Integrand size = 25, antiderivative size = 153 \[ \int \frac {(a+a \sec (c+d x))^2}{\sqrt {e \csc (c+d x)}} \, dx=-\frac {2 a^2 \arctan \left (\sqrt {\sin (c+d x)}\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 a^2 \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a^2 \tan (c+d x)}{d \sqrt {e \csc (c+d x)}} \] Output:
-2*a^2*arctan(sin(d*x+c)^(1/2))/d/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)+2* a^2*arctanh(sin(d*x+c)^(1/2))/d/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)-a^2* EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))/d/(e*csc(d*x+c))^(1/2)/sin(d* x+c)^(1/2)+a^2*tan(d*x+c)/d/(e*csc(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 15.51 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.27 \[ \int \frac {(a+a \sec (c+d x))^2}{\sqrt {e \csc (c+d x)}} \, dx=\frac {2 a^2 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \cot (c+d x) \left (-3 \left (\arctan \left (\sqrt {\csc (c+d x)}\right )+\text {arctanh}\left (\sqrt {\csc (c+d x)}\right )\right ) \sqrt {\cos ^2(c+d x)} \sqrt {-\cot ^2(c+d x)} \csc ^{\frac {3}{2}}(c+d x)+3 \cot ^2(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {3}{2},\frac {3}{4},\csc ^2(c+d x)\right )+2 \cot ^2(c+d x) \csc ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\csc ^2(c+d x)\right )\right ) \sec ^4\left (\frac {1}{2} \csc ^{-1}(\csc (c+d x))\right )}{3 d \left (-\cot ^2(c+d x)\right )^{3/2} \sqrt {e \csc (c+d x)}} \] Input:
Integrate[(a + a*Sec[c + d*x])^2/Sqrt[e*Csc[c + d*x]],x]
Output:
(2*a^2*Cos[(c + d*x)/2]^4*Cot[c + d*x]*(-3*(ArcTan[Sqrt[Csc[c + d*x]]] + A rcTanh[Sqrt[Csc[c + d*x]]])*Sqrt[Cos[c + d*x]^2]*Sqrt[-Cot[c + d*x]^2]*Csc [c + d*x]^(3/2) + 3*Cot[c + d*x]^2*Hypergeometric2F1[-1/4, 3/2, 3/4, Csc[c + d*x]^2] + 2*Cot[c + d*x]^2*Csc[c + d*x]^2*Hypergeometric2F1[3/4, 3/2, 7 /4, Csc[c + d*x]^2])*Sec[ArcCsc[Csc[c + d*x]]/2]^4)/(3*d*(-Cot[c + d*x]^2) ^(3/2)*Sqrt[e*Csc[c + d*x]])
Time = 0.59 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.71, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 4366, 3042, 4360, 3042, 3352, 2009}
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.
| \(\displaystyle \int \frac {(a \sec (c+d x)+a)^2}{\sqrt {e \csc (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}{\sqrt {e \sec \left (c+d x-\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4366 |
| \(\displaystyle \frac {\int (\sec (c+d x) a+a)^2 \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sqrt {\cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \frac {\int (-\cos (c+d x) a-a)^2 \sec ^2(c+d x) \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {-\cos \left (c+d x+\frac {\pi }{2}\right )} \left (-\sin \left (c+d x+\frac {\pi }{2}\right ) a-a\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (\sec ^2(c+d x) \sqrt {\sin (c+d x)} a^2+2 \sec (c+d x) \sqrt {\sin (c+d x)} a^2+\sqrt {\sin (c+d x)} a^2\right )dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {2 a^2 \arctan \left (\sqrt {\sin (c+d x)}\right )}{d}+\frac {2 a^2 \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d}+\frac {a^2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{d}+\frac {a^2 \sin ^{\frac {3}{2}}(c+d x) \sec (c+d x)}{d}}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
Input:
Int[(a + a*Sec[c + d*x])^2/Sqrt[e*Csc[c + d*x]],x]
Output:
((-2*a^2*ArcTan[Sqrt[Sin[c + d*x]]])/d + (2*a^2*ArcTanh[Sqrt[Sin[c + d*x]] ])/d + (a^2*EllipticE[(c - Pi/2 + d*x)/2, 2])/d + (a^2*Sec[c + d*x]*Sin[c + d*x]^(3/2))/d)/(Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*((g_.)*sec[(e_.) + (f_.)*( x_)])^(p_), x_Symbol] :> Simp[g^IntPart[p]*(g*Sec[e + f*x])^FracPart[p]*Cos [e + f*x]^FracPart[p] Int[(a + b*Csc[e + f*x])^m/Cos[e + f*x]^p, x], x] / ; FreeQ[{a, b, e, f, g, m, p}, x] && !IntegerQ[p]
Result contains complex when optimal does not.
