Optimal. Leaf size=127 \[ \frac {\sec (e+f x) (1-\sin (e+f x))^{\frac {p+1}{2}} (a \sin (e+f x)+a)^m (d \sin (e+f x))^{n+1} (g \sec (e+f x))^p (\sin (e+f x)+1)^{\frac {1}{2} (-2 m+p+1)} F_1\left (n+1;\frac {p+1}{2},\frac {1}{2} (-2 m+p+1);n+2;\sin (e+f x),-\sin (e+f x)\right )}{d f (n+1)} \]
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Rubi [A] time = 0.35, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2926, 2886, 135, 133} \[ \frac {\sec (e+f x) (1-\sin (e+f x))^{\frac {p+1}{2}} (a \sin (e+f x)+a)^m (d \sin (e+f x))^{n+1} (g \sec (e+f x))^p (\sin (e+f x)+1)^{\frac {1}{2} (-2 m+p+1)} F_1\left (n+1;\frac {p+1}{2},\frac {1}{2} (-2 m+p+1);n+2;\sin (e+f x),-\sin (e+f x)\right )}{d f (n+1)} \]
Antiderivative was successfully verified.
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Rule 133
Rule 135
Rule 2886
Rule 2926
Rubi steps
\begin {align*} \int (g \sec (e+f x))^p (d \sin (e+f x))^n (a+a \sin (e+f x))^m \, dx &=\left ((g \cos (e+f x))^p (g \sec (e+f x))^p\right ) \int (g \cos (e+f x))^{-p} (d \sin (e+f x))^n (a+a \sin (e+f x))^m \, dx\\ &=\frac {\left (\sec (e+f x) (g \sec (e+f x))^p (a-a \sin (e+f x))^{\frac {1+p}{2}} (a+a \sin (e+f x))^{\frac {1+p}{2}}\right ) \operatorname {Subst}\left (\int (d x)^n (a-a x)^{\frac {1}{2} (-1-p)} (a+a x)^{m+\frac {1}{2} (-1-p)} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\left (\sec (e+f x) (g \sec (e+f x))^p (1-\sin (e+f x))^{\frac {1}{2}+\frac {p}{2}} (a-a \sin (e+f x))^{-\frac {1}{2}-\frac {p}{2}+\frac {1+p}{2}} (a+a \sin (e+f x))^{\frac {1+p}{2}}\right ) \operatorname {Subst}\left (\int (1-x)^{\frac {1}{2} (-1-p)} (d x)^n (a+a x)^{m+\frac {1}{2} (-1-p)} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\left (\sec (e+f x) (g \sec (e+f x))^p (1-\sin (e+f x))^{\frac {1}{2}+\frac {p}{2}} (1+\sin (e+f x))^{\frac {1}{2}-m+\frac {p}{2}} (a-a \sin (e+f x))^{-\frac {1}{2}-\frac {p}{2}+\frac {1+p}{2}} (a+a \sin (e+f x))^{-\frac {1}{2}+m-\frac {p}{2}+\frac {1+p}{2}}\right ) \operatorname {Subst}\left (\int (1-x)^{\frac {1}{2} (-1-p)} (d x)^n (1+x)^{m+\frac {1}{2} (-1-p)} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {F_1\left (1+n;\frac {1+p}{2},\frac {1}{2} (1-2 m+p);2+n;\sin (e+f x),-\sin (e+f x)\right ) \sec (e+f x) (g \sec (e+f x))^p (1-\sin (e+f x))^{\frac {1+p}{2}} (d \sin (e+f x))^{1+n} (1+\sin (e+f x))^{\frac {1}{2} (1-2 m+p)} (a+a \sin (e+f x))^m}{d f (1+n)}\\ \end {align*}
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Mathematica [B] time = 3.32, size = 347, normalized size = 2.73 \[ \frac {g (p-3) (a (\sin (e+f x)+1))^m (d \sin (e+f x))^n (g \sec (e+f x))^{p-1} F_1\left (\frac {1-p}{2};-n,m+n-p+1;\frac {3-p}{2};\cot ^2\left (\frac {1}{4} (2 e+2 f x+\pi )\right ),-\tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )\right )}{f (p-1) \left (2 \tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right ) \left (n F_1\left (\frac {3-p}{2};1-n,m+n-p+1;\frac {5-p}{2};\cot ^2\left (\frac {1}{4} (2 e+2 f x+\pi )\right ),-\tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )\right )+(m+n-p+1) F_1\left (\frac {3-p}{2};-n,m+n-p+2;\frac {5-p}{2};\cot ^2\left (\frac {1}{4} (2 e+2 f x+\pi )\right ),-\tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )\right )\right )+(p-3) F_1\left (\frac {1-p}{2};-n,m+n-p+1;\frac {3-p}{2};\cot ^2\left (\frac {1}{4} (2 e+2 f x+\pi )\right ),-\tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (g \sec \left (f x + e\right )\right )^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \left (d \sin \left (f x + e\right )\right )^{n}, x\right ) \]
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 \[ \int \left (g \sec \left (f x + e\right )\right )^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \left (d \sin \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 19.46, size = 0, normalized size = 0.00 \[ \int \left (g \sec \left (f x +e \right )\right )^{p} \left (d \sin \left (f x +e \right )\right )^{n} \left (a +a \sin \left (f x +e \right )\right )^{m}\, dx \]
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 \[ \int \left (g \sec \left (f x + e\right )\right )^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \left (d \sin \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,\sin \left (e+f\,x\right )\right )}^n\,{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^p\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \]
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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