3.1.57 \(\int \sqrt {x} (a+b \csc (c+d \sqrt {x}))^2 \, dx\) [57]

Optimal. Leaf size=241 \[ -\frac {2 i b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x \cot \left (c+d \sqrt {x}\right )}{d}+\frac {4 b^2 \sqrt {x} \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {8 i a b \sqrt {x} \text {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 i a b \sqrt {x} \text {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {2 i b^2 \text {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {8 a b \text {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {8 a b \text {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3} \]

[Out]

-2*I*b^2*x/d+2/3*a^2*x^(3/2)-8*a*b*x*arctanh(exp(I*(c+d*x^(1/2))))/d-2*b^2*x*cot(c+d*x^(1/2))/d-2*I*b^2*polylo
g(2,exp(2*I*(c+d*x^(1/2))))/d^3-8*a*b*polylog(3,-exp(I*(c+d*x^(1/2))))/d^3+8*a*b*polylog(3,exp(I*(c+d*x^(1/2))
))/d^3+4*b^2*ln(1-exp(2*I*(c+d*x^(1/2))))*x^(1/2)/d^2+8*I*a*b*polylog(2,-exp(I*(c+d*x^(1/2))))*x^(1/2)/d^2-8*I
*a*b*polylog(2,exp(I*(c+d*x^(1/2))))*x^(1/2)/d^2

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Rubi [A]
time = 0.23, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4290, 4275, 4268, 2611, 2320, 6724, 4269, 3798, 2221, 2317, 2438} \begin {gather*} \frac {2}{3} a^2 x^{3/2}-\frac {8 a b \text {Li}_3\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {8 a b \text {Li}_3\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {8 i a b \sqrt {x} \text {Li}_2\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 i a b \sqrt {x} \text {Li}_2\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 a b x \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 i b^2 \text {Li}_2\left (e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {4 b^2 \sqrt {x} \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {2 b^2 x \cot \left (c+d \sqrt {x}\right )}{d}-\frac {2 i b^2 x}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + b*Csc[c + d*Sqrt[x]])^2,x]

[Out]

((-2*I)*b^2*x)/d + (2*a^2*x^(3/2))/3 - (8*a*b*x*ArcTanh[E^(I*(c + d*Sqrt[x]))])/d - (2*b^2*x*Cot[c + d*Sqrt[x]
])/d + (4*b^2*Sqrt[x]*Log[1 - E^((2*I)*(c + d*Sqrt[x]))])/d^2 + ((8*I)*a*b*Sqrt[x]*PolyLog[2, -E^(I*(c + d*Sqr
t[x]))])/d^2 - ((8*I)*a*b*Sqrt[x]*PolyLog[2, E^(I*(c + d*Sqrt[x]))])/d^2 - ((2*I)*b^2*PolyLog[2, E^((2*I)*(c +
d*Sqrt[x]))])/d^3 - (8*a*b*PolyLog[3, -E^(I*(c + d*Sqrt[x]))])/d^3 + (8*a*b*PolyLog[3, E^(I*(c + d*Sqrt[x]))]
)/d^3

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4268

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*
x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[d*(m/f), Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4275

