Optimal. Leaf size=535 \[ \frac {2 \left (58 a^4-41 a^2 b^2+15 b^4\right ) \cot (c+d x) E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{15 a^3 (a-b)^2 (a+b)^{5/2} d}-\frac {2 \left (45 a^4-13 a^3 b-36 a^2 b^2+5 a b^3+15 b^4\right ) \cot (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{15 a^3 (a-b)^2 (a+b)^{5/2} d}-\frac {2 \sqrt {a+b} \cot (c+d x) \Pi \left (\frac {a+b}{a};\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{a^4 d}+\frac {2 b^2 \tan (c+d x)}{5 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{5/2}}+\frac {2 b^2 \left (13 a^2-5 b^2\right ) \tan (c+d x)}{15 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^{3/2}}+\frac {2 b^2 \left (58 a^4-41 a^2 b^2+15 b^4\right ) \tan (c+d x)}{15 a^3 \left (a^2-b^2\right )^3 d \sqrt {a+b \sec (c+d x)}} \]
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Rubi [A]
time = 0.60, antiderivative size = 535, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3870, 4145,
4143, 4006, 3869, 3917, 4089} \begin {gather*} -\frac {2 \sqrt {a+b} \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{a};\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{a^4 d}+\frac {2 b^2 \left (13 a^2-5 b^2\right ) \tan (c+d x)}{15 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^{3/2}}+\frac {2 b^2 \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}+\frac {2 \left (58 a^4-41 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{15 a^3 d (a-b)^2 (a+b)^{5/2}}-\frac {2 \left (45 a^4-13 a^3 b-36 a^2 b^2+5 a b^3+15 b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{15 a^3 d (a-b)^2 (a+b)^{5/2}}+\frac {2 b^2 \left (58 a^4-41 a^2 b^2+15 b^4\right ) \tan (c+d x)}{15 a^3 d \left (a^2-b^2\right )^3 \sqrt {a+b \sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3869
Rule 3870
Rule 3917
Rule 4006
Rule 4089
Rule 4143
Rule 4145
Rubi steps
\begin {align*} \int \frac {1}{(a+b \sec (c+d x))^{7/2}} \, dx &=\frac {2 b^2 \tan (c+d x)}{5 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{5/2}}-\frac {2 \int \frac {-\frac {5}{2} \left (a^2-b^2\right )+\frac {5}{2} a b \sec (c+d x)-\frac {3}{2} b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx}{5 a \left (a^2-b^2\right )}\\ &=\frac {2 b^2 \tan (c+d x)}{5 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{5/2}}+\frac {2 b^2 \left (13 a^2-5 b^2\right ) \tan (c+d x)}{15 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^{3/2}}+\frac {4 \int \frac {\frac {15}{4} \left (a^2-b^2\right )^2-\frac {3}{2} a b \left (5 a^2-b^2\right ) \sec (c+d x)+\frac {1}{4} b^2 \left (13 a^2-5 b^2\right ) \sec ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx}{15 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {2 b^2 \tan (c+d x)}{5 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{5/2}}+\frac {2 b^2 \left (13 a^2-5 b^2\right ) \tan (c+d x)}{15 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^{3/2}}+\frac {2 b^2 \left (58 a^4-41 a^2 b^2+15 b^4\right ) \tan (c+d x)}{15 a^3 \left (a^2-b^2\right )^3 d \sqrt {a+b \sec (c+d x)}}-\frac {8 \int \frac {-\frac {15}{8} \left (a^2-b^2\right )^3+\frac {1}{8} a b \left (45 a^4-23 a^2 b^2+10 b^4\right ) \sec (c+d x)+\frac {1}{8} b^2 \left (58 a^4-41 a^2 b^2+15 b^4\right ) \sec ^2(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{15 a^3 \left (a^2-b^2\right )^3}\\ &=\frac {2 b^2 \tan (c+d x)}{5 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{5/2}}+\frac {2 b^2 \left (13 a^2-5 b^2\right ) \tan (c+d x)}{15 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^{3/2}}+\frac {2 b^2 \left (58 a^4-41 a^2 b^2+15 b^4\right ) \tan (c+d x)}{15 a^3 \left (a^2-b^2\right )^3 d \sqrt {a+b \sec (c+d x)}}-\frac {8 \int \frac {-\frac {15}{8} \left (a^2-b^2\right )^3+\left (\frac {1}{8} a b \left (45 a^4-23 a^2 b^2+10 b^4\right )-\frac {1}{8} b^2 \left (58 a^4-41 a^2 b^2+15 b^4\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{15 a^3 \left (a^2-b^2\right )^3}-\frac {\left (b^2 \left (58 a^4-41 a^2 b^2+15 b^4\right )\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx}{15 a^3 \left (a^2-b^2\right )^3}\\ &=\frac {2 \left (58 a^4-41 a^2 b^2+15 b^4\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{15 a^3 (a-b)^2 (a+b)^{5/2} d}+\frac {2 b^2 \tan (c+d x)}{5 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{5/2}}+\frac {2 b^2 \left (13 a^2-5 b^2\right ) \tan (c+d x)}{15 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^{3/2}}+\frac {2 b^2 \left (58 a^4-41 a^2 b^2+15 b^4\right ) \tan (c+d x)}{15 a^3 \left (a^2-b^2\right )^3 d \sqrt {a+b \sec (c+d x)}}+\frac {\int \frac {1}{\sqrt {a+b \sec (c+d x)}} \, dx}{a^3}-\frac {\left (b \left (45 a^4-13 a^3 b-36 a^2 b^2+5 a b^3+15 b^4\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{15 a^3 (a-b)^2 (a+b)^3}\\ &=\frac {2 \left (58 a^4-41 a^2 b^2+15 b^4\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{15 a^3 (a-b)^2 (a+b)^{5/2} d}-\frac {2 \left (45 a^4-13 a^3 b-36 a^2 b^2+5 a b^3+15 b^4\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{15 a^3 (a-b)^2 (a+b)^{5/2} d}-\frac {2 \sqrt {a+b} \cot (c+d x) \Pi \left (\frac {a+b}{a};\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{a^4 d}+\frac {2 b^2 \tan (c+d x)}{5 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{5/2}}+\frac {2 b^2 \left (13 a^2-5 b^2\right ) \tan (c+d x)}{15 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^{3/2}}+\frac {2 b^2 \left (58 a^4-41 a^2 b^2+15 b^4\right ) \tan (c+d x)}{15 a^3 \left (a^2-b^2\right )^3 d \sqrt {a+b \sec (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 13.56, size = 2346, normalized size = 4.39 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(7837\) vs.
\(2(492)=984\).
time = 0.29, size = 7838, normalized size = 14.65
method | result | size |
default | \(\text {Expression too large to display}\) | \(7838\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \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 {1}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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