Integrand size = 25, antiderivative size = 863 \[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{3/2}} \, dx=\frac {a \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2-b^2\right ) d e^{3/2}}-\frac {b^2 \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a \left (a^2-b^2\right ) d e^{3/2}}-\frac {a \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2-b^2\right ) d e^{3/2}}+\frac {b^2 \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a \left (a^2-b^2\right ) d e^{3/2}}-\frac {a \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2-b^2\right ) d e^{3/2}}+\frac {b^2 \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a \left (a^2-b^2\right ) d e^{3/2}}+\frac {a \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2-b^2\right ) d e^{3/2}}-\frac {b^2 \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a \left (a^2-b^2\right ) d e^{3/2}}-\frac {2 (a-b \sec (c+d x))}{\left (a^2-b^2\right ) d e \sqrt {e \tan (c+d x)}}+\frac {2 \sqrt {2} b^3 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {1+\cos (c+d x)}}\right ),-1\right ) \sqrt {e \tan (c+d x)}}{a (a-b)^{3/2} (a+b)^{3/2} d e^2 \sqrt {\sin (c+d x)}}-\frac {2 \sqrt {2} b^3 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {1+\cos (c+d x)}}\right ),-1\right ) \sqrt {e \tan (c+d x)}}{a (a-b)^{3/2} (a+b)^{3/2} d e^2 \sqrt {\sin (c+d x)}}+\frac {2 b \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {e \tan (c+d x)}}{\left (a^2-b^2\right ) d e^2 \sqrt {\sin (2 c+2 d x)}}-\frac {2 b \cos (c+d x) (e \tan (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e^3} \]
1/2*a*arctan(1-2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))/(a^2-b^2)/d/e^(3/2)*2 ^(1/2)-1/2*b^2*arctan(1-2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))/a/(a^2-b^2)/ d/e^(3/2)*2^(1/2)-1/2*a*arctan(1+2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))/(a^ 2-b^2)/d/e^(3/2)*2^(1/2)+1/2*b^2*arctan(1+2^(1/2)*(e*tan(d*x+c))^(1/2)/e^( 1/2))/a/(a^2-b^2)/d/e^(3/2)*2^(1/2)-1/4*a*ln(e^(1/2)-2^(1/2)*(e*tan(d*x+c) )^(1/2)+e^(1/2)*tan(d*x+c))/(a^2-b^2)/d/e^(3/2)*2^(1/2)+1/4*b^2*ln(e^(1/2) -2^(1/2)*(e*tan(d*x+c))^(1/2)+e^(1/2)*tan(d*x+c))/a/(a^2-b^2)/d/e^(3/2)*2^ (1/2)+1/4*a*ln(e^(1/2)+2^(1/2)*(e*tan(d*x+c))^(1/2)+e^(1/2)*tan(d*x+c))/(a ^2-b^2)/d/e^(3/2)*2^(1/2)-1/4*b^2*ln(e^(1/2)+2^(1/2)*(e*tan(d*x+c))^(1/2)+ e^(1/2)*tan(d*x+c))/a/(a^2-b^2)/d/e^(3/2)*2^(1/2)-2*(a-b*sec(d*x+c))/(a^2- b^2)/d/e/(e*tan(d*x+c))^(1/2)+2*b^3*EllipticPi(sin(d*x+c)^(1/2)/(1+cos(d*x +c))^(1/2),-(a-b)^(1/2)/(a+b)^(1/2),I)*2^(1/2)*cos(d*x+c)^(1/2)*(e*tan(d*x +c))^(1/2)/a/(a-b)^(3/2)/(a+b)^(3/2)/d/e^2/sin(d*x+c)^(1/2)-2*b^3*Elliptic Pi(sin(d*x+c)^(1/2)/(1+cos(d*x+c))^(1/2),(a-b)^(1/2)/(a+b)^(1/2),I)*2^(1/2 )*cos(d*x+c)^(1/2)*(e*tan(d*x+c))^(1/2)/a/(a-b)^(3/2)/(a+b)^(3/2)/d/e^2/si n(d*x+c)^(1/2)-2*b*cos(d*x+c)*(sin(c+1/4*Pi+d*x)^2)^(1/2)/sin(c+1/4*Pi+d*x )*EllipticE(cos(c+1/4*Pi+d*x),2^(1/2))*(e*tan(d*x+c))^(1/2)/(a^2-b^2)/d/e^ 2/sin(2*d*x+2*c)^(1/2)-2*b*cos(d*x+c)*(e*tan(d*x+c))^(3/2)/(a^2-b^2)/d/e^3
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 23.90 (sec) , antiderivative size = 1571, normalized size of antiderivative = 1.82 \[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{3/2}} \, dx =\text {Too large to display} \]
((b + a*Cos[c + d*x])*Sec[c + d*x]*((-2*(b - a*Cos[c + d*x])*Csc[c + d*x]) /(-a^2 + b^2) + (2*b*Sin[c + d*x])/(-a^2 + b^2))*Tan[c + d*x]^2)/(d*(a + b *Sec[c + d*x])*(e*Tan[c + d*x])^(3/2)) + ((b + a*Cos[c + d*x])*Sec[c + d*x ]*Tan[c + d*x]^(3/2)*(-1/12*((-a^2 + 3*b^2)*Sec[c + d*x]*(6*Sqrt[2]*(a^2 - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 6*Sqrt[2]*a^2*ArcTan[1 + Sq rt[2]*Sqrt[Tan[c + d*x]]] + 6*Sqrt[2]*b^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] - (6 + 6*I)*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 - ((1 + I)*Sqrt[b]*S qrt[Tan[c + d*x]])/(a^2 - b^2)^(1/4)] + (6 + 6*I)*Sqrt[b]*(a^2 - b^2)^(3/4 )*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Tan[c + d*x]])/(a^2 - b^2)^(1/4)] - 3*S qrt[2]*a^2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 3*Sqrt[2]* b^2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 3*Sqrt[2]*a^2*Log [1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - 3*Sqrt[2]*b^2*Log[1 + Sq rt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + (3 + 3*I)*Sqrt[b]*(a^2 - b^2)^( 3/4)*Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Tan[c + d*x]] + I*b*Tan[c + d*x]] - (3 + 3*I)*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a ^2 - b^2] + (1 + I)*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Tan[c + d*x]] + I*b*Tan [c + d*x]] + 8*a*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Tan[c + d*x]^(3/2))*(a + b*Sqrt[1 + Tan[c + d*x]^2] ))/((a^3 - a*b^2)*(b + a*Cos[c + d*x])*(1 + Tan[c + d*x]^2)) + (b*Cos[2*(c + d*x)]*Sec[c + d*x]^2*(-84*Sqrt[2]*b*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + ...