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 4290

Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Csc[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx &=2 \text {Subst}\left (\int x^2 (a+b \csc (c+d x))^2 \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (a^2 x^2+2 a b x^2 \csc (c+d x)+b^2 x^2 \csc ^2(c+d x)\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{3} a^2 x^{3/2}+(4 a b) \text {Subst}\left (\int x^2 \csc (c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \text {Subst}\left (\int x^2 \csc ^2(c+d x) \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x \cot \left (c+d \sqrt {x}\right )}{d}-\frac {(8 a b) \text {Subst}\left (\int x \log \left (1-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(8 a b) \text {Subst}\left (\int x \log \left (1+e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {\left (4 b^2\right ) \text {Subst}\left (\int x \cot (c+d x) \, dx,x,\sqrt {x}\right )}{d}\\ &=-\frac {2 i b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x \cot \left (c+d \sqrt {x}\right )}{d}+\frac {8 i a b \sqrt {x} \text {Li}_2\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 i a b \sqrt {x} \text {Li}_2\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {(8 i a b) \text {Subst}\left (\int \text {Li}_2\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(8 i a b) \text {Subst}\left (\int \text {Li}_2\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (8 i b^2\right ) \text {Subst}\left (\int \frac {e^{2 i (c+d x)} x}{1-e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )}{d}\\ &=-\frac {2 i b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x \cot \left (c+d \sqrt {x}\right )}{d}+\frac {4 b^2 \sqrt {x} \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {8 i a b \sqrt {x} \text {Li}_2\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 i a b \sqrt {x} \text {Li}_2\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {(8 a b) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {(8 a b) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {\left (4 b^2\right ) \text {Subst}\left (\int \log \left (1-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}\\ &=-\frac {2 i b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x \cot \left (c+d \sqrt {x}\right )}{d}+\frac {4 b^2 \sqrt {x} \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {8 i a b \sqrt {x} \text {Li}_2\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 i a b \sqrt {x} \text {Li}_2\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 a b \text {Li}_3\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {8 a b \text {Li}_3\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}\\ &=-\frac {2 i b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x \cot \left (c+d \sqrt {x}\right )}{d}+\frac {4 b^2 \sqrt {x} \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {8 i a b \sqrt {x} \text {Li}_2\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 i a b \sqrt {x} \text {Li}_2\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {2 i b^2 \text {Li}_2\left (e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {8 a b \text {Li}_3\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {8 a b \text {Li}_3\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(670\) vs. \(2(241)=482\).
time = 3.66, size = 670, normalized size = 2.78 \begin {gather*} \frac {-12 i b^2 d^2 e^{2 i c} x-2 a^2 d^3 x^{3/2}+2 a^2 d^3 e^{2 i c} x^{3/2}-12 a b d^2 x \log \left (1-e^{i \left (c+d \sqrt {x}\right )}\right )+12 a b d^2 e^{2 i c} x \log \left (1-e^{i \left (c+d \sqrt {x}\right )}\right )+12 a b d^2 x \log \left (1+e^{i \left (c+d \sqrt {x}\right )}\right )-12 a b d^2 e^{2 i c} x \log \left (1+e^{i \left (c+d \sqrt {x}\right )}\right )-12 b^2 d \sqrt {x} \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )+12 b^2 d e^{2 i c} \sqrt {x} \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )+24 i a b d \left (-1+e^{2 i c}\right ) \sqrt {x} \text {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )-24 i a b d \left (-1+e^{2 i c}\right ) \sqrt {x} \text {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )+6 i b^2 \text {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )-6 i b^2 e^{2 i c} \text {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )+24 a b \text {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )-24 a b e^{2 i c} \text {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )-24 a b \text {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )+24 a b e^{2 i c} \text {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )-3 b^2 d^2 x \csc \left (\frac {c}{2}\right ) \csc \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right ) \sin \left (\frac {d \sqrt {x}}{2}\right )+3 b^2 d^2 e^{2 i c} x \csc \left (\frac {c}{2}\right ) \csc \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right ) \sin \left (\frac {d \sqrt {x}}{2}\right )-3 b^2 d^2 x \sec \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right ) \sin \left (\frac {d \sqrt {x}}{2}\right )+3 b^2 d^2 e^{2 i c} x \sec \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right ) \sin \left (\frac {d \sqrt {x}}{2}\right )}{3 d^3 \left (-1+e^{2 i c}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(a + b*Csc[c + d*Sqrt[x]])^2,x]

[Out]

