Integrand size = 13, antiderivative size = 73 \[ \int \frac {\sec ^4(x)}{a+b \cot (x)} \, dx=-\frac {b \left (a^2+b^2\right ) \log (a+b \cot (x))}{a^4}-\frac {b \left (a^2+b^2\right ) \log (\tan (x))}{a^4}+\frac {\left (a^2+b^2\right ) \tan (x)}{a^3}-\frac {b \tan ^2(x)}{2 a^2}+\frac {\tan ^3(x)}{3 a} \] Output:
-b*(a^2+b^2)*ln(a+b*cot(x))/a^4-b*(a^2+b^2)*ln(tan(x))/a^4+(a^2+b^2)*tan(x )/a^3-1/2*b*tan(x)^2/a^2+1/3*tan(x)^3/a
Time = 0.18 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77 \[ \int \frac {\sec ^4(x)}{a+b \cot (x)} \, dx=\frac {-6 b \left (a^2+b^2\right ) \log (b+a \tan (x))+6 a \left (a^2+b^2\right ) \tan (x)-3 a^2 b \tan ^2(x)+2 a^3 \tan ^3(x)}{6 a^4} \] Input:
Integrate[Sec[x]^4/(a + b*Cot[x]),x]
Output:
(-6*b*(a^2 + b^2)*Log[b + a*Tan[x]] + 6*a*(a^2 + b^2)*Tan[x] - 3*a^2*b*Tan [x]^2 + 2*a^3*Tan[x]^3)/(6*a^4)
Time = 0.30 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3042, 3999, 522, 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.
\(\displaystyle \int \frac {\sec ^4(x)}{a+b \cot (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin \left (x+\frac {\pi }{2}\right )^4 \left (a-b \tan \left (x+\frac {\pi }{2}\right )\right )}dx\) |
\(\Big \downarrow \) 3999 |
\(\displaystyle -b \int \frac {\left (\cot ^2(x) b^2+b^2\right ) \tan ^4(x)}{b^4 (a+b \cot (x))}d(b \cot (x))\) |
\(\Big \downarrow \) 522 |
\(\displaystyle -b \int \left (\frac {\tan ^4(x)}{a b^2}-\frac {\tan ^3(x)}{a^2 b}+\frac {\left (a^2+b^2\right ) \tan ^2(x)}{a^3 b^2}+\frac {\left (-a^2-b^2\right ) \tan (x)}{a^4 b}+\frac {a^2+b^2}{a^4 (a+b \cot (x))}\right )d(b \cot (x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -b \left (\frac {\tan ^2(x)}{2 a^2}-\frac {\left (a^2+b^2\right ) \log (b \cot (x))}{a^4}+\frac {\left (a^2+b^2\right ) \log (a+b \cot (x))}{a^4}-\frac {\left (a^2+b^2\right ) \tan (x)}{a^3 b}-\frac {\tan ^3(x)}{3 a b}\right )\) |
Input:
Int[Sec[x]^4/(a + b*Cot[x]),x]
Output:
-(b*(-(((a^2 + b^2)*Log[b*Cot[x]])/a^4) + ((a^2 + b^2)*Log[a + b*Cot[x]])/ a^4 - ((a^2 + b^2)*Tan[x])/(a^3*b) + Tan[x]^2/(2*a^2) - Tan[x]^3/(3*a*b)))
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[b/f Subst[Int[x^m*((a + x)^n/(b^2 + x^2)^(m/2 + 1)), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[m/2]
Time = 4.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77
method | result | size |
default | \(\frac {\frac {\tan \left (x \right )^{3} a^{2}}{3}-\frac {\tan \left (x \right )^{2} a b}{2}+\tan \left (x \right ) a^{2}+b^{2} \tan \left (x \right )}{a^{3}}-\frac {b \left (a^{2}+b^{2}\right ) \ln \left (\tan \left (x \right ) a +b \right )}{a^{4}}\) | \(56\) |
