Optimal. Leaf size=584 \[ \frac {a^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{5/2} \left (-a^2+b^2\right )^{5/4} f g^{3/2}}-\frac {a^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{5/2} \left (-a^2+b^2\right )^{5/4} f g^{3/2}}-\frac {2 b}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a^2 (g \cos (e+f x))^{3/2}}{3 b \left (a^2-b^2\right ) f g^3}-\frac {2 b (g \cos (e+f x))^{3/2}}{3 \left (a^2-b^2\right ) f g^3}-\frac {4 a \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{\left (a^2-b^2\right ) f g^2 \sqrt {\cos (e+f x)}}+\frac {2 a^3 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b^2 \left (a^2-b^2\right ) f g^2 \sqrt {\cos (e+f x)}}-\frac {a^5 \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^3 \left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) f g \sqrt {g \cos (e+f x)}}-\frac {a^5 \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{b^3 \left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a \sin (e+f x)}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}} \]
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Rubi [A]
time = 0.84, antiderivative size = 584, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 15, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used =
{2981, 2646, 2721, 2719, 2645, 14, 2977, 30, 2780, 2886, 2884, 335, 304, 211, 214}
\begin {gather*} \frac {2 a^2 (g \cos (e+f x))^{3/2}}{3 b f g^3 \left (a^2-b^2\right )}-\frac {2 b (g \cos (e+f x))^{3/2}}{3 f g^3 \left (a^2-b^2\right )}-\frac {4 a E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f g^2 \left (a^2-b^2\right ) \sqrt {\cos (e+f x)}}-\frac {2 b}{f g \left (a^2-b^2\right ) \sqrt {g \cos (e+f x)}}+\frac {2 a \sin (e+f x)}{f g \left (a^2-b^2\right ) \sqrt {g \cos (e+f x)}}-\frac {a^5 \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^3 f g \left (a^2-b^2\right ) \left (b-\sqrt {b^2-a^2}\right ) \sqrt {g \cos (e+f x)}}-\frac {a^5 \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^3 f g \left (a^2-b^2\right ) \left (\sqrt {b^2-a^2}+b\right ) \sqrt {g \cos (e+f x)}}+\frac {a^4 \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{b^{5/2} f g^{3/2} \left (b^2-a^2\right )^{5/4}}-\frac {a^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{b^{5/2} f g^{3/2} \left (b^2-a^2\right )^{5/4}}+\frac {2 a^3 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{b^2 f g^2 \left (a^2-b^2\right ) \sqrt {\cos (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 211
Rule 214
Rule 304
Rule 335
Rule 2645
Rule 2646
Rule 2719
Rule 2721
Rule 2780
Rule 2884
Rule 2886
Rule 2977
Rule 2981
Rubi steps
\begin {align*} \int \frac {\sin ^4(e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx &=\frac {a \int \frac {\sin ^2(e+f x)}{(g \cos (e+f x))^{3/2}} \, dx}{a^2-b^2}-\frac {b \int \frac {\sin ^3(e+f x)}{(g \cos (e+f x))^{3/2}} \, dx}{a^2-b^2}-\frac {a^2 \int \frac {\sqrt {g \cos (e+f x)} \sin ^2(e+f x)}{a+b \sin (e+f x)} \, dx}{\left (a^2-b^2\right ) g^2}\\ &=\frac {2 a \sin (e+f x)}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {(2 a) \int \sqrt {g \cos (e+f x)} \, dx}{\left (a^2-b^2\right ) g^2}-\frac {a^2 \int \left (-\frac {a \sqrt {g \cos (e+f x)}}{b^2}+\frac {\sqrt {g \cos (e+f x)} \sin (e+f x)}{b}+\frac {a^2 \sqrt {g \cos (e+f x)}}{b^2 (a+b \sin (e+f x))}\right ) \, dx}{\left (a^2-b^2\right ) g^2}+\frac {b \text {Subst}\left (\int \frac {1-\frac {x^2}{g^2}}{x^{3/2}} \, dx,x,g \cos (e+f x)\right )}{\left (a^2-b^2\right ) f g}\\ &=\frac {2 a \sin (e+f x)}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {a^3 \int \sqrt {g \cos (e+f x)} \, dx}{b^2 \left (a^2-b^2\right ) g^2}-\frac {a^4 \int \frac {\sqrt {g \cos (e+f x)}}{a+b \sin (e+f x)} \, dx}{b^2 \left (a^2-b^2\right ) g^2}-\frac {a^2 \int \sqrt {g \cos (e+f x)} \sin (e+f x) \, dx}{b \left (a^2-b^2\right ) g^2}+\frac {b \text {Subst}\left (\int \left (\frac {1}{x^{3/2}}-\frac {\sqrt {x}}{g^2}\right ) \, dx,x,g \cos (e+f x)\right )}{\left (a^2-b^2\right ) f g}-\frac {\left (2 a \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{\left (a^2-b^2\right ) g^2 \sqrt {\cos (e+f x)}}\\ &=-\frac {2 b}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {2 b (g \cos (e+f x))^{3/2}}{3 \left (a^2-b^2\right ) f g^3}-\frac {4 a \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{\left (a^2-b^2\right ) f g^2 \sqrt {\cos (e+f x)}}+\frac {2 a \sin (e+f