Integrand size = 31, antiderivative size = 103 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{a+b \sec (e+f x)} \, dx=\frac {d (2 b c-a d) \text {arctanh}(\sin (e+f x))}{b^2 f}+\frac {2 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^2 \sqrt {a+b} f}+\frac {d^2 \tan (e+f x)}{b f} \]
d*(-a*d+2*b*c)*arctanh(sin(f*x+e))/b^2/f+2*(-a*d+b*c)^2*arctanh((a-b)^(1/2 )*tan(1/2*f*x+1/2*e)/(a+b)^(1/2))/b^2/f/(a-b)^(1/2)/(a+b)^(1/2)+d^2*tan(f* x+e)/b/f
Time = 2.51 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.31 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{a+b \sec (e+f x)} \, dx=\frac {-\frac {2 (b c-a d)^2 \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+d \left (-\left ((2 b c-a d) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )\right )+b d \tan (e+f x)\right )}{b^2 f} \]
((-2*(b*c - a*d)^2*ArcTanh[((-a + b)*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/S qrt[a^2 - b^2] + d*(-((2*b*c - a*d)*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/ 2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])) + b*d*Tan[e + f*x]))/(b^2 *f)
Time = 0.54 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3042, 4476, 3042, 3431, 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.
| \(\displaystyle \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{a+b \sec (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^2}{a+b \csc \left (e+f x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4476 |
| \(\displaystyle \int \frac {\sec ^2(e+f x) (c \cos (e+f x)+d)^2}{a \cos (e+f x)+b}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (c \sin \left (e+f x+\frac {\pi }{2}\right )+d\right )^2}{\sin \left (e+f x+\frac {\pi }{2}\right )^2 \left (a \sin \left (e+f x+\frac {\pi }{2}\right )+b\right )}dx\) |
| \(\Big \downarrow \) 3431 |
| \(\displaystyle \int \left (\frac {(b c-a d)^2}{b^2 (a \cos (e+f x)+b)}+\frac {d (2 b c-a d) \sec (e+f x)}{b^2}+\frac {d^2 \sec ^2(e+f x)}{b}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d (2 b c-a d) \text {arctanh}(\sin (e+f x))}{b^2 f}+\frac {2 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{b^2 f \sqrt {a-b} \sqrt {a+b}}+\frac {d^2 \tan (e+f x)}{b f}\) |
(d*(2*b*c - a*d)*ArcTanh[Sin[e + f*x]])/(b^2*f) + (2*(b*c - a*d)^2*ArcTanh [(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b] *f) + (d^2*Tan[e + f*x])/(b*f)
3.3.54.3.1 Defintions of rubi rules used
Int[((g_.)*sin[(e_.) + (f_.)*(x_)])^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Int[Exp andTrig[(g*sin[e + f*x])^p*(a + b*sin[e + f*x])^m*(c + d*sin[e + f*x])^n, x ], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[b*c - a*d, 0] && (Int egersQ[m, n] || IntegersQ[m, p] || IntegersQ[n, p]) && NeQ[p, 2]
Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_), x_Symbol] :> Simp[1 /g^(m + n) Int[(g*Csc[e + f*x])^(m + n + p)*(b + a*Sin[e + f*x])^m*(d + c *Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IntegerQ[n]
Time = 0.95 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.60
| method | result | size |
| derivativedivides | \(\frac {-\frac {d^{2}}{b \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {d \left (a d -2 b c \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{b^{2}}-\frac {2 \left (-a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{2} \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {d^{2}}{b \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {d \left (a d -2 b c \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{b^{2}}}{f}\) | \(165\) |
| default | \(\frac {-\frac {d^{2}}{b \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {d \left (a d -2 b c \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{b^{2}}-\frac {2 \left (-a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{2} \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {d^{2}}{b \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {d \left (a d -2 b c \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{b^{2}}}{f}\) | \(165\) |
| risch | \(\frac {2 i d^{2}}{f b \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )}+\frac {d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) a}{b^{2} f}-\frac {2 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c}{b f}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) a^{2} d^{2}}{\sqrt {a^{2}-b^{2}}\, f \,b^{2}}-\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) a c d}{\sqrt {a^{2}-b^{2}}\, f b}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) c^{2}}{\sqrt {a^{2}-b^{2}}\, f}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) a^{2} d^{2}}{\sqrt {a^{2}-b^{2}}\, f \,b^{2}}+\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) a c d}{\sqrt {a^{2}-b^{2}}\, f b}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) c^{2}}{\sqrt {a^{2}-b^{2}}\, f}-\frac {d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) a}{b^{2} f}+\frac {2 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c}{b f}\) | \(570\) |
1/f*(-d^2/b/(tan(1/2*f*x+1/2*e)+1)-d*(a*d-2*b*c)/b^2*ln(tan(1/2*f*x+1/2*e) +1)-2/b^2*(-a^2*d^2+2*a*b*c*d-b^2*c^2)/((a-b)*(a+b))^(1/2)*arctanh((a-b)*t an(1/2*f*x+1/2*e)/((a-b)*(a+b))^(1/2))-d^2/b/(tan(1/2*f*x+1/2*e)-1)+d*(a*d -2*b*c)/b^2*ln(tan(1/2*f*x+1/2*e)-1))
Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (94) = 188\).
