3.1.89 \(\int \frac {x^5}{(a+b \sin (c+d x^3))^2} \, dx\) [89]

Optimal. Leaf size=324 \[ -\frac {i a x^3 \log \left (1-\frac {i b e^{i \left (c+d x^3\right )}}{a-\sqrt {a^2-b^2}}\right )}{3 \left (a^2-b^2\right )^{3/2} d}+\frac {i a x^3 \log \left (1-\frac {i b e^{i \left (c+d x^3\right )}}{a+\sqrt {a^2-b^2}}\right )}{3 \left (a^2-b^2\right )^{3/2} d}-\frac {\log \left (a+b \sin \left (c+d x^3\right )\right )}{3 \left (a^2-b^2\right ) d^2}-\frac {a \text {Li}_2\left (\frac {i b e^{i \left (c+d x^3\right )}}{a-\sqrt {a^2-b^2}}\right )}{3 \left (a^2-b^2\right )^{3/2} d^2}+\frac {a \text {Li}_2\left (\frac {i b e^{i \left (c+d x^3\right )}}{a+\sqrt {a^2-b^2}}\right )}{3 \left (a^2-b^2\right )^{3/2} d^2}+\frac {b x^3 \cos \left (c+d x^3\right )}{3 \left (a^2-b^2\right ) d \left (a+b \sin \left (c+d x^3\right )\right )} \]

[Out]

-1/3*ln(a+b*sin(d*x^3+c))/(a^2-b^2)/d^2-1/3*I*a*x^3*ln(1-I*b*exp(I*(d*x^3+c))/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)^(
3/2)/d+1/3*I*a*x^3*ln(1-I*b*exp(I*(d*x^3+c))/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)^(3/2)/d-1/3*a*polylog(2,I*b*exp(I*
(d*x^3+c))/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)^(3/2)/d^2+1/3*a*polylog(2,I*b*exp(I*(d*x^3+c))/(a+(a^2-b^2)^(1/2)))/
(a^2-b^2)^(3/2)/d^2+1/3*b*x^3*cos(d*x^3+c)/(a^2-b^2)/d/(a+b*sin(d*x^3+c))

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Rubi [A]
time = 0.40, antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3460, 3405, 3404, 2296, 2221, 2317, 2438, 2747, 31} \begin {gather*} -\frac {a \text {PolyLog}\left (2,\frac {i b e^{i \left (c+d x^3\right )}}{a-\sqrt {a^2-b^2}}\right )}{3 d^2 \left (a^2-b^2\right )^{3/2}}+\frac {a \text {PolyLog}\left (2,\frac {i b e^{i \left (c+d x^3\right )}}{\sqrt {a^2-b^2}+a}\right )}{3 d^2 \left (a^2-b^2\right )^{3/2}}-\frac {\log \left (a+b \sin \left (c+d x^3\right )\right )}{3 d^2 \left (a^2-b^2\right )}-\frac {i a x^3 \log \left (1-\frac {i b e^{i \left (c+d x^3\right )}}{a-\sqrt {a^2-b^2}}\right )}{3 d \left (a^2-b^2\right )^{3/2}}+\frac {i a x^3 \log \left (1-\frac {i b e^{i \left (c+d x^3\right )}}{\sqrt {a^2-b^2}+a}\right )}{3 d \left (a^2-b^2\right )^{3/2}}+\frac {b x^3 \cos \left (c+d x^3\right )}{3 d \left (a^2-b^2\right ) \left (a+b \sin \left (c+d x^3\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*Sin[c + d*x^3])^2,x]

[Out]

((-1/3*I)*a*x^3*Log[1 - (I*b*E^(I*(c + d*x^3)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) + ((I/3)*a*x^3*L
og[1 - (I*b*E^(I*(c + d*x^3)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) - Log[a + b*Sin[c + d*x^3]]/(3*(a
^2 - b^2)*d^2) - (a*PolyLog[2, (I*b*E^(I*(c + d*x^3)))/(a - Sqrt[a^2 - b^2])])/(3*(a^2 - b^2)^(3/2)*d^2) + (a*
PolyLog[2, (I*b*E^(I*(c + d*x^3)))/(a + Sqrt[a^2 - b^2])])/(3*(a^2 - b^2)^(3/2)*d^2) + (b*x^3*Cos[c + d*x^3])/
(3*(a^2 - b^2)*d*(a + b*Sin[c + d*x^3]))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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 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 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 3404

