\(\int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx\) [282]

Optimal result

Integrand size = 23, antiderivative size = 169 \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=-\frac {2 a e \sqrt {e \csc (c+d x)}}{d}-\frac {2 a e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {a e \arctan \left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {a e \text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}-\frac {2 a e \sqrt {e \csc (c+d x)} E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d} \]

[Out]

-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*arctanh(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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3963, 3957, 2917, 2644, 331, 335, 304, 209, 212, 2716, 2719} \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=-\frac {a e \sqrt {\sin (c+d x)} \arctan \left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)}}{d}+\frac {a e \sqrt {\sin (c+d x)} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)}}{d}-\frac {2 a e \sqrt {e \csc (c+d x)}}{d}-\frac {2 a e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {2 a e \sqrt {\sin (c+d x)} E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \csc (c+d x)}}{d} \]

[In]

Int[(e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x]),x]

[Out]

(-2*a*e*Sqrt[e*Csc[c + d*x]])/d - (2*a*e*Cos[c + d*x]*Sqrt[e*Csc[c + d*x]])/d - (a*e*ArcTan[Sqrt[Sin[c + d*x]]
]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (a*e*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[
c + d*x]])/d - (2*a*e*Sqrt[e*Csc[c + d*x]]*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/d

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[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] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 2716

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] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]

Rule 2719

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]

Rule 2917

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)
*(x_)]), x_Symbol] :> Dist[a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Dist[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]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 3963

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*((g_.)*sec[(e_.) + (f_.)*(x_)])^(p_), x_Symbol] :> Dist[g^Int
Part[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]

Rubi steps \begin{align*} \text {integral}& = \left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {a+a \sec (c+d x)}{\sin ^{\frac {3}{2}}(c+d x)} \, dx \\ & = -\left (\left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {(-a-a \cos (c+d x)) \sec (c+d x)}{\sin ^{\frac {3}{2}}(c+d x)} \, dx\right ) \\ & = \left (a e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sin ^{\frac {3}{2}}(c+d x)} \, dx+\left (a e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\sec (c+d x)}{\sin ^{\frac {3}{2}}(c+d x)} \, dx \\ & = -\frac {2 a e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\left (a e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx+\frac {\left (a e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{x^{3/2} \left (1-x^2\right )} \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\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 {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}+\frac {\left (a e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\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 {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}+\frac {\left (2 a e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d} \\ & = -\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 {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}+\frac {\left (a e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d}-\frac {\left (a e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d} \\ & = -\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} \\ \end{align*}

Mathematica [C] (verified)

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 \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=\frac {a (e \csc (c+d x))^{3/2} \left (2 \arctan \left (\sqrt {\csc (c+d x)}\right )-4 (1+\cos (c+d x)) \sqrt {\csc (c+d x)}-\log \left (1-\sqrt {\csc (c+d x)}\right )+\log \left (1+\sqrt {\csc (c+d x)}\right )+\frac {2 \csc ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},\csc ^2(c+d x)\right ) \sin (2 (c+d x))}{\sqrt {-\cot ^2(c+d x)}}\right )}{2 d \csc ^{\frac {3}{2}}(c+d x)} \]

[In]

Integrate[(e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x]),x]

[Out]

(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 - Sq
rt[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))

Maple [C] (warning: unable to verify)

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}}}}\)	\(639\)

[In]

int((e*csc(d*x+c))^(3/2)*(a+a*sec(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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)*EllipticE((-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)*Ell
ipticF((-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)-c
sc(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)/(cos(d*x+c)+1)^2)^(1/2)*(cot(d*x+c)+csc(d*x+c)))*csc(d*x+c))

Fricas [C] (verification not implemented)

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 ] \]

[In]

integrate((e*csc(d*x+c))^(3/2)*(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

[-1/8*(2*a*sqrt(-e)*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)) - a*sqrt(-e)*e*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*weie
rstrassZeta(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*co
s(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]

Sympy [F]

\[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=a \left (\int \left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx + \int \left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}} \sec {\left (c + d x \right )}\, dx\right ) \]

[In]

integrate((e*csc(d*x+c))**(3/2)*(a+a*sec(d*x+c)),x)

[Out]

a*(Integral((e*csc(c + d*x))**(3/2), x) + Integral((e*csc(c + d*x))**(3/2)*sec(c + d*x), x))

Maxima [F(-1)]

Timed out. \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate((e*csc(d*x+c))^(3/2)*(a+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=\int { \left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sec \left (d x + c\right ) + a\right )} \,d x } \]

[In]

integrate((e*csc(d*x+c))^(3/2)*(a+a*sec(d*x+c)),x, algorithm="giac")

[Out]

integrate((e*csc(d*x + c))^(3/2)*(a*sec(d*x + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx=\int \left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{3/2} \,d x \]

[In]

int((a + a/cos(c + d*x))*(e/sin(c + d*x))^(3/2),x)

[Out]

int((a + a/cos(c + d*x))*(e/sin(c + d*x))^(3/2), x)