Integrand size = 23, antiderivative size = 126 \[ \int \frac {\cos ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx=-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a-b)^{7/2} d}+\frac {\left (a^2-3 a b+3 b^2\right ) \sin (c+d x)}{(a-b)^3 d}-\frac {(2 a-3 b) \sin ^3(c+d x)}{3 (a-b)^2 d}+\frac {\sin ^5(c+d x)}{5 (a-b) d} \] Output:
-b^3*arctanh((a-b)^(1/2)*sin(d*x+c)/a^(1/2))/a^(1/2)/(a-b)^(7/2)/d+(a^2-3* a*b+3*b^2)*sin(d*x+c)/(a-b)^3/d-1/3*(2*a-3*b)*sin(d*x+c)^3/(a-b)^2/d+1/5*s in(d*x+c)^5/(a-b)/d
Time = 1.17 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.17 \[ \int \frac {\cos ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx=\frac {\frac {120 b^3 \left (\log \left (\sqrt {a}-\sqrt {a-b} \sin (c+d x)\right )-\log \left (\sqrt {a}+\sqrt {a-b} \sin (c+d x)\right )\right )}{\sqrt {a} (a-b)^{7/2}}+\frac {30 \left (5 a^2-16 a b+19 b^2\right ) \sin (c+d x)}{(a-b)^3}+\frac {5 (5 a-9 b) \sin (3 (c+d x))}{(a-b)^2}+\frac {3 \sin (5 (c+d x))}{a-b}}{240 d} \] Input:
Integrate[Cos[c + d*x]^5/(a + b*Tan[c + d*x]^2),x]
Output:
((120*b^3*(Log[Sqrt[a] - Sqrt[a - b]*Sin[c + d*x]] - Log[Sqrt[a] + Sqrt[a - b]*Sin[c + d*x]]))/(Sqrt[a]*(a - b)^(7/2)) + (30*(5*a^2 - 16*a*b + 19*b^ 2)*Sin[c + d*x])/(a - b)^3 + (5*(5*a - 9*b)*Sin[3*(c + d*x)])/(a - b)^2 + (3*Sin[5*(c + d*x)])/(a - b))/(240*d)
Time = 0.34 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4159, 300, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cos ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sec (c+d x)^5 \left (a+b \tan (c+d x)^2\right )}dx\) |
| \(\Big \downarrow \) 4159 |
| \(\displaystyle \frac {\int \frac {\left (1-\sin ^2(c+d x)\right )^3}{a-(a-b) \sin ^2(c+d x)}d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle \frac {\int \left (\frac {\sin ^4(c+d x)}{a-b}-\frac {(2 a-3 b) \sin ^2(c+d x)}{(a-b)^2}+\frac {a^2-3 b a+3 b^2}{(a-b)^3}-\frac {b^3}{(a-b)^3 \left (a-(a-b) \sin ^2(c+d x)\right )}\right )d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\left (a^2-3 a b+3 b^2\right ) \sin (c+d x)}{(a-b)^3}-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a-b)^{7/2}}+\frac {\sin ^5(c+d x)}{5 (a-b)}-\frac {(2 a-3 b) \sin ^3(c+d x)}{3 (a-b)^2}}{d}\) |
Input:
Int[Cos[c + d*x]^5/(a + b*Tan[c + d*x]^2),x]
Output:
(-((b^3*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^(7/2 ))) + ((a^2 - 3*a*b + 3*b^2)*Sin[c + d*x])/(a - b)^3 - ((2*a - 3*b)*Sin[c + d*x]^3)/(3*(a - b)^2) + Sin[c + d*x]^5/(5*(a - b)))/d
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_ ))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ExpandToSum[b*(ff*x)^n + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2 *x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f} , x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