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[(c + d*x)^m*(E
^(I*(e + f*x))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3405

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[b*(c + d*x)^m*(Cos[
e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f*x]))), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[b*d*(m/(f*(a^2 - b^2))), Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/(a + b*Sin[e + f*x])), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3460

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

Rubi steps

\begin {align*} \int \frac {x^5}{\left (a+b \sin \left (c+d x^3\right )\right )^2} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x}{(a+b \sin (c+d x))^2} \, dx,x,x^3\right )\\ &=\frac {b x^3 \cos \left (c+d x^3\right )}{3 \left (a^2-b^2\right ) d \left (a+b \sin \left (c+d x^3\right )\right )}+\frac {a \text {Subst}\left (\int \frac {x}{a+b \sin (c+d x)} \, dx,x,x^3\right )}{3 \left (a^2-b^2\right )}-\frac {b \text {Subst}\left (\int \frac {\cos (c+d x)}{a+b \sin (c+d x)} \, dx,x,x^3\right )}{3 \left (a^2-b^2\right ) d}\\ &=\frac {b x^3 \cos \left (c+d x^3\right )}{3 \left (a^2-b^2\right ) d \left (a+b \sin \left (c+d x^3\right )\right )}+\frac {(2 a) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx,x,x^3\right )}{3 \left (a^2-b^2\right )}-\frac {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sin \left (c+d x^3\right )\right )}{3 \left (a^2-b^2\right ) d^2}\\ &=-\frac {\log \left (a+b \sin \left (c+d x^3\right )\right )}{3 \left (a^2-b^2\right ) d^2}+\frac {b x^3 \cos \left (c+d x^3\right )}{3 \left (a^2-b^2\right ) d \left (a+b \sin \left (c+d x^3\right )\right )}-\frac {(2 i a b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{2 a-2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx,x,x^3\right )}{3 \left (a^2-b^2\right )^{3/2}}+\frac {(2 i a b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{2 a+2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx,x,x^3\right )}{3 \left (a^2-b^2\right )^{3/2}}\\ &=-\frac {i a x^3 \log \left (1-\frac {i b e^{i \left (c+d x^3\right )}}{a-\sqrt {a^2-b^2}}\right )}{3 \left (a^2-b^2\right )^{3/2} d}+\frac {i a x^3 \log \left (1-\frac {i b e^{i \left (c+d x^3\right )}}{a+\sqrt {a^2-b^2}}\right )}{3 \left (a^2-b^2\right )^{3/2} d}-\frac {\log \left (a+b \sin \left (c+d x^3\right )\right )}{3 \left (a^2-b^2\right ) d^2}+\frac {b x^3 \cos \left (c+d x^3\right )}{3 \left (a^2-b^2\right ) d \left (a+b \sin \left (c+d x^3\right )\right )}+\frac {(i a) \text {Subst}\left (\int \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx,x,x^3\right )}{3 \left (a^2-b^2\right )^{3/2} d}-\frac {(i a) \text {Subst}\left (\int \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx,x,x^3\right )}{3 \left (a^2-b^2\right )^{3/2} d}\\ &=-\frac {i a x^3 \log \left (1-\frac {i b e^{i \left (c+d x^3\right )}}{a-\sqrt {a^2-b^2}}\right )}{3 \left (a^2-b^2\right )^{3/2} d}+\frac {i a x^3 \log \left (1-\frac {i b e^{i \left (c+d x^3\right )}}{a+\sqrt {a^2-b^2}}\right )}{3 \left (a^2-b^2\right )^{3/2} d}-\frac {\log \left (a+b \sin \left (c+d x^3\right )\right )}{3 \left (a^2-b^2\right ) d^2}+\frac {b x^3 \cos \left (c+d x^3\right )}{3 \left (a^2-b^2\right ) d \left (a+b \sin \left (c+d x^3\right )\right )}+\frac {a \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i b x}{2 a-2 \sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^3\right )}\right )}{3 \left (a^2-b^2\right )^{3/2} d^2}-\frac {a \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i b x}{2 a+2 \sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^3\right )}\right )}{3 \left (a^2-b^2\right )^{3/2} d^2}\\ &=-\frac {i a x^3 \log \left (1-\frac {i b e^{i \left (c+d x^3\right )}}{a-\sqrt {a^2-b^2}}\right )}{3 \left (a^2-b^2\right )^{3/2} d}+\frac {i a x^3 \log \left (1-\frac {i b e^{i \left (c+d x^3\right )}}{a+\sqrt {a^2-b^2}}\right )}{3 \left (a^2-b^2\right )^{3/2} d}-\frac {\log \left (a+b \sin \left (c+d x^3\right )\right )}{3 \left (a^2-b^2\right ) d^2}-\frac {a \text {Li}_2\left (\frac {i b e^{i \left (c+d x^3\right )}}{a-\sqrt {a^2-b^2}}\right )}{3 \left (a^2-b^2\right )^{3/2} d^2}+\frac {a \text {Li}_2\left (\frac {i b e^{i \left (c+d x^3\right )}}{a+\sqrt {a^2-b^2}}\right )}{3 \left (a^2-b^2\right )^{3/2} d^2}+\frac {b x^3 \cos \left (c+d x^3\right )}{3 \left (a^2-b^2\right ) d \left (a+b \sin \left (c+d x^3\right )\right )}\\ \end {align*}