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.
| \(\displaystyle \int \frac {1}{(e \tan (c+d x))^{3/2} (a+b \sec (c+d x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (-e \cot \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 4381 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {e \tan (c+d x)}}{a+b \sec (c+d x)}dx}{e^2 \left (a^2-b^2\right )}+\frac {\int \frac {a-b \sec (c+d x)}{(e \tan (c+d x))^{3/2}}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {\int \frac {a-b \csc \left (c+d x+\frac {\pi }{2}\right )}{\left (-e \cot \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 4370 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {\frac {2 \int -\frac {1}{2} (a+b \sec (c+d x)) \sqrt {e \tan (c+d x)}dx}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\int (a+b \sec (c+d x)) \sqrt {e \tan (c+d x)}dx}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\int \sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 4372 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {a \int \sqrt {e \tan (c+d x)}dx+b \int \sec (c+d x) \sqrt {e \tan (c+d x)}dx}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {a \int \sqrt {e \tan (c+d x)}dx+b \int \sec (c+d x) \sqrt {e \tan (c+d x)}dx}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 3093 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {a \int \sqrt {e \tan (c+d x)}dx+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-2 \int \cos (c+d x) \sqrt {e \tan (c+d x)}dx\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {a \int \sqrt {e \tan (c+d x)}dx+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-2 \int \frac {\sqrt {e \tan (c+d x)}}{\sec (c+d x)}dx\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 3095 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {a \int \sqrt {e \tan (c+d x)}dx+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \int \sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {a \int \sqrt {e \tan (c+d x)}dx+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \int \sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {a \int \sqrt {e \tan (c+d x)}dx+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) \sqrt {e \tan (c+d x)} \int \sqrt {\sin (2 c+2 d x)}dx}{\sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {a \int \sqrt {e \tan (c+d x)}dx+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) \sqrt {e \tan (c+d x)} \int \sqrt {\sin (2 c+2 d x)}dx}{\sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {a \int \sqrt {e \tan (c+d x)}dx+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {a e \int \frac {\sqrt {e \tan (c+d x)}}{\tan ^2(c+d x) e^2+e^2}d(e \tan (c+d x))}{d}+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {2 a e \int \frac {e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}}{d}+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {2 a e \left (\frac {1}{2} \int \frac {e^2 \tan ^2(c+d x)+e}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}-\frac {1}{2} \int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}\right )}{d}+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {2 a e \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}+\frac {1}{2} \int \frac {1}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}\right )-\frac {1}{2} \int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}\right )}{d}+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {2 a e \left (\frac {1}{2} \left (\frac {\int \frac {1}{-e^2 \tan ^2(c+d x)-1}d\left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e^2 \tan ^2(c+d x)-1}d\left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} \int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}\right )}{d}+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {\frac {2 a e \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} \int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}\right )}{d}+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}+\frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {-\frac {\frac {2 a e \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}+\frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {2 a e \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}+\frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {2 a e \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}+\frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {b^2 \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {2 a e \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d}+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 4378 |
