Integrand size = 28, antiderivative size = 155 \[ \int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\frac {b^3 \cos ^{-2+m}(c+d x)}{d (2-m)}+\frac {3 a b^2 \cos ^{-1+m}(c+d x)}{d (1-m)}-\frac {b \left (3 a^2-b^2\right ) \cos ^m(c+d x)}{d m}-\frac {a \left (a^2-3 b^2\right ) \cos ^{1+m}(c+d x)}{d (1+m)}+\frac {3 a^2 b \cos ^{2+m}(c+d x)}{d (2+m)}+\frac {a^3 \cos ^{3+m}(c+d x)}{d (3+m)} \]
[Out]
Time = 0.47 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {4482, 2916, 962} \[ \int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\frac {a^3 \cos ^{m+3}(c+d x)}{d (m+3)}-\frac {a \left (a^2-3 b^2\right ) \cos ^{m+1}(c+d x)}{d (m+1)}-\frac {b \left (3 a^2-b^2\right ) \cos ^m(c+d x)}{d m}+\frac {3 a^2 b \cos ^{m+2}(c+d x)}{d (m+2)}+\frac {3 a b^2 \cos ^{m-1}(c+d x)}{d (1-m)}+\frac {b^3 \cos ^{m-2}(c+d x)}{d (2-m)} \]
[In]
[Out]
Rule 962
Rule 2916
Rule 4482
Rubi steps \begin{align*} \text {integral}& = \int \cos ^{-3+m}(c+d x) (b+a \cos (c+d x))^3 \sin ^3(c+d x) \, dx \\ & = -\frac {\text {Subst}\left (\int \left (\frac {x}{a}\right )^{-3+m} (b+x)^3 \left (a^2-x^2\right ) \, dx,x,a \cos (c+d x)\right )}{a^3 d} \\ & = -\frac {\text {Subst}\left (\int \left (a^2 b^3 \left (\frac {x}{a}\right )^{-3+m}+3 a^3 b^2 \left (\frac {x}{a}\right )^{-2+m}+a^2 b \left (3 a^2-b^2\right ) \left (\frac {x}{a}\right )^{-1+m}+a^3 \left (a^2-3 b^2\right ) \left (\frac {x}{a}\right )^m-3 a^4 b \left (\frac {x}{a}\right )^{1+m}-a^5 \left (\frac {x}{a}\right )^{2+m}\right ) \, dx,x,a \cos (c+d x)\right )}{a^3 d} \\ & = \frac {b^3 \cos ^{-2+m}(c+d x)}{d (2-m)}+\frac {3 a b^2 \cos ^{-1+m}(c+d x)}{d (1-m)}-\frac {b \left (3 a^2-b^2\right ) \cos ^m(c+d x)}{d m}-\frac {a \left (a^2-3 b^2\right ) \cos ^{1+m}(c+d x)}{d (1+m)}+\frac {3 a^2 b \cos ^{2+m}(c+d x)}{d (2+m)}+\frac {a^3 \cos ^{3+m}(c+d x)}{d (3+m)} \\ \end{align*}
Time = 6.86 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.59 \[ \int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\frac {\cos ^{1+m}(c+d x) \left (-4 b^3 m \left (-6-5 m+5 m^2+5 m^3+m^4\right )-12 a b^2 m \left (-12-16 m-m^2+4 m^3+m^4\right ) \cos (c+d x)-a m \left (4-4 m-m^2+m^3\right ) \left (-12 b^2 (3+m)+a^2 (9+m)\right ) \cos ^3(c+d x)+\left (2-m-2 m^2+m^3\right ) \cos ^2(c+d x) \left (2 b (3+m) \left (2 b^2 (2+m)-3 a^2 (4+m)\right )+6 a^2 b m (3+m) \cos (2 (c+d x))+a^3 m (2+m) \cos (3 (c+d x))\right )\right ) (a+b \sec (c+d x))^3}{4 d (-2+m) (-1+m) m (1+m) (2+m) (3+m) (b+a \cos (c+d x))^3} \]
[In]
[Out]
Time = 1.64 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.37
\[-\frac {a^{3} \cos \left (d x +c \right )^{1+m}}{d \left (1+m \right )}+\frac {a^{3} \cos \left (d x +c \right )^{3} {\mathrm e}^{m \ln \left (\cos \left (d x +c \right )\right )}}{d \left (3+m \right )}+\frac {b^{3} \cos \left (d x +c \right )^{m}}{m d}-\frac {b^{3} {\mathrm e}^{m \ln \left (\cos \left (d x +c \right )\right )}}{d \left (-2+m \right ) \cos \left (d x +c \right )^{2}}-\frac {3 a^{2} b \cos \left (d x +c \right )^{m}}{m d}+\frac {3 a^{2} b \cos \left (d x +c \right )^{2} {\mathrm e}^{m \ln \left (\cos \left (d x +c \right )\right )}}{d \left (2+m \right )}+\frac {3 a \,b^{2} \cos \left (d x +c \right )^{1+m}}{d \left (1+m \right )}-\frac {3 a \,b^{2} {\mathrm e}^{m \ln \left (\cos \left (d x +c \right )\right )}}{d \left (-1+m \right ) \cos \left (d x +c \right )}\]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (152) = 304\).
