Integrand size = 21, antiderivative size = 78 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \, dx=\left (a^2-b^2\right ) x+\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}-\frac {a b \cot ^2(c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {2 a b \log (\sin (c+d x))}{d} \]
(a^2-b^2)*x+(a^2-b^2)*cot(d*x+c)/d-a*b*cot(d*x+c)^2/d-1/3*a^2*cot(d*x+c)^3 /d-2*a*b*ln(sin(d*x+c))/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.77 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.32 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {a^2 \cot ^3(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2(c+d x)\right )}{3 d}-\frac {b^2 \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\tan ^2(c+d x)\right )}{d}-\frac {a b \left (\cot ^2(c+d x)+2 \log (\cos (c+d x))+2 \log (\tan (c+d x))\right )}{d} \]
-1/3*(a^2*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2] )/d - (b^2*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/ d - (a*b*(Cot[c + d*x]^2 + 2*Log[Cos[c + d*x]] + 2*Log[Tan[c + d*x]]))/d
Time = 0.63 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 4025, 3042, 4012, 25, 3042, 4012, 3042, 4014, 3042, 25, 3956}
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.
\(\displaystyle \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+b \tan (c+d x))^2}{\tan (c+d x)^4}dx\) |
\(\Big \downarrow \) 4025 |
\(\displaystyle \int \cot ^3(c+d x) \left (2 a b-\left (a^2-b^2\right ) \tan (c+d x)\right )dx-\frac {a^2 \cot ^3(c+d x)}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\tan (c+d x)^3}dx-\frac {a^2 \cot ^3(c+d x)}{3 d}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle \int -\cot ^2(c+d x) \left (a^2+2 b \tan (c+d x) a-b^2\right )dx-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a b \cot ^2(c+d x)}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \cot ^2(c+d x) \left (a^2+2 b \tan (c+d x) a-b^2\right )dx-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a b \cot ^2(c+d x)}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int \frac {a^2+2 b \tan (c+d x) a-b^2}{\tan (c+d x)^2}dx-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a b \cot ^2(c+d x)}{d}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle -\int \cot (c+d x) \left (2 a b-\left (a^2-b^2\right ) \tan (c+d x)\right )dx+\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a b \cot ^2(c+d x)}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\tan (c+d x)}dx+\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a b \cot ^2(c+d x)}{d}\) |
\(\Big \downarrow \) 4014 |
\(\displaystyle -2 a b \int \cot (c+d x)dx+\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}+x \left (a^2-b^2\right )-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a b \cot ^2(c+d x)}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -2 a b \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}+x \left (a^2-b^2\right )-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a b \cot ^2(c+d x)}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 2 a b \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx+\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}+x \left (a^2-b^2\right )-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a b \cot ^2(c+d x)}{d}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle \frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}+x \left (a^2-b^2\right )-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a b \cot ^2(c+d x)}{d}-\frac {2 a b \log (-\sin (c+d x))}{d}\) |
(a^2 - b^2)*x + ((a^2 - b^2)*Cot[c + d*x])/d - (a*b*Cot[c + d*x]^2)/d - (a ^2*Cot[c + d*x]^3)/(3*d) - (2*a*b*Log[-Sin[c + d*x]])/d
3.5.32.3.1 Defintions of rubi rules used
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[(a*c + b*d)*(x/(a^2 + b^2)), x] + Simp[(b*c - a *d)/(a^2 + b^2) Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && N eQ[a*c + b*d, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
Time = 0.69 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+2 a b \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+b^{2} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(75\) |
default | \(\frac {a^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+2 a b \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+b^{2} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(75\) |
parallelrisch | \(\frac {-\left (\cot ^{3}\left (d x +c \right )\right ) a^{2}-3 \left (\cot ^{2}\left (d x +c \right )\right ) a b +3 a^{2} d x -3 x d \,b^{2}+3 \cot \left (d x +c \right ) a^{2}-3 \cot \left (d x +c \right ) b^{2}-6 a b \ln \left (\tan \left (d x +c \right )\right )+3 a b \ln \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}\) | \(92\) |
