\(\int \frac {\sqrt {e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx\) [311]

Optimal result

Integrand size = 25, antiderivative size = 357 \[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx=-\frac {\sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a d}+\frac {\sqrt {e} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a d}-\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}+\sqrt {e} \tan (c+d x)}\right )}{\sqrt {2} a d}+\frac {2 \sqrt {2} b \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {1+\cos (c+d x)}}\right ),-1\right ) \sqrt {e \tan (c+d x)}}{a \sqrt {a-b} \sqrt {a+b} d \sqrt {\sin (c+d x)}}-\frac {2 \sqrt {2} b \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {1+\cos (c+d x)}}\right ),-1\right ) \sqrt {e \tan (c+d x)}}{a \sqrt {a-b} \sqrt {a+b} d \sqrt {\sin (c+d x)}} \] Output:

-1/2*e^(1/2)*arctan(1-2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))*2^(1/2)/a/d+1/ 
2*e^(1/2)*arctan(1+2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))*2^(1/2)/a/d-1/2*e 
^(1/2)*arctanh(2^(1/2)*(e*tan(d*x+c))^(1/2)/(e^(1/2)+e^(1/2)*tan(d*x+c)))* 
2^(1/2)/a/d+2*2^(1/2)*b*cos(d*x+c)^(1/2)*EllipticPi(sin(d*x+c)^(1/2)/(1+co 
s(d*x+c))^(1/2),-(a-b)^(1/2)/(a+b)^(1/2),I)*(e*tan(d*x+c))^(1/2)/a/(a-b)^( 
1/2)/(a+b)^(1/2)/d/sin(d*x+c)^(1/2)-2*2^(1/2)*b*cos(d*x+c)^(1/2)*EllipticP 
i(sin(d*x+c)^(1/2)/(1+cos(d*x+c))^(1/2),(a-b)^(1/2)/(a+b)^(1/2),I)*(e*tan( 
d*x+c))^(1/2)/a/(a-b)^(1/2)/(a+b)^(1/2)/d/sin(d*x+c)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 5.55 (sec) , antiderivative size = 646, normalized size of antiderivative = 1.81 \[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx=-\frac {\cos (c+d x) \left (a+b \sqrt {\sec ^2(c+d x)}\right ) \sqrt {e \tan (c+d x)} \left (6 \sqrt {2} \left (a^2-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-6 \sqrt {2} a^2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+6 \sqrt {2} b^2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )-(6+6 i) \sqrt {b} \left (a^2-b^2\right )^{3/4} \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )+(6+6 i) \sqrt {b} \left (a^2-b^2\right )^{3/4} \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-3 \sqrt {2} a^2 \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+3 \sqrt {2} b^2 \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+3 \sqrt {2} a^2 \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-3 \sqrt {2} b^2 \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+(3+3 i) \sqrt {b} \left (a^2-b^2\right )^{3/4} \log \left (\sqrt {a^2-b^2}-(1+i) \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\tan (c+d x)}+i b \tan (c+d x)\right )-(3+3 i) \sqrt {b} \left (a^2-b^2\right )^{3/4} \log \left (\sqrt {a^2-b^2}+(1+i) \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\tan (c+d x)}+i b \tan (c+d x)\right )+8 a b \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\tan ^2(c+d x),\frac {b^2 \tan ^2(c+d x)}{a^2-b^2}\right ) \tan ^{\frac {3}{2}}(c+d x)\right )}{12 a \left (a^2-b^2\right ) d (b+a \cos (c+d x)) \sqrt {\tan (c+d x)}} \] Input:

Integrate[Sqrt[e*Tan[c + d*x]]/(a + b*Sec[c + d*x]),x]
 

Output:

-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]])
 

Rubi [A] (warning: unable to verify)

Time = 1.36 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.99, 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

\(\displaystyle \frac {2 e \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{a d}-\frac {4 \sqrt {2} b \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {a-b} \sqrt {a+b}}\right )}{a d \sqrt {\sin (c+d x)}}\)

Input:

Int[Sqrt[e*Tan[c + d*x]]/(a + b*Sec[c + d*x]),x]
 

Output:

(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]])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 266

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]
 

rule 412

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])
 

rule 826

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]]))
 

rule 993

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]
 

rule 1082

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]
 

rule 1103

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]
 

rule 1476

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]
 

rule 1479

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]
 

rule 1537

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3209

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]
 

rule 3212

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]
 

rule 3384

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]
 

rule 3385

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]
 

rule 3957

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]
 

rule 4378

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]
 

Maple [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 909 vs. \(2 (286 ) = 572\).

