Optimal. Leaf size=448 \[ -\frac {a^3 \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{7/2} \sqrt [4]{-a^2+b^2} f}+\frac {a^3 \sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{7/2} \sqrt [4]{-a^2+b^2} f}+\frac {2 a (g \cos (e+f x))^{3/2}}{3 b^2 f g}+\frac {2 a^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b^3 f \sqrt {\cos (e+f x)}}+\frac {4 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 b f \sqrt {\cos (e+f x)}}-\frac {a^4 g \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 \left (b-\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}-\frac {a^4 g \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 \left (b+\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}-\frac {2 (g \cos (e+f x))^{3/2} \sin (e+f x)}{5 b f g} \]
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Rubi [A]
time = 0.60, antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 13, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {2977, 2721,
2719, 2645, 30, 2648, 2780, 2886, 2884, 335, 304, 211, 214} \begin {gather*} \frac {2 a^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{b^3 f \sqrt {\cos (e+f x)}}-\frac {a^4 g \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 f \left (b-\sqrt {b^2-a^2}\right ) \sqrt {g \cos (e+f x)}}-\frac {a^4 g \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 f \left (\sqrt {b^2-a^2}+b\right ) \sqrt {g \cos (e+f x)}}-\frac {a^3 \sqrt {g} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{b^{7/2} f \sqrt [4]{b^2-a^2}}+\frac {a^3 \sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{b^{7/2} f \sqrt [4]{b^2-a^2}}+\frac {2 a (g \cos (e+f x))^{3/2}}{3 b^2 f g}-\frac {2 \sin (e+f x) (g \cos (e+f x))^{3/2}}{5 b f g}+\frac {4 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{5 b f \sqrt {\cos (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 211
Rule 214
Rule 304
Rule 335
Rule 2645
Rule 2648
Rule 2719
Rule 2721
Rule 2780
Rule 2884
Rule 2886
Rule 2977
Rubi steps
\begin {align*} \int \frac {\sqrt {g \cos (e+f x)} \sin ^3(e+f x)}{a+b \sin (e+f x)} \, dx &=\int \left (\frac {a^2 \sqrt {g \cos (e+f x)}}{b^3}-\frac {a \sqrt {g \cos (e+f x)} \sin (e+f x)}{b^2}+\frac {\sqrt {g \cos (e+f x)} \sin ^2(e+f x)}{b}-\frac {a^3 \sqrt {g \cos (e+f x)}}{b^3 (a+b \sin (e+f x))}\right ) \, dx\\ &=\frac {a^2 \int \sqrt {g \cos (e+f x)} \, dx}{b^3}-\frac {a^3 \int \frac {\sqrt {g \cos (e+f x)}}{a+b \sin (e+f x)} \, dx}{b^3}-\frac {a \int \sqrt {g \cos (e+f x)} \sin (e+f x) \, dx}{b^2}+\frac {\int \sqrt {g \cos (e+f x)} \sin ^2(e+f x) \, dx}{b}\\ &=-\frac {2 (g \cos (e+f x))^{3/2} \sin (e+f x)}{5 b f g}+\frac {2 \int \sqrt {g \cos (e+f x)} \, dx}{5 b}+\frac {a \text {Subst}\left (\int \sqrt {x} \, dx,x,g \cos (e+f x)\right )}{b^2 f g}+\frac {\left (a^4 g\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 b^4}-\frac {\left (a^4 g\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 b^4}-\frac {\left (a^3 g\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\left (a^2-b^2\right ) g^2+b^2 x^2} \, dx,x,g \cos (e+f x)\right )}{b^2 f}+\frac {\left (a^2 \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{b^3 \sqrt {\cos (e+f x)}}\\ &=\frac {2 a (g \cos (e+f x))^{3/2}}{3 b^2 f g}+\frac {2 a^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b^3 f \sqrt {\cos (e+f x)}}-\frac {2 (g \cos (e+f x))^{3/2} \sin (e+f x)}{5 b f g}-\frac {\left (2 a^3 g\right ) \text {Subst}\left (\int \frac {x^2}{\left (a^2-b^2\right ) g^2+b^2 x^4} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{b^2 f}+\frac {\left (a^4 g \sqrt {\cos (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)} \left (\sqrt {-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 b^4 \sqrt {g \cos (e+f x)}}-\frac {\left (a^4 g \sqrt {\cos (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)} \left (\sqrt {-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 b^4 \sqrt {g \cos (e+f x)}}+\frac {\left (2 \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{5 b \sqrt {\cos (e+f x)}}\\ &=\frac {2 a (g \cos (e+f x))^{3/2}}{3 b^2 f g}+\frac {2 a^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b^3 f \sqrt {\cos (e+f x)}}+\frac {4 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 b f \sqrt {\cos (e+f x)}}-\frac {a^4 g \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 \left (b-\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}-\frac {a^4 g \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 \left (b+\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}-\frac {2 (g \cos (e+f x))^{3/2} \sin (e+f x)}{5 b f g}+\frac {\left (a^3 g\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} g-b x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{b^3 f}-\frac {\left (a^3 g\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} g+b x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{b^3 f}\\ &=-\frac {a^3 \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{7/2} \sqrt [4]{-a^2+b^2} f}+\frac {a^3 \sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{7/2} \sqrt [4]{-a^2+b^2} f}+\frac {2 a (g \cos (e+f x))^{3/2}}{3 b^2 f g}+\frac {2 a^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b^3 f \sqrt {\cos (e+f x)}}+\frac {4 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 b f \sqrt {\cos (e+f x)}}-\frac {a^4 g \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 \left (b-\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}-\frac {a^4 g \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^4 \left (b+\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}-\frac {2 (g \cos (e+f x))^{3/2} \sin (e+f x)}{5 b f g}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 36.49, size = 789, normalized size = 1.76 \begin {gather*} \frac {\sqrt {g \cos (e+f x)} \left (-\frac {4 a b \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \left (\frac {a F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(e+f x)}{3 \left (a^2-b^2\right )}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \tan ^{-1}\left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \tan ^{-1}\left (1+\frac {(1+i) \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (e+f x)}+i b \cos (e+f x)\right )+\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (e+f x)}+i b \cos (e+f x)\right )\right )}{\sqrt {b} \sqrt [4]{-a^2+b^2}}\right ) \sin (e+f x)}{\sqrt {1-\cos ^2(e+f x)} (a+b \sin (e+f x))}-\frac {\left (5 a^2+2 b^2\right ) \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \left (8 b^{5/2} F_1\left (\frac {3}{4};-\frac {1}{2},1;\frac {7}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(e+f x)+3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (e+f x)}+b \cos (e+f x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (e+f x)}+b \cos (e+f x)\right )\right )\right ) \sin ^2(e+f x)}{12 b^{3/2} \left (-a^2+b^2\right ) \left (1-\cos ^2(e+f x)\right ) (a+b \sin (e+f x))}\right )}{5 b^2 f \sqrt {\cos (e+f x)}}+\frac {\sqrt {g \cos (e+f x)} \left (\frac {2 a \cos (e+f x)}{3 b^2}-\frac {\sin (2 (e+f x))}{5 b}\right )}{f} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 44.56, size = 1345, normalized size = 3.00
method | result | size |
default | \(\text {Expression too large to display}\) | \(1345\) |
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 [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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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 [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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\sin \left (e+f\,x\right )}^3\,\sqrt {g\,\cos \left (e+f\,x\right )}}{a+b\,\sin \left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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