Time = 2.00 (sec) , antiderivative size = 635, normalized size of antiderivative = 4.15
| method | result | size |
| parts | \(-\frac {a^{2} \sqrt {2}\, \left (2 \sqrt {1+i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \sqrt {1-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \left (1+\cos \left (d x +c \right )\right ) \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {1+i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \sqrt {1-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \left (-1-\cos \left (d x +c \right )\right ) \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \left (-1+\cos \left (d x +c \right )\right )\right ) \csc \left (d x +c \right )}{d \sqrt {e \csc \left (d x +c \right )}}-\frac {a^{2} \sqrt {2}\, \left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) \left (-2 \cos \left (d x +c \right )^{2}-2 \cos \left (d x +c \right )\right )+\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) \left (\cos \left (d x +c \right )+\cos \left (d x +c \right )^{2}\right )+\sqrt {2}\, \left (-1+\cos \left (d x +c \right )\right )\right ) \sec \left (d x +c \right ) \csc \left (d x +c \right )}{2 d \sqrt {e \csc \left (d x +c \right )}}-\frac {2 a^{2} \left (\arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}{-1+\cos \left (d x +c \right )}\right )+\operatorname {arctanh}\left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}{-1+\cos \left (d x +c \right )}\right )\right )}{d \left (1+\cos \left (d x +c \right )\right ) \sqrt {e \csc \left (d x +c \right )}\, \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}\) | \(635\) |
| default | \(\frac {a^{2} \sqrt {2}\, \left (i \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \left (2 \cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )\right )+i \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \left (-2 \cos \left (d x +c \right )^{2}-2 \cos \left (d x +c \right )\right )+\sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \left (2 \cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )\right )+\sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \left (2 \cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )\right )+\sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \left (-2 \cos \left (d x +c \right )^{2}-2 \cos \left (d x +c \right )\right )+\sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \left (\cos \left (d x +c \right )+\cos \left (d x +c \right )^{2}\right )+\left (-2 \cos \left (d x +c \right )^{2}+\cos \left (d x +c \right )+1\right ) \sqrt {2}\right ) \sec \left (d x +c \right ) \csc \left (d x +c \right )}{2 d \sqrt {e \csc \left (d x +c \right )}}\) | \(741\) |
Input:
int((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