Time = 3.30 (sec) , antiderivative size = 518, normalized size of antiderivative = 5.03 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{a+b \sec (e+f x)} \, dx=\left [\frac {2 \, {\left (a^{2} b - b^{3}\right )} d^{2} \sin \left (f x + e\right ) + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a^{2} - b^{2}} \cos \left (f x + e\right ) \log \left (\frac {2 \, a b \cos \left (f x + e\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (f x + e\right ) + a\right )} \sin \left (f x + e\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (f x + e\right )^{2} + 2 \, a b \cos \left (f x + e\right ) + b^{2}}\right ) + {\left (2 \, {\left (a^{2} b - b^{3}\right )} c d - {\left (a^{3} - a b^{2}\right )} d^{2}\right )} \cos \left (f x + e\right ) \log \left (\sin \left (f x + e\right ) + 1\right ) - {\left (2 \, {\left (a^{2} b - b^{3}\right )} c d - {\left (a^{3} - a b^{2}\right )} d^{2}\right )} \cos \left (f x + e\right ) \log \left (-\sin \left (f x + e\right ) + 1\right )}{2 \, {\left (a^{2} b^{2} - b^{4}\right )} f \cos \left (f x + e\right )}, \frac {2 \, {\left (a^{2} b - b^{3}\right )} d^{2} \sin \left (f x + e\right ) + 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (f x + e\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (f x + e\right )}\right ) \cos \left (f x + e\right ) + {\left (2 \, {\left (a^{2} b - b^{3}\right )} c d - {\left (a^{3} - a b^{2}\right )} d^{2}\right )} \cos \left (f x + e\right ) \log \left (\sin \left (f x + e\right ) + 1\right ) - {\left (2 \, {\left (a^{2} b - b^{3}\right )} c d - {\left (a^{3} - a b^{2}\right )} d^{2}\right )} \cos \left (f x + e\right ) \log \left (-\sin \left (f x + e\right ) + 1\right )}{2 \, {\left (a^{2} b^{2} - b^{4}\right )} f \cos \left (f x + e\right )}\right ] \]
[1/2*(2*(a^2*b - b^3)*d^2*sin(f*x + e) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*s qrt(a^2 - b^2)*cos(f*x + e)*log((2*a*b*cos(f*x + e) - (a^2 - 2*b^2)*cos(f* x + e)^2 + 2*sqrt(a^2 - b^2)*(b*cos(f*x + e) + a)*sin(f*x + e) + 2*a^2 - b ^2)/(a^2*cos(f*x + e)^2 + 2*a*b*cos(f*x + e) + b^2)) + (2*(a^2*b - b^3)*c* d - (a^3 - a*b^2)*d^2)*cos(f*x + e)*log(sin(f*x + e) + 1) - (2*(a^2*b - b^ 3)*c*d - (a^3 - a*b^2)*d^2)*cos(f*x + e)*log(-sin(f*x + e) + 1))/((a^2*b^2 - b^4)*f*cos(f*x + e)), 1/2*(2*(a^2*b - b^3)*d^2*sin(f*x + e) + 2*(b^2*c^ 2 - 2*a*b*c*d + a^2*d^2)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos( f*x + e) + a)/((a^2 - b^2)*sin(f*x + e)))*cos(f*x + e) + (2*(a^2*b - b^3)* c*d - (a^3 - a*b^2)*d^2)*cos(f*x + e)*log(sin(f*x + e) + 1) - (2*(a^2*b - b^3)*c*d - (a^3 - a*b^2)*d^2)*cos(f*x + e)*log(-sin(f*x + e) + 1))/((a^2*b ^2 - b^4)*f*cos(f*x + e))]
\[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{a+b \sec (e+f x)} \, dx=\int \frac {\left (c + d \sec {\left (e + f x \right )}\right )^{2} \sec {\left (e + f x \right )}}{a + b \sec {\left (e + f x \right )}}\, dx \]
Exception generated. \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{a+b \sec (e+f x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (94) = 188\).
