Optimal. Leaf size=575 \[ \frac {11 a e^6 \left (45 a^2-37 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{20 b^6 d \sqrt {\cos (c+d x)}}+\frac {11 e^5 (e \cos (c+d x))^{3/2} \left (5 \left (9 a^2-2 b^2\right )-27 a b \sin (c+d x)\right )}{60 b^5 d}-\frac {11 e^{13/2} \left (9 a^4-11 a^2 b^2+2 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 b^{13/2} d \sqrt [4]{b^2-a^2}}+\frac {11 e^{13/2} \left (9 a^4-11 a^2 b^2+2 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 b^{13/2} d \sqrt [4]{b^2-a^2}}-\frac {11 a e^7 \left (9 a^4-11 a^2 b^2+2 b^4\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b^7 d \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}-\frac {11 a e^7 \left (9 a^4-11 a^2 b^2+2 b^4\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b^7 d \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}-\frac {11 e^3 (e \cos (c+d x))^{7/2} (9 a+2 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))}-\frac {e (e \cos (c+d x))^{11/2}}{2 b d (a+b \sin (c+d x))^2} \]
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Rubi [A] time = 1.41, antiderivative size = 575, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {2693, 2863, 2865, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208} \[ \frac {11 e^5 (e \cos (c+d x))^{3/2} \left (5 \left (9 a^2-2 b^2\right )-27 a b \sin (c+d x)\right )}{60 b^5 d}-\frac {11 e^{13/2} \left (-11 a^2 b^2+9 a^4+2 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 b^{13/2} d \sqrt [4]{b^2-a^2}}+\frac {11 e^{13/2} \left (-11 a^2 b^2+9 a^4+2 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 b^{13/2} d \sqrt [4]{b^2-a^2}}+\frac {11 a e^6 \left (45 a^2-37 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{20 b^6 d \sqrt {\cos (c+d x)}}-\frac {11 a e^7 \left (-11 a^2 b^2+9 a^4+2 b^4\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b^7 d \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}-\frac {11 a e^7 \left (-11 a^2 b^2+9 a^4+2 b^4\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b^7 d \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}-\frac {11 e^3 (e \cos (c+d x))^{7/2} (9 a+2 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))}-\frac {e (e \cos (c+d x))^{11/2}}{2 b d (a+b \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 298
Rule 329
Rule 2639
Rule 2640
Rule 2693
Rule 2701
Rule 2805
Rule 2807
Rule 2863
Rule 2865
Rule 2867
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^{13/2}}{(a+b \sin (c+d x))^3} \, dx &=-\frac {e (e \cos (c+d x))^{11/2}}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (11 e^2\right ) \int \frac {(e \cos (c+d x))^{9/2} \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{4 b}\\ &=-\frac {e (e \cos (c+d x))^{11/2}}{2 b d (a+b \sin (c+d x))^2}-\frac {11 e^3 (e \cos (c+d x))^{7/2} (9 a+2 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))}+\frac {\left (11 e^4\right ) \int \frac {(e \cos (c+d x))^{5/2} \left (-b-\frac {9}{2} a \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 b^3}\\ &=-\frac {e (e \cos (c+d x))^{11/2}}{2 b d (a+b \sin (c+d x))^2}-\frac {11 e^3 (e \cos (c+d x))^{7/2} (9 a+2 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))}+\frac {11 e^5 (e \cos (c+d x))^{3/2} \left (5 \left (9 a^2-2 b^2\right )-27 a b \sin (c+d x)\right )}{60 b^5 d}+\frac {\left (11 e^6\right ) \int \frac {\sqrt {e \cos (c+d x)} \left (\frac {1}{2} b \left (9 a^2-5 b^2\right )+\frac {1}{4} a \left (45 a^2-37 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{10 b^5}\\ &=-\frac {e (e \cos (c+d x))^{11/2}}{2 b d (a+b \sin (c+d x))^2}-\frac {11 e^3 (e \cos (c+d x))^{7/2} (9 a+2 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))}+\frac {11 e^5 (e \cos (c+d x))^{3/2} \left (5 \left (9 a^2-2 b^2\right )-27 a b \sin (c+d x)\right )}{60 b^5 d}+\frac {\left (11 a \left (45 a^2-37 b^2\right ) e^6\right ) \int \sqrt {e \cos (c+d x)} \, dx}{40 b^6}-\frac {\left (11 \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^6\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx}{8 b^6}\\ &=-\frac {e (e \cos (c+d x))^{11/2}}{2 b d (a+b \sin (c+d x))^2}-\frac {11 e^3 (e \cos (c+d x))^{7/2} (9 a+2 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))}+\frac {11 e^5 (e \cos (c+d x))^{3/2} \left (5 \left (9 a^2-2 b^2\right )-27 a b \sin (c+d x)\right )}{60 b^5 d}+\frac {\left (11 a \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{16 b^7}-\frac {\left (11 a \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{16 b^7}-\frac {\left (11 \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{\left (a^2-b^2\right ) e^2+b^2 x^2} \, dx,x,e \cos (c+d x)\right )}{8 b^5 d}+\frac {\left (11 a \left (45 a^2-37 b^2\right ) e^6 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{40 b^6 \sqrt {\cos (c+d x)}}\\ &=\frac {11 a \left (45 a^2-37 b^2\right ) e^6 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{20 b^6 d \sqrt {\cos (c+d x)}}-\frac {e (e \cos (c+d x))^{11/2}}{2 b d (a+b \sin (c+d x))^2}-\frac {11 e^3 (e \cos (c+d x))^{7/2} (9 a+2 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))}+\frac {11 e^5 (e \cos (c+d x))^{3/2} \left (5 \left (9 a^2-2 b^2\right )-27 a b \sin (c+d x)\right )}{60 b^5 d}-\frac {\left (11 \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a^2-b^2\right ) e^2+b^2 