\(\int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx\) [288]

Optimal result

Integrand size = 25, antiderivative size = 240 \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx=-\frac {4 a^2 e \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \sqrt {e \csc (c+d x)} \sec (c+d x)}{d}-\frac {2 a^2 e \arctan \left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {2 a^2 e \text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}-\frac {5 a^2 e \sqrt {e \csc (c+d x)} E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d}+\frac {3 a^2 e \sqrt {e \csc (c+d x)} \sin (c+d x) \tan (c+d x)}{d} \]

[Out]

-4*a^2*e*(e*csc(d*x+c))^(1/2)/d-2*a^2*e*cos(d*x+c)*(e*csc(d*x+c))^(1/2)/d-2*a^2*e*sec(d*x+c)*(e*csc(d*x+c))^(1
/2)/d-2*a^2*e*arctan(sin(d*x+c)^(1/2))*(e*csc(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/d+2*a^2*e*arctanh(sin(d*x+c)^(1/2
))*(e*csc(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/d+5*a^2*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+3*a^2*e*sin(d*x+c)*(e*
csc(d*x+c))^(1/2)*tan(d*x+c)/d

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3963, 3957, 2952, 2716, 2719, 2644, 331, 335, 304, 209, 212, 2650, 2651} \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx=-\frac {2 a^2 e \sqrt {\sin (c+d x)} \arctan \left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)}}{d}+\frac {2 a^2 e \sqrt {\sin (c+d x)} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)}}{d}-\frac {4 a^2 e \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \sec (c+d x) \sqrt {e \csc (c+d x)}}{d}+\frac {3 a^2 e \sin (c+d x) \tan (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {5 a^2 e \sqrt {\sin (c+d x)} E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \csc (c+d x)}}{d} \]

[In]

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

[Out]

(-4*a^2*e*Sqrt[e*Csc[c + d*x]])/d - (2*a^2*e*Cos[c + d*x]*Sqrt[e*Csc[c + d*x]])/d - (2*a^2*e*Sqrt[e*Csc[c + d*
x]]*Sec[c + d*x])/d - (2*a^2*e*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (2*a^2*
e*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d - (5*a^2*e*Sqrt[e*Csc[c + d*x]]*Ellip
ticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/d + (3*a^2*e*Sqrt[e*Csc[c + d*x]]*Sin[c + d*x]*Tan[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 2650

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*Cos[e + f*
x])^(n + 1)*((a*Sin[e + f*x])^(m + 1)/(a*b*f*(m + 1))), x] + Dist[(m + n + 2)/(a^2*(m + 1)), Int[(b*Cos[e + f*
x])^n*(a*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n]

Rule 2651

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b*Sin[e +
f*x])^(n + 1))*((a*Cos[e + f*x])^(m + 1)/(a*b*f*(m + 1))), x] + Dist[(m + n + 2)/(a^2*(m + 1)), Int[(b*Sin[e +
f*x])^n*(a*Cos[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*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 2952

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]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]

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))^2}{\sin ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {(-a-a \cos (c+d x))^2 \sec ^2(c+d x)}{\sin ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \left (\frac {a^2}{\sin ^{\frac {3}{2}}(c+d x)}+\frac {2 a^2 \sec (c+d x)}{\sin ^{\frac {3}{2}}(c+d x)}+\frac {a^2 \sec ^2(c+d x)}{\sin ^{\frac {3}{2}}(c+d x)}\right ) \, dx \\ & = \left (a^2 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^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\sec ^2(c+d x)}{\sin ^{\frac {3}{2}}(c+d x)} \, dx+\left (2 a^2 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^2 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \sqrt {e \csc (c+d x)} \sec (c+d x)}{d}-\left (a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx+\left (3 a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \sec ^2(c+d x) \sqrt {\sin (c+d x)} \, dx+\frac {\left (2 a^2 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 {4 a^2 e \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \sqrt {e \csc (c+d x)} \sec (c+d x)}{d}-\frac {2 a^2 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 {3 a^2 e \sqrt {e \csc (c+d x)} \sin (c+d x) \tan (c+d x)}{d}-\frac {1}{2} \left (3 a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx+\frac {\left (2 a^2 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 {4 a^2 e \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \sqrt {e \csc (c+d x)} \sec (c+d x)}{d}-\frac {5 a^2 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 {3 a^2 e \sqrt {e \csc (c+d x)} \sin (c+d x) \tan (c+d x)}{d}+\frac {\left (4 a^2 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 {4 a^2 e \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \sqrt {e \csc (c+d x)} \sec (c+d x)}{d}-\frac {5 a^2 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 {3 a^2 e \sqrt {e \csc (c+d x)} \sin (c+d x) \tan (c+d x)}{d}+\frac {\left (2 a^2 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 (2 a^2 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 {4 a^2 e \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \sqrt {e \csc (c+d x)} \sec (c+d x)}{d}-\frac {2 a^2 e \arctan \left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {2 a^2 e \text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}-\frac {5 a^2 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 {3 a^2 e \sqrt {e \csc (c+d x)} \sin (c+d x) \tan (c+d x)}{d} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 14.91 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.81 \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx=\frac {2 a^2 \cos ^4\left (\frac {1}{2} (c+d x)\right ) (e \csc (c+d x))^{3/2} \left (3 \arctan \left (\sqrt {\csc (c+d x)}\right ) \sqrt {\cos ^2(c+d x)}+3 \text {arctanh}\left (\sqrt {\csc (c+d x)}\right ) \sqrt {\cos ^2(c+d x)}-6 \sqrt {\csc (c+d x)}-6 \sqrt {\cos ^2(c+d x)} \sqrt {\csc (c+d x)}+5 \sqrt {-\cot ^2(c+d x)} \sqrt {\csc (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\csc ^2(c+d x)\right )\right ) \sec (c+d x) \sec ^4\left (\frac {1}{2} \csc ^{-1}(\csc (c+d x))\right )}{3 d \csc ^{\frac {3}{2}}(c+d x)} \]

