Integrand size = 14, antiderivative size = 106 \[ \int (a+b \sin (c+d x))^{4/3} \, dx=-\frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {4}{3},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x)),\frac {b (1-\sin (c+d x))}{a+b}\right ) \cos (c+d x) (a+b \sin (c+d x))^{4/3}}{d \sqrt {1+\sin (c+d x)} \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{4/3}} \] Output:
-2^(1/2)*AppellF1(1/2,-4/3,1/2,3/2,b*(1-sin(d*x+c))/(a+b),1/2-1/2*sin(d*x+ c))*cos(d*x+c)*(a+b*sin(d*x+c))^(4/3)/d/(1+sin(d*x+c))^(1/2)/((a+b*sin(d*x +c))/(a+b))^(4/3)
Leaf count is larger than twice the leaf count of optimal. \(244\) vs. \(2(106)=212\).
Time = 1.88 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.30 \[ \int (a+b \sin (c+d x))^{4/3} \, dx=-\frac {3 \sec (c+d x) \sqrt [3]{a+b \sin (c+d x)} \left (4 b^2 \cos ^2(c+d x)+4 \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right ) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {\frac {b (1+\sin (c+d x))}{-a+b}}-5 a \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right ) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {\frac {b (1+\sin (c+d x))}{-a+b}} (a+b \sin (c+d x))\right )}{16 b d} \] Input:
Integrate[(a + b*Sin[c + d*x])^(4/3),x]
Output:
(-3*Sec[c + d*x]*(a + b*Sin[c + d*x])^(1/3)*(4*b^2*Cos[c + d*x]^2 + 4*(a^2 - b^2)*AppellF1[1/3, 1/2, 1/2, 4/3, (a + b*Sin[c + d*x])/(a - b), (a + b* Sin[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)] - 5*a*AppellF1[4/3, 1/2, 1/2, 7/3, (a + b*Sin[ c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Sin[c + d *x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]*(a + b*Sin[c + d*x]) ))/(16*b*d)
Time = 0.26 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3144, 156, 155}
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.
\(\displaystyle \int (a+b \sin (c+d x))^{4/3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a+b \sin (c+d x))^{4/3}dx\) |
\(\Big \downarrow \) 3144 |
\(\displaystyle \frac {\cos (c+d x) \int \frac {(a+b \sin (c+d x))^{4/3}}{\sqrt {1-\sin (c+d x)} \sqrt {\sin (c+d x)+1}}d\sin (c+d x)}{d \sqrt {1-\sin (c+d x)} \sqrt {\sin (c+d x)+1}}\) |
\(\Big \downarrow \) 156 |
\(\displaystyle \frac {(a+b) \cos (c+d x) \sqrt [3]{a+b \sin (c+d x)} \int \frac {\left (\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}\right )^{4/3}}{\sqrt {1-\sin (c+d x)} \sqrt {\sin (c+d x)+1}}d\sin (c+d x)}{d \sqrt {1-\sin (c+d x)} \sqrt {\sin (c+d x)+1} \sqrt [3]{\frac {a+b \sin (c+d x)}{a+b}}}\) |
\(\Big \downarrow \) 155 |
\(\displaystyle -\frac {\sqrt {2} (a+b) \cos (c+d x) \sqrt [3]{a+b \sin (c+d x)} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {4}{3},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x)),\frac {b (1-\sin (c+d x))}{a+b}\right )}{d \sqrt {\sin (c+d x)+1} \sqrt [3]{\frac {a+b \sin (c+d x)}{a+b}}}\) |
Input:
Int[(a + b*Sin[c + d*x])^(4/3),x]
Output:
-((Sqrt[2]*(a + b)*AppellF1[1/2, 1/2, -4/3, 3/2, (1 - Sin[c + d*x])/2, (b* (1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^(1/3))/(d*S qrt[1 + Sin[c + d*x]]*((a + b*Sin[c + d*x])/(a + b))^(1/3)))
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* Simplify[b/(b*e - a*f)]^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/ (b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] && GtQ[Sim plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] && !(GtQ[Simpl ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d *x, a + b*x]) && !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c - e*d)], 0] && SimplerQ[e + f*x, a + b*x])
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[b/(b*e - a*f)]^IntPart[p ]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p]) Int[(a + b*x)^m*(c + d*x)^n*Si mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] & & GtQ[Simplify[b/(b*c - a*d)], 0] && !GtQ[Simplify[b/(b*e - a*f)], 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]/(d*Sqrt[1 + Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]]) Subst[Int[(a + b*x )^n/(Sqrt[1 + x]*Sqrt[1 - x]), x], x, Sin[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0] && !IntegerQ[2*n]
\[\int \left (a +b \sin \left (d x +c \right )\right )^{\frac {4}{3}}d x\]
Input:
int((a+b*sin(d*x+c))^(4/3),x)
Output:
int((a+b*sin(d*x+c))^(4/3),x)
\[ \int (a+b \sin (c+d x))^{4/3} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \,d x } \] Input:
integrate((a+b*sin(d*x+c))^(4/3),x, algorithm="fricas")
Output:
integral((b*sin(d*x + c) + a)^(4/3), x)
\[ \int (a+b \sin (c+d x))^{4/3} \, dx=\int \left (a + b \sin {\left (c + d x \right )}\right )^{\frac {4}{3}}\, dx \] Input:
integrate((a+b*sin(d*x+c))**(4/3),x)
Output:
Integral((a + b*sin(c + d*x))**(4/3), x)
\[ \int (a+b \sin (c+d x))^{4/3} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \,d x } \] Input:
integrate((a+b*sin(d*x+c))^(4/3),x, algorithm="maxima")
Output:
integrate((b*sin(d*x + c) + a)^(4/3), x)
\[ \int (a+b \sin (c+d x))^{4/3} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \,d x } \] Input:
integrate((a+b*sin(d*x+c))^(4/3),x, algorithm="giac")
Output:
integrate((b*sin(d*x + c) + a)^(4/3), x)
Timed out. \[ \int (a+b \sin (c+d x))^{4/3} \, dx=\int {\left (a+b\,\sin \left (c+d\,x\right )\right )}^{4/3} \,d x \] Input:
int((a + b*sin(c + d*x))^(4/3),x)
Output:
int((a + b*sin(c + d*x))^(4/3), x)
\[ \int (a+b \sin (c+d x))^{4/3} \, dx=\left (\int \left (\sin \left (d x +c \right ) b +a \right )^{\frac {1}{3}}d x \right ) a +\left (\int \left (\sin \left (d x +c \right ) b +a \right )^{\frac {1}{3}} \sin \left (d x +c \right )d x \right ) b \] Input:
int((a+b*sin(d*x+c))^(4/3),x)
Output:
int((sin(c + d*x)*b + a)**(1/3),x)*a + int((sin(c + d*x)*b + a)**(1/3)*sin (c + d*x),x)*b