| \(\displaystyle \frac {b^2 \left (\frac {\int \sqrt {e \tan (c+d x)}dx}{a}-\frac {b \int \frac {\sqrt {e \tan (c+d x)}}{b+a \cos (c+d x)}dx}{a}\right )}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {2 a e \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d}+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \left (\frac {\int \sqrt {e \tan (c+d x)}dx}{a}-\frac {b \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}\right )}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {2 a e \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d}+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 3212 |
| \(\displaystyle \frac {b^2 \left (\frac {\int \sqrt {e \tan (c+d x)}dx}{a}-\frac {b \sqrt {e \tan (c+d x)} \sqrt {e \cot (c+d x)} \int \frac {1}{(b+a \cos (c+d x)) \sqrt {e \cot (c+d x)}}dx}{a}\right )}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {2 a e \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d}+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \left (\frac {\int \sqrt {e \tan (c+d x)}dx}{a}-\frac {b \sqrt {e \tan (c+d x)} \sqrt {e \cot (c+d x)} \int \frac {1}{\left (b-a \sin \left (c+d x-\frac {\pi }{2}\right )\right ) \sqrt {-e \tan \left (c+d x-\frac {\pi }{2}\right )}}dx}{a}\right )}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {2 a e \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d}+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
| \(\Big \downarrow \) 3209 |
| \(\displaystyle \frac {b^2 \left (\frac {\int \sqrt {e \tan (c+d x)}dx}{a}-\frac {b \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)} \int \frac {\sqrt {\sin (c+d x)}}{\sqrt {-\cos (c+d x)} (b+a \cos (c+d x))}dx}{a \sqrt {\sin (c+d x)}}\right )}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {2 a e \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d}+b \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-b \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}}{a^2-b^2}\) |
3.4.16.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]] , x_Symbol] :> Simp[Sqrt[a*Sin[e + f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]) Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f}, x]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[a^2*(a*Sec[e + f*x])^(m - 2)*((b*Tan[e + f*x])^(n + 1)/(b*f*(m + n - 1))), x] + Simp[a^2*((m - 2)/(m + n - 1)) Int[(a*Sec[e + f*x])^(m - 2)*(b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && ( GtQ[m, 1] || (EqQ[m, 1] && EqQ[n, 1/2])) && NeQ[m + n - 1, 0] && IntegersQ[ 2*m, 2*n]
Int[Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]]/sec[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[Sqrt[Cos[e + f*x]]*(Sqrt[b*Tan[e + f*x]]/Sqrt[Sin[e + f*x]]) Int[ Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]], x], x] /; FreeQ[{b, e, f}, x]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/(((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(g_)*tan[(e_.) + (f_.)*( x_)]]), x_Symbol] :> Simp[Sqrt[Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*Sqrt[g*Tan [e + f*x]]) Int[Sqrt[Cos[e + f*x]]/(Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x ])), x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Int[(cot[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.), x_Symbol] :> Simp[g^(2*IntPart[p])*(g*Cot[e + f*x])^FracPart[p] *(g*Tan[e + f*x])^FracPart[p] Int[(a + b*Sin[e + f*x])^m/(g*Tan[e + f*x]) ^p, x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && !IntegerQ[p]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Subst[Int [x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[(-(e*Cot[c + d*x])^(m + 1))*((a + b*Csc[c + d*x])/( d*e*(m + 1))), x] - Simp[1/(e^2*(m + 1)) Int[(e*Cot[c + d*x])^(m + 2)*(a* (m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && L tQ[m, -1]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(e*Cot[c + d*x])^m, x], x] + Simp[b Int[ (e*Cot[c + d*x])^m*Csc[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, m}, x]
Int[Sqrt[cot[(c_.) + (d_.)*(x_)]*(e_.)]/(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a _)), x_Symbol] :> Simp[1/a Int[Sqrt[e*Cot[c + d*x]], x], x] - Simp[b/a Int[Sqrt[e*Cot[c + d*x]]/(b + a*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2, 0]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)/(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[1/(a^2 - b^2) Int[(e*Cot[c + d*x])^m*(a - b*Csc[c + d*x]), x], x] + Simp[b^2/(e^2*(a^2 - b^2)) Int[(e*Cot[c + d*x])^(m + 2 )/(a + b*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^ 2, 0] && ILtQ[m + 1/2, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4015 vs. \(2 (746 ) = 1492\).