Time = 23.64 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.31
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {a^{2} \sin \left (d x +c \right )^{5}}{5}-\frac {2 a b \sin \left (d x +c \right )^{5}}{5}+\frac {\sin \left (d x +c \right )^{5} b^{2}}{5}-\frac {2 \sin \left (d x +c \right )^{3} a^{2}}{3}+\frac {5 a b \sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )^{3} b^{2}+\sin \left (d x +c \right ) a^{2}-3 a b \sin \left (d x +c \right )+3 \sin \left (d x +c \right ) b^{2}}{\left (a -b \right )^{3}}-\frac {b^{3} \operatorname {arctanh}\left (\frac {\left (a -b \right ) \sin \left (d x +c \right )}{\sqrt {a \left (a -b \right )}}\right )}{\left (a -b \right )^{3} \sqrt {a \left (a -b \right )}}}{d}\) | \(165\) |
| default | \(\frac {\frac {\frac {a^{2} \sin \left (d x +c \right )^{5}}{5}-\frac {2 a b \sin \left (d x +c \right )^{5}}{5}+\frac {\sin \left (d x +c \right )^{5} b^{2}}{5}-\frac {2 \sin \left (d x +c \right )^{3} a^{2}}{3}+\frac {5 a b \sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )^{3} b^{2}+\sin \left (d x +c \right ) a^{2}-3 a b \sin \left (d x +c \right )+3 \sin \left (d x +c \right ) b^{2}}{\left (a -b \right )^{3}}-\frac {b^{3} \operatorname {arctanh}\left (\frac {\left (a -b \right ) \sin \left (d x +c \right )}{\sqrt {a \left (a -b \right )}}\right )}{\left (a -b \right )^{3} \sqrt {a \left (a -b \right )}}}{d}\) | \(165\) |
| risch | \(-\frac {5 i {\mathrm e}^{i \left (d x +c \right )} a^{2}}{16 \left (a -b \right )^{3} d}+\frac {i {\mathrm e}^{i \left (d x +c \right )} a b}{\left (a -b \right )^{3} d}-\frac {19 i {\mathrm e}^{i \left (d x +c \right )} b^{2}}{16 \left (a -b \right )^{3} d}+\frac {5 i {\mathrm e}^{-i \left (d x +c \right )} a^{2}}{16 \left (a -b \right ) \left (a^{2}-2 a b +b^{2}\right ) d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )} a b}{\left (a -b \right ) \left (a^{2}-2 a b +b^{2}\right ) d}+\frac {19 i {\mathrm e}^{-i \left (d x +c \right )} b^{2}}{16 \left (a -b \right ) \left (a^{2}-2 a b +b^{2}\right ) d}+\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{\sqrt {a^{2}-a b}}-1\right )}{2 \sqrt {a^{2}-a b}\, \left (a -b \right )^{3} d}-\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{\sqrt {a^{2}-a b}}-1\right )}{2 \sqrt {a^{2}-a b}\, \left (a -b \right )^{3} d}+\frac {\sin \left (5 d x +5 c \right )}{80 \left (a -b \right ) d}+\frac {5 \sin \left (3 d x +3 c \right ) a}{48 \left (a -b \right )^{2} d}-\frac {3 \sin \left (3 d x +3 c \right ) b}{16 \left (a -b \right )^{2} d}\) | \(374\) |
Input:
int(cos(d*x+c)^5/(a+b*tan(d*x+c)^2),x,method=_RETURNVERBOSE)
Output:
1/d*(1/(a-b)^3*(1/5*a^2*sin(d*x+c)^5-2/5*a*b*sin(d*x+c)^5+1/5*sin(d*x+c)^5 *b^2-2/3*sin(d*x+c)^3*a^2+5/3*a*b*sin(d*x+c)^3-sin(d*x+c)^3*b^2+sin(d*x+c) *a^2-3*a*b*sin(d*x+c)+3*sin(d*x+c)*b^2)-b^3/(a-b)^3/(a*(a-b))^(1/2)*arctan h((a-b)*sin(d*x+c)/(a*(a-b))^(1/2)))
Time = 0.14 (sec) , antiderivative size = 395, normalized size of antiderivative = 3.13 \[ \int \frac {\cos ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx=\left [-\frac {15 \, \sqrt {a^{2} - a b} b^{3} \log \left (-\frac {{\left (a - b\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - a b} \sin \left (d x + c\right ) - 2 \, a + b}{{\left (a - b\right )} \cos \left (d x + c\right )^{2} + b}\right ) - 2 \, {\left (3 \, {\left (a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right )} \cos \left (d x + c\right )^{4} + 8 \, a^{4} - 34 \, a^{3} b + 59 \, a^{2} b^{2} - 33 \, a b^{3} + {\left (4 \, a^{4} - 17 \, a^{3} b + 22 \, a^{2} b^{2} - 9 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{30 \, {\left (a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right )} d}, \frac {15 \, \sqrt {-a^{2} + a b} b^{3} \arctan \left (\frac {\sqrt {-a^{2} + a b} \sin \left (d x + c\right )}{a}\right ) + {\left (3 \, {\left (a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right )} \cos \left (d x + c\right )^{4} + 8 \, a^{4} - 34 \, a^{3} b + 59 \, a^{2} b^{2} - 33 \, a b^{3} + {\left (4 \, a^{4} - 17 \, a^{3} b + 22 \, a^{2} b^{2} - 9 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{15 \, {\left (a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right )} d}\right ] \] Input:
integrate(cos(d*x+c)^5/(a+b*tan(d*x+c)^2),x, algorithm="fricas")
Output:
[-1/30*(15*sqrt(a^2 - a*b)*b^3*log(-((a - b)*cos(d*x + c)^2 - 2*sqrt(a^2 - a*b)*sin(d*x + c) - 2*a + b)/((a - b)*cos(d*x + c)^2 + b)) - 2*(3*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*cos(d*x + c)^4 + 8*a^4 - 34*a^3*b + 59*a^2*b^ 2 - 33*a*b^3 + (4*a^4 - 17*a^3*b + 22*a^2*b^2 - 9*a*b^3)*cos(d*x + c)^2)*s in(d*x + c))/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d), 1/15*(15 *sqrt(-a^2 + a*b)*b^3*arctan(sqrt(-a^2 + a*b)*sin(d*x + c)/a) + (3*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*cos(d*x + c)^4 + 8*a^4 - 34*a^3*b + 59*a^2*b^ 2 - 33*a*b^3 + (4*a^4 - 17*a^3*b + 22*a^2*b^2 - 9*a*b^3)*cos(d*x + c)^2)*s in(d*x + c))/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)]
Timed out. \[ \int \frac {\cos ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**5/(a+b*tan(d*x+c)**2),x)
Output:
Timed out
Exception generated. \[ \int \frac {\cos ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cos(d*x+c)^5/(a+b*tan(d*x+c)^2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for more details)Is
Leaf count of result is larger than twice the leaf count of optimal. 319 vs. \(2 (114) = 228\).
