Integrand size = 19, antiderivative size = 109 \[ \int \sin ^m(c+d x) (a+b \tan (c+d x)) \, dx=\frac {a \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(c+d x)\right ) \sin ^{1+m}(c+d x)}{d (1+m) \sqrt {\cos ^2(c+d x)}}+\frac {b \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},\sin ^2(c+d x)\right ) \sin ^{2+m}(c+d x)}{d (2+m)} \] Output:
a*cos(d*x+c)*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],sin(d*x+c)^2)*sin(d*x+ c)^(1+m)/d/(1+m)/(cos(d*x+c)^2)^(1/2)+b*hypergeom([1, 1+1/2*m],[2+1/2*m],s in(d*x+c)^2)*sin(d*x+c)^(2+m)/d/(2+m)
Time = 0.21 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00 \[ \int \sin ^m(c+d x) (a+b \tan (c+d x)) \, dx=\frac {a \sqrt {\cos ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(c+d x)\right ) \sec (c+d x) \sin ^{1+m}(c+d x)}{d (1+m)}+\frac {b \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},\sin ^2(c+d x)\right ) \sin ^{2+m}(c+d x)}{d (2+m)} \] Input:
Integrate[Sin[c + d*x]^m*(a + b*Tan[c + d*x]),x]
Output:
(a*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*Sin[c + d*x]^(1 + m))/(d*(1 + m)) + (b*Hypergeomet ric2F1[1, (2 + m)/2, (4 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + m))/(d*( 2 + m))
Time = 0.36 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3042, 4901, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sin ^m(c+d x) (a+b \tan (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (c+d x)^m (a+b \tan (c+d x))dx\) |
\(\Big \downarrow \) 4901 |
\(\displaystyle \int \left (a \sin ^m(c+d x)+b \sec (c+d x) \sin ^{m+1}(c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a \cos (c+d x) \sin ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(c+d x)\right )}{d (m+1) \sqrt {\cos ^2(c+d x)}}+\frac {b \sin ^{m+2}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},\sin ^2(c+d x)\right )}{d (m+2)}\) |
Input:
Int[Sin[c + d*x]^m*(a + b*Tan[c + d*x]),x]
Output:
(a*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^ 2]*Sin[c + d*x]^(1 + m))/(d*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (b*Hypergeomet ric2F1[1, (2 + m)/2, (4 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + m))/(d*( 2 + m))
Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /; !InertTrigFreeQ[u]
\[\int \sin \left (d x +c \right )^{m} \left (a +b \tan \left (d x +c \right )\right )d x\]
Input:
int(sin(d*x+c)^m*(a+b*tan(d*x+c)),x)
Output:
int(sin(d*x+c)^m*(a+b*tan(d*x+c)),x)
\[ \int \sin ^m(c+d x) (a+b \tan (c+d x)) \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )^{m} \,d x } \] Input:
integrate(sin(d*x+c)^m*(a+b*tan(d*x+c)),x, algorithm="fricas")
Output:
integral((b*tan(d*x + c) + a)*sin(d*x + c)^m, x)
\[ \int \sin ^m(c+d x) (a+b \tan (c+d x)) \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right ) \sin ^{m}{\left (c + d x \right )}\, dx \] Input:
integrate(sin(d*x+c)**m*(a+b*tan(d*x+c)),x)
Output:
Integral((a + b*tan(c + d*x))*sin(c + d*x)**m, x)
\[ \int \sin ^m(c+d x) (a+b \tan (c+d x)) \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )^{m} \,d x } \] Input:
integrate(sin(d*x+c)^m*(a+b*tan(d*x+c)),x, algorithm="maxima")
Output:
integrate((b*tan(d*x + c) + a)*sin(d*x + c)^m, x)
\[ \int \sin ^m(c+d x) (a+b \tan (c+d x)) \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )^{m} \,d x } \] Input:
integrate(sin(d*x+c)^m*(a+b*tan(d*x+c)),x, algorithm="giac")
Output:
integrate((b*tan(d*x + c) + a)*sin(d*x + c)^m, x)
Timed out. \[ \int \sin ^m(c+d x) (a+b \tan (c+d x)) \, dx=\int {\sin \left (c+d\,x\right )}^m\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right ) \,d x \] Input:
int(sin(c + d*x)^m*(a + b*tan(c + d*x)),x)
Output:
int(sin(c + d*x)^m*(a + b*tan(c + d*x)), x)
\[ \int \sin ^m(c+d x) (a+b \tan (c+d x)) \, dx=\left (\int \sin \left (d x +c \right )^{m}d x \right ) a +\left (\int \sin \left (d x +c \right )^{m} \tan \left (d x +c \right )d x \right ) b \] Input:
int(sin(d*x+c)^m*(a+b*tan(d*x+c)),x)
Output:
int(sin(c + d*x)**m,x)*a + int(sin(c + d*x)**m*tan(c + d*x),x)*b