norman | \(\frac {\left (a^{2}-b^{2}\right ) x \left (\tan ^{3}\left (d x +c \right )\right )+\frac {\left (a^{2}-b^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{d}-\frac {a^{2}}{3 d}-\frac {a b \tan \left (d x +c \right )}{d}}{\tan \left (d x +c \right )^{3}}+\frac {a b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}-\frac {2 a b \ln \left (\tan \left (d x +c \right )\right )}{d}\) | \(104\) |
risch | \(2 i a b x +a^{2} x -b^{2} x +\frac {4 i a b c}{d}-\frac {2 i \left (6 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}-6 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+3 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+6 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-4 a^{2}+3 b^{2}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {2 a b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(161\) |
1/d*(a^2*(-1/3*cot(d*x+c)^3+cot(d*x+c)+d*x+c)+2*a*b*(-1/2*cot(d*x+c)^2-ln( sin(d*x+c)))+b^2*(-cot(d*x+c)-d*x-c))
Time = 0.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.37 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {3 \, a b \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{3} - 3 \, {\left ({\left (a^{2} - b^{2}\right )} d x - a b\right )} \tan \left (d x + c\right )^{3} + 3 \, a b \tan \left (d x + c\right ) - 3 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} + a^{2}}{3 \, d \tan \left (d x + c\right )^{3}} \]
-1/3*(3*a*b*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c)^3 - 3*(( a^2 - b^2)*d*x - a*b)*tan(d*x + c)^3 + 3*a*b*tan(d*x + c) - 3*(a^2 - b^2)* tan(d*x + c)^2 + a^2)/(d*tan(d*x + c)^3)
Time = 0.82 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.59 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \, dx=\begin {cases} \tilde {\infty } a^{2} x & \text {for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan {\left (c \right )}\right )^{2} \cot ^{4}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } a^{2} x & \text {for}\: c = - d x \\a^{2} x + \frac {a^{2}}{d \tan {\left (c + d x \right )}} - \frac {a^{2}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {a b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - \frac {2 a b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {a b}{d \tan ^{2}{\left (c + d x \right )}} - b^{2} x - \frac {b^{2}}{d \tan {\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*a**2*x, Eq(c, 0) & Eq(d, 0)), (x*(a + b*tan(c))**2*cot(c)** 4, Eq(d, 0)), (zoo*a**2*x, Eq(c, -d*x)), (a**2*x + a**2/(d*tan(c + d*x)) - a**2/(3*d*tan(c + d*x)**3) + a*b*log(tan(c + d*x)**2 + 1)/d - 2*a*b*log(t an(c + d*x))/d - a*b/(d*tan(c + d*x)**2) - b**2*x - b**2/(d*tan(c + d*x)), True))
Time = 0.40 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.17 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {3 \, a b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \, a b \log \left (\tan \left (d x + c\right )\right ) + 3 \, {\left (a^{2} - b^{2}\right )} {\left (d x + c\right )} - \frac {3 \, a b \tan \left (d x + c\right ) - 3 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} + a^{2}}{\tan \left (d x + c\right )^{3}}}{3 \, d} \]
1/3*(3*a*b*log(tan(d*x + c)^2 + 1) - 6*a*b*log(tan(d*x + c)) + 3*(a^2 - b^ 2)*(d*x + c) - (3*a*b*tan(d*x + c) - 3*(a^2 - b^2)*tan(d*x + c)^2 + a^2)/t an(d*x + c)^3)/d
Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (76) = 152\).
Time = 0.76 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.45 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 48 \, a b \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 48 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, {\left (a^{2} - b^{2}\right )} {\left (d x + c\right )} + \frac {88 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
1/24*(a^2*tan(1/2*d*x + 1/2*c)^3 - 6*a*b*tan(1/2*d*x + 1/2*c)^2 + 48*a*b*l og(tan(1/2*d*x + 1/2*c)^2 + 1) - 48*a*b*log(abs(tan(1/2*d*x + 1/2*c))) - 1 5*a^2*tan(1/2*d*x + 1/2*c) + 12*b^2*tan(1/2*d*x + 1/2*c) + 24*(a^2 - b^2)* (d*x + c) + (88*a*b*tan(1/2*d*x + 1/2*c)^3 + 15*a^2*tan(1/2*d*x + 1/2*c)^2 - 12*b^2*tan(1/2*d*x + 1/2*c)^2 - 6*a*b*tan(1/2*d*x + 1/2*c) - a^2)/tan(1 /2*d*x + 1/2*c)^3)/d
Time = 4.91 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.44 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {{\mathrm {cot}\left (c+d\,x\right )}^3\,\left (\frac {a^2}{3}-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a^2-b^2\right )+a\,b\,\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {2\,a\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (a-b\,1{}\mathrm {i}\right )}^2\,1{}\mathrm {i}}{2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^2\,1{}\mathrm {i}}{2\,d} \]