x)}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {a^2 \text {Subst}\left (\int \sqrt {x} \, dx,x,g \cos (e+f x)\right )}{b \left (a^2-b^2\right ) f g^3}+\frac {a^5 \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 b^3 \left (a^2-b^2\right ) g}-\frac {a^5 \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 b^3 \left (a^2-b^2\right ) g}-\frac {a^4 \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 \left (a^2-b^2\right ) f g}+\frac {\left (a^3 \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{b^2 \left (a^2-b^2\right ) g^2 \sqrt {\cos (e+f x)}}\\ &=-\frac {2 b}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a^2 (g \cos (e+f x))^{3/2}}{3 b \left (a^2-b^2\right ) f g^3}-\frac {2 b (g \cos (e+f x))^{3/2}}{3 \left (a^2-b^2\right ) f g^3}-\frac {4 a \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{\left (a^2-b^2\right ) f g^2 \sqrt {\cos (e+f x)}}+\frac {2 a^3 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b^2 \left (a^2-b^2\right ) f g^2 \sqrt {\cos (e+f x)}}+\frac {2 a \sin (e+f x)}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {\left (2 a^4\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 \left (a^2-b^2\right ) f g}+\frac {\left (a^5 \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^3 \left (a^2-b^2\right ) g \sqrt {g \cos (e+f x)}}-\frac {\left (a^5 \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^3 \left (a^2-b^2\right ) g \sqrt {g \cos (e+f x)}}\\ &=-\frac {2 b}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a^2 (g \cos (e+f x))^{3/2}}{3 b \left (a^2-b^2\right ) f g^3}-\frac {2 b (g \cos (e+f x))^{3/2}}{3 \left (a^2-b^2\right ) f g^3}-\frac {4 a \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{\left (a^2-b^2\right ) f g^2 \sqrt {\cos (e+f x)}}+\frac {2 a^3 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b^2 \left (a^2-b^2\right ) f g^2 \sqrt {\cos (e+f x)}}-\frac {a^5 \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^3 \left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) f g \sqrt {g \cos (e+f x)}}-\frac {a^5 \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^3 \left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a \sin (e+f x)}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {a^4 \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^2 \left (a^2-b^2\right ) f g}-\frac {a^4 \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^2 \left (a^2-b^2\right ) f g}\\ &=\frac {a^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{5/2} \left (-a^2+b^2\right )^{5/4} f g^{3/2}}-\frac {a^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{5/2} \left (-a^2+b^2\right )^{5/4} f g^{3/2}}-\frac {2 b}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a^2 (g \cos (e+f x))^{3/2}}{3 b \left (a^2-b^2\right ) f g^3}-\frac {2 b (g \cos (e+f x))^{3/2}}{3 \left (a^2-b^2\right ) f g^3}-\frac {4 a \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{\left (a^2-b^2\right ) f g^2 \sqrt {\cos (e+f x)}}+\frac {2 a^3 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b^2 \left (a^2-b^2\right ) f g^2 \sqrt {\cos (e+f x)}}-\frac {a^5 \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^3 \left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) f g \sqrt {g \cos (e+f x)}}-\frac {a^5 \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^3 \left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a \sin (e+f x)}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 37.11, size = 820, normalized size = 1.40 \begin {gather*} \frac {\cos ^2(e+f x) \left (\frac {2 \cos (e+f x)}{3 b}+\frac {2 \sec (e+f x) (-b+a \sin (e+f x))}{a^2-b^2}\right )}{f (g \cos (e+f x))^{3/2}}+\frac {a \cos ^{\frac {3}{2}}(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 (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 )}{(a-b) b (a+b) f (g \cos (e+f x))^{3/2}} \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 = 57.96, size = 1216, normalized size = 2.08
method | result | size |
default | \(\text {Expression too large to display}\) | \(1216\) |
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(-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.
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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 )}^4}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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