2) + (-a^2 + b^2 + 2*a*Sqrt[-b^2])*Log[Sqrt[-b^2]
- b*Tan[c + d*x]]*(a + b*Tan[c + d*x]) - 4*a*Sqrt[-b^2]*Log[a + b*Tan[c + d*x]]*(a + b*Tan[c + d*x]) + (a^2 -
b^2 + 2*a*Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]]*(a + b*Tan[c + d*x]))))/(a^2 + b^2)^3)/(16*b*(a^2 + b^2
)*d*(a + b*Tan[c + d*x]))

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Maple [B]  time = 0.103, size = 661, normalized size = 2.8 \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/(a+b*tan(d*x+c))^2,x)

[Out]

3/8/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)^3*a^6+15/8/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)^3*a^4*b
^2+5/8/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)^3*a^2*b^4-7/8/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)^3
*b^6+2/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)^2*a^3*b^3+2/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)^2*a
*b^5+17/8/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)*a^4*b^2+3/8/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)*
a^2*b^4-9/8/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)*b^6+5/8/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)*a^
6+1/2/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*a^5*b+3/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*a^3*b^3+5/2/d/(a^2+b^2)^4/(1+t
an(d*x+c)^2)^2*a*b^5-3/d/(a^2+b^2)^4*a*b^5*ln(1+tan(d*x+c)^2)+45/8/d/(a^2+b^2)^4*arctan(tan(d*x+c))*a^2*b^4-15
/8/d/(a^2+b^2)^4*arctan(tan(d*x+c))*b^6+3/8/d/(a^2+b^2)^4*arctan(tan(d*x+c))*a^6+15/8/d/(a^2+b^2)^4*arctan(tan
(d*x+c))*a^4*b^2-1/d*b^5/(a^2+b^2)^3/(a+b*tan(d*x+c))+6/d*b^5/(a^2+b^2)^4*a*ln(a+b*tan(d*x+c))

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Maxima [B]  time = 1.73441, size = 678, normalized size = 2.89 \begin{align*} \frac{\frac{48 \, a b^{5} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac{24 \, a b^{5} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac{3 \,{\left (a^{6} + 5 \, a^{4} b^{2} + 15 \, a^{2} b^{4} - 5 \, b^{6}\right )}{\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac{4 \, a^{4} b + 20 \, a^{2} b^{3} - 8 \, b^{5} + 3 \,{\left (a^{4} b + 4 \, a^{2} b^{3} - 5 \, b^{5}\right )} \tan \left (d x + c\right )^{4} + 3 \,{\left (a^{5} + 4 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \tan \left (d x + c\right )^{3} +{\left (5 \, a^{4} b + 28 \, a^{2} b^{3} - 25 \, b^{5}\right )} \tan \left (d x + c\right )^{2} +{\left (5 \, a^{5} + 16 \, a^{3} b^{2} + 11 \, a b^{4}\right )} \tan \left (d x + c\right )}{a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6} +{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{5} +{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{4} + 2 \,{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{3} + 2 \,{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{2} +{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4/(a+b*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

1/8*(48*a*b^5*log(b*tan(d*x + c) + a)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 24*a*b^5*log(tan(d*x +
c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 3*(a^6 + 5*a^4*b^2 + 15*a^2*b^4 - 5*b^6)*(d*x + c
)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + (4*a^4*b + 20*a^2*b^3 - 8*b^5 + 3*(a^4*b + 4*a^2*b^3 - 5*b
^5)*tan(d*x + c)^4 + 3*(a^5 + 4*a^3*b^2 + 3*a*b^4)*tan(d*x + c)^3 + (5*a^4*b + 28*a^2*b^3 - 25*b^5)*tan(d*x +
c)^2 + (5*a^5 + 16*a^3*b^2 + 11*a*b^4)*tan(d*x + c))/(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6 + (a^6*b + 3*a^4*b^3
+ 3*a^2*b^5 + b^7)*tan(d*x + c)^5 + (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*tan(d*x + c)^4 + 2*(a^6*b + 3*a^4*b
^3 + 3*a^2*b^5 + b^7)*tan(d*x + c)^3 + 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*tan(d*x + c)^2 + (a^6*b + 3*a^4
*b^3 + 3*a^2*b^5 + b^7)*tan(d*x + c)))/d

