Optimal. Leaf size=201 \[ -\frac {6 (a \sin (c+d x)+a)^{m+3} (e \cos (c+d x))^{-m-3}}{a^3 d e \left (m^4-10 m^2+9\right )}+\frac {6 (a \sin (c+d x)+a)^{m+2} (e \cos (c+d x))^{-m-3}}{a^2 d e (3-m) \left (1-m^2\right )}-\frac {(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m-3}}{d e (3-m)}-\frac {3 (a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-m-3}}{a d e (1-m) (3-m)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.32, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ \frac {6 (a \sin (c+d x)+a)^{m+2} (e \cos (c+d x))^{-m-3}}{a^2 d e (3-m) \left (1-m^2\right )}-\frac {6 (a \sin (c+d x)+a)^{m+3} (e \cos (c+d x))^{-m-3}}{a^3 d e \left (m^4-10 m^2+9\right )}-\frac {(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m-3}}{d e (3-m)}-\frac {3 (a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-m-3}}{a d e (1-m) (3-m)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2671
Rule 2672
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{-4-m} (a+a \sin (c+d x))^m \, dx &=-\frac {(e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^m}{d e (3-m)}+\frac {3 \int (e \cos (c+d x))^{-4-m} (a+a \sin (c+d x))^{1+m} \, dx}{a (3-m)}\\ &=-\frac {(e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^m}{d e (3-m)}-\frac {3 (e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^{1+m}}{a d e (1-m) (3-m)}+\frac {6 \int (e \cos (c+d x))^{-4-m} (a+a \sin (c+d x))^{2+m} \, dx}{a^2 (1-m) (3-m)}\\ &=-\frac {(e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^m}{d e (3-m)}-\frac {3 (e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^{1+m}}{a d e (1-m) (3-m)}+\frac {6 (e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^{2+m}}{a^2 d e (1-m) (3-m) (1+m)}-\frac {6 \int (e \cos (c+d x))^{-4-m} (a+a \sin (c+d x))^{3+m} \, dx}{a^3 (1-m) (3-m) (1+m)}\\ &=-\frac {(e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^m}{d e (3-m)}-\frac {3 (e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^{1+m}}{a d e (1-m) (3-m)}+\frac {6 (e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^{2+m}}{a^2 d e (1-m) (3-m) (1+m)}-\frac {6 (e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^{3+m}}{a^3 d e \left (9-10 m^2+m^4\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.19, size = 101, normalized size = 0.50 \[ \frac {\sec ^3(c+d x) \left (-3 \left (m^2-3\right ) \sin (c+d x)+6 m \sin ^2(c+d x)-6 \sin ^3(c+d x)+m \left (m^2-7\right )\right ) (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-m}}{d e^4 (m-3) (m-1) (m+1) (m+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 104, normalized size = 0.52 \[ -\frac {{\left (6 \, m \cos \left (d x + c\right )^{3} - {\left (m^{3} - m\right )} \cos \left (d x + c\right ) - 3 \, {\left (2 \, \cos \left (d x + c\right )^{3} - {\left (m^{2} - 1\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \left (e \cos \left (d x + c\right )\right )^{-m - 4} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{d m^{4} - 10 \, d m^{2} + 9 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x + c\right )\right )^{-m - 4} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x +c \right )\right )^{-4-m} \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x + c\right )\right )^{-m - 4} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 6.83, size = 137, normalized size = 0.68 \[ \frac {2\,{\left (a\,\left (\sin \left (c+d\,x\right )+1\right )\right )}^m\,\left (12\,\sin \left (2\,c+2\,d\,x\right )+3\,\sin \left (4\,c+4\,d\,x\right )-22\,m\,\cos \left (c+d\,x\right )-6\,m\,\cos \left (3\,c+3\,d\,x\right )+4\,m^3\,\cos \left (c+d\,x\right )-6\,m^2\,\sin \left (2\,c+2\,d\,x\right )\right )}{d\,e^4\,{\left (e\,\cos \left (c+d\,x\right )\right )}^m\,\left (4\,\cos \left (2\,c+2\,d\,x\right )+\cos \left (4\,c+4\,d\,x\right )+3\right )\,\left (m^4-10\,m^2+9\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________