[In]

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

[Out]

(2*a^2*Cos[(c + d*x)/2]^4*(e*Csc[c + d*x])^(3/2)*(3*ArcTan[Sqrt[Csc[c + d*x]]]*Sqrt[Cos[c + d*x]^2] + 3*ArcTan
h[Sqrt[Csc[c + d*x]]]*Sqrt[Cos[c + d*x]^2] - 6*Sqrt[Csc[c + d*x]] - 6*Sqrt[Cos[c + d*x]^2]*Sqrt[Csc[c + d*x]]
+ 5*Sqrt[-Cot[c + d*x]^2]*Sqrt[Csc[c + d*x]]*Hypergeometric2F1[3/4, 3/2, 7/4, Csc[c + d*x]^2])*Sec[c + d*x]*Se
c[ArcCsc[Csc[c + d*x]]/2]^4)/(3*d*Csc[c + d*x]^(3/2))

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 13.20 (sec) , antiderivative size = 1069, normalized size of antiderivative = 4.45

method	result	size
parts	\(\text {Expression too large to display}\)	\(1069\)
default	\(\text {Expression too large to display}\)	\(1231\)

[In]

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

[Out]

a^2/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)*E
llipticF((-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))+1/2*a^2/d*2^(1/2)*e*(
e*csc(d*x+c))^(1/2)*(6*(-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)-3*(-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)+6*(-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))-3*(-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))-3*2^(1/2)+2^(1/2)*sec(d*x+c))-2*a^2/d*(e*cs
c(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 = 0.19 (sec) , antiderivative size = 729, normalized size of antiderivative = 3.04 \[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx=\left [-\frac {2 \, a^{2} \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 ) \cos \left (d x + c\right ) - a^{2} \sqrt {-e} e \cos \left (d x + c\right ) \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 ) + 10 \, a^{2} \sqrt {2 i \, e} e \cos \left (d x + c\right ) {\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 ) + 10 \, a^{2} \sqrt {-2 i \, e} e \cos \left (d x + c\right ) {\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 ) + 4 \, {\left (5 \, a^{2} e \cos \left (d x + c\right )^{2} + 4 \, a^{2} e \cos \left (d x + c\right ) - a^{2} e\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{4 \, d \cos \left (d x + c\right )}, \frac {2 \, a^{2} 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 ) \cos \left (d x + c\right ) + a^{2} e^{\frac {3}{2}} \cos \left (d x + c\right ) \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 ) - 10 \, a^{2} \sqrt {2 i \, e} e \cos \left (d x + c\right ) {\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 ) - 10 \, a^{2} \sqrt {-2 i \, e} e \cos \left (d x + c\right ) {\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 ) - 4 \, {\left (5 \, a^{2} e \cos \left (d x + c\right )^{2} + 4 \, a^{2} e \cos \left (d x + c\right ) - a^{2} e\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{4 \, d \cos \left (d x + c\right )}\right ] \]

[In]

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

[Out]

[-1/4*(2*a^2*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))*cos(d*x + c) - a^2*sqrt(-e)*e*cos(d*x + c)*log((e*cos(d*x + c)^4 - 72*e*cos(d*x + c)^2 - 8*(co
s(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)) + 10*a^2*sqrt(2*I*e)*e*cos(d*x + c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c
) + I*sin(d*x + c))) + 10*a^2*sqrt(-2*I*e)*e*cos(d*x + c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(
d*x + c) - I*sin(d*x + c))) + 4*(5*a^2*e*cos(d*x + c)^2 + 4*a^2*e*cos(d*x + c) - a^2*e)*sqrt(e/sin(d*x + c)))/
(d*cos(d*x + c)), 1/4*(2*a^2*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))*cos(d*x + c) + a^2*e^(3/2)*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)) - 10*a^2*sqrt(2*I*e)*e*cos(d*x + c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(
d*x + c) + I*sin(d*x + c))) - 10*a^2*sqrt(-2*I*e)*e*cos(d*x + c)*weierstrassZeta(4, 0, weierstrassPInverse(4,
0, cos(d*x + c) - I*sin(d*x + c))) - 4*(5*a^2*e*cos(d*x + c)^2 + 4*a^2*e*cos(d*x + c) - a^2*e)*sqrt(e/sin(d*x
+ c)))/(d*cos(d*x + c))]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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