Time = 1.05 (sec) , antiderivative size = 910, normalized size of antiderivative = 2.55

method	result	size
default	\(\text {Expression too large to display}\)	\(910\)

Input:

int((e*tan(d*x+c))^(1/2)/(a+b*sec(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

(1/2+1/2*I)/d*b/((a^2-b^2)^(1/2)-a+b)/((a^2-b^2)^(1/2)+a-b)/a*(-csc(d*x+c) 
+cot(d*x+c))^(1/2)*(2*cot(d*x+c)-2*csc(d*x+c)+2)^(1/2)*(-cot(d*x+c)+csc(d* 
x+c)+1)^(1/2)*(e*tan(d*x+c))^(1/2)*(-b*EllipticPi((-cot(d*x+c)+csc(d*x+c)+ 
1)^(1/2),-(a-b)/(-a+b+((a+b)*(a-b))^(1/2)),1/2*2^(1/2))-2*EllipticPi((-cot 
(d*x+c)+csc(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*a-b*EllipticPi((-cot(d* 
x+c)+csc(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)*EllipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),-(a-b)/(-a+b+((a+b 
)*(a-b))^(1/2)),1/2*2^(1/2))+EllipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),( 
a-b)/(a-b+((a+b)*(a-b))^(1/2)),1/2*2^(1/2))*a+EllipticPi((-cot(d*x+c)+csc( 
d*x+c)+1)^(1/2),-(a-b)/(-a+b+((a+b)*(a-b))^(1/2)),1/2*2^(1/2))*a+2*Ellipti 
cPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*b-(a^2-b^2)^(1 
/2)*EllipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),(a-b)/(a-b+((a+b)*(a-b))^( 
1/2)),1/2*2^(1/2))-I*EllipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),(a-b)/(a- 
b+((a+b)*(a-b))^(1/2)),1/2*2^(1/2))*a+I*b*EllipticPi((-cot(d*x+c)+csc(d*x+ 
c)+1)^(1/2),(a-b)/(a-b+((a+b)*(a-b))^(1/2)),1/2*2^(1/2))+2*I*EllipticPi((- 
cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*a+I*b*EllipticPi((-c 
ot(d*x+c)+csc(d*x+c)+1)^(1/2),-(a-b)/(-a+b+((a+b)*(a-b))^(1/2)),1/2*2^(1/2 
))-I*(a^2-b^2)^(1/2)*EllipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),-(a-b)/(- 
a+b+((a+b)*(a-b))^(1/2)),1/2*2^(1/2))-2*I*EllipticPi((-cot(d*x+c)+csc(d*x+ 
c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*b+I*(a^2-b^2)^(1/2)*EllipticPi((-cot...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((e*tan(d*x+c))^(1/2)/(a+b*sec(d*x+c)),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx=\int \frac {\sqrt {e \tan {\left (c + d x \right )}}}{a + b \sec {\left (c + d x \right )}}\, dx \] Input:

integrate((e*tan(d*x+c))**(1/2)/(a+b*sec(d*x+c)),x)
 

Output:

Integral(sqrt(e*tan(c + d*x))/(a + b*sec(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx=\int { \frac {\sqrt {e \tan \left (d x + c\right )}}{b \sec \left (d x + c\right ) + a} \,d x } \] Input:

integrate((e*tan(d*x+c))^(1/2)/(a+b*sec(d*x+c)),x, algorithm="maxima")
 

Output:

integrate(sqrt(e*tan(d*x + c))/(b*sec(d*x + c) + a), x)
 

Giac [F]

\[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx=\int { \frac {\sqrt {e \tan \left (d x + c\right )}}{b \sec \left (d x + c\right ) + a} \,d x } \] Input:

integrate((e*tan(d*x+c))^(1/2)/(a+b*sec(d*x+c)),x, algorithm="giac")
 

Output:

integrate(sqrt(e*tan(d*x + c))/(b*sec(d*x + c) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )\,\sqrt {e\,\mathrm {tan}\left (c+d\,x\right )}}{b+a\,\cos \left (c+d\,x\right )} \,d x \] Input:

int((e*tan(c + d*x))^(1/2)/(a + b/cos(c + d*x)),x)
 

Output:

int((cos(c + d*x)*(e*tan(c + d*x))^(1/2))/(b + a*cos(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {\tan \left (d x +c \right )}}{\sec \left (d x +c \right ) b +a}d x \right ) \] Input:

int((e*tan(d*x+c))^(1/2)/(a+b*sec(d*x+c)),x)
 

Output:

sqrt(e)*int(sqrt(tan(c + d*x))/(sec(c + d*x)*b + a),x)