Optimal. Leaf size=145 \[ -\frac {a \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{32 \sqrt {2} c^{7/2} f}+\frac {a \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \cos (e+f x)}{24 c f (c-c \sin (e+f x))^{5/2}}-\frac {a \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{3/2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2815, 2759,
2729, 2728, 212} \begin {gather*} -\frac {a \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{32 \sqrt {2} c^{7/2} f}-\frac {a \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{3/2}}-\frac {a \cos (e+f x)}{24 c f (c-c \sin (e+f x))^{5/2}}+\frac {a \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2728
Rule 2729
Rule 2759
Rule 2815
Rubi steps
\begin {align*} \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx &=(a c) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{9/2}} \, dx\\ &=\frac {a \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \int \frac {1}{(c-c \sin (e+f x))^{5/2}} \, dx}{6 c}\\ &=\frac {a \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \cos (e+f x)}{24 c f (c-c \sin (e+f x))^{5/2}}-\frac {a \int \frac {1}{(c-c \sin (e+f x))^{3/2}} \, dx}{16 c^2}\\ &=\frac {a \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \cos (e+f x)}{24 c f (c-c \sin (e+f x))^{5/2}}-\frac {a \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{3/2}}-\frac {a \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{64 c^3}\\ &=\frac {a \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \cos (e+f x)}{24 c f (c-c \sin (e+f x))^{5/2}}-\frac {a \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac {a \text {Subst}\left (\int \frac {1}{2 c-x^2} \, dx,x,-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{32 c^3 f}\\ &=-\frac {a \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{32 \sqrt {2} c^{7/2} f}+\frac {a \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \cos (e+f x)}{24 c f (c-c \sin (e+f x))^{5/2}}-\frac {a \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.70, size = 189, normalized size = 1.30 \begin {gather*} \frac {a \left (12 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {-c (1+\sin (e+f x))}}{\sqrt {2} \sqrt {c}}\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \sqrt {-c (1+\sin (e+f x))}+2 \sqrt {c} (-14 \cos (2 (e+f x))+131 \sin (e+f x)+3 (38+\sin (3 (e+f x))))\right )}{768 c^{7/2} f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c-c \sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.57, size = 243, normalized size = 1.68
method | result | size |
default | \(\frac {a \left (24 \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, c^{\frac {9}{2}}+32 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} c^{\frac {7}{2}}-6 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}} c^{\frac {5}{2}}+3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \left (\sin ^{3}\left (f x +e \right )\right ) c^{5}-9 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \left (\sin ^{2}\left (f x +e \right )\right ) c^{5}+9 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right ) c^{5}-3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{5}\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}}{192 c^{\frac {17}{2}} \left (\sin \left (f x +e \right )-1\right )^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(243\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 442 vs.
\(2 (130) = 260\).
time = 0.35, size = 442, normalized size = 3.05 \begin {gather*} \frac {3 \, \sqrt {2} {\left (a \cos \left (f x + e\right )^{4} - 3 \, a \cos \left (f x + e\right )^{3} - 8 \, a \cos \left (f x + e\right )^{2} + 4 \, a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right )^{3} + 4 \, a \cos \left (f x + e\right )^{2} - 4 \, a \cos \left (f x + e\right ) - 8 \, a\right )} \sin \left (f x + e\right ) + 8 \, a\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) + 4 \, {\left (3 \, a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} + 22 \, a \cos \left (f x + e\right ) + {\left (3 \, a \cos \left (f x + e\right )^{2} + 10 \, a \cos \left (f x + e\right ) + 32 \, a\right )} \sin \left (f x + e\right ) + 32 \, a\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{384 \, {\left (c^{4} f \cos \left (f x + e\right )^{4} - 3 \, c^{4} f \cos \left (f x + e\right )^{3} - 8 \, c^{4} f \cos \left (f x + e\right )^{2} + 4 \, c^{4} f \cos \left (f x + e\right ) + 8 \, c^{4} f + {\left (c^{4} f \cos \left (f x + e\right )^{3} + 4 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f \cos \left (f x + e\right ) - 8 \, c^{4} f\right )} \sin \left (f x + e\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 391 vs.
\(2 (130) = 260\).
time = 0.61, size = 391, normalized size = 2.70 \begin {gather*} -\frac {\frac {12 \, \sqrt {2} a \log \left (-\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1}\right )}{c^{\frac {7}{2}} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\sqrt {2} {\left (a \sqrt {c} - \frac {3 \, a \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} - \frac {3 \, a \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} + \frac {22 \, a \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{3}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}}\right )} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}}{c^{4} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\sqrt {2} {\left (\frac {3 \, a c^{\frac {17}{2}} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {3 \, a c^{\frac {17}{2}} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} - \frac {a c^{\frac {17}{2}} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{3}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}}\right )}}{c^{12} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{1536 \, f} \end {gather*}
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 \begin {gather*} \int \frac {a+a\,\sin \left (e+f\,x\right )}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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