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Fricas [A]  time = 2.54553, size = 944, normalized size = 4.02 \begin{align*} \frac{4 \,{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{5} - 2 \,{\left (a^{6} b - 3 \, a^{4} b^{3} - 9 \, a^{2} b^{5} - 5 \, b^{7}\right )} \cos \left (d x + c\right )^{3} +{\left (3 \, a^{6} b + 8 \, a^{4} b^{3} - 9 \, a^{2} b^{5} - 30 \, b^{7} + 6 \,{\left (a^{7} + 5 \, a^{5} b^{2} + 15 \, a^{3} b^{4} - 5 \, a b^{6}\right )} d x\right )} \cos \left (d x + c\right ) + 48 \,{\left (a^{2} b^{5} \cos \left (d x + c\right ) + a b^{6} \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) -{\left (3 \, a^{5} b^{2} + 22 \, a^{3} b^{4} + 3 \, a b^{6} - 4 \,{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{4} - 6 \,{\left (a^{6} b + 5 \, a^{4} b^{3} + 15 \, a^{2} b^{5} - 5 \, b^{7}\right )} d x - 6 \,{\left (a^{7} + 5 \, a^{5} b^{2} + 7 \, a^{3} b^{4} + 3 \, a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \,{\left ({\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} d \cos \left (d x + c\right ) +{\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} d \sin \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(4*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cos(d*x + c)^5 - 2*(a^6*b - 3*a^4*b^3 - 9*a^2*b^5 - 5*b^7)*cos(d
*x + c)^3 + (3*a^6*b + 8*a^4*b^3 - 9*a^2*b^5 - 30*b^7 + 6*(a^7 + 5*a^5*b^2 + 15*a^3*b^4 - 5*a*b^6)*d*x)*cos(d*
x + c) + 48*(a^2*b^5*cos(d*x + c) + a*b^6*sin(d*x + c))*log(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(
d*x + c)^2 + b^2) - (3*a^5*b^2 + 22*a^3*b^4 + 3*a*b^6 - 4*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cos(d*x + c)^4
- 6*(a^6*b + 5*a^4*b^3 + 15*a^2*b^5 - 5*b^7)*d*x - 6*(a^7 + 5*a^5*b^2 + 7*a^3*b^4 + 3*a*b^6)*cos(d*x + c)^2)*
sin(d*x + c))/((a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*d*cos(d*x + c) + (a^8*b + 4*a^6*b^3 + 6*a^4*b
^5 + 4*a^2*b^7 + b^9)*d*sin(d*x + c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**4/(a+b*tan(d*x+c))**2,x)

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.38562, size = 626, normalized size = 2.66 \begin{align*} \frac{\frac{48 \, a b^{6} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} - \frac{24 \, a b^{5} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac{3 \,{\left (a^{6} + 5 \, a^{4} b^{2} + 15 \, a^{2} b^{4} - 5 \, b^{6}\right )}{\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac{8 \,{\left (6 \, a b^{6} \tan \left (d x + c\right ) + 7 \, a^{2} b^{5} + b^{7}\right )}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}{\left (b \tan \left (d x + c\right ) + a\right )}} + \frac{36 \, a b^{5} \tan \left (d x + c\right )^{4} + 3 \, a^{6} \tan \left (d x + c\right )^{3} + 15 \, a^{4} b^{2} \tan \left (d x + c\right )^{3} + 5 \, a^{2} b^{4} \tan \left (d x + c\right )^{3} - 7 \, b^{6} \tan \left (d x + c\right )^{3} + 16 \, a^{3} b^{3} \tan \left (d x + c\right )^{2} + 88 \, a b^{5} \tan \left (d x + c\right )^{2} + 5 \, a^{6} \tan \left (d x + c\right ) + 17 \, a^{4} b^{2} \tan \left (d x + c\right ) + 3 \, a^{2} b^{4} \tan \left (d x + c\right ) - 9 \, b^{6} \tan \left (d x + c\right ) + 4 \, a^{5} b + 24 \, a^{3} b^{3} + 56 \, a b^{5}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}{\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2}}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(48*a*b^6*log(abs(b*tan(d*x + c) + a))/(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9) - 24*a*b^5*log(ta
n(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 3*(a^6 + 5*a^4*b^2 + 15*a^2*b^4 - 5*b^6)*(
d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 8*(6*a*b^6*tan(d*x + c) + 7*a^2*b^5 + b^7)/((a^8 +
4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*(b*tan(d*x + c) + a)) + (36*a*b^5*tan(d*x + c)^4 + 3*a^6*tan(d*x + c)
^3 + 15*a^4*b^2*tan(d*x + c)^3 + 5*a^2*b^4*tan(d*x + c)^3 - 7*b^6*tan(d*x + c)^3 + 16*a^3*b^3*tan(d*x + c)^2 +
88*a*b^5*tan(d*x + c)^2 + 5*a^6*tan(d*x + c) + 17*a^4*b^2*tan(d*x + c) + 3*a^2*b^4*tan(d*x + c) - 9*b^6*tan(d
*x + c) + 4*a^5*b + 24*a^3*b^3 + 56*a*b^5)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*(tan(d*x + c)^2 +
1)^2))/d