\(\int \sin ^2(c+d x) (a+b \tan (c+d x))^n \, dx\) [85]

Optimal result

Integrand size = 21, antiderivative size = 276 \[ \int \sin ^2(c+d x) (a+b \tan (c+d x))^n \, dx=-\frac {\left (a b^2 n+\sqrt {-b^2} \left (a^2+b^2 (1+n)\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a-\sqrt {-b^2}}\right ) (a+b \tan (c+d x))^{1+n}}{4 b \left (a^2+b^2\right ) \left (a-\sqrt {-b^2}\right ) d (1+n)}-\frac {\left (a b^2 n-\sqrt {-b^2} \left (a^2+b^2 (1+n)\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a+\sqrt {-b^2}}\right ) (a+b \tan (c+d x))^{1+n}}{4 b \left (a^2+b^2\right ) \left (a+\sqrt {-b^2}\right ) d (1+n)}-\frac {\cos ^2(c+d x) (b+a \tan (c+d x)) (a+b \tan (c+d x))^{1+n}}{2 \left (a^2+b^2\right ) d} \]

[Out]

-1/4*hypergeom([1, 1+n],[2+n],(a+b*tan(d*x+c))/(a+(-b^2)^(1/2)))*(a*b^2*n-(a^2+b^2*(1+n))*(-b^2)^(1/2))*(a+b*t
an(d*x+c))^(1+n)/b/(a^2+b^2)/d/(1+n)/(a+(-b^2)^(1/2))-1/4*hypergeom([1, 1+n],[2+n],(a+b*tan(d*x+c))/(a-(-b^2)^
(1/2)))*(a*b^2*n+(a^2+b^2*(1+n))*(-b^2)^(1/2))*(a+b*tan(d*x+c))^(1+n)/b/(a^2+b^2)/d/(1+n)/(a-(-b^2)^(1/2))-1/2
*cos(d*x+c)^2*(b+a*tan(d*x+c))*(a+b*tan(d*x+c))^(1+n)/(a^2+b^2)/d

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3597, 1663, 845, 70} \[ \int \sin ^2(c+d x) (a+b \tan (c+d x))^n \, dx=-\frac {\left (\sqrt {-b^2} \left (a^2+b^2 (n+1)\right )+a b^2 n\right ) (a+b \tan (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {a+b \tan (c+d x)}{a-\sqrt {-b^2}}\right )}{4 b d (n+1) \left (a^2+b^2\right ) \left (a-\sqrt {-b^2}\right )}-\frac {\left (a b^2 n-\sqrt {-b^2} \left (a^2+b^2 (n+1)\right )\right ) (a+b \tan (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {a+b \tan (c+d x)}{a+\sqrt {-b^2}}\right )}{4 b d (n+1) \left (a^2+b^2\right ) \left (a+\sqrt {-b^2}\right )}-\frac {\cos ^2(c+d x) (a \tan (c+d x)+b) (a+b \tan (c+d x))^{n+1}}{2 d \left (a^2+b^2\right )} \]

[In]

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

[Out]

-1/4*((a*b^2*n + Sqrt[-b^2]*(a^2 + b^2*(1 + n)))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a -
Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(b*(a^2 + b^2)*(a - Sqrt[-b^2])*d*(1 + n)) - ((a*b^2*n - Sqrt[-b^2]
*(a^2 + b^2*(1 + n)))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])]*(a + b*Tan[c +
d*x])^(1 + n))/(4*b*(a^2 + b^2)*(a + Sqrt[-b^2])*d*(1 + n)) - (Cos[c + d*x]^2*(b + a*Tan[c + d*x])*(a + b*Tan
[c + d*x])^(1 + n))/(2*(a^2 + b^2)*d)

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/(b^(
n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rule 845

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x)^m, (f + g*x)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !Ration
alQ[m]