risch | \(-\frac {2 \left (-3 i b^{2} {\mathrm e}^{4 i x}+3 a b \,{\mathrm e}^{4 i x}-6 i a^{2} {\mathrm e}^{2 i x}-6 i b^{2} {\mathrm e}^{2 i x}+3 a b \,{\mathrm e}^{2 i x}-2 i a^{2}-3 i b^{2}\right )}{3 a^{3} \left ({\mathrm e}^{2 i x}+1\right )^{3}}-\frac {b \ln \left ({\mathrm e}^{2 i x}+\frac {i b -a}{i b +a}\right )}{a^{2}}-\frac {b^{3} \ln \left ({\mathrm e}^{2 i x}+\frac {i b -a}{i b +a}\right )}{a^{4}}+\frac {b \ln \left ({\mathrm e}^{2 i x}+1\right )}{a^{2}}+\frac {b^{3} \ln \left ({\mathrm e}^{2 i x}+1\right )}{a^{4}}\) | \(170\) |
Input:
int(sec(x)^4/(a+b*cot(x)),x,method=_RETURNVERBOSE)
Output:
1/a^3*(1/3*tan(x)^3*a^2-1/2*tan(x)^2*a*b+tan(x)*a^2+b^2*tan(x))-b*(a^2+b^2 )/a^4*ln(tan(x)*a+b)
Time = 0.10 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.45 \[ \int \frac {\sec ^4(x)}{a+b \cot (x)} \, dx=-\frac {3 \, {\left (a^{2} b + b^{3}\right )} \cos \left (x\right )^{3} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) - {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + a^{2}\right ) - 3 \, {\left (a^{2} b + b^{3}\right )} \cos \left (x\right )^{3} \log \left (\cos \left (x\right )^{2}\right ) + 3 \, a^{2} b \cos \left (x\right ) - 2 \, {\left (a^{3} + {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{6 \, a^{4} \cos \left (x\right )^{3}} \] Input:
integrate(sec(x)^4/(a+b*cot(x)),x, algorithm="fricas")
Output:
-1/6*(3*(a^2*b + b^3)*cos(x)^3*log(2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x )^2 + a^2) - 3*(a^2*b + b^3)*cos(x)^3*log(cos(x)^2) + 3*a^2*b*cos(x) - 2*( a^3 + (2*a^3 + 3*a*b^2)*cos(x)^2)*sin(x))/(a^4*cos(x)^3)
\[ \int \frac {\sec ^4(x)}{a+b \cot (x)} \, dx=\int \frac {\sec ^{4}{\left (x \right )}}{a + b \cot {\left (x \right )}}\, dx \] Input:
integrate(sec(x)**4/(a+b*cot(x)),x)
Output:
Integral(sec(x)**4/(a + b*cot(x)), x)
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77 \[ \int \frac {\sec ^4(x)}{a+b \cot (x)} \, dx=\frac {2 \, a^{2} \tan \left (x\right )^{3} - 3 \, a b \tan \left (x\right )^{2} + 6 \, {\left (a^{2} + b^{2}\right )} \tan \left (x\right )}{6 \, a^{3}} - \frac {{\left (a^{2} b + b^{3}\right )} \log \left (a \tan \left (x\right ) + b\right )}{a^{4}} \] Input:
integrate(sec(x)^4/(a+b*cot(x)),x, algorithm="maxima")
Output:
1/6*(2*a^2*tan(x)^3 - 3*a*b*tan(x)^2 + 6*(a^2 + b^2)*tan(x))/a^3 - (a^2*b + b^3)*log(a*tan(x) + b)/a^4
Time = 0.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int \frac {\sec ^4(x)}{a+b \cot (x)} \, dx=\frac {2 \, a^{2} \tan \left (x\right )^{3} - 3 \, a b \tan \left (x\right )^{2} + 6 \, a^{2} \tan \left (x\right ) + 6 \, b^{2} \tan \left (x\right )}{6 \, a^{3}} - \frac {{\left (a^{2} b + b^{3}\right )} \log \left ({\left | a \tan \left (x\right ) + b \right |}\right )}{a^{4}} \] Input:
integrate(sec(x)^4/(a+b*cot(x)),x, algorithm="giac")
Output:
1/6*(2*a^2*tan(x)^3 - 3*a*b*tan(x)^2 + 6*a^2*tan(x) + 6*b^2*tan(x))/a^3 - (a^2*b + b^3)*log(abs(a*tan(x) + b))/a^4
