Integrand size = 25, antiderivative size = 836 \[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{5/2}} \, dx=\frac {a \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2-b^2\right ) d e^{5/2}}-\frac {b^2 \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a \left (a^2-b^2\right ) d e^{5/2}}-\frac {a \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2-b^2\right ) d e^{5/2}}+\frac {b^2 \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a \left (a^2-b^2\right ) d e^{5/2}}+\frac {a \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2-b^2\right ) d e^{5/2}}-\frac {b^2 \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a \left (a^2-b^2\right ) d e^{5/2}}-\frac {a \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2-b^2\right ) d e^{5/2}}+\frac {b^2 \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a \left (a^2-b^2\right ) d e^{5/2}}-\frac {2 (a-b \sec (c+d x))}{3 \left (a^2-b^2\right ) d e (e \tan (c+d x))^{3/2}}-\frac {2 \sqrt {2} b^3 \operatorname {EllipticPi}\left (\frac {b}{a-\sqrt {a^2-b^2}},\arcsin \left (\frac {\sqrt {-\cos (c+d x)}}{\sqrt {1+\sin (c+d x)}}\right ),-1\right ) \sqrt {\sin (c+d x)}}{a \left (a^2-b^2\right )^{3/2} d e^2 \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}+\frac {2 \sqrt {2} b^3 \operatorname {EllipticPi}\left (\frac {b}{a+\sqrt {a^2-b^2}},\arcsin \left (\frac {\sqrt {-\cos (c+d x)}}{\sqrt {1+\sin (c+d x)}}\right ),-1\right ) \sqrt {\sin (c+d x)}}{a \left (a^2-b^2\right )^{3/2} d e^2 \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}+\frac {b \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 \left (a^2-b^2\right ) d e^2 \sqrt {e \tan (c+d x)}} \]
1/2*a*arctan(1-2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))/(a^2-b^2)/d/e^(5/2)*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^(5/2)*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^(5/2)*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^(5/2)*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^(5/2)*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^(5/2)*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^(5/2)*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^(5/2)*2^(1/2)-2*b^3*EllipticPi((-cos(d *x+c))^(1/2)/(1+sin(d*x+c))^(1/2),b/(a-(a^2-b^2)^(1/2)),I)*2^(1/2)*sin(d*x +c)^(1/2)/a/(a^2-b^2)^(3/2)/d/e^2/(-cos(d*x+c))^(1/2)/(e*tan(d*x+c))^(1/2) +2*b^3*EllipticPi((-cos(d*x+c))^(1/2)/(1+sin(d*x+c))^(1/2),b/(a+(a^2-b^2)^ (1/2)),I)*2^(1/2)*sin(d*x+c)^(1/2)/a/(a^2-b^2)^(3/2)/d/e^2/(-cos(d*x+c))^( 1/2)/(e*tan(d*x+c))^(1/2)-1/3*b*(sin(c+1/4*Pi+d*x)^2)^(1/2)/sin(c+1/4*Pi+d *x)*EllipticF(cos(c+1/4*Pi+d*x),2^(1/2))*sec(d*x+c)*sin(2*d*x+2*c)^(1/2)/( a^2-b^2)/d/e^2/(e*tan(d*x+c))^(1/2)-2/3*(a-b*sec(d*x+c))/(a^2-b^2)/d/e/(e* tan(d*x+c))^(3/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 33.62 (sec) , antiderivative size = 2169, normalized size of antiderivative = 2.59 \[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{5/2}} \, dx=\text {Result too large to show} \]
((b + a*Cos[c + d*x])*((2*a)/(3*(a^2 - b^2)) - (2*(-a + b*Cos[c + d*x])*Cs c[c + d*x]^2)/(3*(-a^2 + b^2)))*Sec[c + d*x]*Tan[c + d*x]^3)/(d*(a + b*Sec [c + d*x])*(e*Tan[c + d*x])^(5/2)) - ((b + a*Cos[c + d*x])*Sec[c + d*x]*Ta n[c + d*x]^(5/2)*((2*(3*a^2 - 5*b^2)*Sec[c + d*x]^3*(a + b*Sqrt[1 + Tan[c + d*x]^2])*(((-1/8 + I/8)*a*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Tan[c + d* x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Tan[c + d*x]] )/(a^2 - b^2)^(1/4)] + 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]] - 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]]))/(Sqrt [b]*(a^2 - b^2)^(3/4)) + (5*b*(-a^2 + b^2)*AppellF1[1/4, -1/2, 1, 5/4, -Ta n[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Tan[c + d*x]]*Sqrt[1 + Tan[c + d*x]^2])/((5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, -Tan[c + d* x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (a^2 - b^2)*Appe llF1[5/4, 1/2, 1, 9/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]) *Tan[c + d*x]^2)*(a^2 - b^2*(1 + Tan[c + d*x]^2)))))/((b + a*Cos[c + d*x]) *(1 + Tan[c + d*x]^2)^2) + (8*a*b*Sec[c + d*x]^2*(a + b*Sqrt[1 + Tan[c + d *x]^2])*((Sqrt[b]*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Tan[c + d*x]])/(-a^ 2 + b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Tan[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[b]*(-a^2 + b^2)^(1/...
