\(\int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [1338]

Optimal result

Integrand size = 29, antiderivative size = 268 \[ \int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {3 b x}{2 \left (a^2-b^2\right )}-\frac {a^2 \left (2 a^2+b^2\right ) x}{2 b^3 \left (a^2-b^2\right )}+\frac {2 a^5 \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^3 \left (a^2-b^2\right )^{3/2} d}+\frac {a \cos (c+d x)}{\left (a^2-b^2\right ) d}-\frac {a^3 \cos (c+d x)}{b^2 \left (a^2-b^2\right ) d}+\frac {a \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d}-\frac {3 b \tan (c+d x)}{2 \left (a^2-b^2\right ) d}+\frac {b \sin ^2(c+d x) \tan (c+d x)}{2 \left (a^2-b^2\right ) d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {2981, 2670, 14, 2671, 294, 327, 209, 2872, 3102, 2814, 2739, 632, 210} \[ \int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {a \cos (c+d x)}{d \left (a^2-b^2\right )}-\frac {3 b \tan (c+d x)}{2 d \left (a^2-b^2\right )}+\frac {a \sec (c+d x)}{d \left (a^2-b^2\right )}+\frac {b \sin ^2(c+d x) \tan (c+d x)}{2 d \left (a^2-b^2\right )}+\frac {a^2 \sin (c+d x) \cos (c+d x)}{2 b d \left (a^2-b^2\right )}+\frac {3 b x}{2 \left (a^2-b^2\right )}-\frac {a^2 x \left (2 a^2+b^2\right )}{2 b^3 \left (a^2-b^2\right )}+\frac {2 a^5 \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^3 d \left (a^2-b^2\right )^{3/2}}-\frac {a^3 \cos (c+d x)}{b^2 d \left (a^2-b^2\right )} \]

[In]

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

[Out]

(3*b*x)/(2*(a^2 - b^2)) - (a^2*(2*a^2 + b^2)*x)/(2*b^3*(a^2 - b^2)) + (2*a^5*ArcTan[(b + a*Tan[(c + d*x)/2])/S
qrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)*d) + (a*Cos[c + d*x])/((a^2 - b^2)*d) - (a^3*Cos[c + d*x])/(b^2*(a^2 -
b^2)*d) + (a*Sec[c + d*x])/((a^2 - b^2)*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d) - (3*b*Tan[c
+ d*x])/(2*(a^2 - b^2)*d) + (b*Sin[c + d*x]^2*Tan[c + d*x])/(2*(a^2 - b^2)*d)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 210

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

Rule 294

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

Rule 327

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

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2670

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-f^(-1), Subst[Int[(1 - x^2
)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]

Rule 2671

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> With[{ff = FreeFactors[Ta
n[e + f*x], x]}, Dist[b*(ff/f), Subst[Int[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, b*(Tan[e + f*x]/ff
)], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 2814

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

Rule 2872

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Dist[1/
(d*(m + n)), Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d*(m + n) + b^2*(b*c*(m - 2) + a
*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n
- 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
&& NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m]
|| (EqQ[a, 0] && NeQ[c, 0])))

Rule 2981

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_))/((a_) + (b_.)*sin[(e_.) + (f_.
)*(x_)]), x_Symbol] :> Dist[a*(d^2/(a^2 - b^2)), Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 2), x], x] + (-D
ist[b*(d/(a^2 - b^2)), Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x], x] - Dist[a^2*(d^2/(g^2*(a^2 - b^2
))), Int[(g*Cos[e + f*x])^(p + 2)*((d*Sin[e + f*x])^(n - 2)/(a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, d,
e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[p, -1] && GtQ[n, 1]

