Integrand size = 23, antiderivative size = 169 \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=-\frac {2 a e \sqrt {e \csc (c+d x)}}{d}-\frac {2 a e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {a e \arctan \left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {a e \text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}-\frac {2 a e \sqrt {e \csc (c+d x)} E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d} \]
-2*a*e*(e*csc(d*x+c))^(1/2)/d-2*a*e*cos(d*x+c)*(e*csc(d*x+c))^(1/2)/d-a*e* arctan(sin(d*x+c)^(1/2))*(e*csc(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/d+a*e*arcta nh(sin(d*x+c)^(1/2))*(e*csc(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/d+2*a*e*(sin(1/ 2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticE(cos(1/2*c +1/4*Pi+1/2*d*x),2^(1/2))*(e*csc(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 11.20 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.86 \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=\frac {a (e \csc (c+d x))^{3/2} \left (2 \arctan \left (\sqrt {\csc (c+d x)}\right )-4 (1+\cos (c+d x)) \sqrt {\csc (c+d x)}-\log \left (1-\sqrt {\csc (c+d x)}\right )+\log \left (1+\sqrt {\csc (c+d x)}\right )+\frac {2 \csc ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},\csc ^2(c+d x)\right ) \sin (2 (c+d x))}{\sqrt {-\cot ^2(c+d x)}}\right )}{2 d \csc ^{\frac {3}{2}}(c+d x)} \]
(a*(e*Csc[c + d*x])^(3/2)*(2*ArcTan[Sqrt[Csc[c + d*x]]] - 4*(1 + Cos[c + d *x])*Sqrt[Csc[c + d*x]] - Log[1 - Sqrt[Csc[c + d*x]]] + Log[1 + Sqrt[Csc[c + d*x]]] + (2*Csc[c + d*x]^(3/2)*Hypergeometric2F1[-1/4, 1/2, 3/4, Csc[c + d*x]^2]*Sin[2*(c + d*x)])/Sqrt[-Cot[c + d*x]^2]))/(2*d*Csc[c + d*x]^(3/2 ))
Time = 0.66 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.73, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.870, Rules used = {3042, 4366, 3042, 4360, 25, 25, 3042, 25, 3317, 25, 3042, 3044, 264, 266, 827, 216, 219, 3116, 3042, 3119}
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.
| \(\displaystyle \int (a \sec (c+d x)+a) (e \csc (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right ) \left (e \sec \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2}dx\) |
| \(\Big \downarrow \) 4366 |
| \(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\sec (c+d x) a+a}{\sin ^{\frac {3}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {a-a \csc \left (c+d x-\frac {\pi }{2}\right )}{\cos \left (c+d x-\frac {\pi }{2}\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int -\frac {(-\cos (c+d x) a-a) \sec (c+d x)}{\sin ^{\frac {3}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int -\frac {(\cos (c+d x) a+a) \sec (c+d x)}{\sin ^{\frac {3}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {(\cos (c+d x) a+a) \sec (c+d x)}{\sin ^{\frac {3}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int -\frac {a-a \sin \left (c+d x-\frac {\pi }{2}\right )}{\cos \left (c+d x-\frac {\pi }{2}\right )^{3/2} \sin \left (c+d x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^{3/2} \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}dx\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle -e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (a \int -\frac {\sec (c+d x)}{\sin ^{\frac {3}{2}}(c+d x)}dx-a \int \frac {1}{\sin ^{\frac {3}{2}}(c+d x)}dx\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-a \int \frac {1}{\sin ^{\frac {3}{2}}(c+d x)}dx-a \int \frac {\sec (c+d x)}{\sin ^{\frac {3}{2}}(c+d x)}dx\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-a \int \frac {1}{\sin (c+d x)^{3/2}}dx-a \int \frac {1}{\cos (c+d x) \sin (c+d x)^{3/2}}dx\right )\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle -e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-\frac {a \int \frac {1}{\sin ^{\frac {3}{2}}(c+d x) \left (1-\sin ^2(c+d x)\right )}d\sin (c+d x)}{d}-a \int \frac {1}{\sin (c+d x)^{3/2}}dx\right )\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle -e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-\frac {a \left (\int \frac {\sqrt {\sin (c+d x)}}{1-\sin ^2(c+d x)}d\sin (c+d x)-\frac {2}{\sqrt {\sin (c+d x)}}\right )}{d}-a \int \frac {1}{\sin (c+d x)^{3/2}}dx\right )\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-\frac {a \left (2 \int \frac {\sin (c+d x)}{1-\sin ^2(c+d x)}d\sqrt {\sin (c+d x)}-\frac {2}{\sqrt {\sin (c+d x)}}\right )}{d}-a \int \frac {1}{\sin (c+d x)^{3/2}}dx\right )\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-a \int \frac {1}{\sin (c+d x)^{3/2}}dx-\frac {a \left (2 \left (\frac {1}{2} \int \frac {1}{1-\sin (c+d x)}d\sqrt {\sin (c+d x)}-\frac {1}{2} \int \frac {1}{\sin (c+d x)+1}d\sqrt {\sin (c+d x)}\right )-\frac {2}{\sqrt {\sin (c+d x)}}\right )}{d}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-\frac {a \left (2 \left (\frac {1}{2} \int \frac {1}{1-\sin (c+d x)}d\sqrt {\sin (c+d x)}-\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )\right )-\frac {2}{\sqrt {\sin (c+d x)}}\right )}{d}-a \int \frac {1}{\sin (c+d x)^{3/2}}dx\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-a \int \frac {1}{\sin (c+d x)^{3/2}}dx-\frac {a \left (2 \left (\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )-\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )\right )-\frac {2}{\sqrt {\sin (c+d x)}}\right )}{d}\right )\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle -e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-a \left (-\int \sqrt {\sin (c+d x)}dx-\frac {2 \cos (c+d x)}{d \sqrt {\sin (c+d x)}}\right )-\frac {a \left (2 \left (\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )-\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )\right )-\frac {2}{\sqrt {\sin (c+d x)}}\right )}{d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-a \left (-\int \sqrt {\sin (c+d x)}dx-\frac {2 \cos (c+d x)}{d \sqrt {\sin (c+d x)}}\right )-\frac {a \left (2 \left (\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )-\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )\right )-\frac {2}{\sqrt {\sin (c+d x)}}\right )}{d}\right )\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle -e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-\frac {a \left (2 \left (\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )-\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )\right )-\frac {2}{\sqrt {\sin (c+d x)}}\right )}{d}-a \left (-\frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{d}-\frac {2 \cos (c+d x)}{d \sqrt {\sin (c+d x)}}\right )\right )\) |
-(e*Sqrt[e*Csc[c + d*x]]*(-((a*(2*(-1/2*ArcTan[Sqrt[Sin[c + d*x]]] + ArcTa nh[Sqrt[Sin[c + d*x]]]/2) - 2/Sqrt[Sin[c + d*x]]))/d) - a*((-2*EllipticE[( c - Pi/2 + d*x)/2, 2])/d - (2*Cos[c + d*x])/(d*Sqrt[Sin[c + d*x]])))*Sqrt[ Sin[c + d*x]])
3.3.82.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
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 = 2.33 (sec) , antiderivative size = 639, normalized size of antiderivative = 3.78
| method | result | size |
| default | \(\frac {a \sqrt {2}\, e \sqrt {e \csc \left (d x +c \right )}\, \left (2 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )-\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )+2 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\right )}{d}-\frac {a \sqrt {e \csc \left (d x +c \right )}\, e \left (2 \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}+\arctan \left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right ) \cot \left (d x +c \right )+\operatorname {arctanh}\left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right ) \cot \left (d x +c \right )-\arctan \left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right ) \csc \left (d x +c \right )-\operatorname {arctanh}\left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right ) \csc \left (d x +c \right )\right )}{d \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}}\) | \(639\) |
| parts | \(\frac {a \sqrt {2}\, e \sqrt {e \csc \left (d x +c \right )}\, \left (2 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )-\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )+2 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\right )}{d}-\frac {a \sqrt {e \csc \left (d x +c \right )}\, e \left (2 \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}+\arctan \left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right ) \cot \left (d x +c \right )+\operatorname {arctanh}\left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right ) \cot \left (d x +c \right )-\arctan \left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right ) \csc \left (d x +c \right )-\operatorname {arctanh}\left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right ) \csc \left (d x +c \right )\right )}{d \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}}\) | \(639\) |