Rule 1663

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + c
*x^2, x], x, 1]}, Simp[(-(d + e*x)^(m + 1))*(a + c*x^2)^(p + 1)*((a*(e*f - d*g) + (c*d*f + a*e*g)*x)/(2*a*(p +
1)*(c*d^2 + a*e^2))), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandTo
Sum[2*a*(p + 1)*(c*d^2 + a*e^2)*Q + c*d^2*f*(2*p + 3) - a*e*(d*g*m - e*f*(m + 2*p + 3)) + e*(c*d*f + a*e*g)*(m
+ 2*p + 4)*x, x], x], x]] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1]
&&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rule 3597

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

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

Mathematica [A] (verified)

Time = 1.33 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.98 \[ \int \sin ^2(c+d x) (a+b \tan (c+d x))^n \, dx=\frac {\left (\left (a^3 \sqrt {-b^2}+a^2 b^2 (-1+n)-b^4 (1+n)-a \left (-b^2\right )^{3/2} (1+2 n)\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a-\sqrt {-b^2}}\right )-\left (a^3 \sqrt {-b^2}-a^2 b^2 (-1+n)+b^4 (1+n)-a \left (-b^2\right )^{3/2} (1+2 n)\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a+\sqrt {-b^2}}\right )+2 b \left (a^2+b^2\right ) (1+n) \cos (c+d x) (b \cos (c+d x)+a \sin (c+d x))\right ) (a+b \tan (c+d x))^{1+n}}{4 b \left (a^2+b^2\right ) \left (-a+\sqrt {-b^2}\right ) \left (a+\sqrt {-b^2}\right ) d (1+n)} \]

[In]

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

[Out]

(((a^3*Sqrt[-b^2] + a^2*b^2*(-1 + n) - b^4*(1 + n) - a*(-b^2)^(3/2)*(1 + 2*n))*Hypergeometric2F1[1, 1 + n, 2 +
n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])] - (a^3*Sqrt[-b^2] - a^2*b^2*(-1 + n) + b^4*(1 + n) - a*(-b^2)^(3/2)
*(1 + 2*n))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])] + 2*b*(a^2 + b^2)*(1 + n
)*Cos[c + d*x]*(b*Cos[c + d*x] + a*Sin[c + d*x]))*(a + b*Tan[c + d*x])^(1 + n))/(4*b*(a^2 + b^2)*(-a + Sqrt[-b
^2])*(a + Sqrt[-b^2])*d*(1 + n))

Maple [F]

\[\int \left (\sin ^{2}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{n}d x\]

[In]

int(sin(d*x+c)^2*(a+b*tan(d*x+c))^n,x)

[Out]

int(sin(d*x+c)^2*(a+b*tan(d*x+c))^n,x)

Fricas [F]

\[ \int \sin ^2(c+d x) (a+b \tan (c+d x))^n \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(sin(d*x+c)^2*(a+b*tan(d*x+c))^n,x, algorithm="fricas")

[Out]

integral(-(cos(d*x + c)^2 - 1)*(b*tan(d*x + c) + a)^n, x)

Sympy [F]

\[ \int \sin ^2(c+d x) (a+b \tan (c+d x))^n \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{n} \sin ^{2}{\left (c + d x \right )}\, dx \]

[In]

integrate(sin(d*x+c)**2*(a+b*tan(d*x+c))**n,x)

[Out]

Integral((a + b*tan(c + d*x))**n*sin(c + d*x)**2, x)

Maxima [F]

\[ \int \sin ^2(c+d x) (a+b \tan (c+d x))^n \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(sin(d*x+c)^2*(a+b*tan(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((b*tan(d*x + c) + a)^n*sin(d*x + c)^2, x)

Giac [F]

\[ \int \sin ^2(c+d x) (a+b \tan (c+d x))^n \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(sin(d*x+c)^2*(a+b*tan(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((b*tan(d*x + c) + a)^n*sin(d*x + c)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \sin ^2(c+d x) (a+b \tan (c+d x))^n \, dx=\int {\sin \left (c+d\,x\right )}^2\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n \,d x \]

[In]

int(sin(c + d*x)^2*(a + b*tan(c + d*x))^n,x)

[Out]

int(sin(c + d*x)^2*(a + b*tan(c + d*x))^n, x)