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {a \int \sin (c+d x) \tan ^2(c+d x) \, dx}{a^2-b^2}-\frac {a^2 \int \frac {\sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx}{a^2-b^2}-\frac {b \int \sin ^2(c+d x) \tan ^2(c+d x) \, dx}{a^2-b^2} \\ & = \frac {a^2 \cos (c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d}-\frac {a^2 \int \frac {a+b \sin (c+d x)-2 a \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{2 b \left (a^2-b^2\right )}-\frac {a \text {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,\cos (c+d x)\right )}{\left (a^2-b^2\right ) d}-\frac {b \text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2-b^2\right ) d} \\ & = -\frac {a^3 \cos (c+d x)}{b^2 \left (a^2-b^2\right ) d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d}+\frac {b \sin ^2(c+d x) \tan (c+d x)}{2 \left (a^2-b^2\right ) d}-\frac {a^2 \int \frac {a b+\left (2 a^2+b^2\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )}-\frac {a \text {Subst}\left (\int \left (-1+\frac {1}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{\left (a^2-b^2\right ) d}-\frac {(3 b) \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 \left (a^2-b^2\right ) d} \\ & = -\frac {a^2 \left (2 a^2+b^2\right ) x}{2 b^3 \left (a^2-b^2\right )}+\frac {a \cos (c+d x)}{\left (a^2-b^2\right ) d}-\frac {a^3 \cos (c+d x)}{b^2 \left (a^2-b^2\right ) d}+\frac {a \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d}-\frac {3 b \tan (c+d x)}{2 \left (a^2-b^2\right ) d}+\frac {b \sin ^2(c+d x) \tan (c+d x)}{2 \left (a^2-b^2\right ) d}+\frac {a^5 \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^3 \left (a^2-b^2\right )}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 \left (a^2-b^2\right ) d} \\ & = \frac {3 b x}{2 \left (a^2-b^2\right )}-\frac {a^2 \left (2 a^2+b^2\right ) x}{2 b^3 \left (a^2-b^2\right )}+\frac {a \cos (c+d x)}{\left (a^2-b^2\right ) d}-\frac {a^3 \cos (c+d x)}{b^2 \left (a^2-b^2\right ) d}+\frac {a \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d}-\frac {3 b \tan (c+d x)}{2 \left (a^2-b^2\right ) d}+\frac {b \sin ^2(c+d x) \tan (c+d x)}{2 \left (a^2-b^2\right ) d}+\frac {\left (2 a^5\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^3 \left (a^2-b^2\right ) d} \\ & = \frac {3 b x}{2 \left (a^2-b^2\right )}-\frac {a^2 \left (2 a^2+b^2\right ) x}{2 b^3 \left (a^2-b^2\right )}+\frac {a \cos (c+d x)}{\left (a^2-b^2\right ) d}-\frac {a^3 \cos (c+d x)}{b^2 \left (a^2-b^2\right ) d}+\frac {a \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d}-\frac {3 b \tan (c+d x)}{2 \left (a^2-b^2\right ) d}+\frac {b \sin ^2(c+d x) \tan (c+d x)}{2 \left (a^2-b^2\right ) d}-\frac {\left (4 a^5\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^3 \left (a^2-b^2\right ) d} \\ & = \frac {3 b x}{2 \left (a^2-b^2\right )}-\frac {a^2 \left (2 a^2+b^2\right ) x}{2 b^3 \left (a^2-b^2\right )}+\frac {2 a^5 \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^3 \left (a^2-b^2\right )^{3/2} d}+\frac {a \cos (c+d x)}{\left (a^2-b^2\right ) d}-\frac {a^3 \cos (c+d x)}{b^2 \left (a^2-b^2\right ) d}+\frac {a \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d}-\frac {3 b \tan (c+d x)}{2 \left (a^2-b^2\right ) d}+\frac {b \sin ^2(c+d x) \tan (c+d x)}{2 \left (a^2-b^2\right ) d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.35 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.82 \[ \int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {-4 a b^3+4 a^4 (c+d x)+2 a^2 b^2 (c+d x)-6 b^4 (c+d x)}{-a^2 b^3+b^5}+\frac {8 a^5 \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^3 \left (a^2-b^2\right )^{3/2}}-\frac {4 a \cos (c+d x)}{b^2}+\frac {4 \sin \left (\frac {1}{2} (c+d x)\right )}{(a+b) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {4 \sin \left (\frac {1}{2} (c+d x)\right )}{(a-b) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {\sin (2 (c+d x))}{b}}{4 d} \]

[In]

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

[Out]

((-4*a*b^3 + 4*a^4*(c + d*x) + 2*a^2*b^2*(c + d*x) - 6*b^4*(c + d*x))/(-(a^2*b^3) + b^5) + (8*a^5*ArcTan[(b +
a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)) - (4*a*Cos[c + d*x])/b^2 + (4*Sin[(c + d*x)/2])/
((a + b)*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (4*Sin[(c + d*x)/2])/((a - b)*(Cos[(c + d*x)/2] + Sin[(c + d
*x)/2])) + Sin[2*(c + d*x)]/b)/(4*d)

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{2}+\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{2}+a b}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (2 a^{2}+3 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{3}}-\frac {64}{\left (64 a +64 b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2 a^{5} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right ) \left (a +b \right ) b^{3} \sqrt {a^{2}-b^{2}}}+\frac {64}{\left (64 a -64 b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\)	\(208\)
default	\(\frac {-\frac {2 \left (\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{2}+\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{2}+a b}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (2 a^{2}+3 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{3}}-\frac {64}{\left (64 a +64 b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2 a^{5} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right ) \left (a +b \right ) b^{3} \sqrt {a^{2}-b^{2}}}+\frac {64}{\left (64 a -64 b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\)	\(208\)
risch	\(-\frac {x \,a^{2}}{b^{3}}-\frac {3 x}{2 b}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 b d}-\frac {a \,{\mathrm e}^{i \left (d x +c \right )}}{2 d \,b^{2}}-\frac {a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 b^{2} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 b d}+\frac {2 i \left (i a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}{d \left (-a^{2}+b^{2}\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {i a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{3}}+\frac {i a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{3}}\)	\(304\)