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.
\(\displaystyle \int \frac {1}{(e \tan (c+d x))^{5/2} (a+b \sec (c+d x))} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (-e \cot \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx\) |
\(\Big \downarrow \) 4381 |
\(\displaystyle \frac {b^2 \int \frac {1}{(a+b \sec (c+d x)) \sqrt {e \tan (c+d x)}}dx}{e^2 \left (a^2-b^2\right )}+\frac {\int \frac {a-b \sec (c+d x)}{(e \tan (c+d x))^{5/2}}dx}{a^2-b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {\int \frac {a-b \csc \left (c+d x+\frac {\pi }{2}\right )}{\left (-e \cot \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{a^2-b^2}\) |
\(\Big \downarrow \) 4370 |
\(\displaystyle \frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {\frac {2 \int -\frac {3 a-b \sec (c+d x)}{2 \sqrt {e \tan (c+d x)}}dx}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\int \frac {3 a-b \sec (c+d x)}{\sqrt {e \tan (c+d x)}}dx}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\int \frac {3 a-b \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 4372 |
\(\displaystyle \frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {3 a \int \frac {1}{\sqrt {e \tan (c+d x)}}dx-b \int \frac {\sec (c+d x)}{\sqrt {e \tan (c+d x)}}dx}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {3 a \int \frac {1}{\sqrt {e \tan (c+d x)}}dx-b \int \frac {\sec (c+d x)}{\sqrt {e \tan (c+d x)}}dx}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 3094 |
\(\displaystyle \frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {3 a \int \frac {1}{\sqrt {e \tan (c+d x)}}dx-\frac {b \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}}dx}{\sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {3 a \int \frac {1}{\sqrt {e \tan (c+d x)}}dx-\frac {b \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}}dx}{\sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 3053 |
\(\displaystyle \frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {3 a \int \frac {1}{\sqrt {e \tan (c+d x)}}dx-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \int \frac {1}{\sqrt {\sin (2 c+2 d x)}}dx}{\sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {3 a \int \frac {1}{\sqrt {e \tan (c+d x)}}dx-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \int \frac {1}{\sqrt {\sin (2 c+2 d x)}}dx}{\sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {3 a \int \frac {1}{\sqrt {e \tan (c+d x)}}dx-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle \frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {3 a e \int \frac {1}{\sqrt {e \tan (c+d x)} \left (\tan ^2(c+d x) e^2+e^2\right )}d(e \tan (c+d x))}{d}-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {6 a e \int \frac {1}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}}{d}-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 755 |
\(\displaystyle \frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {6 a e \left (\frac {\int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}}{2 e}+\frac {\int \frac {e^2 \tan ^2(c+d x)+e}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}}{2 e}\right )}{d}-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {6 a e \left (\frac {\frac {1}{2} \int \frac {1}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}+\frac {1}{2} \int \frac {1}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 e}+\frac {\int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}}{2 e}\right )}{d}-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {6 a e \left (\frac {\frac {\int \frac {1}{-e^2 \tan ^2(c+d x)-1}d\left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e^2 \tan ^2(c+d x)-1}d\left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}}{2 e}\right )}{d}-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {-\frac {\frac {6 a e \left (\frac {\int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}+\frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {-\frac {\frac {6 a e \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}+\frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {-\frac {\frac {6 a e \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}+\frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {-\frac {\frac {6 a e \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {e}}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}+\frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {b^2 \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {6 a e \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 4380 |
\(\displaystyle \frac {b^2 \left (\frac {\int \frac {1}{\sqrt {e \tan (c+d x)}}dx}{a}-\frac {b \int \frac {1}{(b+a \cos (c+d x)) \sqrt {e \tan (c+d x)}}dx}{a}\right )}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {6 a e \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b^2 \left (\frac {\int \frac {1}{\sqrt {e \tan (c+d x)}}dx}{a}-\frac {b \int \frac {1}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}\right )}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {6 a e \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 3212 |