a/d*2^(1/2)*e*(e*csc(d*x+c))^(1/2)*(2*(-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2) *(-I*(I+cot(d*x+c)-csc(d*x+c)))^(1/2)*(-I*(cot(d*x+c)-csc(d*x+c)))^(1/2)*E llipticE((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/2))*cos(d*x+c)-(-I* (I-cot(d*x+c)+csc(d*x+c)))^(1/2)*(-I*(I+cot(d*x+c)-csc(d*x+c)))^(1/2)*(-I* (cot(d*x+c)-csc(d*x+c)))^(1/2)*EllipticF((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1 /2),1/2*2^(1/2))*cos(d*x+c)+2*(-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2)*(-I*(I+ cot(d*x+c)-csc(d*x+c)))^(1/2)*(-I*(cot(d*x+c)-csc(d*x+c)))^(1/2)*EllipticE ((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/2))-(-I*(I-cot(d*x+c)+csc(d *x+c)))^(1/2)*(-I*(I+cot(d*x+c)-csc(d*x+c)))^(1/2)*(-I*(cot(d*x+c)-csc(d*x +c)))^(1/2)*EllipticF((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/2))-2^ (1/2))-a/d*(e*csc(d*x+c))^(1/2)*e/(sin(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(2*( sin(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)+arctan((sin(d*x+c)/(cos(d*x+c)+1)^2)^(1 /2)*(cot(d*x+c)+csc(d*x+c)))*cot(d*x+c)+arctanh((sin(d*x+c)/(cos(d*x+c)+1) ^2)^(1/2)*(cot(d*x+c)+csc(d*x+c)))*cot(d*x+c)-arctan((sin(d*x+c)/(cos(d*x+ c)+1)^2)^(1/2)*(cot(d*x+c)+csc(d*x+c)))*csc(d*x+c)-arctanh((sin(d*x+c)/(co s(d*x+c)+1)^2)^(1/2)*(cot(d*x+c)+csc(d*x+c)))*csc(d*x+c))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.22 (sec) , antiderivative size = 609, normalized size of antiderivative = 3.60 \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=\left [-\frac {2 \, a \sqrt {-e} e \arctan \left (-\frac {{\left (\cos \left (d x + c\right )^{2} - 6 \, \sin \left (d x + c\right ) - 2\right )} \sqrt {-e} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{4 \, {\left (e \sin \left (d x + c\right ) + e\right )}}\right ) - a \sqrt {-e} e \log \left (\frac {e \cos \left (d x + c\right )^{4} - 72 \, e \cos \left (d x + c\right )^{2} - 8 \, {\left (\cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{2} + {\left (7 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) + 8\right )} \sqrt {-e} \sqrt {\frac {e}{\sin \left (d x + c\right )}} + 28 \, {\left (e \cos \left (d x + c\right )^{2} - 2 \, e\right )} \sin \left (d x + c\right ) + 72 \, e}{\cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 8}\right ) + 8 \, a \sqrt {2 i \, e} e {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 8 \, a \sqrt {-2 i \, e} e {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 16 \, {\left (a e \cos \left (d x + c\right ) + a e\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{8 \, d}, \frac {2 \, a e^{\frac {3}{2}} \arctan \left (\frac {{\left (\cos \left (d x + c\right )^{2} + 6 \, \sin \left (d x + c\right ) - 2\right )} \sqrt {e} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{4 \, {\left (e \sin \left (d x + c\right ) - e\right )}}\right ) + a e^{\frac {3}{2}} \log \left (\frac {e \cos \left (d x + c\right )^{4} - 72 \, e \cos \left (d x + c\right )^{2} + 8 \, {\left (\cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{2} - {\left (7 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) + 8\right )} \sqrt {e} \sqrt {\frac {e}{\sin \left (d x + c\right )}} - 28 \, {\left (e \cos \left (d x + c\right )^{2} - 2 \, e\right )} \sin \left (d x + c\right ) + 72 \, e}{\cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 8}\right ) - 8 \, a \sqrt {2 i \, e} e {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 8 \, a \sqrt {-2 i \, e} e {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 16 \, {\left (a e \cos \left (d x + c\right ) + a e\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{8 \, d}\right ] \]
[-1/8*(2*a*sqrt(-e)*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)) - a*sqrt(-e)*e*log((e*co s(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)) + 8*a*sqrt(2*I*e)*e *weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) + 8*a*sqrt(-2*I*e)*e*weierstrassZeta(4, 0, weierstrassPInverse(4, 0 , cos(d*x + c) - I*sin(d*x + c))) + 16*(a*e*cos(d*x + c) + a*e)*sqrt(e/sin (d*x + c)))/d, 1/8*(2*a*e^(3/2)*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)) + a*e^(3/2)*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)) - 8*a*sqrt(2*I* e)*e*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) + I*sin( d*x + c))) - 8*a*sqrt(-2*I*e)*e*weierstrassZeta(4, 0, weierstrassPInverse( 4, 0, cos(d*x + c) - I*sin(d*x + c))) - 16*(a*e*cos(d*x + c) + a*e)*sqrt(e /sin(d*x + c)))/d]
\[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=a \left (\int \left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx + \int \left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}} \sec {\left (c + d x \right )}\, dx\right ) \]
a*(Integral((e*csc(c + d*x))**(3/2), x) + Integral((e*csc(c + d*x))**(3/2) *sec(c + d*x), x))
Timed out. \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=\text {Timed out} \]
\[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=\int { \left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sec \left (d x + c\right ) + a\right )} \,d x } \]
Timed out. \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=\int \left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{3/2} \,d x \]