\(\int \frac {\sin ^m(c+d x)}{a+b \tan (c+d x)} \, dx\) [82]

Optimal result

Integrand size = 21, antiderivative size = 765 \[ \int \frac {\sin ^m(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {2^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},1+m,\frac {3+m}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \tan \left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m}{a d (1+m)}+\frac {2^{1+m} b \operatorname {AppellF1}\left (\frac {2+m}{2},1+m,1,\frac {4+m}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right ),\frac {a^2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}{\left (b-\sqrt {a^2+b^2}\right )^2}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m}{\sqrt {a^2+b^2} \left (b-\sqrt {a^2+b^2}\right ) d (2+m)}-\frac {2^{1+m} b \operatorname {AppellF1}\left (\frac {2+m}{2},1+m,1,\frac {4+m}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right ),\frac {a^2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}{\left (b+\sqrt {a^2+b^2}\right )^2}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m}{\sqrt {a^2+b^2} \left (b+\sqrt {a^2+b^2}\right ) d (2+m)}+\frac {2^{1+m} a b \operatorname {AppellF1}\left (\frac {3+m}{2},1+m,1,\frac {5+m}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right ),\frac {a^2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}{\left (b-\sqrt {a^2+b^2}\right )^2}\right ) \tan ^3\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m}{\sqrt {a^2+b^2} \left (b-\sqrt {a^2+b^2}\right )^2 d (3+m)}-\frac {2^{1+m} a b \operatorname {AppellF1}\left (\frac {3+m}{2},1+m,1,\frac {5+m}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right ),\frac {a^2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}{\left (b+\sqrt {a^2+b^2}\right )^2}\right ) \tan ^3\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m}{\sqrt {a^2+b^2} \left (b+\sqrt {a^2+b^2}\right )^2 d (3+m)} \]

[Out]

2^(1+m)*hypergeom([1+m, 1/2+1/2*m],[3/2+1/2*m],-tan(1/2*d*x+1/2*c)^2)*tan(1/2*d*x+1/2*c)*(tan(1/2*d*x+1/2*c)/(
1+tan(1/2*d*x+1/2*c)^2))^m*(1+tan(1/2*d*x+1/2*c)^2)^m/a/d/(1+m)+2^(1+m)*b*AppellF1(1+1/2*m,1,1+m,2+1/2*m,a^2*t
an(1/2*d*x+1/2*c)^2/(b-(a^2+b^2)^(1/2))^2,-tan(1/2*d*x+1/2*c)^2)*tan(1/2*d*x+1/2*c)^2*(tan(1/2*d*x+1/2*c)/(1+t
an(1/2*d*x+1/2*c)^2))^m*(1+tan(1/2*d*x+1/2*c)^2)^m/d/(2+m)/(b-(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2)-2^(1+m)*b*Appel
lF1(1+1/2*m,1,1+m,2+1/2*m,a^2*tan(1/2*d*x+1/2*c)^2/(b+(a^2+b^2)^(1/2))^2,-tan(1/2*d*x+1/2*c)^2)*tan(1/2*d*x+1/
2*c)^2*(tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2))^m*(1+tan(1/2*d*x+1/2*c)^2)^m/d/(2+m)/(a^2+b^2)^(1/2)/(b+(
a^2+b^2)^(1/2))+2^(1+m)*a*b*AppellF1(3/2+1/2*m,1,1+m,5/2+1/2*m,a^2*tan(1/2*d*x+1/2*c)^2/(b-(a^2+b^2)^(1/2))^2,
-tan(1/2*d*x+1/2*c)^2)*tan(1/2*d*x+1/2*c)^3*(tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2))^m*(1+tan(1/2*d*x+1/2
*c)^2)^m/d/(3+m)/(b-(a^2+b^2)^(1/2))^2/(a^2+b^2)^(1/2)-2^(1+m)*a*b*AppellF1(3/2+1/2*m,1,1+m,5/2+1/2*m,a^2*tan(
1/2*d*x+1/2*c)^2/(b+(a^2+b^2)^(1/2))^2,-tan(1/2*d*x+1/2*c)^2)*tan(1/2*d*x+1/2*c)^3*(tan(1/2*d*x+1/2*c)/(1+tan(
1/2*d*x+1/2*c)^2))^m*(1+tan(1/2*d*x+1/2*c)^2)^m/d/(3+m)/(a^2+b^2)^(1/2)/(b+(a^2+b^2)^(1/2))^2

Rubi [A] (verified)

