Integrand size = 40, antiderivative size = 106 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx=-\frac {2^{\frac {9}{4}+m} (g \cos (e+f x))^{7/2} \operatorname {Hypergeometric2F1}\left (\frac {7}{4},-\frac {1}{4}-m,\frac {11}{4},\frac {1}{2} (1-\sin (e+f x))\right ) \sec (e+f x) (1+\sin (e+f x))^{-\frac {5}{4}-m} (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)}}{7 f g^2} \] Output:
-1/7*2^(9/4+m)*(g*cos(f*x+e))^(7/2)*hypergeom([7/4, -1/4-m],[11/4],1/2-1/2 *sin(f*x+e))*sec(f*x+e)*(1+sin(f*x+e))^(-5/4-m)*(a+a*sin(f*x+e))^m*(c-c*si n(f*x+e))^(1/2)/f/g^2
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx=\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx \] Input:
Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]],x]
Output:
Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]], x]
Time = 0.61 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3042, 3332, 3042, 3168, 80, 79}
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.
\(\displaystyle \int \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} (a \sin (e+f x)+a)^m \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} (a \sin (e+f x)+a)^mdx\) |
\(\Big \downarrow \) 3332 |
\(\displaystyle \frac {\sec (e+f x) \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)} \int (g \cos (e+f x))^{5/2} (\sin (e+f x) a+a)^{m-\frac {1}{2}}dx}{g}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sec (e+f x) \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)} \int (g \cos (e+f x))^{5/2} (\sin (e+f x) a+a)^{m-\frac {1}{2}}dx}{g}\) |
\(\Big \downarrow \) 3168 |
\(\displaystyle \frac {a^2 \sec (e+f x) \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{7/2} \int (a-a \sin (e+f x))^{3/4} (\sin (e+f x) a+a)^{m+\frac {1}{4}}d\sin (e+f x)}{f g^2 (a-a \sin (e+f x))^{7/4} (a \sin (e+f x)+a)^{5/4}}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {a^2 2^{m+\frac {1}{4}} \sec (e+f x) \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{7/2} (\sin (e+f x)+1)^{-m-\frac {1}{4}} (a \sin (e+f x)+a)^{m-1} \int \left (\frac {1}{2} \sin (e+f x)+\frac {1}{2}\right )^{m+\frac {1}{4}} (a-a \sin (e+f x))^{3/4}d\sin (e+f x)}{f g^2 (a-a \sin (e+f x))^{7/4}}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {a 2^{m+\frac {9}{4}} \sec (e+f x) \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{7/2} (\sin (e+f x)+1)^{-m-\frac {1}{4}} (a \sin (e+f x)+a)^{m-1} \operatorname {Hypergeometric2F1}\left (\frac {7}{4},-m-\frac {1}{4},\frac {11}{4},\frac {1}{2} (1-\sin (e+f x))\right )}{7 f g^2}\) |
Input:
Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]] ,x]
Output:
-1/7*(2^(9/4 + m)*a*(g*Cos[e + f*x])^(7/2)*Hypergeometric2F1[7/4, -1/4 - m , 11/4, (1 - Sin[e + f*x])/2]*Sec[e + f*x]*(1 + Sin[e + f*x])^(-1/4 - m)*( a + a*Sin[e + f*x])^(-1 + m)*Sqrt[c - c*Sin[e + f*x]])/(f*g^2)
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.), x_Symbol] :> Simp[a^2*((g*Cos[e + f*x])^(p + 1)/(f*g*(a + b*Sin [e + f*x])^((p + 1)/2)*(a - b*Sin[e + f*x])^((p + 1)/2))) Subst[Int[(a + b*x)^(m + (p - 1)/2)*(a - b*x)^((p - 1)/2), x], x, Sin[e + f*x]], x] /; Fre eQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a^ IntPart[m]*c^IntPart[m]*(a + b*Sin[e + f*x])^FracPart[m]*((c + d*Sin[e + f* x])^FracPart[m]/(g^(2*IntPart[m])*(g*Cos[e + f*x])^(2*FracPart[m]))) Int[ (g*Cos[e + f*x])^(2*m + p)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a , b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && (FractionQ[m] || !FractionQ[n])
\[\int \left (g \cos \left (f x +e \right )\right )^{\frac {3}{2}} \left (a +a \sin \left (f x +e \right )\right )^{m} \sqrt {c -c \sin \left (f x +e \right )}d x\]
Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(1/2),x)
Output:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(1/2),x)
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(1/2),x , algorithm="fricas")
Output:
integral(sqrt(g*cos(f*x + e))*sqrt(-c*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m*g*cos(f*x + e), x)
Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**m*(c-c*sin(f*x+e))**(1/2 ),x)
Output:
Timed out
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(1/2),x , algorithm="maxima")
Output:
integrate((g*cos(f*x + e))^(3/2)*sqrt(-c*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m, x)
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(1/2),x , algorithm="giac")
Output:
integrate((g*cos(f*x + e))^(3/2)*sqrt(-c*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m, x)
Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx=\int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\sqrt {c-c\,\sin \left (e+f\,x\right )} \,d x \] Input:
int((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(1/ 2),x)
Output:
int((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(1/ 2), x)
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx=\sqrt {g}\, \sqrt {c}\, \left (\int \left (a +a \sin \left (f x +e \right )\right )^{m} \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right )d x \right ) g \] Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(1/2),x)
Output:
sqrt(g)*sqrt(c)*int((sin(e + f*x)*a + a)**m*sqrt( - sin(e + f*x) + 1)*sqrt (cos(e + f*x))*cos(e + f*x),x)*g