-a^2/d*2^(1/2)*(2*(1+I*(-csc(d*x+c)+cot(d*x+c)))^(1/2)*(1-I*(-csc(d*x+c)+c ot(d*x+c)))^(1/2)*(-I*(-csc(d*x+c)+cot(d*x+c)))^(1/2)*(1+cos(d*x+c))*Ellip ticE((1+I*cot(d*x+c)-I*csc(d*x+c))^(1/2),1/2*2^(1/2))+(1+I*(-csc(d*x+c)+co t(d*x+c)))^(1/2)*(1-I*(-csc(d*x+c)+cot(d*x+c)))^(1/2)*(-I*(-csc(d*x+c)+cot (d*x+c)))^(1/2)*(-1-cos(d*x+c))*EllipticF((1+I*cot(d*x+c)-I*csc(d*x+c))^(1 /2),1/2*2^(1/2))+2^(1/2)*(-1+cos(d*x+c)))/(e*csc(d*x+c))^(1/2)*csc(d*x+c)- 1/2*a^2/d*2^(1/2)*((1+I*cot(d*x+c)-I*csc(d*x+c))^(1/2)*(1-I*cot(d*x+c)+I*c sc(d*x+c))^(1/2)*(-I*(-csc(d*x+c)+cot(d*x+c)))^(1/2)*EllipticE((1+I*cot(d* x+c)-I*csc(d*x+c))^(1/2),1/2*2^(1/2))*(-2*cos(d*x+c)^2-2*cos(d*x+c))+(1+I* cot(d*x+c)-I*csc(d*x+c))^(1/2)*(1-I*cot(d*x+c)+I*csc(d*x+c))^(1/2)*(-I*(-c sc(d*x+c)+cot(d*x+c)))^(1/2)*EllipticF((1+I*cot(d*x+c)-I*csc(d*x+c))^(1/2) ,1/2*2^(1/2))*(cos(d*x+c)+cos(d*x+c)^2)+2^(1/2)*(-1+cos(d*x+c)))/(e*csc(d* x+c))^(1/2)*sec(d*x+c)*csc(d*x+c)-2*a^2/d*(arctan(sin(d*x+c)*(sin(d*x+c)/( 1+cos(d*x+c))^2)^(1/2)/(-1+cos(d*x+c)))+arctanh(sin(d*x+c)*(sin(d*x+c)/(1+ cos(d*x+c))^2)^(1/2)/(-1+cos(d*x+c))))/(1+cos(d*x+c))/(e*csc(d*x+c))^(1/2) /(sin(d*x+c)/(1+cos(d*x+c))^2)^(1/2)
Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 698, normalized size of antiderivative = 4.56 \[ \int \frac {(a+a \sec (c+d x))^2}{\sqrt {e \csc (c+d x)}} \, dx =\text {Too large to display} \] Input:
integrate((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(1/2),x, algorithm="fricas")
Output:
[-1/4*(2*a^2*sqrt(-e)*arctan(-1/4*(cos(d*x + c)^2 - 6*sin(d*x + c) - 2)*sq rt(-e)*sqrt(e/sin(d*x + c))/(e*sin(d*x + c) + e))*cos(d*x + c) + a^2*sqrt( -e)*cos(d*x + c)*log((e*cos(d*x + c)^4 - 72*e*cos(d*x + c)^2 + 8*(cos(d*x + c)^4 - 9*cos(d*x + c)^2 + (7*cos(d*x + c)^2 - 8)*sin(d*x + c) + 8)*sqrt( -e)*sqrt(e/sin(d*x + c)) + 28*(e*cos(d*x + c)^2 - 2*e)*sin(d*x + c) + 72*e )/(cos(d*x + c)^4 - 8*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 - 2)*sin(d*x + c) + 8)) - 2*a^2*sqrt(2*I*e)*cos(d*x + c)*weierstrassZeta(4, 0, weierstrassP Inverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) - 2*a^2*sqrt(-2*I*e)*cos(d*x + c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) - I*sin (d*x + c))) + 4*(a^2*cos(d*x + c)^2 - a^2)*sqrt(e/sin(d*x + c)))/(d*e*cos( d*x + c)), 1/4*(2*a^2*sqrt(e)*arctan(1/4*(cos(d*x + c)^2 + 6*sin(d*x + c) - 2)*sqrt(e)*sqrt(e/sin(d*x + c))/(e*sin(d*x + c) - e))*cos(d*x + c) + a^2 *sqrt(e)*cos(d*x + c)*log((e*cos(d*x + c)^4 - 72*e*cos(d*x + c)^2 + 8*(cos (d*x + c)^4 - 9*cos(d*x + c)^2 - (7*cos(d*x + c)^2 - 8)*sin(d*x + c) + 8)* sqrt(e)*sqrt(e/sin(d*x + c)) - 28*(e*cos(d*x + c)^2 - 2*e)*sin(d*x + c) + 72*e)/(cos(d*x + c)^4 - 8*cos(d*x + c)^2 + 4*(cos(d*x + c)^2 - 2)*sin(d*x + c) + 8)) + 2*a^2*sqrt(2*I*e)*cos(d*x + c)*weierstrassZeta(4, 0, weierstr assPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) + 2*a^2*sqrt(-2*I*e)*cos (d*x + c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) - I *sin(d*x + c))) - 4*(a^2*cos(d*x + c)^2 - a^2)*sqrt(e/sin(d*x + c)))/(d...