[In]

int(sec(d*x+c)^2*sin(d*x+c)^5/(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(-2/b^3*((1/2*tan(1/2*d*x+1/2*c)^3*b^2+tan(1/2*d*x+1/2*c)^2*a*b-1/2*tan(1/2*d*x+1/2*c)*b^2+a*b)/(1+tan(1/2
*d*x+1/2*c)^2)^2+1/2*(2*a^2+3*b^2)*arctan(tan(1/2*d*x+1/2*c)))-64/(64*a+64*b)/(tan(1/2*d*x+1/2*c)-1)+2/(a-b)/(
a+b)*a^5/b^3/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))+64/(64*a-64*b)/(tan(1/2*
d*x+1/2*c)+1))

Fricas [A] (verification not implemented)

none

Time = 0.46 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.94 \[ \int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\left [\frac {\sqrt {-a^{2} + b^{2}} a^{5} \cos \left (d x + c\right ) \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) + 2 \, a^{3} b^{3} - 2 \, a b^{5} - {\left (2 \, a^{6} - a^{4} b^{2} - 4 \, a^{2} b^{4} + 3 \, b^{6}\right )} d x \cos \left (d x + c\right ) - 2 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{2} - {\left (2 \, a^{2} b^{4} - 2 \, b^{6} - {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} d \cos \left (d x + c\right )}, -\frac {2 \, \sqrt {a^{2} - b^{2}} a^{5} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) \cos \left (d x + c\right ) - 2 \, a^{3} b^{3} + 2 \, a b^{5} + {\left (2 \, a^{6} - a^{4} b^{2} - 4 \, a^{2} b^{4} + 3 \, b^{6}\right )} d x \cos \left (d x + c\right ) + 2 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, a^{2} b^{4} - 2 \, b^{6} - {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} d \cos \left (d x + c\right )}\right ] \]

[In]

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

[Out]

[1/2*(sqrt(-a^2 + b^2)*a^5*cos(d*x + c)*log(-((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 -
2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) -
a^2 - b^2)) + 2*a^3*b^3 - 2*a*b^5 - (2*a^6 - a^4*b^2 - 4*a^2*b^4 + 3*b^6)*d*x*cos(d*x + c) - 2*(a^5*b - 2*a^3*
b^3 + a*b^5)*cos(d*x + c)^2 - (2*a^2*b^4 - 2*b^6 - (a^4*b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^2)*sin(d*x + c))/(
(a^4*b^3 - 2*a^2*b^5 + b^7)*d*cos(d*x + c)), -1/2*(2*sqrt(a^2 - b^2)*a^5*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^
2 - b^2)*cos(d*x + c)))*cos(d*x + c) - 2*a^3*b^3 + 2*a*b^5 + (2*a^6 - a^4*b^2 - 4*a^2*b^4 + 3*b^6)*d*x*cos(d*x
+ c) + 2*(a^5*b - 2*a^3*b^3 + a*b^5)*cos(d*x + c)^2 + (2*a^2*b^4 - 2*b^6 - (a^4*b^2 - 2*a^2*b^4 + b^6)*cos(d*
x + c)^2)*sin(d*x + c))/((a^4*b^3 - 2*a^2*b^5 + b^7)*d*cos(d*x + c))]

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]

[In]

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

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.78 \[ \int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {4 \, {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} a^{5}}{{\left (a^{2} b^{3} - b^{5}\right )} \sqrt {a^{2} - b^{2}}} + \frac {4 \, {\left (b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )}}{{\left (a^{2} - b^{2}\right )} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}} - \frac {{\left (2 \, a^{2} + 3 \, b^{2}\right )} {\left (d x + c\right )}}{b^{3}} - \frac {2 \, {\left (b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} b^{2}}}{2 \, d} \]

[In]

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

[Out]

1/2*(4*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))*a^5/((
a^2*b^3 - b^5)*sqrt(a^2 - b^2)) + 4*(b*tan(1/2*d*x + 1/2*c) - a)/((a^2 - b^2)*(tan(1/2*d*x + 1/2*c)^2 - 1)) -
(2*a^2 + 3*b^2)*(d*x + c)/b^3 - 2*(b*tan(1/2*d*x + 1/2*c)^3 + 2*a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2
*c) + 2*a)/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*b^2))/d

Mupad [B] (verification not implemented)