Time = 0.35 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.89 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{a+b \sec (e+f x)} \, dx=-\frac {\frac {2 \, d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} b} - \frac {{\left (2 \, b c d - a d^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{b^{2}} + \frac {{\left (2 \, b c d - a d^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{b^{2}} + \frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{\sqrt {-a^{2} + b^{2}} b^{2}}}{f} \]
-(2*d^2*tan(1/2*f*x + 1/2*e)/((tan(1/2*f*x + 1/2*e)^2 - 1)*b) - (2*b*c*d - a*d^2)*log(abs(tan(1/2*f*x + 1/2*e) + 1))/b^2 + (2*b*c*d - a*d^2)*log(abs (tan(1/2*f*x + 1/2*e) - 1))/b^2 + 2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(pi*fl oor(1/2*(f*x + e)/pi + 1/2)*sgn(2*a - 2*b) + arctan((a*tan(1/2*f*x + 1/2*e ) - b*tan(1/2*f*x + 1/2*e))/sqrt(-a^2 + b^2)))/(sqrt(-a^2 + b^2)*b^2))/f
Time = 18.71 (sec) , antiderivative size = 3559, normalized size of antiderivative = 34.55 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{a+b \sec (e+f x)} \, dx=\text {Too large to display} \]
- (2*d^2*tan(e/2 + (f*x)/2))/(b*f*(tan(e/2 + (f*x)/2)^2 - 1)) - (atan((((( a + b)*(a - b))^(1/2)*((32*tan(e/2 + (f*x)/2)*(2*a^5*d^4 - b^5*c^4 + a*b^4 *c^4 - 4*a^4*b*d^4 - a^2*b^3*d^4 + 3*a^3*b^2*d^4 - 4*b^5*c^2*d^2 + 12*a*b^ 4*c^2*d^2 - 12*a^2*b^3*c*d^3 - 4*a^2*b^3*c^3*d + 16*a^3*b^2*c*d^3 - 18*a^2 *b^3*c^2*d^2 + 10*a^3*b^2*c^2*d^2 + 4*a*b^4*c*d^3 + 4*a*b^4*c^3*d - 8*a^4* b*c*d^3))/b^2 + (((a + b)*(a - b))^(1/2)*((32*(b^7*c^2 - 2*a*b^6*c^2 - a*b ^6*d^2 + a^2*b^5*c^2 + 2*a^2*b^5*d^2 - a^3*b^4*d^2 + 2*b^7*c*d - 4*a*b^6*c *d + 2*a^2*b^5*c*d))/b^3 - (32*tan(e/2 + (f*x)/2)*((a + b)*(a - b))^(1/2)* (a*d - b*c)^2*(2*a*b^6 - 4*a^2*b^5 + 2*a^3*b^4))/(b^2*(b^4 - a^2*b^2)))*(a *d - b*c)^2)/(b^4 - a^2*b^2))*(a*d - b*c)^2*1i)/(b^4 - a^2*b^2) + (((a + b )*(a - b))^(1/2)*((32*tan(e/2 + (f*x)/2)*(2*a^5*d^4 - b^5*c^4 + a*b^4*c^4 - 4*a^4*b*d^4 - a^2*b^3*d^4 + 3*a^3*b^2*d^4 - 4*b^5*c^2*d^2 + 12*a*b^4*c^2 *d^2 - 12*a^2*b^3*c*d^3 - 4*a^2*b^3*c^3*d + 16*a^3*b^2*c*d^3 - 18*a^2*b^3* c^2*d^2 + 10*a^3*b^2*c^2*d^2 + 4*a*b^4*c*d^3 + 4*a*b^4*c^3*d - 8*a^4*b*c*d ^3))/b^2 - (((a + b)*(a - b))^(1/2)*((32*(b^7*c^2 - 2*a*b^6*c^2 - a*b^6*d^ 2 + a^2*b^5*c^2 + 2*a^2*b^5*d^2 - a^3*b^4*d^2 + 2*b^7*c*d - 4*a*b^6*c*d + 2*a^2*b^5*c*d))/b^3 + (32*tan(e/2 + (f*x)/2)*((a + b)*(a - b))^(1/2)*(a*d - b*c)^2*(2*a*b^6 - 4*a^2*b^5 + 2*a^3*b^4))/(b^2*(b^4 - a^2*b^2)))*(a*d - b*c)^2)/(b^4 - a^2*b^2))*(a*d - b*c)^2*1i)/(b^4 - a^2*b^2))/((64*(a^4*b*d^ 6 - a^5*d^6 - 2*b^5*c^5*d + 4*b^5*c^4*d^2 - 12*a*b^4*c^3*d^3 + a*b^4*c^...