Time = 6.24 (sec) , antiderivative size = 765, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {12, 6851, 6860, 371, 973, 524} \[ \int \frac {\sin ^m(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {b 2^{m+1} \tan ^2\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\tan ^2\left (\frac {1}{2} (c+d x)\right )+1}\right )^m \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )+1\right )^m \operatorname {AppellF1}\left (\frac {m+2}{2},m+1,1,\frac {m+4}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right ),\frac {a^2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}{\left (b-\sqrt {a^2+b^2}\right )^2}\right )}{d (m+2) \sqrt {a^2+b^2} \left (b-\sqrt {a^2+b^2}\right )}-\frac {b 2^{m+1} \tan ^2\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\tan ^2\left (\frac {1}{2} (c+d x)\right )+1}\right )^m \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )+1\right )^m \operatorname {AppellF1}\left (\frac {m+2}{2},m+1,1,\frac {m+4}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right ),\frac {a^2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}{\left (b+\sqrt {a^2+b^2}\right )^2}\right )}{d (m+2) \sqrt {a^2+b^2} \left (\sqrt {a^2+b^2}+b\right )}+\frac {a b 2^{m+1} \tan ^3\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\tan ^2\left (\frac {1}{2} (c+d x)\right )+1}\right )^m \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )+1\right )^m \operatorname {AppellF1}\left (\frac {m+3}{2},m+1,1,\frac {m+5}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right ),\frac {a^2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}{\left (b-\sqrt {a^2+b^2}\right )^2}\right )}{d (m+3) \sqrt {a^2+b^2} \left (b-\sqrt {a^2+b^2}\right )^2}-\frac {a b 2^{m+1} \tan ^3\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\tan ^2\left (\frac {1}{2} (c+d x)\right )+1}\right )^m \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )+1\right )^m \operatorname {AppellF1}\left (\frac {m+3}{2},m+1,1,\frac {m+5}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right ),\frac {a^2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}{\left (b+\sqrt {a^2+b^2}\right )^2}\right )}{d (m+3) \sqrt {a^2+b^2} \left (\sqrt {a^2+b^2}+b\right )^2}+\frac {2^{m+1} \tan \left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\tan ^2\left (\frac {1}{2} (c+d x)\right )+1}\right )^m \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )+1\right )^m \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},m+1,\frac {m+3}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}{a d (m+1)} \]

[In]

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

[Out]

(2^(1 + m)*Hypergeometric2F1[(1 + m)/2, 1 + m, (3 + m)/2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]*(Tan[(c + d*x)
/2]/(1 + Tan[(c + d*x)/2]^2))^m*(1 + Tan[(c + d*x)/2]^2)^m)/(a*d*(1 + m)) + (2^(1 + m)*b*AppellF1[(2 + m)/2, 1
+ m, 1, (4 + m)/2, -Tan[(c + d*x)/2]^2, (a^2*Tan[(c + d*x)/2]^2)/(b - Sqrt[a^2 + b^2])^2]*Tan[(c + d*x)/2]^2*
(Tan[(c + d*x)/2]/(1 + Tan[(c + d*x)/2]^2))^m*(1 + Tan[(c + d*x)/2]^2)^m)/(Sqrt[a^2 + b^2]*(b - Sqrt[a^2 + b^2
])*d*(2 + m)) - (2^(1 + m)*b*AppellF1[(2 + m)/2, 1 + m, 1, (4 + m)/2, -Tan[(c + d*x)/2]^2, (a^2*Tan[(c + d*x)/
2]^2)/(b + Sqrt[a^2 + b^2])^2]*Tan[(c + d*x)/2]^2*(Tan[(c + d*x)/2]/(1 + Tan[(c + d*x)/2]^2))^m*(1 + Tan[(c +
d*x)/2]^2)^m)/(Sqrt[a^2 + b^2]*(b + Sqrt[a^2 + b^2])*d*(2 + m)) + (2^(1 + m)*a*b*AppellF1[(3 + m)/2, 1 + m, 1,
(5 + m)/2, -Tan[(c + d*x)/2]^2, (a^2*Tan[(c + d*x)/2]^2)/(b - Sqrt[a^2 + b^2])^2]*Tan[(c + d*x)/2]^3*(Tan[(c
+ d*x)/2]/(1 + Tan[(c + d*x)/2]^2))^m*(1 + Tan[(c + d*x)/2]^2)^m)/(Sqrt[a^2 + b^2]*(b - Sqrt[a^2 + b^2])^2*d*(
3 + m)) - (2^(1 + m)*a*b*AppellF1[(3 + m)/2, 1 + m, 1, (5 + m)/2, -Tan[(c + d*x)/2]^2, (a^2*Tan[(c + d*x)/2]^2
)/(b + Sqrt[a^2 + b^2])^2]*Tan[(c + d*x)/2]^3*(Tan[(c + d*x)/2]/(1 + Tan[(c + d*x)/2]^2))^m*(1 + Tan[(c + d*x)
/2]^2)^m)/(Sqrt[a^2 + b^2]*(b + Sqrt[a^2 + b^2])^2*d*(3 + m))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 973

