Integrand size = 25, antiderivative size = 415 \[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx=-\frac {\sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a d}+\frac {\sqrt {e} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a d}+\frac {\sqrt {e} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a d}-\frac {\sqrt {e} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a d}+\frac {2 \sqrt {2} b \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {1+\cos (c+d x)}}\right ),-1\right ) \sqrt {e \tan (c+d x)}}{a \sqrt {a-b} \sqrt {a+b} d \sqrt {\sin (c+d x)}}-\frac {2 \sqrt {2} b \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {1+\cos (c+d x)}}\right ),-1\right ) \sqrt {e \tan (c+d x)}}{a \sqrt {a-b} \sqrt {a+b} d \sqrt {\sin (c+d x)}} \]
-1/2*arctan(1-2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))*e^(1/2)/a/d*2^(1/2)+1/ 2*arctan(1+2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))*e^(1/2)/a/d*2^(1/2)+1/4*l n(e^(1/2)-2^(1/2)*(e*tan(d*x+c))^(1/2)+e^(1/2)*tan(d*x+c))*e^(1/2)/a/d*2^( 1/2)-1/4*ln(e^(1/2)+2^(1/2)*(e*tan(d*x+c))^(1/2)+e^(1/2)*tan(d*x+c))*e^(1/ 2)/a/d*2^(1/2)+2*b*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/d/(a -b)^(1/2)/(a+b)^(1/2)/sin(d*x+c)^(1/2)-2*b*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*t an(d*x+c))^(1/2)/a/d/(a-b)^(1/2)/(a+b)^(1/2)/sin(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 5.98 (sec) , antiderivative size = 646, normalized size of antiderivative = 1.56 \[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx=-\frac {\cos (c+d x) \left (a+b \sqrt {\sec ^2(c+d x)}\right ) \sqrt {e \tan (c+d x)} \left (6 \sqrt {2} \left (a^2-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-6 \sqrt {2} a^2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+6 \sqrt {2} b^2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )-(6+6 i) \sqrt {b} \left (a^2-b^2\right )^{3/4} \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )+(6+6 i) \sqrt {b} \left (a^2-b^2\right )^{3/4} \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-3 \sqrt {2} a^2 \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+3 \sqrt {2} b^2 \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+3 \sqrt {2} a^2 \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-3 \sqrt {2} b^2 \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+(3+3 i) \sqrt {b} \left (a^2-b^2\right )^{3/4} \log \left (\sqrt {a^2-b^2}-(1+i) \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\tan (c+d x)}+i b \tan (c+d x)\right )-(3+3 i) \sqrt {b} \left (a^2-b^2\right )^{3/4} \log \left (\sqrt {a^2-b^2}+(1+i) \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\tan (c+d x)}+i b \tan (c+d x)\right )+8 a b \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\tan ^2(c+d x),\frac {b^2 \tan ^2(c+d x)}{a^2-b^2}\right ) \tan ^{\frac {3}{2}}(c+d x)\right )}{12 a \left (a^2-b^2\right ) d (b+a \cos (c+d x)) \sqrt {\tan (c+d x)}} \]
-1/12*(Cos[c + d*x]*(a + b*Sqrt[Sec[c + d*x]^2])*Sqrt[e*Tan[c + d*x]]*(6*S qrt[2]*(a^2 - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 6*Sqrt[2]*a^2* ArcTan[1 + Sqrt[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]*Sqrt[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*Sqrt[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*Sq rt[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 + 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*(a^2 - b^2)*d* (b + a*Cos[c + d*x])*Sqrt[Tan[c + d*x]])
Time = 1.35 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.85, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.920, Rules used = {3042, 4378, 3042, 3212, 3042, 3209, 3042, 3385, 3042, 3384, 993, 1537, 412, 3957, 266, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
\(\displaystyle \int \frac {\sqrt {e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 4378 |
\(\displaystyle \frac {\int \sqrt {e \tan (c+d x)}dx}{a}-\frac {b \int \frac {\sqrt {e \tan (c+d x)}}{b+a \cos (c+d x)}dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sqrt {e \tan (c+d x)}dx}{a}-\frac {b \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}\) |
\(\Big \downarrow \) 3212 |
\(\displaystyle \frac {\int \sqrt {e \tan (c+d x)}dx}{a}-\frac {b \sqrt {e \tan (c+d x)} \sqrt {e \cot (c+d x)} \int \frac {1}{(b+a \cos (c+d x)) \sqrt {e \cot (c+d x)}}dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sqrt {e \tan (c+d x)}dx}{a}-\frac {b \sqrt {e \tan (c+d x)} \sqrt {e \cot (c+d x)} \int \frac {1}{\left (b-a \sin \left (c+d x-\frac {\pi }{2}\right )\right ) \sqrt {-e \tan \left (c+d x-\frac {\pi }{2}\right )}}dx}{a}\) |
\(\Big \downarrow \) 3209 |
