Optimal. Leaf size=231 \[ -\frac {3 i f^3 x}{8 a d^3}-\frac {i (e+f x)^3}{4 a d}-\frac {6 f^3 \cosh (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}+\frac {3 i f^3 \cosh (c+d x) \sinh (c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac {3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \sinh ^2(c+d x)}{2 a d} \]
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Rubi [A]
time = 0.18, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {5682, 3377,
2718, 5554, 3392, 32, 2715, 8} \begin {gather*} -\frac {6 f^3 \cosh (c+d x)}{a d^4}+\frac {3 i f^3 \sinh (c+d x) \cosh (c+d x)}{8 a d^4}-\frac {3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac {3 i f (e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{4 a d^2}-\frac {i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}-\frac {3 i f^3 x}{8 a d^3}-\frac {i (e+f x)^3}{4 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 32
Rule 2715
Rule 2718
Rule 3377
Rule 3392
Rule 5554
Rule 5682
Rubi steps
\begin {align*} \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac {i \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x) \, dx}{a}+\frac {\int (e+f x)^3 \cosh (c+d x) \, dx}{a}\\ &=\frac {(e+f x)^3 \sinh (c+d x)}{a d}-\frac {i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}+\frac {(3 i f) \int (e+f x)^2 \sinh ^2(c+d x) \, dx}{2 a d}-\frac {(3 f) \int (e+f x)^2 \sinh (c+d x) \, dx}{a d}\\ &=-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}+\frac {3 i f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac {3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}-\frac {(3 i f) \int (e+f x)^2 \, dx}{4 a d}+\frac {\left (6 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{a d^2}+\frac {\left (3 i f^3\right ) \int \sinh ^2(c+d x) \, dx}{4 a d^3}\\ &=-\frac {i (e+f x)^3}{4 a d}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}+\frac {3 i f^3 \cosh (c+d x) \sinh (c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac {3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}-\frac {\left (3 i f^3\right ) \int 1 \, dx}{8 a d^3}-\frac {\left (6 f^3\right ) \int \sinh (c+d x) \, dx}{a d^3}\\ &=-\frac {3 i f^3 x}{8 a d^3}-\frac {i (e+f x)^3}{4 a d}-\frac {6 f^3 \cosh (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}+\frac {3 i f^3 \cosh (c+d x) \sinh (c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac {3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}\\ \end {align*}
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Mathematica [A]
time = 0.71, size = 134, normalized size = 0.58 \begin {gather*} \frac {-96 f \left (2 f^2+d^2 (e+f x)^2\right ) \cosh (c+d x)-4 i d (e+f x) \left (3 f^2+2 d^2 (e+f x)^2\right ) \cosh (2 (c+d x))+4 \left (8 d (e+f x) \left (6 f^2+d^2 (e+f x)^2\right )+3 i f \left (f^2+2 d^2 (e+f x)^2\right ) \cosh (c+d x)\right ) \sinh (c+d x)}{32 a d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 725 vs. \(2 (213 ) = 426\).
time = 1.21, size = 726, normalized size = 3.14 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 401, normalized size = 1.74 \begin {gather*} \frac {{\left (-4 i \, d^{3} f^{3} x^{3} - 6 i \, d^{2} f^{3} x^{2} - 6 i \, d f^{3} x - 4 i \, d^{3} e^{3} - 3 i \, f^{3} - 6 \, {\left (2 i \, d^{3} f x + i \, d^{2} f\right )} e^{2} - 6 \, {\left (2 i \, d^{3} f^{2} x^{2} + 2 i \, d^{2} f^{2} x + i \, d f^{2}\right )} e + {\left (-4 i \, d^{3} f^{3} x^{3} + 6 i \, d^{2} f^{3} x^{2} - 6 i \, d f^{3} x - 4 i \, d^{3} e^{3} + 3 i \, f^{3} - 6 \, {\left (2 i \, d^{3} f x - i \, d^{2} f\right )} e^{2} - 6 \, {\left (2 i \, d^{3} f^{2} x^{2} - 2 i \, d^{2} f^{2} x + i \, d f^{2}\right )} e\right )} e^{\left (4 \, d x + 4 \, c\right )} + 16 \, {\left (d^{3} f^{3} x^{3} - 3 \, d^{2} f^{3} x^{2} + 6 \, d f^{3} x + d^{3} e^{3} - 6 \, f^{3} + 3 \, {\left (d^{3} f x - d^{2} f\right )} e^{2} + 3 \, {\left (d^{3} f^{2} x^{2} - 2 \, d^{2} f^{2} x + 2 \, d f^{2}\right )} e\right )} e^{\left (3 \, d x + 3 \, c\right )} - 16 \, {\left (d^{3} f^{3} x^{3} + 3 \, d^{2} f^{3} x^{2} + 6 \, d f^{3} x + d^{3} e^{3} + 6 \, f^{3} + 3 \, {\left (d^{3} f x + d^{2} f\right )} e^{2} + 3 \, {\left (d^{3} f^{2} x^{2} + 2 \, d^{2} f^{2} x + 2 \, d f^{2}\right )} e\right )} e^{\left (d x + c\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{32 \, a d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1040 vs. \(2 (214) = 428\).