Time = 0.29 (sec) , antiderivative size = 412, normalized size of antiderivative = 2.66 \[ \int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=-\frac {{\left (b^{3} m^{5} + 5 \, b^{3} m^{4} + 5 \, b^{3} m^{3} - {\left (a^{3} m^{5} - 5 \, a^{3} m^{3} + 4 \, a^{3} m\right )} \cos \left (d x + c\right )^{5} - 5 \, b^{3} m^{2} - 3 \, {\left (a^{2} b m^{5} + a^{2} b m^{4} - 7 \, a^{2} b m^{3} - a^{2} b m^{2} + 6 \, a^{2} b m\right )} \cos \left (d x + c\right )^{4} - 6 \, b^{3} m + {\left ({\left (a^{3} - 3 \, a b^{2}\right )} m^{5} + 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} m^{4} - 7 \, {\left (a^{3} - 3 \, a b^{2}\right )} m^{3} - 8 \, {\left (a^{3} - 3 \, a b^{2}\right )} m^{2} + 12 \, {\left (a^{3} - 3 \, a b^{2}\right )} m\right )} \cos \left (d x + c\right )^{3} + {\left ({\left (3 \, a^{2} b - b^{3}\right )} m^{5} + 3 \, {\left (3 \, a^{2} b - b^{3}\right )} m^{4} - 5 \, {\left (3 \, a^{2} b - b^{3}\right )} m^{3} + 36 \, a^{2} b - 12 \, b^{3} - 15 \, {\left (3 \, a^{2} b - b^{3}\right )} m^{2} + 4 \, {\left (3 \, a^{2} b - b^{3}\right )} m\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (a b^{2} m^{5} + 4 \, a b^{2} m^{4} - a b^{2} m^{3} - 16 \, a b^{2} m^{2} - 12 \, a b^{2} m\right )} \cos \left (d x + c\right )\right )} \cos \left (d x + c\right )^{m}}{{\left (d m^{6} + 3 \, d m^{5} - 5 \, d m^{4} - 15 \, d m^{3} + 4 \, d m^{2} + 12 \, d m\right )} \cos \left (d x + c\right )^{2}} \]
[In]
[Out]
\[ \int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\int \left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right )^{3} \cos ^{m}{\left (c + d x \right )}\, dx \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.16 \[ \int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\frac {\frac {{\left ({\left (m + 1\right )} \cos \left (d x + c\right )^{3} - {\left (m + 3\right )} \cos \left (d x + c\right )\right )} a^{3} \cos \left (d x + c\right )^{m}}{m^{2} + 4 \, m + 3} + \frac {3 \, {\left (m \cos \left (d x + c\right )^{2} - m - 2\right )} a^{2} b \cos \left (d x + c\right )^{m}}{m^{2} + 2 \, m} + \frac {3 \, {\left ({\left (m - 1\right )} \cos \left (d x + c\right )^{2} - m - 1\right )} a b^{2} \cos \left (d x + c\right )^{m}}{{\left (m^{2} - 1\right )} \cos \left (d x + c\right )} + \frac {{\left ({\left (m - 2\right )} \cos \left (d x + c\right )^{2} - m\right )} b^{3} \cos \left (d x + c\right )^{m}}{{\left (m^{2} - 2 \, m\right )} \cos \left (d x + c\right )^{2}}}{d} \]
[In]
[Out]
