Integrand size = 41, antiderivative size = 235 \[ \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx=\frac {3 C \cos ^m(c+d x) \sin (c+d x)}{b d (2+3 m) \sqrt [3]{b \cos (c+d x)}}-\frac {3 (C (1-3 m)-A (2+3 m)) \cos ^m(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (-1+3 m),\frac {1}{6} (5+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{b d (1-3 m) (2+3 m) \sqrt [3]{b \cos (c+d x)} \sqrt {\sin ^2(c+d x)}}-\frac {3 B \cos ^{1+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (2+3 m),\frac {1}{6} (8+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{b d (2+3 m) \sqrt [3]{b \cos (c+d x)} \sqrt {\sin ^2(c+d x)}} \]
3*C*cos(d*x+c)^m*sin(d*x+c)/b/d/(2+3*m)/(b*cos(d*x+c))^(1/3)-3*(C*(1-3*m)- A*(2+3*m))*cos(d*x+c)^m*hypergeom([1/2, -1/6+1/2*m],[5/6+1/2*m],cos(d*x+c) ^2)*sin(d*x+c)/b/d/(-9*m^2-3*m+2)/(b*cos(d*x+c))^(1/3)/(sin(d*x+c)^2)^(1/2 )-3*B*cos(d*x+c)^(1+m)*hypergeom([1/2, 1/3+1/2*m],[4/3+1/2*m],cos(d*x+c)^2 )*sin(d*x+c)/b/d/(2+3*m)/(b*cos(d*x+c))^(1/3)/(sin(d*x+c)^2)^(1/2)
Time = 0.40 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.74 \[ \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx=-\frac {3 \cos ^{1+m}(c+d x) \csc (c+d x) \left ((C (-1+3 m)+A (2+3 m)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (-1+3 m),\frac {1}{6} (5+3 m),\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}-(-1+3 m) \left (C \sin ^2(c+d x)-B \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (2+3 m),\frac {1}{6} (8+3 m),\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )\right )}{d (-1+3 m) (2+3 m) (b \cos (c+d x))^{4/3}} \]
(-3*Cos[c + d*x]^(1 + m)*Csc[c + d*x]*((C*(-1 + 3*m) + A*(2 + 3*m))*Hyperg eometric2F1[1/2, (-1 + 3*m)/6, (5 + 3*m)/6, Cos[c + d*x]^2]*Sqrt[Sin[c + d *x]^2] - (-1 + 3*m)*(C*Sin[c + d*x]^2 - B*Cos[c + d*x]*Hypergeometric2F1[1 /2, (2 + 3*m)/6, (8 + 3*m)/6, Cos[c + d*x]^2]*Sqrt[Sin[c + d*x]^2])))/(d*( -1 + 3*m)*(2 + 3*m)*(b*Cos[c + d*x])^(4/3))
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.
| \(\displaystyle \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx\) |
| \(\Big \downarrow \) 2034 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \int \cos ^{m-\frac {4}{3}}(c+d x) \left (C \cos ^2(c+d x)+B \cos (c+d x)+A\right )dx}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{m-\frac {4}{3}} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+A\right )dx}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {3 \int -\frac {1}{3} \cos ^{m-\frac {4}{3}}(c+d x) \left (3 C \left (\frac {1}{3}-m\right )-3 A \left (m+\frac {2}{3}\right )-B (3 m+2) \cos (c+d x)\right )dx}{3 m+2}+\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}-\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (3 m A+2 A-C+3 C m+B (3 m+2) \cos (c+d x))dx}{3 m+2}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (C (1-3 m)-A (3 m+2)-B (3 m+2) \cos (c+d x))dx}{3 m+2}+\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}-\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (3 m A+2 A-C+3 C m+B (3 m+2) \cos (c+d x))dx}{3 m+2}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (C (1-3 m)-A (3 m+2)-B (3 m+2) \cos (c+d x))dx}{3 m+2}+\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}-\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (3 m A+2 A-C+3 C m+B (3 m+2) \cos (c+d x))dx}{3 m+2}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (C (1-3 m)-A (3 m+2)-B (3 m+2) \cos (c+d x))dx}{3 m+2}+\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}-\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (3 m A+2 A-C+3 C m+B (3 m+2) \cos (c+d x))dx}{3 m+2}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (C (1-3 m)-A (3 m+2)-B (3 m+2) \cos (c+d x))dx}{3 m+2}+\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}-\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (3 m A+2 A-C+3 C m+B (3 m+2) \cos (c+d x))dx}{3 m+2}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (C (1-3 m)-A (3 m+2)-B (3 m+2) \cos (c+d x))dx}{3 m+2}+\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}-\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (3 