Int[(((g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[d*((g*x)^n/x^n), In
t[(x^n*(a + c*x^2)^p)/(d^2 - e^2*x^2), x], x] - Dist[e*((g*x)^n/x^n), Int[(x^(n + 1)*(a + c*x^2)^p)/(d^2 - e^2
*x^2), x], x] /; FreeQ[{a, c, d, e, g, n, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] &&  !IntegersQ[n, 2
*p]

Rule 6851

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m*w^n)^FracPart[p]/(v^(m*Fr
acPart[p])*w^(n*FracPart[p]))), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {2^m \left (1-x^2\right ) \left (\frac {x}{1+x^2}\right )^{1+m}}{x \left (a+2 b x-a x^2\right )} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{d} \\ & = \frac {2^{1+m} \text {Subst}\left (\int \frac {\left (1-x^2\right ) \left (\frac {x}{1+x^2}\right )^{1+m}}{x \left (a+2 b x-a x^2\right )} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{d} \\ & = \frac {\left (2^{1+m} \tan ^{-m}\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m\right ) \text {Subst}\left (\int \frac {x^m \left (1-x^2\right ) \left (1+x^2\right )^{-1-m}}{a+2 b x-a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{d} \\ & = \frac {\left (2^{1+m} \tan ^{-m}\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m\right ) \text {Subst}\left (\int \left (\frac {x^m \left (1+x^2\right )^{-1-m}}{a}-\frac {2 b x^{1+m} \left (1+x^2\right )^{-1-m}}{a \left (a+2 b x-a x^2\right )}\right ) \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{d} \\ & = \frac {\left (2^{1+m} \tan ^{-m}\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m\right ) \text {Subst}\left (\int x^m \left (1+x^2\right )^{-1-m} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}-\frac {\left (2^{2+m} b \tan ^{-m}\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m\right ) \text {Subst}\left (\int \frac {x^{1+m} \left (1+x^2\right )^{-1-m}}{a+2 b x-a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d} \\ & = \frac {2^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},1+m,\frac {3+m}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \tan \left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m}{a d (1+m)}-\frac {\left (2^{2+m} b \tan ^{-m}\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m\right ) \text {Subst}\left (\int \left (-\frac {a x^{1+m} \left (1+x^2\right )^{-1-m}}{\sqrt {a^2+b^2} \left (2 b-2 \sqrt {a^2+b^2}-2 a x\right )}+\frac {a x^{1+m} \left (1+x^2\right )^{-1-m}}{\sqrt {a^2+b^2} \left (2 b+2 \sqrt {a^2+b^2}-2 a x\right )}\right ) \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d} \\ & = \frac {2^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},1+m,\frac {3+m}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \tan \left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m}{a d (1+m)}+\frac {\left (2^{2+m} b \tan ^{-m}\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m\right ) \text {Subst}\left (\int \frac {x^{1+m} \left (1+x^2\right )^{-1-m}}{2 b-2 \sqrt {a^2+b^2}-2 a x} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2+b^2} d}-\frac {\left (2^{2+m} b \tan ^{-m}\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m\right ) \text {Subst}\left (\int \frac {x^{1+m} \left (1+x^2\right )^{-1-m}}{2 b+2 \sqrt {a^2+b^2}-2 a x} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2+b^2} d} \\ & = \frac {2^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},1+m,\frac {3+m}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \tan \left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m}{a d (1+m)}+\frac {\left (2^{3+m} a b \tan ^{-m}\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m\right ) \text {Subst}\left (\int \frac {x^{2+m} \left (1+x^2\right )^{-1-m}}{\left (2 b-2 \sqrt {a^2+b^2}\right )^2-4 a^2 x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2+b^2} d}-\frac {\left (2^{3+m} a b \tan ^{-m}\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m\right ) \text {Subst}\left (\int \frac {x^{2+m} \left (1+x^2\right )^{-1-m}}{\left (2 b+2 \sqrt {a^2+b^2}\right )^2-4 a^2 x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2+b^2} d}+\frac {\left (2^{3+m} b \left (b-\sqrt {a^2+b^2}\right ) \tan ^{-m}\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m\right ) \text {Subst}\left (\int \frac {x^{1+m} \left (1+x^2\right )^{-1-m}}{\left (2 b-2 \sqrt {a^2+b^2}\right )^2-4 a^2 x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2+b^2} d}-\frac {\left (2^{3+m} b \left (b+\sqrt {a^2+b^2}\right ) \tan ^{-m}\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m\right ) \text {Subst}\left (\int \frac {x^{1+m} \left (1+x^2\right )^{-1-m}}{\left (2 b+2 \sqrt {a^2+b^2}\right )^2-4 a^2 x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2+b^2} d} \\ & = \frac {2^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},1+m,\frac {3+m}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \tan \left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m}{a d (1+m)}+\frac {2^{1+m} b \operatorname {AppellF1}\left (\frac {2+m}{2},1+m,1,\frac {4+m}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right ),\frac {a^2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}{\left (b-\sqrt {a^2+b^2}\right )^2}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m}{\sqrt {a^2+b^2} \left (b-\sqrt {a^2+b^2}\right ) d (2+m)}-\frac {2^{1+m} b \operatorname {AppellF1}\left (\frac {2+m}{2},1+m,1,\frac {4+m}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right ),\frac {a^2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}{\left (b+\sqrt {a^2+b^2}\right )^2}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m}{\sqrt {a^2+b^2} \left (b+\sqrt {a^2+b^2}\right ) d (2+m)}+\frac {2^{1+m} a b \operatorname {AppellF1}\left (\frac {3+m}{2},1+m,1,\frac {5+m}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right ),\frac {a^2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}{\left (b-\sqrt {a^2+b^2}\right )^2}\right ) \tan ^3\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m}{\sqrt {a^2+b^2} \left (b-\sqrt {a^2+b^2}\right )^2 d (3+m)}-\frac {2^{1+m} a b \operatorname {AppellF1}\left (\frac {3+m}{2},1+m,1,\frac {5+m}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right ),\frac {a^2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}{\left (b+\sqrt {a^2+b^2}\right )^2}\right ) \tan ^3\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^m}{\sqrt {a^2+b^2} \left (b+\sqrt {a^2+b^2}\right )^2 d (3+m)} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\sin ^{m}\left (d x +c \right )}{a +b \tan \left (d x +c \right )}d x\]