\(\displaystyle \frac {\int \sqrt {e \tan (c+d x)}dx}{a}-\frac {b \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)} \int \frac {\sqrt {\sin (c+d x)}}{\sqrt {-\cos (c+d x)} (b+a \cos (c+d x))}dx}{a \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sqrt {e \tan (c+d x)}dx}{a}-\frac {b \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)} \int \frac {\sqrt {-\cos \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {-\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 3385 |
\(\displaystyle \frac {\int \sqrt {e \tan (c+d x)}dx}{a}-\frac {b \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \int \frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{a \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sqrt {e \tan (c+d x)}dx}{a}-\frac {b \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \int \frac {\sqrt {-\cos \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 3384 |
\(\displaystyle \frac {\int \sqrt {e \tan (c+d x)}dx}{a}-\frac {4 \sqrt {2} b \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \int \frac {\sin (c+d x)}{(\cos (c+d x)+1) \sqrt {1-\frac {\sin ^2(c+d x)}{(\cos (c+d x)+1)^2}} \left (-\frac {(a-b) \sin ^2(c+d x)}{(\cos (c+d x)+1)^2}+a+b\right )}d\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}}{a d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 993 |
\(\displaystyle \frac {\int \sqrt {e \tan (c+d x)}dx}{a}-\frac {4 \sqrt {2} b \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \left (\frac {\int \frac {1}{\left (\sqrt {a+b}-\frac {\sqrt {a-b} \sin (c+d x)}{\cos (c+d x)+1}\right ) \sqrt {1-\frac {\sin ^2(c+d x)}{(\cos (c+d x)+1)^2}}}d\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}}{2 \sqrt {a-b}}-\frac {\int \frac {1}{\left (\frac {\sqrt {a-b} \sin (c+d x)}{\cos (c+d x)+1}+\sqrt {a+b}\right ) \sqrt {1-\frac {\sin ^2(c+d x)}{(\cos (c+d x)+1)^2}}}d\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}}{2 \sqrt {a-b}}\right )}{a d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 1537 |
\(\displaystyle \frac {\int \sqrt {e \tan (c+d x)}dx}{a}-\frac {4 \sqrt {2} b \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \left (\frac {\int \frac {1}{\sqrt {1-\frac {\sin (c+d x)}{\cos (c+d x)+1}} \sqrt {\frac {\sin (c+d x)}{\cos (c+d x)+1}+1} \left (\sqrt {a+b}-\frac {\sqrt {a-b} \sin (c+d x)}{\cos (c+d x)+1}\right )}d\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}}{2 \sqrt {a-b}}-\frac {\int \frac {1}{\sqrt {1-\frac {\sin (c+d x)}{\cos (c+d x)+1}} \sqrt {\frac {\sin (c+d x)}{\cos (c+d x)+1}+1} \left (\frac {\sqrt {a-b} \sin (c+d x)}{\cos (c+d x)+1}+\sqrt {a+b}\right )}d\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}}{2 \sqrt {a-b}}\right )}{a d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle \frac {\int \sqrt {e \tan (c+d x)}dx}{a}-\frac {4 \sqrt {2} b \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}\right )}{a d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle \frac {e \int \frac {\sqrt {e \tan (c+d x)}}{\tan ^2(c+d x) e^2+e^2}d(e \tan (c+d x))}{a d}-\frac {4 \sqrt {2} b \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}\right )}{a d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {2 e \int \frac {e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}}{a d}-\frac {4 \sqrt {2} b \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}\right )}{a d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 826 |
\(\displaystyle \frac {2 e \left (\frac {1}{2} \int \frac {e^2 \tan ^2(c+d x)+e}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}-\frac {1}{2} \int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}\right )}{a d}-\frac {4 \sqrt {2} b \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}\right )}{a d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {2 e \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}+\frac {1}{2} \int \frac {1}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}\right )-\frac {1}{2} \int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}\right )}{a d}-\frac {4 \sqrt {2} b \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}\right )}{a d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {2 e \left (\frac {1}{2} \left (\frac {\int \frac {1}{-e^2 \tan ^2(c+d x)-1}d\left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e^2 \tan ^2(c+d x)-1}d\left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} \int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}\right )}{a d}-\frac {4 \sqrt {2} b \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}\right )}{a d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {2 e \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} \int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}\right )}{a d}-\frac {4 \sqrt {2} b \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}\right )}{a d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {2 e \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{a d}-\frac {4 \sqrt {2} b \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}\right )}{a d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 e \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{a d}-\frac {4 \sqrt {2} b \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}\right )}{a d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 e \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{a d}-\frac {4 \sqrt {2} b \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}\right )}{a d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {2 e \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{a d}-\frac {4 \sqrt {2} b \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}\right )}{a d \sqrt {\sin (c+d x)}}\) |
