Optimal. Leaf size=277 \[ -\frac {2^{\frac {1}{2}+m} a (A-B) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right ) (a+a \sin (e+f x))^{-1+m} \left (\frac {(c+d) (1+\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac {1}{2}-m} (c+d \sin (e+f x))^{-m}}{(c+d) f}+\frac {\sqrt {2} B F_1\left (\frac {3}{2}+m;\frac {1}{2},1+m;\frac {5}{2}+m;\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^{-m} \left (\frac {c+d \sin (e+f x)}{c-d}\right )^m}{a (c-d) f (3+2 m) \sqrt {1-\sin (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.33, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3066, 2867,
134, 145, 144, 143} \begin {gather*} \frac {\sqrt {2} B \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^{-m} \left (\frac {c+d \sin (e+f x)}{c-d}\right )^m F_1\left (m+\frac {3}{2};\frac {1}{2},m+1;m+\frac {5}{2};\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a f (2 m+3) (c-d) \sqrt {1-\sin (e+f x)}}-\frac {a 2^{m+\frac {1}{2}} (A-B) \cos (e+f x) (a \sin (e+f x)+a)^{m-1} \left (\frac {(c+d) (\sin (e+f x)+1)}{c+d \sin (e+f x)}\right )^{\frac {1}{2}-m} (c+d \sin (e+f x))^{-m} \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right )}{f (c+d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 134
Rule 143
Rule 144
Rule 145
Rule 2867
Rule 3066
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^{-1-m} \, dx &=(A-B) \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m} \, dx+\frac {B \int (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^{-1-m} \, dx}{a}\\ &=\frac {\left (a^2 (A-B) \cos (e+f x)\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m} (c+d x)^{-1-m}}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}+\frac {(a B \cos (e+f x)) \text {Subst}\left (\int \frac {(a+a x)^{\frac {1}{2}+m} (c+d x)^{-1-m}}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {2^{\frac {1}{2}+m} a (A-B) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right ) (a+a \sin (e+f x))^{-1+m} \left (\frac {(c+d) (1+\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac {1}{2}-m} (c+d \sin (e+f x))^{-m}}{(c+d) f}+\frac {\left (a B \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {(a+a x)^{\frac {1}{2}+m} (c+d x)^{-1-m}}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} f (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {2^{\frac {1}{2}+m} a (A-B) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right ) (a+a \sin (e+f x))^{-1+m} \left (\frac {(c+d) (1+\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac {1}{2}-m} (c+d \sin (e+f x))^{-m}}{(c+d) f}+\frac {\left (a^2 B \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}} (c+d \sin (e+f x))^{-m} \left (\frac {a (c+d \sin (e+f x))}{a c-a d}\right )^m\right ) \text {Subst}\left (\int \frac {(a+a x)^{\frac {1}{2}+m} \left (\frac {a c}{a c-a d}+\frac {a d x}{a c-a d}\right )^{-1-m}}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} (a c-a d) f (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {2^{\frac {1}{2}+m} a (A-B) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right ) (a+a \sin (e+f x))^{-1+m} \left (\frac {(c+d) (1+\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac {1}{2}-m} (c+d \sin (e+f x))^{-m}}{(c+d) f}+\frac {\sqrt {2} B F_1\left (\frac {3}{2}+m;\frac {1}{2},1+m;\frac {5}{2}+m;\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) \sqrt {1-\sin (e+f x)} (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^{-m} \left (\frac {c+d \sin (e+f x)}{c-d}\right )^m}{(c-d) f (3+2 m) (a-a \sin (e+f x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(573\) vs. \(2(277)=554\).
time = 4.52, size = 573, normalized size = 2.07 \begin {gather*} \frac {2 \cos ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )^{-\frac {1}{2}+m} \cot \left (\frac {1}{4} (2 e+\pi +2 f x)\right ) (a (1+\sin (e+f x)))^m (c+d \sin (e+f x))^{-m} \left (-\frac {3 B (c+d)^2 F_1\left (\frac {1}{2};\frac {1}{2}-m,m;\frac {3}{2};\cos ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),\frac {2 d \sin ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{c+d}\right )}{d \left (3 (c+d) F_1\left (\frac {1}{2};\frac {1}{2}-m,m;\frac {3}{2};\cos ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),\frac {2 d \sin ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{c+d}\right )-\left (-4 d m F_1\left (\frac {3}{2};\frac {1}{2}-m,1+m;\frac {5}{2};\cos ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),\frac {2 d \sin ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{c+d}\right )+(c+d) (-1+2 m) F_1\left (\frac {3}{2};\frac {3}{2}-m,m;\frac {5}{2};\cos ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),\frac {2 d \sin ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{c+d}\right )\right ) \cos ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right )\right )}-A \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {(c-d) \sin ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{c+d \sin (e+f x)}\right ) \left (\frac {(c+d) \cos ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{c+d \sin (e+f x)}\right )^{\frac {1}{2}-m}+\frac {B c \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {(c-d) \sin ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{c+d \sin (e+f x)}\right ) \left (\frac {(c+d) \cos ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{c+d \sin (e+f x)}\right )^{\frac {1}{2}-m}}{d}\right ) \sin ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right )^{\frac {1}{2}-m}}{(c+d) f} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.52, size = 0, normalized size = 0.00 \[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{-1-m}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{m+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________