time = 0.65, size = 1040, normalized size = 4.50 \begin {gather*} \begin {cases} \frac {\left (\left (- 2048 a^{3} d^{15} e^{3} e^{2 c} - 6144 a^{3} d^{15} e^{2} f x e^{2 c} - 6144 a^{3} d^{15} e f^{2} x^{2} e^{2 c} - 2048 a^{3} d^{15} f^{3} x^{3} e^{2 c} - 6144 a^{3} d^{14} e^{2} f e^{2 c} - 12288 a^{3} d^{14} e f^{2} x e^{2 c} - 6144 a^{3} d^{14} f^{3} x^{2} e^{2 c} - 12288 a^{3} d^{13} e f^{2} e^{2 c} - 12288 a^{3} d^{13} f^{3} x e^{2 c} - 12288 a^{3} d^{12} f^{3} e^{2 c}\right ) e^{- d x} + \left (2048 a^{3} d^{15} e^{3} e^{4 c} + 6144 a^{3} d^{15} e^{2} f x e^{4 c} + 6144 a^{3} d^{15} e f^{2} x^{2} e^{4 c} + 2048 a^{3} d^{15} f^{3} x^{3} e^{4 c} - 6144 a^{3} d^{14} e^{2} f e^{4 c} - 12288 a^{3} d^{14} e f^{2} x e^{4 c} - 6144 a^{3} d^{14} f^{3} x^{2} e^{4 c} + 12288 a^{3} d^{13} e f^{2} e^{4 c} + 12288 a^{3} d^{13} f^{3} x e^{4 c} - 12288 a^{3} d^{12} f^{3} e^{4 c}\right ) e^{d x} + \left (- 512 i a^{3} d^{15} e^{3} e^{c} - 1536 i a^{3} d^{15} e^{2} f x e^{c} - 1536 i a^{3} d^{15} e f^{2} x^{2} e^{c} - 512 i a^{3} d^{15} f^{3} x^{3} e^{c} - 768 i a^{3} d^{14} e^{2} f e^{c} - 1536 i a^{3} d^{14} e f^{2} x e^{c} - 768 i a^{3} d^{14} f^{3} x^{2} e^{c} - 768 i a^{3} d^{13} e f^{2} e^{c} - 768 i a^{3} d^{13} f^{3} x e^{c} - 384 i a^{3} d^{12} f^{3} e^{c}\right ) e^{- 2 d x} + \left (- 512 i a^{3} d^{15} e^{3} e^{5 c} - 1536 i a^{3} d^{15} e^{2} f x e^{5 c} - 1536 i a^{3} d^{15} e f^{2} x^{2} e^{5 c} - 512 i a^{3} d^{15} f^{3} x^{3} e^{5 c} + 768 i a^{3} d^{14} e^{2} f e^{5 c} + 1536 i a^{3} d^{14} e f^{2} x e^{5 c} + 768 i a^{3} d^{14} f^{3} x^{2} e^{5 c} - 768 i a^{3} d^{13} e f^{2} e^{5 c} - 768 i a^{3} d^{13} f^{3} x e^{5 c} + 384 i a^{3} d^{12} f^{3} e^{5 c}\right ) e^{2 d x}\right ) e^{- 3 c}}{4096 a^{4} d^{16}} & \text {for}\: a^{4} d^{16} e^{3 c} \neq 0 \\\frac {x^{4} \left (- i f^{3} e^{4 c} + 2 f^{3} e^{3 c} + 2 f^{3} e^{c} + i f^{3}\right ) e^{- 2 c}}{16 a} + \frac {x^{3} \left (- i e f^{2} e^{4 c} + 2 e f^{2} e^{3 c} + 2 e f^{2} e^{c} + i e f^{2}\right ) e^{- 2 c}}{4 a} + \frac {x^{2} \left (- 3 i e^{2} f e^{4 c} + 6 e^{2} f e^{3 c} + 6 e^{2} f e^{c} + 3 i e^{2} f\right ) e^{- 2 c}}{8 a} + \frac {x \left (- i e^{3} e^{4 c} + 2 e^{3} e^{3 c} + 2 e^{3} e^{c} + i e^{3}\right ) e^{- 2 c}}{4 a} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 618 vs. \(2 (207) = 414\).
