Optimal. Leaf size=311 \[ -\frac {\left (a^2 (m+1)-b^2\right ) (\sin (c+d x)+1)^3 \sec ^4(c+d x) (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^{m+1} \left (\frac {(a+b) (\sin (c+d x)+1)}{(a-b) (\sin (c+d x)-1)}\right )^{\frac {m-2}{2}} \, _2F_1\left (\frac {m}{2},m+1;m+2;-\frac {2 (a+b \sin (c+d x))}{(a-b) (\sin (c+d x)-1)}\right )}{d e^3 m (m+1) (a-b)^3}+\frac {(a (m+2)-2 b) (\sin (c+d x)-1) (\sin (c+d x)+1)^2 \sec ^4(c+d x) (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^{m+1}}{d e^3 m (m+2) (a-b)^2}+\frac {(\sin (c+d x)-1) (\sin (c+d x)+1) \sec ^4(c+d x) (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^{m+1}}{d e^3 (m+2) (a-b)} \]
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Rubi [A] time = 0.51, antiderivative size = 420, normalized size of antiderivative = 1.35, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2700, 2698, 2920, 96, 132} \[ -\frac {b (1-\sin (c+d x)) (e \cos (c+d x))^{-m-2} \left (-\frac {(a-b) (1-\sin (c+d x))}{(a+b) (\sin (c+d x)+1)}\right )^{m/2} (a+b \sin (c+d x))^{m+1} \, _2F_1\left (m+1,\frac {m+2}{2};m+2;\frac {2 (a+b \sin (c+d x))}{(a+b) (\sin (c+d x)+1)}\right )}{d e (m+1) (m+2) \left (a^2-b^2\right )}+\frac {a (\sin (c+d x)+1) (e \cos (c+d x))^{-m-2} (a+b \sin (c+d x))^{m+1}}{d e (m+2) \left (a^2-b^2\right )}+\frac {a 2^{-m/2} (a m+a+b) (1-\sin (c+d x)) (e \cos (c+d x))^{-m-2} \left (\frac {(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right )^{\frac {m+2}{2}} (a+b \sin (c+d x))^{m+1} \, _2F_1\left (-\frac {m}{2},\frac {m+2}{2};\frac {2-m}{2};\frac {(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right )}{d e m (m+2) (a-b) (a+b)^2}-\frac {(e \cos (c+d x))^{-m-2} (a+b \sin (c+d x))^{m+1}}{d e (m+2) (a-b)} \]
Warning: Unable to verify antiderivative.
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Rule 96
Rule 132
Rule 2698
Rule 2700
Rule 2920
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{-3-m} (a+b \sin (c+d x))^m \, dx &=-\frac {(e \cos (c+d x))^{-2-m} (a+b \sin (c+d x))^{1+m}}{(a-b) d e (2+m)}+\frac {a \int \frac {(e \cos (c+d x))^{-1-m} (a+b \sin (c+d x))^m}{1-\sin (c+d x)} \, dx}{(a-b) e^2}-\frac {b \int (e \cos (c+d x))^{-1-m} (a+b \sin (c+d x))^m \, dx}{(a-b) e^2 (2+m)}\\ &=-\frac {(e \cos (c+d x))^{-2-m} (a+b \sin (c+d x))^{1+m}}{(a-b) d e (2+m)}-\frac {b (e \cos (c+d x))^{-2-m} \, _2F_1\left (1+m,\frac {2+m}{2};2+m;\frac {2 (a+b \sin (c+d x))}{(a+b) (1+\sin (c+d x))}\right ) (1-\sin (c+d x)) \left (-\frac {(a-b) (1-\sin (c+d x))}{(a+b) (1+\sin (c+d x))}\right )^{m/2} (a+b \sin (c+d x))^{1+m}}{\left (a^2-b^2\right ) d e (1+m) (2+m)}+\frac {\left (a (e \cos (c+d x))^{-2-m} (1-\sin (c+d x))^{\frac {2+m}{2}} (1+\sin (c+d x))^{\frac {2+m}{2}}\right ) \operatorname {Subst}\left (\int (1-x)^{-1+\frac {1}{2} (-2-m)} (1+x)^{\frac {1}{2} (-2-m)} (a+b x)^m \, dx,x,\sin (c+d x)\right )}{(a-b) d e}\\ &=-\frac {(e \cos (c+d x))^{-2-m} (a+b \sin (c+d x))^{1+m}}{(a-b) d e (2+m)}-\frac {b (e \cos (c+d