Time = 8.90 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.75 \[ \int \frac {\sec ^4(x)}{a+b \cot (x)} \, dx=\mathrm {tan}\left (x\right )\,\left (\frac {1}{a}+\frac {b^2}{a^3}\right )+\frac {{\mathrm {tan}\left (x\right )}^3}{3\,a}-\frac {\ln \left (b+a\,\mathrm {tan}\left (x\right )\right )\,\left (a^2\,b+b^3\right )}{a^4}-\frac {b\,{\mathrm {tan}\left (x\right )}^2}{2\,a^2} \] Input:
int(1/(cos(x)^4*(a + b*cot(x))),x)
Output:
tan(x)*(1/a + b^2/a^3) + tan(x)^3/(3*a) - (log(b + a*tan(x))*(a^2*b + b^3) )/a^4 - (b*tan(x)^2)/(2*a^2)
Time = 0.17 (sec) , antiderivative size = 322, normalized size of antiderivative = 4.41 \[ \int \frac {\sec ^4(x)}{a+b \cot (x)} \, dx=\frac {6 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )-1\right ) \sin \left (x \right )^{2} a^{2} b +6 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )-1\right ) \sin \left (x \right )^{2} b^{3}-6 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )-1\right ) a^{2} b -6 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )-1\right ) b^{3}+6 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )+1\right ) \sin \left (x \right )^{2} a^{2} b +6 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )+1\right ) \sin \left (x \right )^{2} b^{3}-6 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )+1\right ) a^{2} b -6 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )+1\right ) b^{3}-6 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} b -2 \tan \left (\frac {x}{2}\right ) a -b \right ) \sin \left (x \right )^{2} a^{2} b -6 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} b -2 \tan \left (\frac {x}{2}\right ) a -b \right ) \sin \left (x \right )^{2} b^{3}+6 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} b -2 \tan \left (\frac {x}{2}\right ) a -b \right ) a^{2} b +6 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} b -2 \tan \left (\frac {x}{2}\right ) a -b \right ) b^{3}-\cos \left (x \right ) \sin \left (x \right )^{2} a^{2} b +4 \cos \left (x \right ) a^{2} b +4 \sin \left (x \right )^{3} a^{3}+6 \sin \left (x \right )^{3} a \,b^{2}-6 \sin \left (x \right ) a^{3}-6 \sin \left (x \right ) a \,b^{2}}{6 \cos \left (x \right ) a^{4} \left (\sin \left (x \right )^{2}-1\right )} \] Input:
int(sec(x)^4/(a+b*cot(x)),x)
Output:
(6*cos(x)*log(tan(x/2) - 1)*sin(x)**2*a**2*b + 6*cos(x)*log(tan(x/2) - 1)* sin(x)**2*b**3 - 6*cos(x)*log(tan(x/2) - 1)*a**2*b - 6*cos(x)*log(tan(x/2) - 1)*b**3 + 6*cos(x)*log(tan(x/2) + 1)*sin(x)**2*a**2*b + 6*cos(x)*log(ta n(x/2) + 1)*sin(x)**2*b**3 - 6*cos(x)*log(tan(x/2) + 1)*a**2*b - 6*cos(x)* log(tan(x/2) + 1)*b**3 - 6*cos(x)*log(tan(x/2)**2*b - 2*tan(x/2)*a - b)*si n(x)**2*a**2*b - 6*cos(x)*log(tan(x/2)**2*b - 2*tan(x/2)*a - b)*sin(x)**2* b**3 + 6*cos(x)*log(tan(x/2)**2*b - 2*tan(x/2)*a - b)*a**2*b + 6*cos(x)*lo g(tan(x/2)**2*b - 2*tan(x/2)*a - b)*b**3 - cos(x)*sin(x)**2*a**2*b + 4*cos (x)*a**2*b + 4*sin(x)**3*a**3 + 6*sin(x)**3*a*b**2 - 6*sin(x)*a**3 - 6*sin (x)*a*b**2)/(6*cos(x)*a**4*(sin(x)**2 - 1))