((-12*I)*b^2*d^2*E^((2*I)*c)*x - 2*a^2*d^3*x^(3/2) + 2*a^2*d^3*E^((2*I)*c)*x^(3/2) - 12*a*b*d^2*x*Log[1 - E^(I
*(c + d*Sqrt[x]))] + 12*a*b*d^2*E^((2*I)*c)*x*Log[1 - E^(I*(c + d*Sqrt[x]))] + 12*a*b*d^2*x*Log[1 + E^(I*(c +
d*Sqrt[x]))] - 12*a*b*d^2*E^((2*I)*c)*x*Log[1 + E^(I*(c + d*Sqrt[x]))] - 12*b^2*d*Sqrt[x]*Log[1 - E^((2*I)*(c
+ d*Sqrt[x]))] + 12*b^2*d*E^((2*I)*c)*Sqrt[x]*Log[1 - E^((2*I)*(c + d*Sqrt[x]))] + (24*I)*a*b*d*(-1 + E^((2*I)
*c))*Sqrt[x]*PolyLog[2, -E^(I*(c + d*Sqrt[x]))] - (24*I)*a*b*d*(-1 + E^((2*I)*c))*Sqrt[x]*PolyLog[2, E^(I*(c +
d*Sqrt[x]))] + (6*I)*b^2*PolyLog[2, E^((2*I)*(c + d*Sqrt[x]))] - (6*I)*b^2*E^((2*I)*c)*PolyLog[2, E^((2*I)*(c
+ d*Sqrt[x]))] + 24*a*b*PolyLog[3, -E^(I*(c + d*Sqrt[x]))] - 24*a*b*E^((2*I)*c)*PolyLog[3, -E^(I*(c + d*Sqrt[
x]))] - 24*a*b*PolyLog[3, E^(I*(c + d*Sqrt[x]))] + 24*a*b*E^((2*I)*c)*PolyLog[3, E^(I*(c + d*Sqrt[x]))] - 3*b^
2*d^2*x*Csc[c/2]*Csc[(c + d*Sqrt[x])/2]*Sin[(d*Sqrt[x])/2] + 3*b^2*d^2*E^((2*I)*c)*x*Csc[c/2]*Csc[(c + d*Sqrt[
x])/2]*Sin[(d*Sqrt[x])/2] - 3*b^2*d^2*x*Sec[c/2]*Sec[(c + d*Sqrt[x])/2]*Sin[(d*Sqrt[x])/2] + 3*b^2*d^2*E^((2*I
)*c)*x*Sec[c/2]*Sec[(c + d*Sqrt[x])/2]*Sin[(d*Sqrt[x])/2])/(3*d^3*(-1 + E^((2*I)*c)))

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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \left (a +b \csc \left (c +d \sqrt {x}\right )\right )^{2} \sqrt {x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*csc(c+d*x^(1/2)))^2*x^(1/2),x)

[Out]

int((a+b*csc(c+d*x^(1/2)))^2*x^(1/2),x)