x))^{-2-m} \, _2F_1\left (1+m,\frac {2+m}{2};2+m;\frac {2 (a+b \sin (c+d x))}{(a+b) (1+\sin (c+d x))}\right ) (1-\sin (c+d x)) \left (-\frac {(a-b) (1-\sin (c+d x))}{(a+b) (1+\sin (c+d x))}\right )^{m/2} (a+b \sin (c+d x))^{1+m}}{\left (a^2-b^2\right ) d e (1+m) (2+m)}+\frac {a (e \cos (c+d x))^{-2-m} (1+\sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{\left (a^2-b^2\right ) d e (2+m)}+\frac {\left (a (a+b+a m) (e \cos (c+d x))^{-2-m} (1-\sin (c+d x))^{\frac {2+m}{2}} (1+\sin (c+d x))^{\frac {2+m}{2}}\right ) \operatorname {Subst}\left (\int (1-x)^{\frac {1}{2} (-2-m)} (1+x)^{\frac {1}{2} (-2-m)} (a+b x)^m \, dx,x,\sin (c+d x)\right )}{(a-b) (a+b) d e (2+m)}\\ &=-\frac {(e \cos (c+d x))^{-2-m} (a+b \sin (c+d x))^{1+m}}{(a-b) d e (2+m)}-\frac {b (e \cos (c+d x))^{-2-m} \, _2F_1\left (1+m,\frac {2+m}{2};2+m;\frac {2 (a+b \sin (c+d x))}{(a+b) (1+\sin (c+d x))}\right ) (1-\sin (c+d x)) \left (-\frac {(a-b) (1-\sin (c+d x))}{(a+b) (1+\sin (c+d x))}\right )^{m/2} (a+b \sin (c+d x))^{1+m}}{\left (a^2-b^2\right ) d e (1+m) (2+m)}+\frac {a (e \cos (c+d x))^{-2-m} (1+\sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{\left (a^2-b^2\right ) d e (2+m)}+\frac {2^{-m/2} a (a+b+a m) (e \cos (c+d x))^{-2-m} \, _2F_1\left (-\frac {m}{2},\frac {2+m}{2};\frac {2-m}{2};\frac {(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right ) (1-\sin (c+d x)) \left (\frac {(a+b) (1+\sin (c+d x))}{a+b \sin (c+d x)}\right )^{\frac {2+m}{2}} (a+b \sin (c+d x))^{1+m}}{(a-b) (a+b)^2 d e m (2+m)}\\ \end {align*}
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Mathematica [A] time = 5.05, size = 319, normalized size = 1.03 \[ \frac {\sec ^2(c+d x) (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^m \left (\frac {b (\sin (c+d x)+1) (a+b \sin (c+d x)) \left (\frac {(a+b) (\sin (c+d x)+1)}{(a-b) (\sin (c+d x)-1)}\right )^{m/2} \, _2F_1\left (m+1,\frac {m+2}{2};m+2;-\frac {2 (a+b \sin (c+d x))}{(a-b) (\sin (c+d x)-1)}\right )}{(m+1) (a-b)}+\frac {a (1-\sin (c+d x)) (\sin (c+d x)+1) \left (2^{-m/2} (a m+a+b) \left (\frac {(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right )^{m/2} \, _2F_1\left (-\frac {m}{2},\frac {m+2}{2};1-\frac {m}{2};-\frac {(a-b) (\sin (c+d x)-1)}{2 (a+b \sin (c+d x))}\right )-\frac {m (a+b \sin (c+d x))}{\sin (c+d x)-1}\right )}{m (a+b)}-a-b \sin (c+d x)\right )}{d e^3 (m+2) (a-b)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e \cos \left (d x + c\right )\right )^{-m - 3} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x + c\right )\right )^{-m - 3} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x +c \right )\right )^{-3-m} \left (a +b \sin \left (d x +c \right )\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x + c\right )\right )^{-m - 3} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^m}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{m+3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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