[In]

int(sin(d*x+c)^m/(a+b*tan(d*x+c)),x)

[Out]

int(sin(d*x+c)^m/(a+b*tan(d*x+c)),x)

Fricas [F]

\[ \int \frac {\sin ^m(c+d x)}{a+b \tan (c+d x)} \, dx=\int { \frac {\sin \left (d x + c\right )^{m}}{b \tan \left (d x + c\right ) + a} \,d x } \]

[In]

integrate(sin(d*x+c)^m/(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

integral(sin(d*x + c)^m/(b*tan(d*x + c) + a), x)

Sympy [F]

\[ \int \frac {\sin ^m(c+d x)}{a+b \tan (c+d x)} \, dx=\int \frac {\sin ^{m}{\left (c + d x \right )}}{a + b \tan {\left (c + d x \right )}}\, dx \]

[In]

integrate(sin(d*x+c)**m/(a+b*tan(d*x+c)),x)

[Out]

Integral(sin(c + d*x)**m/(a + b*tan(c + d*x)), x)

Maxima [F]

\[ \int \frac {\sin ^m(c+d x)}{a+b \tan (c+d x)} \, dx=\int { \frac {\sin \left (d x + c\right )^{m}}{b \tan \left (d x + c\right ) + a} \,d x } \]

[In]

integrate(sin(d*x+c)^m/(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

integrate(sin(d*x + c)^m/(b*tan(d*x + c) + a), x)

Giac [F]

\[ \int \frac {\sin ^m(c+d x)}{a+b \tan (c+d x)} \, dx=\int { \frac {\sin \left (d x + c\right )^{m}}{b \tan \left (d x + c\right ) + a} \,d x } \]

[In]

integrate(sin(d*x+c)^m/(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

integrate(sin(d*x + c)^m/(b*tan(d*x + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin ^m(c+d x)}{a+b \tan (c+d x)} \, dx=\int \frac {{\sin \left (c+d\,x\right )}^m}{a+b\,\mathrm {tan}\left (c+d\,x\right )} \,d x \]

[In]

int(sin(c + d*x)^m/(a + b*tan(c + d*x)),x)

[Out]

int(sin(c + d*x)^m/(a + b*tan(c + d*x)), x)