Time = 0.47 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.53 \[ \int \frac {\cos ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx=-\frac {\frac {15 \, b^{3} \arctan \left (-\frac {a \sin \left (d x + c\right ) - b \sin \left (d x + c\right )}{\sqrt {-a^{2} + a b}}\right )}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \sqrt {-a^{2} + a b}} - \frac {3 \, a^{4} \sin \left (d x + c\right )^{5} - 12 \, a^{3} b \sin \left (d x + c\right )^{5} + 18 \, a^{2} b^{2} \sin \left (d x + c\right )^{5} - 12 \, a b^{3} \sin \left (d x + c\right )^{5} + 3 \, b^{4} \sin \left (d x + c\right )^{5} - 10 \, a^{4} \sin \left (d x + c\right )^{3} + 45 \, a^{3} b \sin \left (d x + c\right )^{3} - 75 \, a^{2} b^{2} \sin \left (d x + c\right )^{3} + 55 \, a b^{3} \sin \left (d x + c\right )^{3} - 15 \, b^{4} \sin \left (d x + c\right )^{3} + 15 \, a^{4} \sin \left (d x + c\right ) - 75 \, a^{3} b \sin \left (d x + c\right ) + 150 \, a^{2} b^{2} \sin \left (d x + c\right ) - 135 \, a b^{3} \sin \left (d x + c\right ) + 45 \, b^{4} \sin \left (d x + c\right )}{a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}}}{15 \, d} \] Input:
integrate(cos(d*x+c)^5/(a+b*tan(d*x+c)^2),x, algorithm="giac")
Output:
-1/15*(15*b^3*arctan(-(a*sin(d*x + c) - b*sin(d*x + c))/sqrt(-a^2 + a*b))/ ((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*sqrt(-a^2 + a*b)) - (3*a^4*sin(d*x + c)^5 - 12*a^3*b*sin(d*x + c)^5 + 18*a^2*b^2*sin(d*x + c)^5 - 12*a*b^3*sin(d*x + c)^5 + 3*b^4*sin(d*x + c)^5 - 10*a^4*sin(d*x + c)^3 + 45*a^3*b*sin(d*x + c)^3 - 75*a^2*b^2*sin(d*x + c)^3 + 55*a*b^3*sin(d*x + c)^3 - 15*b^4*sin(d *x + c)^3 + 15*a^4*sin(d*x + c) - 75*a^3*b*sin(d*x + c) + 150*a^2*b^2*sin( d*x + c) - 135*a*b^3*sin(d*x + c) + 45*b^4*sin(d*x + c))/(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5))/d
Time = 10.81 (sec) , antiderivative size = 1493, normalized size of antiderivative = 11.85 \[ \int \frac {\cos ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx=\text {Too large to display} \] Input:
int(cos(c + d*x)^5/(a + b*tan(c + d*x)^2),x)
Output:
((2*tan(c/2 + (d*x)/2)*(a^2 - 3*a*b + 3*b^2))/(3*a*b^2 - 3*a^2*b + a^3 - b ^3) + (tan(c/2 + (d*x)/2)^9*(2*a^2 - 6*a*b + 6*b^2))/(3*a*b^2 - 3*a^2*b + a^3 - b^3) + (tan(c/2 + (d*x)/2)^3*((8*a^2)/3 - (32*a*b)/3 + 16*b^2))/(3*a *b^2 - 3*a^2*b + a^3 - b^3) + (tan(c/2 + (d*x)/2)^7*((8*a^2)/3 - (32*a*b)/ 3 + 16*b^2))/(3*a*b^2 - 3*a^2*b + a^3 - b^3) + (tan(c/2 + (d*x)/2)^5*((116 *a^2)/15 - (332*a*b)/15 + (132*b^2)/5))/(3*a*b^2 - 3*a^2*b + a^3 - b^3))/( d*(5*tan(c/2 + (d*x)/2)^2 + 10*tan(c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x)/2 )^6 + 5*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 + 1)) - (b^3*atan(((b ^3*((tan(c/2 + (d*x)/2)*(16*a*b^10 - 96*a^2*b^9 + 240*a^3*b^8 - 320*a^4*b^ 7 + 240*a^5*b^6 - 96*a^6*b^5 + 16*a^7*b^4))/2 + (b^3*(tan(c/2 + (d*x)/2)^2 *(4*a^12 - 44*a^11*b + 8*a^2*b^10 - 76*a^3*b^9 + 324*a^4*b^8 - 816*a^5*b^7 + 1344*a^6*b^6 - 1512*a^7*b^5 + 1176*a^8*b^4 - 624*a^9*b^3 + 216*a^10*b^2 ) + 36*a^11*b - 4*a^12 + 4*a^3*b^9 - 36*a^4*b^8 + 144*a^5*b^7 - 336*a^6*b^ 6 + 504*a^7*b^5 - 504*a^8*b^4 + 336*a^9*b^3 - 144*a^10*b^2))/(a^(1/2)*(a - b)^(7/2)))*1i)/(a^(1/2)*(a - b)^(7/2)) + (b^3*((tan(c/2 + (d*x)/2)*(16*a* b^10 - 96*a^2*b^9 + 240*a^3*b^8 - 320*a^4*b^7 + 240*a^5*b^6 - 96*a^6*b^5 + 16*a^7*b^4))/2 - (b^3*(tan(c/2 + (d*x)/2)^2*(4*a^12 - 44*a^11*b + 8*a^2*b ^10 - 76*a^3*b^9 + 324*a^4*b^8 - 816*a^5*b^7 + 1344*a^6*b^6 - 1512*a^7*b^5 + 1176*a^8*b^4 - 624*a^9*b^3 + 216*a^10*b^2) + 36*a^11*b - 4*a^12 + 4*a^3 *b^9 - 36*a^4*b^8 + 144*a^5*b^7 - 336*a^6*b^6 + 504*a^7*b^5 - 504*a^8*b...