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1217 vs. \(2 (190) = 380\).
time = 0.35, size = 1217, normalized size = 5.05 \begin {gather*} \frac {2 \, {\left ({\left (d \sqrt {x} + c\right )}^{3} a^{2} - 3 \, {\left (d \sqrt {x} + c\right )}^{2} a^{2} c + 3 \, {\left (d \sqrt {x} + c\right )} a^{2} c^{2} - 6 \, a b c^{2} \log \left (\cot \left (d \sqrt {x} + c\right ) + \csc \left (d \sqrt {x} + c\right )\right ) - \frac {3 \, {\left (2 \, b^{2} c^{2} - 2 \, {\left ({\left (d \sqrt {x} + c\right )}^{2} a b + b^{2} c - {\left (2 \, a b c + b^{2}\right )} {\left (d \sqrt {x} + c\right )} - {\left ({\left (d \sqrt {x} + c\right )}^{2} a b + b^{2} c - {\left (2 \, a b c + b^{2}\right )} {\left (d \sqrt {x} + c\right )}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (-i \, {\left (d \sqrt {x} + c\right )}^{2} a b - i \, b^{2} c + {\left (2 i \, a b c + i \, b^{2}\right )} {\left (d \sqrt {x} + c\right )}\right )} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )\right )} \arctan \left (\sin \left (d \sqrt {x} + c\right ), \cos \left (d \sqrt {x} + c\right ) + 1\right ) + 2 \, {\left (b^{2} c \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + i \, b^{2} c \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) - b^{2} c\right )} \arctan \left (\sin \left (d \sqrt {x} + c\right ), \cos \left (d \sqrt {x} + c\right ) - 1\right ) - 2 \, {\left ({\left (d \sqrt {x} + c\right )}^{2} a b - {\left (2 \, a b c - b^{2}\right )} {\left (d \sqrt {x} + c\right )} - {\left ({\left (d \sqrt {x} + c\right )}^{2} a b - {\left (2 \, a b c - b^{2}\right )} {\left (d \sqrt {x} + c\right )}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (-i \, {\left (d \sqrt {x} + c\right )}^{2} a b + {\left (2 i \, a b c - i \, b^{2}\right )} {\left (d \sqrt {x} + c\right )}\right )} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )\right )} \arctan \left (\sin \left (d \sqrt {x} + c\right ), -\cos \left (d \sqrt {x} + c\right ) + 1\right ) + 2 \, {\left ({\left (d \sqrt {x} + c\right )}^{2} b^{2} - 2 \, {\left (d \sqrt {x} + c\right )} b^{2} c\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + 2 \, {\left (2 \, {\left (d \sqrt {x} + c\right )} a b - 2 \, a b c - b^{2} - {\left (2 \, {\left (d \sqrt {x} + c\right )} a b - 2 \, a b c - b^{2}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) - {\left (2 i \, {\left (d \sqrt {x} + c\right )} a b - 2 i \, a b c - i \, b^{2}\right )} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )\right )} {\rm Li}_2\left (-e^{\left (i \, d \sqrt {x} + i \, c\right )}\right ) - 2 \, {\left (2 \, {\left (d \sqrt {x} + c\right )} a b - 2 \, a b c + b^{2} - {\left (2 \, {\left (d \sqrt {x} + c\right )} a b - 2 \, a b c + b^{2}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (-2 i \, {\left (d \sqrt {x} + c\right )} a b + 2 i \, a b c - i \, b^{2}\right )} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )\right )} {\rm Li}_2\left (e^{\left (i \, d \sqrt {x} + i \, c\right )}\right ) + {\left (i \, {\left (d \sqrt {x} + c\right )}^{2} a b + i \, b^{2} c + {\left (-2 i \, a b c - i \, b^{2}\right )} {\left (d \sqrt {x} + c\right )} + {\left (-i \, {\left (d \sqrt {x} + c\right )}^{2} a b - i \, b^{2} c + {\left (2 i \, a b c + i \, b^{2}\right )} {\left (d \sqrt {x} + c\right )}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left ({\left (d \sqrt {x} + c\right )}^{2} a b + b^{2} c - {\left (2 \, a b c + b^{2}\right )} {\left (d \sqrt {x} + c\right )}\right )} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )\right )} \log \left (\cos \left (d \sqrt {x} + c\right )^{2} + \sin \left (d \sqrt {x} + c\right )^{2} + 2 \, \cos \left (d \sqrt {x} + c\right ) + 1\right ) + {\left (-i \, {\left (d \sqrt {x} + c\right )}^{2} a b + i \, b^{2} c + {\left (2 i \, a b c - i \, b^{2}\right )} {\left (d \sqrt {x} + c\right )} + {\left (i \, {\left (d \sqrt {x} + c\right )}^{2} a b - i \, b^{2} c + {\left (-2 i \, a b c + i \, b^{2}\right )} {\left (d \sqrt {x} + c\right )}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) - {\left ({\left (d \sqrt {x} + c\right )}^{2} a b - b^{2} c - {\left (2 \, a b c - b^{2}\right )} {\left (d \sqrt {x} + c\right )}\right )} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )\right )} \log \left (\cos \left (d \sqrt {x} + c\right )^{2} + \sin \left (d \sqrt {x} + c\right )^{2} - 2 \, \cos \left (d \sqrt {x} + c\right ) + 1\right ) - 4 \, {\left (i \, a b \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) - a b \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) - i \, a b\right )} {\rm Li}_{3}(-e^{\left (i \, d \sqrt {x} + i \, c\right )}) - 4 \, {\left (-i \, a b \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + a b \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + i \, a b\right )} {\rm Li}_{3}(e^{\left (i \, d \sqrt {x} + i \, c\right )}) - 2 \, {\left (-i \, {\left (d \sqrt {x} + c\right )}^{2} b^{2} + 2 i \, {\left (d \sqrt {x} + c\right )} b^{2} c\right )} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )\right )}}{-i \, \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + i}\right )}}{3 \, d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*((d*sqrt(x) + c)^3*a^2 - 3*(d*sqrt(x) + c)^2*a^2*c + 3*(d*sqrt(x) + c)*a^2*c^2 - 6*a*b*c^2*log(cot(d*sqrt(
x) + c) + csc(d*sqrt(x) + c)) - 3*(2*b^2*c^2 - 2*((d*sqrt(x) + c)^2*a*b + b^2*c - (2*a*b*c + b^2)*(d*sqrt(x) +
c) - ((d*sqrt(x) + c)^2*a*b + b^2*c - (2*a*b*c + b^2)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) + (-I*(d*sqrt(x
) + c)^2*a*b - I*b^2*c + (2*I*a*b*c + I*b^2)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c))*arctan2(sin(d*sqrt(x) +
c), cos(d*sqrt(x) + c) + 1) + 2*(b^2*c*cos(2*d*sqrt(x) + 2*c) + I*b^2*c*sin(2*d*sqrt(x) + 2*c) - b^2*c)*arctan
2(sin(d*sqrt(x) + c), cos(d*sqrt(x) + c) - 1) - 2*((d*sqrt(x) + c)^2*a*b - (2*a*b*c - b^2)*(d*sqrt(x) + c) - (
(d*sqrt(x) + c)^2*a*b - (2*a*b*c - b^2)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) + (-I*(d*sqrt(x) + c)^2*a*b +
(2*I*a*b*c - I*b^2)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c))*arctan2(sin(d*sqrt(x) + c), -cos(d*sqrt(x) + c) +
1) + 2*((d*sqrt(x) + c)^2*b^2 - 2*(d*sqrt(x) + c)*b^2*c)*cos(2*d*sqrt(x) + 2*c) + 2*(2*(d*sqrt(x) + c)*a*b -
2*a*b*c - b^2 - (2*(d*sqrt(x) + c)*a*b - 2*a*b*c - b^2)*cos(2*d*sqrt(x) + 2*c) - (2*I*(d*sqrt(x) + c)*a*b - 2*
I*a*b*c - I*b^2)*sin(2*d*sqrt(x) + 2*c))*dilog(-e^(I*d*sqrt(x) + I*c)) - 2*(2*(d*sqrt(x) + c)*a*b - 2*a*b*c +
b^2 - (2*(d*sqrt(x) + c)*a*b - 2*a*b*c + b^2)*cos(2*d*sqrt(x) + 2*c) + (-2*I*(d*sqrt(x) + c)*a*b + 2*I*a*b*c -
I*b^2)*sin(2*d*sqrt(x) + 2*c))*dilog(e^(I*d*sqrt(x) + I*c)) + (I*(d*sqrt(x) + c)^2*a*b + I*b^2*c + (-2*I*a*b*
c - I*b^2)*(d*sqrt(x) + c) + (-I*(d*sqrt(x) + c)^2*a*b - I*b^2*c + (2*I*a*b*c + I*b^2)*(d*sqrt(x) + c))*cos(2*
d*sqrt(x) + 2*c) + ((d*sqrt(x) + c)^2*a*b + b^2*c - (2*a*b*c + b^2)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c))*l
og(cos(d*sqrt(x) + c)^2 + sin(d*sqrt(x) + c)^2 + 2*cos(d*sqrt(x) + c) + 1) + (-I*(d*sqrt(x) + c)^2*a*b + I*b^2
*c + (2*I*a*b*c - I*b^2)*(d*sqrt(x) + c) + (I*(d*sqrt(x) + c)^2*a*b - I*b^2*c + (-2*I*a*b*c + I*b^2)*(d*sqrt(x
) + c))*cos(2*d*sqrt(x) + 2*c) - ((d*sqrt(x) + c)^2*a*b - b^2*c - (2*a*b*c - b^2)*(d*sqrt(x) + c))*sin(2*d*sqr
t(x) + 2*c))*log(cos(d*sqrt(x) + c)^2 + sin(d*sqrt(x) + c)^2 - 2*cos(d*sqrt(x) + c) + 1) - 4*(I*a*b*cos(2*d*sq
rt(x) + 2*c) - a*b*sin(2*d*sqrt(x) + 2*c) - I*a*b)*polylog(3, -e^(I*d*sqrt(x) + I*c)) - 4*(-I*a*b*cos(2*d*sqrt
(x) + 2*c) + a*b*sin(2*d*sqrt(x) + 2*c) + I*a*b)*polylog(3, e^(I*d*sqrt(x) + I*c)) - 2*(-I*(d*sqrt(x) + c)^2*b
^2 + 2*I*(d*sqrt(x) + c)*b^2*c)*sin(2*d*sqrt(x) + 2*c))/(-I*cos(2*d*sqrt(x) + 2*c) + sin(2*d*sqrt(x) + 2*c) +
I))/d^3

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(b^2*sqrt(x)*csc(d*sqrt(x) + c)^2 + 2*a*b*sqrt(x)*csc(d*sqrt(x) + c) + a^2*sqrt(x), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x} \left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*csc(c+d*x**(1/2)))**2*x**(1/2),x)

[Out]

Integral(sqrt(x)*(a + b*csc(c + d*sqrt(x)))**2, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*csc(d*sqrt(x) + c) + a)^2*sqrt(x), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sqrt {x}\,{\left (a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(a + b/sin(c + d*x^(1/2)))^2,x)

[Out]

int(x^(1/2)*(a + b/sin(c + d*x^(1/2)))^2, x)

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