\(\displaystyle \frac {b^2 \left (\frac {\int \frac {1}{\sqrt {e \tan (c+d x)}}dx}{a}-\frac {b \int \frac {\sqrt {e \cot (c+d x)}}{b+a \cos (c+d x)}dx}{a \sqrt {e \tan (c+d x)} \sqrt {e \cot (c+d x)}}\right )}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {6 a e \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b^2 \left (\frac {\int \frac {1}{\sqrt {e \tan (c+d x)}}dx}{a}-\frac {b \int \frac {\sqrt {-e \tan \left (c+d x-\frac {\pi }{2}\right )}}{b-a \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{a \sqrt {e \tan (c+d x)} \sqrt {e \cot (c+d x)}}\right )}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {6 a e \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 3208 |
\(\displaystyle \frac {b^2 \left (\frac {\int \frac {1}{\sqrt {e \tan (c+d x)}}dx}{a}-\frac {b \sqrt {\sin (c+d x)} \int \frac {\sqrt {-\cos (c+d x)}}{(b+a \cos (c+d x)) \sqrt {\sin (c+d x)}}dx}{a \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}\right )}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {6 a e \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b^2 \left (\frac {\int \frac {1}{\sqrt {e \tan (c+d x)}}dx}{a}-\frac {b \sqrt {\sin (c+d x)} \int \frac {\sqrt {\sin \left (c+d x-\frac {\pi }{2}\right )}}{\sqrt {\cos \left (c+d x-\frac {\pi }{2}\right )} \left (b-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{a \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}\right )}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {6 a e \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
\(\Big \downarrow \) 3386 |
\(\displaystyle \frac {b^2 \left (\frac {\int \frac {1}{\sqrt {e \tan (c+d x)}}dx}{a}-\frac {b \sqrt {\sin (c+d x)} \left (\frac {2 \sqrt {2} \left (1-\frac {a}{\sqrt {a^2-b^2}}\right ) \int -\frac {1}{\sqrt {1-\frac {\cos ^2(c+d x)}{(\sin (c+d x)+1)^2}} \left (a-\sqrt {a^2-b^2}+\frac {b \cos (c+d x)}{\sin (c+d x)+1}\right )}d\frac {\sqrt {-\cos (c+d x)}}{\sqrt {\sin (c+d x)+1}}}{d}+\frac {2 \sqrt {2} \left (\frac {a}{\sqrt {a^2-b^2}}+1\right ) \int -\frac {1}{\sqrt {1-\frac {\cos ^2(c+d x)}{(\sin (c+d x)+1)^2}} \left (a+\sqrt {a^2-b^2}+\frac {b \cos (c+d x)}{\sin (c+d x)+1}\right )}d\frac {\sqrt {-\cos (c+d x)}}{\sqrt {\sin (c+d x)+1}}}{d}\right )}{a \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}\right )}{e^2 \left (a^2-b^2\right )}+\frac {-\frac {\frac {6 a e \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {b \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-b \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}}{a^2-b^2}\) |
3.4.17.3.1 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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) 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[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_ )]]), x_Symbol] :> Simp[Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b *Cos[e + f*x]]) Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f }, x]
Int[sec[(e_.) + (f_.)*(x_)]/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[Sqrt[Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]]) Int[ 1/(Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]]), x], x] /; FreeQ[{b, e, f}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[Sqrt[(g_.)*tan[(e_.) + (f_.)*(x_)]]/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_ )]), x_Symbol] :> Simp[Sqrt[Cos[e + f*x]]*(Sqrt[g*Tan[e + f*x]]/Sqrt[Sin[e + f*x]]) Int[Sqrt[Sin[e + f*x]]/(Sqrt[Cos[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[Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[cos[(e_.) + (f_.)*(x_)]]*((a_ ) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> With[{q = Rt[-a^2 + b^2, 2]}, Simp[2*Sqrt[2]*d*((b + q)/(f*q)) Subst[Int[1/((d*(b + q) + a*x^2)*Sq rt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]], x] - Simp[2*Sqrt[2]*d*((b - q)/(f*q)) Subst[Int[1/((d*(b - q) + a*x^2)*Sqrt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]], x]] /; F reeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
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[1/(Sqrt[cot[(c_.) + (d_.)*(x_)]*(e_.)]*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))), x_Symbol] :> Simp[1/a Int[1/Sqrt[e*Cot[c + d*x]], x], x] - Simp[ b/a Int[1/(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. 7580 vs. \(2 (725 ) = 1450\).
Time = 4.79 (sec) , antiderivative size = 7581, normalized size of antiderivative = 9.07
Timed out. \[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{5/2}} \, dx=\int \frac {1}{\left (e \tan {\left (c + d x \right )}\right )^{\frac {5}{2}} \left (a + b \sec {\left (c + d x \right )}\right )}\, dx \]
\[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )} \left (e \tan \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )} \left (e \tan \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{5/2}} \, dx=\int \frac {\cos \left (c+d\,x\right )}{{\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}\,\left (b+a\,\cos \left (c+d\,x\right )\right )} \,d x \]