Time = 0.24 (sec) , antiderivative size = 297, normalized size of antiderivative = 2.36 \[ \int \frac {\cos ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx=\frac {15 \sqrt {a}\, \sqrt {a -b}\, \mathrm {log}\left (-2 \sqrt {a -b}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\sqrt {a}\right ) b^{3}-15 \sqrt {a}\, \sqrt {a -b}\, \mathrm {log}\left (2 \sqrt {a -b}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\sqrt {a}\right ) b^{3}+6 \sin \left (d x +c \right )^{5} a^{4}-18 \sin \left (d x +c \right )^{5} a^{3} b +18 \sin \left (d x +c \right )^{5} a^{2} b^{2}-6 \sin \left (d x +c \right )^{5} a \,b^{3}-20 \sin \left (d x +c \right )^{3} a^{4}+70 \sin \left (d x +c \right )^{3} a^{3} b -80 \sin \left (d x +c \right )^{3} a^{2} b^{2}+30 \sin \left (d x +c \right )^{3} a \,b^{3}+30 \sin \left (d x +c \right ) a^{4}-120 \sin \left (d x +c \right ) a^{3} b +180 \sin \left (d x +c \right ) a^{2} b^{2}-90 \sin \left (d x +c \right ) a \,b^{3}}{30 a d \left (a^{4}-4 a^{3} b +6 a^{2} b^{2}-4 a \,b^{3}+b^{4}\right )} \] Input:
int(cos(d*x+c)^5/(a+b*tan(d*x+c)^2),x)
Output:
(15*sqrt(a)*sqrt(a - b)*log( - 2*sqrt(a - b)*tan((c + d*x)/2) + sqrt(a)*ta n((c + d*x)/2)**2 + sqrt(a))*b**3 - 15*sqrt(a)*sqrt(a - b)*log(2*sqrt(a - b)*tan((c + d*x)/2) + sqrt(a)*tan((c + d*x)/2)**2 + sqrt(a))*b**3 + 6*sin( c + d*x)**5*a**4 - 18*sin(c + d*x)**5*a**3*b + 18*sin(c + d*x)**5*a**2*b** 2 - 6*sin(c + d*x)**5*a*b**3 - 20*sin(c + d*x)**3*a**4 + 70*sin(c + d*x)** 3*a**3*b - 80*sin(c + d*x)**3*a**2*b**2 + 30*sin(c + d*x)**3*a*b**3 + 30*s in(c + d*x)*a**4 - 120*sin(c + d*x)*a**3*b + 180*sin(c + d*x)*a**2*b**2 - 90*sin(c + d*x)*a*b**3)/(30*a*d*(a**4 - 4*a**3*b + 6*a**2*b**2 - 4*a*b**3 + b**4))