\[ \int \frac {(a+a \sec (c+d x))^2}{\sqrt {e \csc (c+d x)}} \, dx=a^{2} \left (\int \frac {1}{\sqrt {e \csc {\left (c + d x \right )}}}\, dx + \int \frac {2 \sec {\left (c + d x \right )}}{\sqrt {e \csc {\left (c + d x \right )}}}\, dx + \int \frac {\sec ^{2}{\left (c + d x \right )}}{\sqrt {e \csc {\left (c + d x \right )}}}\, dx\right ) \] Input:
integrate((a+a*sec(d*x+c))**2/(e*csc(d*x+c))**(1/2),x)
Output:
a**2*(Integral(1/sqrt(e*csc(c + d*x)), x) + Integral(2*sec(c + d*x)/sqrt(e *csc(c + d*x)), x) + Integral(sec(c + d*x)**2/sqrt(e*csc(c + d*x)), x))
\[ \int \frac {(a+a \sec (c+d x))^2}{\sqrt {e \csc (c+d x)}} \, dx=\int { \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\sqrt {e \csc \left (d x + c\right )}} \,d x } \] Input:
integrate((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate((a*sec(d*x + c) + a)^2/sqrt(e*csc(d*x + c)), x)
\[ \int \frac {(a+a \sec (c+d x))^2}{\sqrt {e \csc (c+d x)}} \, dx=\int { \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\sqrt {e \csc \left (d x + c\right )}} \,d x } \] Input:
integrate((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(1/2),x, algorithm="giac")
Output:
integrate((a*sec(d*x + c) + a)^2/sqrt(e*csc(d*x + c)), x)
Timed out. \[ \int \frac {(a+a \sec (c+d x))^2}{\sqrt {e \csc (c+d x)}} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2}{\sqrt {\frac {e}{\sin \left (c+d\,x\right )}}} \,d x \] Input:
int((a + a/cos(c + d*x))^2/(e/sin(c + d*x))^(1/2),x)
Output:
int((a + a/cos(c + d*x))^2/(e/sin(c + d*x))^(1/2), x)
\[ \int \frac {(a+a \sec (c+d x))^2}{\sqrt {e \csc (c+d x)}} \, dx=\frac {\sqrt {e}\, a^{2} \left (\int \frac {\sqrt {\csc \left (d x +c \right )}}{\csc \left (d x +c \right )}d x +\int \frac {\sqrt {\csc \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}}{\csc \left (d x +c \right )}d x +2 \left (\int \frac {\sqrt {\csc \left (d x +c \right )}\, \sec \left (d x +c \right )}{\csc \left (d x +c \right )}d x \right )\right )}{e} \] Input:
int((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(1/2),x)
Output:
(sqrt(e)*a**2*(int(sqrt(csc(c + d*x))/csc(c + d*x),x) + int((sqrt(csc(c + d*x))*sec(c + d*x)**2)/csc(c + d*x),x) + 2*int((sqrt(csc(c + d*x))*sec(c + d*x))/csc(c + d*x),x)))/e