(2*e*((-(ArcTan[1 - Sqrt[2]*Sqrt[e]*Tan[c + d*x]]/(Sqrt[2]*Sqrt[e])) + Arc Tan[1 + Sqrt[2]*Sqrt[e]*Tan[c + d*x]]/(Sqrt[2]*Sqrt[e]))/2 + (Log[e - Sqrt [2]*e^(3/2)*Tan[c + d*x] + e^2*Tan[c + d*x]^2]/(2*Sqrt[2]*Sqrt[e]) - Log[e + Sqrt[2]*e^(3/2)*Tan[c + d*x] + e^2*Tan[c + d*x]^2]/(2*Sqrt[2]*Sqrt[e])) /2))/(a*d) - (4*Sqrt[2]*b*Sqrt[Cos[c + d*x]]*(-1/2*EllipticPi[-(Sqrt[a - b ]/Sqrt[a + b]), ArcSin[Sqrt[Sin[c + d*x]]/Sqrt[1 + Cos[c + d*x]]], -1]/(Sq rt[a - b]*Sqrt[a + b]) + EllipticPi[Sqrt[a - b]/Sqrt[a + b], ArcSin[Sqrt[S in[c + d*x]]/Sqrt[1 + Cos[c + d*x]]], -1]/(2*Sqrt[a - b]*Sqrt[a + b]))*Sqr t[e*Tan[c + d*x]])/(a*d*Sqrt[Sin[c + d*x]])
3.4.14.3.1 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[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
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[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* b) Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b) Int[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
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/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[(-a)*c, 2]}, Simp[Sqrt[-c] Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqr t[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] & & GtQ[a, 0] && LtQ[c, 0]
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[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/(Sqrt[sin[(e_.) + (f_.)*(x_)]]*((a_ ) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[-4*Sqrt[2]*(g/f) S ubst[Int[x^2/(((a + b)*g^2 + (a - b)*x^4)*Sqrt[1 - x^4/g^2]), x], x, Sqrt[g *Cos[e + f*x]]/Sqrt[1 + Sin[e + f*x]]], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/(Sqrt[(d_)*sin[(e_.) + (f_.)*(x_)]] *((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[Sqrt[Sin[e + f* x]]/Sqrt[d*Sin[e + f*x]] Int[Sqrt[g*Cos[e + f*x]]/(Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f, g}, 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[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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 819 vs. \(2 (328 ) = 656\).
Time = 2.54 (sec) , antiderivative size = 820, normalized size of antiderivative = 1.98
1/d*2^(1/2)*b/((a^2-b^2)^(1/2)-a+b)/((a^2-b^2)^(1/2)+a-b)/a*(I*EllipticPi( (csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*a-I*EllipticPi((csc (d*x+c)-cot(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*b-I*EllipticPi((csc(d*x +c)-cot(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*a+I*EllipticPi((csc(d*x+c)- cot(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*b-EllipticPi((csc(d*x+c)-cot(d* x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*a+EllipticPi((csc(d*x+c)-cot(d*x+c)+1 )^(1/2),1/2-1/2*I,1/2*2^(1/2))*b-EllipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2 ),1/2+1/2*I,1/2*2^(1/2))*a+EllipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2+ 1/2*I,1/2*2^(1/2))*b-(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))+(a^2-b^2)^(1/2)*Ellipti cPi((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))+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*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))+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*EllipticPi((c sc(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 )))*(cot(d*x+c)-csc(d*x+c))^(1/2)*(2-2*csc(d*x+c)+2*cot(d*x+c))^(1/2)*(csc (d*x+c)-cot(d*x+c)+1)^(1/2)*(-e/((1-cos(d*x+c))^2*csc(d*x+c)^2-1)*(-cot(d* x+c)+csc(d*x+c)))^(1/2)*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)/((1-cos(d*x+c))^ 3*csc(d*x+c)^3+cot(d*x+c)-csc(d*x+c))^(1/2)/((1-cos(d*x+c))*((1-cos(d*x...
Timed out. \[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx=\int \frac {\sqrt {e \tan {\left (c + d x \right )}}}{a + b \sec {\left (c + d x \right )}}\, dx \]
\[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx=\int { \frac {\sqrt {e \tan \left (d x + c\right )}}{b \sec \left (d x + c\right ) + a} \,d x } \]
\[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx=\int { \frac {\sqrt {e \tan \left (d x + c\right )}}{b \sec \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )\,\sqrt {e\,\mathrm {tan}\left (c+d\,x\right )}}{b+a\,\cos \left (c+d\,x\right )} \,d x \]