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Mathematica [A]
time = 0.65, size = 302, normalized size = 0.93 \begin {gather*} \frac {-\frac {i a d x^3 \log \left (1+\frac {i b e^{i \left (c+d x^3\right )}}{-a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+\frac {i a d x^3 \log \left (1-\frac {i b e^{i \left (c+d x^3\right )}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {\log \left (a+b \sin \left (c+d x^3\right )\right )}{a^2-b^2}-\frac {a \text {Li}_2\left (-\frac {i b e^{i \left (c+d x^3\right )}}{-a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+\frac {a \text {Li}_2\left (\frac {i b e^{i \left (c+d x^3\right )}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+\frac {b d x^3 \cos \left (c+d x^3\right )}{\left (a^2-b^2\right ) \left (a+b \sin \left (c+d x^3\right )\right )}}{3 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*Sin[c + d*x^3])^2,x]

[Out]

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

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Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int \frac {x^{5}}{\left (a +b \sin \left (d \,x^{3}+c \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a+b*sin(d*x^3+c))^2,x)

[Out]

int(x^5/(a+b*sin(d*x^3+c))^2,x)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b*sin(d*x^3+c))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1509 vs. \(2 (274) = 548\).
time = 0.61, size = 1509, normalized size = 4.66 \begin {gather*} \frac {2 \, {\left (a^{2} b - b^{3}\right )} d x^{3} \cos \left (d x^{3} + c\right ) + {\left (i \, a b^{2} \sin \left (d x^{3} + c\right ) + i \, a^{2} b\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {i \, a \cos \left (d x^{3} + c\right ) - a \sin \left (d x^{3} + c\right ) + {\left (b \cos \left (d x^{3} + c\right ) + i \, b \sin \left (d x^{3} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) + {\left (-i \, a b^{2} \sin \left (d x^{3} + c\right ) - i \, a^{2} b\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {i \, a \cos \left (d x^{3} + c\right ) - a \sin \left (d x^{3} + c\right ) - {\left (b \cos \left (d x^{3} + c\right ) + i \, b \sin \left (d x^{3} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) + {\left (-i \, a b^{2} \sin \left (d x^{3} + c\right ) - i \, a^{2} b\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {-i \, a \cos \left (d x^{3} + c\right ) - a \sin \left (d x^{3} + c\right ) + {\left (b \cos \left (d x^{3} + c\right ) - i \, b \sin \left (d x^{3} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) + {\left (i \, a b^{2} \sin \left (d x^{3} + c\right ) + i \, a^{2} b\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {-i \, a \cos \left (d x^{3} + c\right ) - a \sin \left (d x^{3} + c\right ) - {\left (b \cos \left (d x^{3} + c\right ) - i \, b \sin \left (d x^{3} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) - {\left (a^{2} b d x^{3} + a^{2} b c + {\left (a b^{2} d x^{3} + a b^{2} c\right )} \sin \left (d x^{3} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (-\frac {i \, a \cos \left (d