Integrand size = 39, antiderivative size = 149 \[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\frac {2 \sqrt {a} g^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{d f}-\frac {2 \sqrt {a} \sqrt {c} g^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{d \sqrt {c+d} f} \]
2*g^(3/2)*arctanh(a^(1/2)*g^(1/2)*tan(f*x+e)/(g*sec(f*x+e))^(1/2)/(a+a*sec (f*x+e))^(1/2))*a^(1/2)/d/f-2*g^(3/2)*arctanh(a^(1/2)*c^(1/2)*g^(1/2)*tan( f*x+e)/(c+d)^(1/2)/(g*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(1/2))*a^(1/2)*c^ (1/2)/d/f/(c+d)^(1/2)
Time = 2.03 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.26 \[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=-\frac {g^2 \left (\sqrt {c+d} \log \left (\sqrt {2}-2 \sin \left (\frac {1}{2} (e+f x)\right )\right )-\sqrt {c+d} \log \left (\sqrt {2}+2 \sin \left (\frac {1}{2} (e+f x)\right )\right )+\sqrt {c} \left (-\log \left (\sqrt {2} \sqrt {c+d}-2 \sqrt {c} \sin \left (\frac {1}{2} (e+f x)\right )\right )+\log \left (\sqrt {2} \sqrt {c+d}+2 \sqrt {c} \sin \left (\frac {1}{2} (e+f x)\right )\right )\right )\right ) \sec \left (\frac {1}{2} (e+f x)\right ) \sqrt {a (1+\sec (e+f x))}}{\sqrt {2} d \sqrt {c+d} f \sqrt {g \sec (e+f x)}} \]
-((g^2*(Sqrt[c + d]*Log[Sqrt[2] - 2*Sin[(e + f*x)/2]] - Sqrt[c + d]*Log[Sq rt[2] + 2*Sin[(e + f*x)/2]] + Sqrt[c]*(-Log[Sqrt[2]*Sqrt[c + d] - 2*Sqrt[c ]*Sin[(e + f*x)/2]] + Log[Sqrt[2]*Sqrt[c + d] + 2*Sqrt[c]*Sin[(e + f*x)/2] ]))*Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])])/(Sqrt[2]*d*Sqrt[c + d]*f* Sqrt[g*Sec[e + f*x]]))
Time = 0.94 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3042, 4458, 3042, 4289, 221, 4453, 221}
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.
| \(\displaystyle \int \frac {\sqrt {a \sec (e+f x)+a} (g \sec (e+f x))^{3/2}}{c+d \sec (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a \csc \left (e+f x+\frac {\pi }{2}\right )+a} \left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{c+d \csc \left (e+f x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4458 |
| \(\displaystyle \frac {g \int \sqrt {g \sec (e+f x)} \sqrt {\sec (e+f x) a+a}dx}{d}-\frac {c g \int \frac {\sqrt {g \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}{c+d \sec (e+f x)}dx}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g \int \sqrt {g \csc \left (e+f x+\frac {\pi }{2}\right )} \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}dx}{d}-\frac {c g \int \frac {\sqrt {g \csc \left (e+f x+\frac {\pi }{2}\right )} \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{c+d \csc \left (e+f x+\frac {\pi }{2}\right )}dx}{d}\) |
| \(\Big \downarrow \) 4289 |
| \(\displaystyle -\frac {2 a g^2 \int \frac {1}{a-\frac {a^2 \sin (e+f x) \tan (e+f x)}{\sec (e+f x) a+a}}d\left (-\frac {a \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}\right )}{d f}-\frac {c g \int \frac {\sqrt {g \csc \left (e+f x+\frac {\pi }{2}\right )} \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{c+d \csc \left (e+f x+\frac {\pi }{2}\right )}dx}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 \sqrt {a} g^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{d f}-\frac {c g \int \frac {\sqrt {g \csc \left (e+f x+\frac {\pi }{2}\right )} \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{c+d \csc \left (e+f x+\frac {\pi }{2}\right )}dx}{d}\) |
| \(\Big \downarrow \) 4453 |
| \(\displaystyle \frac {2 a c g^2 \int \frac {1}{a (c+d)-\frac {a^2 c \sin (e+f x) \tan (e+f x)}{\sec (e+f x) a+a}}d\left (-\frac {a \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}\right )}{d f}+\frac {2 \sqrt {a} g^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{d f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 \sqrt {a} g^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{d f}-\frac {2 \sqrt {a} \sqrt {c} g^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{d f \sqrt {c+d}}\) |