x^4} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{4 b^5 d}+\frac {\left (11 a \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{16 b^7 \sqrt {e \cos (c+d x)}}-\frac {\left (11 a \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{16 b^7 \sqrt {e \cos (c+d x)}}\\ &=\frac {11 a \left (45 a^2-37 b^2\right ) e^6 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{20 b^6 d \sqrt {\cos (c+d x)}}-\frac {11 a \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7 \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b^7 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {11 a \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7 \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b^7 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {e (e \cos (c+d x))^{11/2}}{2 b d (a+b \sin (c+d x))^2}-\frac {11 e^3 (e \cos (c+d x))^{7/2} (9 a+2 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))}+\frac {11 e^5 (e \cos (c+d x))^{3/2} \left (5 \left (9 a^2-2 b^2\right )-27 a b \sin (c+d x)\right )}{60 b^5 d}+\frac {\left (11 \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e-b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{8 b^6 d}-\frac {\left (11 \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e+b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{8 b^6 d}\\ &=-\frac {11 \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^{13/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 b^{13/2} \sqrt [4]{-a^2+b^2} d}+\frac {11 \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^{13/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 b^{13/2} \sqrt [4]{-a^2+b^2} d}+\frac {11 a \left (45 a^2-37 b^2\right ) e^6 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{20 b^6 d \sqrt {\cos (c+d x)}}-\frac {11 a \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7 \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b^7 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {11 a \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7 \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b^7 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {e (e \cos (c+d x))^{11/2}}{2 b d (a+b \sin (c+d x))^2}-\frac {11 e^3 (e \cos (c+d x))^{7/2} (9 a+2 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))}+\frac {11 e^5 (e \cos (c+d x))^{3/2} \left (5 \left (9 a^2-2 b^2\right )-27 a b \sin (c+d x)\right )}{60 b^5 d}\\ \end {align*}
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Mathematica [C] time = 27.05, size = 932, normalized size = 1.62 \[ \frac {11 \left (-\frac {\left (45 a^3-37 a b^2\right ) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (8 F_1\left (\frac {3}{4};-\frac {1}{2},1;\frac {7}{4};\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{b^2-a^2}\right ) \cos ^{\frac {3}{2}}(c+d x) b^{5/2}+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 (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}+1\right )-\log \left (b \cos (c+d x)-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+\sqrt {a^2-b^2}\right )+\log \left (b \cos (c+d x)+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+\sqrt {a^2-b^2}\right )\right )\right ) \sin ^2(c+d x)}{12 b^{3/2} \left (b^2-a^2\right ) \left (1-\cos ^2(c+d x)\right ) (a+b \sin (c+d x))}-\frac {2 \left (18 a^2 b-10 b^3\right ) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (\frac {a F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{b^2-a^2}\right ) \cos ^{\frac {3}{2}}(c+d 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 (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )-2 \tan ^{-1}\left (\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{b^2-a^2}}+1\right )-\log \left (i b \cos (c+d x)-(1+i) \sqrt {b} \sqrt [4]{b^2-a^2} \sqrt {\cos (c+d x)}+\sqrt {b^2-a^2}\right )+\log \left (i b \cos (c+d x)+(1+i) \sqrt {b} \sqrt [4]{b^2-a^2} \sqrt {\cos (c+d x)}+\sqrt {b^2-a^2}\right )\right )}{\sqrt {b} \sqrt [4]{b^2-a^2}}\right ) \sin (c+d x)}{\sqrt {1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right ) (e \cos (c+d x))^{13/2}}{40 b^5 d \cos ^{\frac {13}{2}}(c+d x)}+\frac {\sec ^6(c+d x) \left (-\frac {\left (65 b^2-168 a^2\right ) \cos (c+d x)}{42 b^5}-\frac {\cos (3 (c+d x))}{14 b^3}-\frac {3 a \sin (2 (c+d x))}{5 b^4}+\frac {19 \left (a^3 \cos (c+d x)-a b^2 \cos (c+d x)\right )}{4 b^5 (a+b \sin (c+d x))}+\frac {-\cos (c+d x) a^4+2 b^2 \cos (c+d x) a^2-b^4 \cos (c+d x)}{2 b^5 (a+b \sin (c+d x))^2}\right ) (e \cos (c+d x))^{13/2}}{d} \]
Warning: Unable to verify antiderivative.
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fricas [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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giac [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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maple [C] time = 36.49, size = 111631, normalized size = 194.14 \[ \text {output too large to display} \]
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 \[ \int \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {13}{2}}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{13/2}}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3} \,d x \]
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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