x^{3} + c\right ) - a \sin \left (d x^{3} + c\right ) + {\left (b \cos \left (d x^{3} + c\right ) + i \, b \sin \left (d x^{3} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) + {\left (a^{2} b d x^{3} + a^{2} b c + {\left (a b^{2} d x^{3} + a b^{2} c\right )} \sin \left (d x^{3} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (-\frac {i \, a \cos \left (d x^{3} + c\right ) - a \sin \left (d x^{3} + c\right ) - {\left (b \cos \left (d x^{3} + c\right ) + i \, b \sin \left (d x^{3} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) - {\left (a^{2} b d x^{3} + a^{2} b c + {\left (a b^{2} d x^{3} + a b^{2} c\right )} \sin \left (d x^{3} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (-\frac {-i \, a \cos \left (d x^{3} + c\right ) - a \sin \left (d x^{3} + c\right ) + {\left (b \cos \left (d x^{3} + c\right ) - i \, b \sin \left (d x^{3} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) + {\left (a^{2} b d x^{3} + a^{2} b c + {\left (a b^{2} d x^{3} + a b^{2} c\right )} \sin \left (d x^{3} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (-\frac {-i \, a \cos \left (d x^{3} + c\right ) - a \sin \left (d x^{3} + c\right ) - {\left (b \cos \left (d x^{3} + c\right ) - i \, b \sin \left (d x^{3} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) - {\left (a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} \sin \left (d x^{3} + c\right ) + {\left (a b^{2} c \sin \left (d x^{3} + c\right ) + a^{2} b c\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}}\right )} \log \left (2 \, b \cos \left (d x^{3} + c\right ) + 2 i \, b \sin \left (d x^{3} + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} + 2 i \, a\right ) - {\left (a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} \sin \left (d x^{3} + c\right ) + {\left (a b^{2} c \sin \left (d x^{3} + c\right ) + a^{2} b c\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}}\right )} \log \left (2 \, b \cos \left (d x^{3} + c\right ) - 2 i \, b \sin \left (d x^{3} + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - 2 i \, a\right ) - {\left (a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} \sin \left (d x^{3} + c\right ) - {\left (a b^{2} c \sin \left (d x^{3} + c\right ) + a^{2} b c\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}}\right )} \log \left (-2 \, b \cos \left (d x^{3} + c\right ) + 2 i \, b \sin \left (d x^{3} + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} + 2 i \, a\right ) - {\left (a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} \sin \left (d x^{3} + c\right ) - {\left (a b^{2} c \sin \left (d x^{3} + c\right ) + a^{2} b c\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}}\right )} \log \left (-2 \, b \cos \left (d x^{3} + c\right ) - 2 i \, b \sin \left (d x^{3} + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - 2 i \, a\right )}{6 \, {\left ({\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d^{2} \sin \left (d x^{3} + c\right ) + {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b*sin(d*x^3+c))^2,x, algorithm="fricas")

[Out]

