Optimal. Leaf size=106 \[ \frac {2 a e \sqrt {a \sec (c+d x)+a} (1-\cos (c+d x))^{\frac {1-m}{2}} (\cos (c+d x)+1)^{-m/2} F_1\left (-\frac {1}{2};\frac {1-m}{2},\frac {1}{2} (-m-2);\frac {1}{2};\cos (c+d x),-\cos (c+d x)\right ) (e \sin (c+d x))^{m-1}}{d} \]
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Rubi [A] time = 0.37, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3876, 2886, 135, 133} \[ \frac {2 a e \sqrt {a \sec (c+d x)+a} (1-\cos (c+d x))^{\frac {1-m}{2}} (\cos (c+d x)+1)^{-m/2} F_1\left (-\frac {1}{2};\frac {1-m}{2},\frac {1}{2} (-m-2);\frac {1}{2};\cos (c+d x),-\cos (c+d x)\right ) (e \sin (c+d x))^{m-1}}{d} \]
Antiderivative was successfully verified.
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Rule 133
Rule 135
Rule 2886
Rule 3876
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^{3/2} (e \sin (c+d x))^m \, dx &=\frac {\left (\sqrt {-\cos (c+d x)} \sqrt {a+a \sec (c+d x)}\right ) \int \frac {(-a-a \cos (c+d x))^{3/2} (e \sin (c+d x))^m}{(-\cos (c+d x))^{3/2}} \, dx}{\sqrt {-a-a \cos (c+d x)}}\\ &=-\frac {\left (e \sqrt {-\cos (c+d x)} (-a-a \cos (c+d x))^{-\frac {1}{2}+\frac {1-m}{2}} (-a+a \cos (c+d x))^{\frac {1-m}{2}} \sqrt {a+a \sec (c+d x)} (e \sin (c+d x))^{-1+m}\right ) \operatorname {Subst}\left (\int \frac {(-a-a x)^{\frac {3}{2}+\frac {1}{2} (-1+m)} (-a+a x)^{\frac {1}{2} (-1+m)}}{(-x)^{3/2}} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {\left (a e \sqrt {-\cos (c+d x)} (1+\cos (c+d x))^{-m/2} (-a-a \cos (c+d x))^{-\frac {1}{2}+\frac {1-m}{2}+\frac {m}{2}} (-a+a \cos (c+d x))^{\frac {1-m}{2}} \sqrt {a+a \sec (c+d x)} (e \sin (c+d x))^{-1+m}\right ) \operatorname {Subst}\left (\int \frac {(1+x)^{\frac {3}{2}+\frac {1}{2} (-1+m)} (-a+a x)^{\frac {1}{2} (-1+m)}}{(-x)^{3/2}} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {\left (a e (1-\cos (c+d x))^{\frac {1}{2}-\frac {m}{2}} \sqrt {-\cos (c+d x)} (1+\cos (c+d x))^{-m/2} (-a-a \cos (c+d x))^{-\frac {1}{2}+\frac {1-m}{2}+\frac {m}{2}} (-a+a \cos (c+d x))^{-\frac {1}{2}+\frac {1-m}{2}+\frac {m}{2}} \sqrt {a+a \sec (c+d x)} (e \sin (c+d x))^{-1+m}\right ) \operatorname {Subst}\left (\int \frac {(1-x)^{\frac {1}{2} (-1+m)} (1+x)^{\frac {3}{2}+\frac {1}{2} (-1+m)}}{(-x)^{3/2}} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {2 a e F_1\left (-\frac {1}{2};\frac {1-m}{2},\frac {1}{2} (-2-m);\frac {1}{2};\cos (c+d x),-\cos (c+d x)\right ) (1-\cos (c+d x))^{\frac {1-m}{2}} (1+\cos (c+d x))^{-m/2} \sqrt {a+a \sec (c+d x)} (e \sin (c+d x))^{-1+m}}{d}\\ \end {align*}
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Mathematica [B] time = 9.75, size = 1243, normalized size = 11.73 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \left (e \sin \left (d x + c\right )\right )^{m}, x\right ) \]
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 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \left (e \sin \left (d x + c\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.17, size = 0, normalized size = 0.00 \[ \int \left (a +a \sec \left (d x +c \right )\right )^{\frac {3}{2}} \left (e \sin \left (d x +c \right )\right )^{m}\, dx \]
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 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \left (e \sin \left (d x + c\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (e\,\sin \left (c+d\,x\right )\right )}^m\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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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