Time = 17.09 (sec) , antiderivative size = 2098, normalized size of antiderivative = 7.83 \[ \int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(4*a^5*cos(c + d*x) + (5*a^5)/2 + (3*a^5*cos(2*c + 2*d*x))/2)/(d*cos(c + d*x)*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b
^2)) - (2*a^8*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/(b^3*d*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) - (b*((
11*a^4*sin(c + d*x))/8 + (3*a^4*sin(3*c + 3*d*x))/8 - 3*a^4*cos(c + d*x)*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*
x)/2))))/(d*cos(c + d*x)*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) + (b^4*((3*a)/2 + 2*a*cos(c + d*x) + (a*cos(2*c
+ 2*d*x))/2))/(d*cos(c + d*x)*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) - (b^5*((9*sin(c + d*x))/8 + sin(3*c + 3*d*
x)/8 - 3*cos(c + d*x)*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))))/(d*cos(c + d*x)*(a^2 - b^2)*(a^4 + b^4 - 2
*a^2*b^2)) - (a^7*cos(c + d*x) + a^7/2 + (a^7*cos(2*c + 2*d*x))/2)/(b^2*d*cos(c + d*x)*(a^2 - b^2)*(a^4 + b^4
- 2*a^2*b^2)) - (b^2*(5*a^3*cos(c + d*x) + (7*a^3)/2 + (3*a^3*cos(2*c + 2*d*x))/2))/(d*cos(c + d*x)*(a^2 - b^2
)*(a^4 + b^4 - 2*a^2*b^2)) + ((a^6*sin(c + d*x))/8 + (a^6*sin(3*c + 3*d*x))/8 + 3*a^6*cos(c + d*x)*atan(sin(c/
2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/(b*d*cos(c + d*x)*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) + (b^3*((19*a^2*sin(c
+ d*x))/8 + (3*a^2*sin(3*c + 3*d*x))/8 - 7*a^2*cos(c + d*x)*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))))/(d*
cos(c + d*x)*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) + (a^5*atan((a^12*sin(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4
+ 3*a^4*b^2)^(3/2)*8i + a^18*sin(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2)*8i - b^18*sin(c/2 +
(d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2)*18i + a^3*b^15*cos(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 +
3*a^4*b^2)^(1/2)*42i - a^5*b^13*cos(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2)*67i + a^7*b^11*co
s(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2)*24i + a^9*b^9*cos(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2
*b^4 + 3*a^4*b^2)^(1/2)*45i - a^11*b^7*cos(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2)*46i + a^13
*b^5*cos(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2)*3i + a^15*b^3*cos(c/2 + (d*x)/2)*(b^6 - a^6
- 3*a^2*b^4 + 3*a^4*b^2)^(1/2)*12i - a^10*b^2*sin(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(3/2)*12i
+ a^2*b^16*sin(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2)*93i - a^4*b^14*sin(c/2 + (d*x)/2)*(b^
6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2)*176i + a^6*b^12*sin(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^
(1/2)*115i + a^8*b^10*sin(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2)*66i - a^10*b^8*sin(c/2 + (d
*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2)*133i + a^12*b^6*sin(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3
*a^4*b^2)^(1/2)*36i + a^14*b^4*sin(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2)*45i - a^16*b^2*sin
(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2)*36i - a^11*b*cos(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b
^4 + 3*a^4*b^2)^(3/2)*4i - a*b^17*cos(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2)*9i - a^17*b*cos
(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2)*4i)/(18*b^21*sin(c/2 + (d*x)/2) + 9*a*b^20*cos(c/2 +
(d*x)/2) - 60*a^3*b^18*cos(c/2 + (d*x)/2) + 160*a^5*b^16*cos(c/2 + (d*x)/2) - 200*a^7*b^14*cos(c/2 + (d*x)/2)
+ 70*a^9*b^12*cos(c/2 + (d*x)/2) + 116*a^11*b^10*cos(c/2 + (d*x)/2) - 160*a^13*b^8*cos(c/2 + (d*x)/2) + 80*a^
15*b^6*cos(c/2 + (d*x)/2) - 15*a^17*b^4*cos(c/2 + (d*x)/2) - 120*a^2*b^19*sin(c/2 + (d*x)/2) + 320*a^4*b^17*si
n(c/2 + (d*x)/2) - 400*a^6*b^15*sin(c/2 + (d*x)/2) + 140*a^8*b^13*sin(c/2 + (d*x)/2) + 232*a^10*b^11*sin(c/2 +
(d*x)/2) - 320*a^12*b^9*sin(c/2 + (d*x)/2) + 160*a^14*b^7*sin(c/2 + (d*x)/2) - 30*a^16*b^5*sin(c/2 + (d*x)/2)
))*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2)*2i)/(b^3*d*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2))