m A+2 A-C+3 C m+B (3 m+2) \cos (c+d x))dx}{3 m+2}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (C (1-3 m)-A (3 m+2)-B (3 m+2) \cos (c+d x))dx}{3 m+2}+\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}-\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (3 m A+2 A-C+3 C m+B (3 m+2) \cos (c+d x))dx}{3 m+2}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (C (1-3 m)-A (3 m+2)-B (3 m+2) \cos (c+d x))dx}{3 m+2}+\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}-\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (3 m A+2 A-C+3 C m+B (3 m+2) \cos (c+d x))dx}{3 m+2}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (C (1-3 m)-A (3 m+2)-B (3 m+2) \cos (c+d x))dx}{3 m+2}+\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}-\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (3 m A+2 A-C+3 C m+B (3 m+2) \cos (c+d x))dx}{3 m+2}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (C (1-3 m)-A (3 m+2)-B (3 m+2) \cos (c+d x))dx}{3 m+2}+\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}-\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (3 m A+2 A-C+3 C m+B (3 m+2) \cos (c+d x))dx}{3 m+2}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (C (1-3 m)-A (3 m+2)-B (3 m+2) \cos (c+d x))dx}{3 m+2}+\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}-\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (3 m A+2 A-C+3 C m+B (3 m+2) \cos (c+d x))dx}{3 m+2}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (C (1-3 m)-A (3 m+2)-B (3 m+2) \cos (c+d x))dx}{3 m+2}+\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}-\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (3 m A+2 A-C+3 C m+B (3 m+2) \cos (c+d x))dx}{3 m+2}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (C (1-3 m)-A (3 m+2)-B (3 m+2) \cos (c+d x))dx}{3 m+2}+\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}-\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (3 m A+2 A-C+3 C m+B (3 m+2) \cos (c+d x))dx}{3 m+2}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (C (1-3 m)-A (3 m+2)-B (3 m+2) \cos (c+d x))dx}{3 m+2}+\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{\cos (c+d x)} \left (\frac {3 C \sin (c+d x) \cos ^{m-\frac {1}{3}}(c+d x)}{d (3 m+2)}-\frac {\int -\cos ^{m-\frac {4}{3}}(c+d x) (3 m A+2 A-C+3 C m+B (3 m+2) \cos (c+d x))dx}{3 m+2}\right )}{b \sqrt [3]{b \cos (c+d x)}}\) |
3.4.68.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Simp[b^IntPart [n]*((b*v)^FracPart[n]/(a^IntPart[n]*(a*v)^FracPart[n])) Int[(a*v)^(m + n )*Fx, x], x] /; FreeQ[{a, b, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[m + n]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
\[\int \frac {\left (\cos ^{m}\left (d x +c \right )\right ) \left (A +B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right )}{\left (\cos \left (d x +c \right ) b \right )^{\frac {4}{3}}}d x\]
\[ \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}}} \,d x } \]
integrate(cos(d*x+c)^m*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(b*cos(d*x+c))^(4/3 ),x, algorithm="fricas")
integral((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c))^(2/3)*co s(d*x + c)^m/(b^2*cos(d*x + c)^2), x)
\[ \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx=\int \frac {\left (A + B \cos {\left (c + d x \right )} + C \cos ^{2}{\left (c + d x \right )}\right ) \cos ^{m}{\left (c + d x \right )}}{\left (b \cos {\left (c + d x \right )}\right )^{\frac {4}{3}}}\, dx \]
Integral((A + B*cos(c + d*x) + C*cos(c + d*x)**2)*cos(c + d*x)**m/(b*cos(c + d*x))**(4/3), x)
\[ \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}}} \,d x } \]
integrate(cos(d*x+c)^m*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(b*cos(d*x+c))^(4/3 ),x, algorithm="maxima")
\[ \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}}} \,d x } \]
integrate(cos(d*x+c)^m*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(b*cos(d*x+c))^(4/3 ),x, algorithm="giac")
Timed out. \[ \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^m\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{4/3}} \,d x \]