(2*Sqrt[a]*g^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[g]*Tan[e + f*x])/(Sqrt[g*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])])/(d*f) - (2*Sqrt[a]*Sqrt[c]*g^(3/2)*ArcTa nh[(Sqrt[a]*Sqrt[c]*Sqrt[g]*Tan[e + f*x])/(Sqrt[c + d]*Sqrt[g*Sec[e + f*x] ]*Sqrt[a + a*Sec[e + f*x]])])/(d*Sqrt[c + d]*f)
3.3.39.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*b*(d/f) Subst[Int[1/(b - d*x^2), x], x, b*( Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]]))], x] /; FreeQ [{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && !GtQ[a*(d/b), 0]
Int[(Sqrt[csc[(e_.) + (f_.)*(x_)]*(g_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)])/(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[-2*b*(g /f) Subst[Int[1/(b*c + a*d - c*g*x^2), x], x, b*(Cot[e + f*x]/(Sqrt[g*Csc [e + f*x]]*Sqrt[a + b*Csc[e + f*x]]))], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
Int[((csc[(e_.) + (f_.)*(x_)]*(g_.))^(3/2)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)])/(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[g/d Int[Sqrt[g*Csc[e + f*x]]*Sqrt[a + b*Csc[e + f*x]], x], x] - Simp[c*(g/d) Int[Sqrt[g*Csc[e + f*x]]*(Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(684\) vs. \(2(117)=234\).
Time = 22.71 (sec) , antiderivative size = 685, normalized size of antiderivative = 4.60
| method | result | size |
| default | \(-\frac {g \sqrt {2}\, \left (c -d \right ) \sqrt {-\frac {g \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1\right )}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right ) \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (\sqrt {\left (c +d \right ) \left (c -d \right )}\, \operatorname {arctanh}\left (\frac {\left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right ) \sqrt {2}}{2 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1}}\right ) \sqrt {\frac {c}{c -d}}+\sqrt {\left (c +d \right ) \left (c -d \right )}\, \operatorname {arctanh}\left (\frac {\left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right ) \sqrt {2}}{2 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1}}\right ) \sqrt {\frac {c}{c -d}}-c \ln \left (-\frac {2 \left (\sqrt {2}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1}\, \sqrt {\frac {c}{c -d}}\, c -\sqrt {2}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1}\, \sqrt {\frac {c}{c -d}}\, d +\sqrt {\left (c +d \right ) \left (c -d \right )}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+c -d \right )}{-c \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+\left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) d +\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )+c \ln \left (\frac {2 \sqrt {2}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1}\, \sqrt {\frac {c}{c -d}}\, c -2 \sqrt {2}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1}\, \sqrt {\frac {c}{c -d}}\, d -2 \sqrt {\left (c +d \right ) \left (c -d \right )}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+2 c -2 d}{c \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-\left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) d +\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )\right )}{f \sqrt {\left (c +d \right ) \left (c -d \right )}\, \left (c -d +\sqrt {\left (c +d \right ) \left (c -d \right )}\right ) \left (-c +d +\sqrt {\left (c +d \right ) \left (c -d \right )}\right ) \sqrt {\frac {c}{c -d}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1}}\) | \(685\) |