Time = 3.27 (sec) , antiderivative size = 4016, normalized size of antiderivative = 4.65
-1/d*2^(1/2)*b/((a^2-b^2)^(1/2)-a+b)/((a^2-b^2)^(1/2)+a-b)/(a-b)/a/(a+b)*( 1-cos(d*x+c))*(-((1-cos(d*x+c))*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)*csc(d*x+ c))^(1/2)*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(2-2*csc(d*x+c)+2*cot(d*x+c))^(1 /2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*EllipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/ 2),1/2-1/2*I,1/2*2^(1/2))*a^3-2*((1-cos(d*x+c))^3*csc(d*x+c)^3+cot(d*x+c)- csc(d*x+c))^(1/2)*a*b^2*(1-cos(d*x+c))^2*csc(d*x+c)^2+4*((1-cos(d*x+c))^3* csc(d*x+c)^3+cot(d*x+c)-csc(d*x+c))^(1/2)*a^2*b*(1-cos(d*x+c))^2*csc(d*x+c )^2+((1-cos(d*x+c))*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)*csc(d*x+c))^(1/2)*(c sc(d*x+c)-cot(d*x+c)+1)^(1/2)*(2-2*csc(d*x+c)+2*cot(d*x+c))^(1/2)*(cot(d*x +c)-csc(d*x+c))^(1/2)*(a^2-b^2)^(1/2)*EllipticPi((csc(d*x+c)-cot(d*x+c)+1) ^(1/2),(a-b)/(a-b+((a-b)*(a+b))^(1/2)),1/2*2^(1/2))*b^2-((1-cos(d*x+c))*(( 1-cos(d*x+c))^2*csc(d*x+c)^2-1)*csc(d*x+c))^(1/2)*(csc(d*x+c)-cot(d*x+c)+1 )^(1/2)*(2-2*csc(d*x+c)+2*cot(d*x+c))^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)* (a^2-b^2)^(1/2)*EllipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),-(a-b)/(-a+b+(( a-b)*(a+b))^(1/2)),1/2*2^(1/2))*b^2-((1-cos(d*x+c))*((1-cos(d*x+c))^2*csc( d*x+c)^2-1)*csc(d*x+c))^(1/2)*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(2-2*csc(d*x +c)+2*cot(d*x+c))^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*EllipticPi((csc(d*x+ c)-cot(d*x+c)+1)^(1/2),(a-b)/(a-b+((a-b)*(a+b))^(1/2)),1/2*2^(1/2))*a*b^2+ ((1-cos(d*x+c))*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)*csc(d*x+c))^(1/2)*(csc(d *x+c)-cot(d*x+c)+1)^(1/2)*(2-2*csc(d*x+c)+2*cot(d*x+c))^(1/2)*(cot(d*x+...
Timed out. \[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{3/2}} \, dx=\int \frac {1}{\left (e \tan {\left (c + d x \right )}\right )^{\frac {3}{2}} \left (a + b \sec {\left (c + d x \right )}\right )}\, dx \]
\[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )} \left (e \tan \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )} \left (e \tan \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{3/2}} \, dx=\int \frac {\cos \left (c+d\,x\right )}{{\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}\,\left (b+a\,\cos \left (c+d\,x\right )\right )} \,d x \]