1/6*(2*(a^2*b - b^3)*d*x^3*cos(d*x^3 + c) + (I*a*b^2*sin(d*x^3 + c) + I*a^2*b)*sqrt(-(a^2 - b^2)/b^2)*dilog((I
*a*cos(d*x^3 + c) - a*sin(d*x^3 + c) + (b*cos(d*x^3 + c) + I*b*sin(d*x^3 + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b +
1) + (-I*a*b^2*sin(d*x^3 + c) - I*a^2*b)*sqrt(-(a^2 - b^2)/b^2)*dilog((I*a*cos(d*x^3 + c) - a*sin(d*x^3 + c)
- (b*cos(d*x^3 + c) + I*b*sin(d*x^3 + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1) + (-I*a*b^2*sin(d*x^3 + c) - I*a^
2*b)*sqrt(-(a^2 - b^2)/b^2)*dilog((-I*a*cos(d*x^3 + c) - a*sin(d*x^3 + c) + (b*cos(d*x^3 + c) - I*b*sin(d*x^3
+ c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1) + (I*a*b^2*sin(d*x^3 + c) + I*a^2*b)*sqrt(-(a^2 - b^2)/b^2)*dilog((-I
*a*cos(d*x^3 + c) - a*sin(d*x^3 + c) - (b*cos(d*x^3 + c) - I*b*sin(d*x^3 + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b +
1) - (a^2*b*d*x^3 + a^2*b*c + (a*b^2*d*x^3 + a*b^2*c)*sin(d*x^3 + c))*sqrt(-(a^2 - b^2)/b^2)*log(-(I*a*cos(d*
x^3 + c) - a*sin(d*x^3 + c) + (b*cos(d*x^3 + c) + I*b*sin(d*x^3 + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b) + (a^2*b*
d*x^3 + a^2*b*c + (a*b^2*d*x^3 + a*b^2*c)*sin(d*x^3 + c))*sqrt(-(a^2 - b^2)/b^2)*log(-(I*a*cos(d*x^3 + c) - a*
sin(d*x^3 + c) - (b*cos(d*x^3 + c) + I*b*sin(d*x^3 + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b) - (a^2*b*d*x^3 + a^2*b
*c + (a*b^2*d*x^3 + a*b^2*c)*sin(d*x^3 + c))*sqrt(-(a^2 - b^2)/b^2)*log(-(-I*a*cos(d*x^3 + c) - a*sin(d*x^3 +
c) + (b*cos(d*x^3 + c) - I*b*sin(d*x^3 + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b) + (a^2*b*d*x^3 + a^2*b*c + (a*b^2*
d*x^3 + a*b^2*c)*sin(d*x^3 + c))*sqrt(-(a^2 - b^2)/b^2)*log(-(-I*a*cos(d*x^3 + c) - a*sin(d*x^3 + c) - (b*cos(
d*x^3 + c) - I*b*sin(d*x^3 + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b) - (a^3 - a*b^2 + (a^2*b - b^3)*sin(d*x^3 + c)
+ (a*b^2*c*sin(d*x^3 + c) + a^2*b*c)*sqrt(-(a^2 - b^2)/b^2))*log(2*b*cos(d*x^3 + c) + 2*I*b*sin(d*x^3 + c) + 2
*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a) - (a^3 - a*b^2 + (a^2*b - b^3)*sin(d*x^3 + c) + (a*b^2*c*sin(d*x^3 + c) + a
^2*b*c)*sqrt(-(a^2 - b^2)/b^2))*log(2*b*cos(d*x^3 + c) - 2*I*b*sin(d*x^3 + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 2
*I*a) - (a^3 - a*b^2 + (a^2*b - b^3)*sin(d*x^3 + c) - (a*b^2*c*sin(d*x^3 + c) + a^2*b*c)*sqrt(-(a^2 - b^2)/b^2
))*log(-2*b*cos(d*x^3 + c) + 2*I*b*sin(d*x^3 + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a) - (a^3 - a*b^2 + (a^2*
b - b^3)*sin(d*x^3 + c) - (a*b^2*c*sin(d*x^3 + c) + a^2*b*c)*sqrt(-(a^2 - b^2)/b^2))*log(-2*b*cos(d*x^3 + c) -
2*I*b*sin(d*x^3 + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*a))/((a^4*b - 2*a^2*b^3 + b^5)*d^2*sin(d*x^3 + c) + (
a^5 - 2*a^3*b^2 + a*b^4)*d^2)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(a+b*sin(d*x**3+c))**2,x)

[Out]

Timed out

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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(x^5/(a+b*sin(d*x^3+c))^2,x, algorithm="giac")

[Out]

integrate(x^5/(b*sin(d*x^3 + c) + a)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5}{{\left (a+b\,\sin \left (d\,x^3+c\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a + b*sin(c + d*x^3))^2,x)

[Out]

int(x^5/(a + b*sin(c + d*x^3))^2, x)

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