time = 0.46, size = 618, normalized size = 2.68 \begin {gather*} -\frac {{\left (4 i \, d^{3} f^{3} x^{3} e^{\left (4 \, d x + 4 \, c\right )} - 16 \, d^{3} f^{3} x^{3} e^{\left (3 \, d x + 3 \, c\right )} + 16 \, d^{3} f^{3} x^{3} e^{\left (d x + c\right )} + 4 i \, d^{3} f^{3} x^{3} + 12 i \, d^{3} e f^{2} x^{2} e^{\left (4 \, d x + 4 \, c\right )} - 48 \, d^{3} e f^{2} x^{2} e^{\left (3 \, d x + 3 \, c\right )} + 48 \, d^{3} e f^{2} x^{2} e^{\left (d x + c\right )} + 12 i \, d^{3} e f^{2} x^{2} + 12 i \, d^{3} e^{2} f x e^{\left (4 \, d x + 4 \, c\right )} - 6 i \, d^{2} f^{3} x^{2} e^{\left (4 \, d x + 4 \, c\right )} - 48 \, d^{3} e^{2} f x e^{\left (3 \, d x + 3 \, c\right )} + 48 \, d^{2} f^{3} x^{2} e^{\left (3 \, d x + 3 \, c\right )} + 48 \, d^{3} e^{2} f x e^{\left (d x + c\right )} + 48 \, d^{2} f^{3} x^{2} e^{\left (d x + c\right )} + 12 i \, d^{3} e^{2} f x + 6 i \, d^{2} f^{3} x^{2} + 4 i \, d^{3} e^{3} e^{\left (4 \, d x + 4 \, c\right )} - 12 i \, d^{2} e f^{2} x e^{\left (4 \, d x + 4 \, c\right )} - 16 \, d^{3} e^{3} e^{\left (3 \, d x + 3 \, c\right )} + 96 \, d^{2} e f^{2} x e^{\left (3 \, d x + 3 \, c\right )} + 16 \, d^{3} e^{3} e^{\left (d x + c\right )} + 96 \, d^{2} e f^{2} x e^{\left (d x + c\right )} + 4 i \, d^{3} e^{3} + 12 i \, d^{2} e f^{2} x - 6 i \, d^{2} e^{2} f e^{\left (4 \, d x + 4 \, c\right )} + 6 i \, d f^{3} x e^{\left (4 \, d x + 4 \, c\right )} + 48 \, d^{2} e^{2} f e^{\left (3 \, d x + 3 \, c\right )} - 96 \, d f^{3} x e^{\left (3 \, d x + 3 \, c\right )} + 48 \, d^{2} e^{2} f e^{\left (d x + c\right )} + 96 \, d f^{3} x e^{\left (d x + c\right )} + 6 i \, d^{2} e^{2} f + 6 i \, d f^{3} x + 6 i \, d e f^{2} e^{\left (4 \, d x + 4 \, c\right )} - 96 \, d e f^{2} e^{\left (3 \, d x + 3 \, c\right )} + 96 \, d e f^{2} e^{\left (d x + c\right )} + 6 i \, d e f^{2} - 3 i \, f^{3} e^{\left (4 \, d x + 4 \, c\right )} + 96 \, f^{3} e^{\left (3 \, d x + 3 \, c\right )} + 96 \, f^{3} e^{\left (d x + c\right )} + 3 i \, f^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{32 \, a d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.28, size = 449, normalized size = 1.94 \begin {gather*} -{\mathrm {e}}^{c+d\,x}\,\left (\frac {-d^3\,e^3+3\,d^2\,e^2\,f-6\,d\,e\,f^2+6\,f^3}{2\,a\,d^4}-\frac {f^3\,x^3}{2\,a\,d}+\frac {3\,f^2\,x^2\,\left (f-d\,e\right )}{2\,a\,d^2}-\frac {3\,f\,x\,\left (d^2\,e^2-2\,d\,e\,f+2\,f^2\right )}{2\,a\,d^3}\right )-{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (\frac {\left (4\,d^3\,e^3+6\,d^2\,e^2\,f+6\,d\,e\,f^2+3\,f^3\right )\,1{}\mathrm {i}}{32\,a\,d^4}+\frac {f^3\,x^3\,1{}\mathrm {i}}{8\,a\,d}+\frac {f\,x\,\left (2\,d^2\,e^2+2\,d\,e\,f+f^2\right )\,3{}\mathrm {i}}{16\,a\,d^3}+\frac {f^2\,x^2\,\left (f+2\,d\,e\right )\,3{}\mathrm {i}}{16\,a\,d^2}\right )+{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (\frac {\left (-4\,d^3\,e^3+6\,d^2\,e^2\,f-6\,d\,e\,f^2+3\,f^3\right )\,1{}\mathrm {i}}{32\,a\,d^4}-\frac {f^3\,x^3\,1{}\mathrm {i}}{8\,a\,d}-\frac {f\,x\,\left (2\,d^2\,e^2-2\,d\,e\,f+f^2\right )\,3{}\mathrm {i}}{16\,a\,d^3}+\frac {f^2\,x^2\,\left (f-2\,d\,e\right )\,3{}\mathrm {i}}{16\,a\,d^2}\right )-{\mathrm {e}}^{-c-d\,x}\,\left (\frac {d^3\,e^3+3\,d^2\,e^2\,f+6\,d\,e\,f^2+6\,f^3}{2\,a\,d^4}+\frac {f^3\,x^3}{2\,a\,d}+\frac {3\,f^2\,x^2\,\left (f+d\,e\right )}{2\,a\,d^2}+\frac {3\,f\,x\,\left (d^2\,e^2+2\,d\,e\,f+2\,f^2\right )}{2\,a\,d^3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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