Exception generated. \[ \int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Time = 33.54 (sec) , antiderivative size = 861, normalized size of antiderivative = 5.55 \[ \int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\frac {{\left (\frac {1}{2}\right )}^m\,{\left ({\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\right )}^m\,\left (\frac {a^3\,\left (\frac {m^4}{8}-\frac {5\,m^2}{8}+\frac {1}{2}\right )}{d\,\left (m^5+3\,m^4-5\,m^3-15\,m^2+4\,m+12\right )}+\frac {a^3\,{\mathrm {e}}^{c\,10{}\mathrm {i}+d\,x\,10{}\mathrm {i}}\,\left (m^4-5\,m^2+4\right )}{8\,d\,\left (m^5+3\,m^4-5\,m^3-15\,m^2+4\,m+12\right )}-\frac {a\,{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,\left (-m^3+m^2+4\,m-4\right )\,\left (a^2\,m+12\,b^2\,m-7\,a^2+36\,b^2\right )}{8\,d\,\left (m^5+3\,m^4-5\,m^3-15\,m^2+4\,m+12\right )}-\frac {a\,{\mathrm {e}}^{c\,8{}\mathrm {i}+d\,x\,8{}\mathrm {i}}\,\left (-m^3+m^2+4\,m-4\right )\,\left (a^2\,m+12\,b^2\,m-7\,a^2+36\,b^2\right )}{8\,d\,\left (m^5+3\,m^4-5\,m^3-15\,m^2+4\,m+12\right )}+\frac {3\,a^2\,b\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (m^4+m^3-7\,m^2-m+6\right )}{4\,d\,\left (m^5+3\,m^4-5\,m^3-15\,m^2+4\,m+12\right )}+\frac {3\,a^2\,b\,{\mathrm {e}}^{c\,9{}\mathrm {i}+d\,x\,9{}\mathrm {i}}\,\left (m^4+m^3-7\,m^2-m+6\right )}{4\,d\,\left (m^5+3\,m^4-5\,m^3-15\,m^2+4\,m+12\right )}-\frac {a\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\left (m^2-4\right )\,\left (a^2\,m^2+12\,a^2\,m-13\,a^2+6\,b^2\,m^2+60\,b^2\,m+126\,b^2\right )}{4\,d\,\left (m^5+3\,m^4-5\,m^3-15\,m^2+4\,m+12\right )}-\frac {a\,{\mathrm {e}}^{c\,6{}\mathrm {i}+d\,x\,6{}\mathrm {i}}\,\left (m^2-4\right )\,\left (a^2\,m^2+12\,a^2\,m-13\,a^2+6\,b^2\,m^2+60\,b^2\,m+126\,b^2\right )}{4\,d\,\left (m^5+3\,m^4-5\,m^3-15\,m^2+4\,m+12\right )}+\frac {b\,{\mathrm {e}}^{c\,3{}\mathrm {i}+d\,x\,3{}\mathrm {i}}\,\left (b^2\,m-6\,a^2+2\,b^2\right )\,\left (m^4+m^3-7\,m^2-m+6\right )}{d\,m\,\left (m^5+3\,m^4-5\,m^3-15\,m^2+4\,m+12\right )}+\frac {b\,{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\left (b^2\,m-6\,a^2+2\,b^2\right )\,\left (m^4+m^3-7\,m^2-m+6\right )}{d\,m\,\left (m^5+3\,m^4-5\,m^3-15\,m^2+4\,m+12\right )}+\frac {b\,{\mathrm {e}}^{c\,5{}\mathrm {i}+d\,x\,5{}\mathrm {i}}\,\left (-m^3-3\,m^2+m+3\right )\,\left (3\,a^2\,m^2+18\,a^2\,m-48\,a^2+4\,b^2\,m^2+16\,b^2\,m+16\,b^2\right )}{2\,d\,m\,\left (m^5+3\,m^4-5\,m^3-15\,m^2+4\,m+12\right )}\right )}{{\mathrm {e}}^{c\,3{}\mathrm {i}+d\,x\,3{}\mathrm {i}}+2\,{\mathrm {e}}^{c\,5{}\mathrm {i}+d\,x\,5{}\mathrm {i}}+{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}} \]
[In]
[Out]