-g/f*2^(1/2)*(c-d)/((c+d)*(c-d))^(1/2)/(c-d+((c+d)*(c-d))^(1/2))/(-c+d+((c +d)*(c-d))^(1/2))/(c/(c-d))^(1/2)*(-g*((1-cos(f*x+e))^2*csc(f*x+e)^2+1)/(( 1-cos(f*x+e))^2*csc(f*x+e)^2-1))^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)*( -2*a/((1-cos(f*x+e))^2*csc(f*x+e)^2-1))^(1/2)*(((c+d)*(c-d))^(1/2)*arctanh (1/2*(-cot(f*x+e)+csc(f*x+e)+1)*2^(1/2)/((1-cos(f*x+e))^2*csc(f*x+e)^2+1)^ (1/2))*(c/(c-d))^(1/2)+((c+d)*(c-d))^(1/2)*arctanh(1/2*(-cot(f*x+e)+csc(f* x+e)-1)*2^(1/2)/((1-cos(f*x+e))^2*csc(f*x+e)^2+1)^(1/2))*(c/(c-d))^(1/2)-c *ln(-2*(2^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2+1)^(1/2)*(c/(c-d))^(1/2)*c- 2^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2+1)^(1/2)*(c/(c-d))^(1/2)*d+((c+d)*( c-d))^(1/2)*(-cot(f*x+e)+csc(f*x+e))+c-d)/(-c*(-cot(f*x+e)+csc(f*x+e))+(-c ot(f*x+e)+csc(f*x+e))*d+((c+d)*(c-d))^(1/2)))+c*ln(2*(2^(1/2)*((1-cos(f*x+ e))^2*csc(f*x+e)^2+1)^(1/2)*(c/(c-d))^(1/2)*c-2^(1/2)*((1-cos(f*x+e))^2*cs c(f*x+e)^2+1)^(1/2)*(c/(c-d))^(1/2)*d-((c+d)*(c-d))^(1/2)*(-cot(f*x+e)+csc (f*x+e))+c-d)/(c*(-cot(f*x+e)+csc(f*x+e))-(-cot(f*x+e)+csc(f*x+e))*d+((c+d )*(c-d))^(1/2))))/((1-cos(f*x+e))^2*csc(f*x+e)^2+1)^(1/2)
Time = 4.81 (sec) , antiderivative size = 1126, normalized size of antiderivative = 7.56 \[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\text {Too large to display} \]
[1/2*(sqrt(a*c*g/(c + d))*g*log((a*c^2*g*cos(f*x + e)^3 - (7*a*c^2 + 6*a*c *d)*g*cos(f*x + e)^2 + 4*((c^2 + c*d)*cos(f*x + e)^2 - (2*c^2 + 3*c*d + d^ 2)*cos(f*x + e))*sqrt(a*c*g/(c + d))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e ))*sqrt(g/cos(f*x + e))*sin(f*x + e) + (2*a*c*d + a*d^2)*g*cos(f*x + e) + (8*a*c^2 + 8*a*c*d + a*d^2)*g)/(c^2*cos(f*x + e)^3 + (c^2 + 2*c*d)*cos(f*x + e)^2 + d^2 + (2*c*d + d^2)*cos(f*x + e))) + sqrt(a*g)*g*log((a*g*cos(f* x + e)^3 - 7*a*g*cos(f*x + e)^2 - 4*sqrt(a*g)*(cos(f*x + e)^2 - 2*cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt(g/cos(f*x + e))*sin(f*x + e) + 8*a*g)/(cos(f*x + e)^3 + cos(f*x + e)^2)))/(d*f), -1/2*(2*sqrt(-a* c*g/(c + d))*g*arctan(1/2*(c*cos(f*x + e)^2 - (2*c + d)*cos(f*x + e))*sqrt (-a*c*g/(c + d))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt(g/cos(f*x + e))/(a*c*g*sin(f*x + e))) - sqrt(a*g)*g*log((a*g*cos(f*x + e)^3 - 7*a*g*co s(f*x + e)^2 - 4*sqrt(a*g)*(cos(f*x + e)^2 - 2*cos(f*x + e))*sqrt((a*cos(f *x + e) + a)/cos(f*x + e))*sqrt(g/cos(f*x + e))*sin(f*x + e) + 8*a*g)/(cos (f*x + e)^3 + cos(f*x + e)^2)))/(d*f), 1/2*(2*sqrt(-a*g)*g*arctan(2*sqrt(- a*g)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt(g/cos(f*x + e))*cos(f*x + e)*sin(f*x + e)/(a*g*cos(f*x + e)^2 - a*g*cos(f*x + e) - 2*a*g)) + sqrt( a*c*g/(c + d))*g*log((a*c^2*g*cos(f*x + e)^3 - (7*a*c^2 + 6*a*c*d)*g*cos(f *x + e)^2 + 4*((c^2 + c*d)*cos(f*x + e)^2 - (2*c^2 + 3*c*d + d^2)*cos(f*x + e))*sqrt(a*c*g/(c + d))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt(...
\[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\int \frac {\sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )} \left (g \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}{c + d \sec {\left (e + f x \right )}}\, dx \]
Exception generated. \[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\int { \frac {\sqrt {a \sec \left (f x + e\right ) + a} \left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{d \sec \left (f x + e\right ) + c} \,d x } \]
Timed out. \[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\int \frac {\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{c+\frac {d}{\cos \left (e+f\,x\right )}} \,d x \]